Bilevel Mixed-integer Nonlinear Programming Research Articles

Economic predictive control-based sizing and energy management for grid-connected hybrid renewable energy systems

Hybrid renewable energy systems (HRES) comprise a group of different energy sources and storage units that feed a specific load, efficiently. This research paper aims to develop a new methodology that provides the optimal size of the proposed HRES and runs it efficiently. Bi-level mixed-integer nonlinear programming (BMINLP) optimization approach is used to combine the sizing task and the energy management strategy (EMS) established utilizing the economic model predictive control (EMPC) method. The sizing task (upper layer) is formulated as mixed integer nonlinear programming (MINLP) optimization that has been implemented by the solver: multi-objective genetic algorithm (MOGA). EMS task (lower layer) is represented as a constraint embedded within the upper layer and executed as mixed integer linear programming (MILP) per each applied solution of the sizing task. The principal findings indicate that the total cost of the system is about 114,224 $. In detail, the annual fixed operation and maintenance, investment, and operating costs are 15,090 $, 28,351 $, and 70,783 $, correspondingly. Furthermore, around 90 % of the overall produced power is imported from the grid, while the load power represents about 95 % of the total demanded power.

Sizing and energy management of grid-connected hybrid renewable energy systems based on techno-economic predictive technique

This study intends to construct and optimally run a grid-connected hybrid renewable energy system (HRES) powering a residential building. A special techno-economic multi-objective optimization technique is used for sizing the HRES. In addition, an energy management strategy (EMS) is established as a distinct multi-objective optimization problem to minimize degradation and operating costs while improving system performance. A Bi-level mixed integer non-linear programming (BMINLP) is the framework used to formulate the two-optimization problems. Sizing problem is characterized by the higher level, on contrast; the nested EMS problem within the constraints of the size problem is presented on the lower level. For this investigation, the problem is executed via the MATLAB program. The global optimization toolbox is applied via the Multi-Objective Genetic Algorithm (MOGA) to solve the sizing optimization. While the EMS is executed using the "Intlinprg" mixed integer linear programming (MILP) solver. The technical and economic performance parameters such as degradation, renewable energy fraction (REF), operating, maintenance, and investment costs are evaluated in this study. The primary results reveal that the summer week encounters the lowest total cost and the highest REF, at roughly 99484 $ and 60 %, respectively. Furthermore, the investigation shows that the lowest degradation value attained in the same week with approximately 50.

A novel approach to optimize an integrated network design and pricing of a healthcare supply chain

Adopting various pricing policies has been highly regarded in recent years for setting prices and increasing firms' profits. One of the most common steps in pricing is to identify costs. Since a significant part of costs is related to the corresponding supply chain, many researchers in different fields have used decision-making for simultaneous pricing and network design. However, there is no such approach in the field of healthcare. This paper tries to fill this gap by formulating a mixed-integer nonlinear bi-level programming model examining the interaction of hospitals and their medicines suppliers. At the upper level, there is a competitive market where a new firm (entrant) intends to enter the market and faces the challenge of pricing medicines and making network design decisions. At the lower level, there is a private hospital competing with a public hospital, and it also struggles with healthcare services pricing and supplier selection. A comprehensive utility function that considers healthcare services prices, quality, waiting time, health insurance, readmission rate, and referral rate is extended at this level. Three novel meta-heuristic algorithms are recommended, including bi-level, improved fruit fly, jellyfish optimization, and forensic-based investigation optimization algorithms to solve the presented complex mathematical problem.

Bi-level planning approach for incorporating the demand-side flexibility of cloud data centers under electricity-carbon markets

With the prevalence of cloud computing applications, cloud data centers (CDCs) are proliferating around the world. However, in the context of hybrid energy‑carbon markets environment, the high energy costs and environmental expenses caused by CDC operation are pressing CDC owners to restructure the future development of CDC in a more economical and low-carbon manner. The potential spatio-temporal transferability and reducibility of workloads provides CDCs with significant flexibility in their operations and thus may interact with the power grid as active demand users. Nonetheless, other than private data centers, public CDCs have no direct control over the workloads submitted by terminal cloud users. As such, this paper presents a bi-level model for CDC allocation planning, so as to incorporate cloud service-demand response (CS-DR) from the perspective of a hybrid electricity-carbon market. The upper level pertains to multi-domain resource collaborative planning model, which determines the optimum siting and sizing of CDCs as well as the incentive design for CS-DR program, with the objective of maximizing the total expected benefits of CDC. The lower level models correspond to the market clearing (electricity-carbon tariff model) of Independent system operator (ISO) and the decision-making of cloud users regarding CS-DR participation. The proposed model belongs to a bi-level mixed integer nonlinear programming problem with two non-convex lower levels, which can be intractable in mathematics. To solve such difficult problem, a hybrid solution method combining multiple linearization techniques and reformulation and decomposition (R&D) strategy based on column-and-constraint generation (C&CG) algorithm is developed. The proposed model is demonstrated on a modified IEEE 30-bus test case, and the simulation results verified the effectiveness of the proposed approach.

Sizing and energy management of standalone hybrid renewable energy systems based on economic predictive control

This paper aims to design and run a standalone hybrid renewable energy System (HRES) feeding residential building load. The suggested HRES encompasses photovoltaic panels and wind turbines as energy sources, combined with a hydrogen subsystem and a bank of batteries for energy storage. The meteorological input data includes irradiance, temperature, and wind speed for four seasonal weeks. These data sets are collected for a rural site situated in the town of Ras Munif, northern Jordan. The load profiles characterize a selected benchmark residential building. Consequently, a bi-level mixed integer nonlinear programming (BMINLP) problem is formulated to incorporate the designing problem with the energy management process. The upper level represents the sizing problem, while the lower level problem denotes the energy management strategy (EMS). Each problem has its independent objective function, constraints, and optimization variables. The EMS is an economic-based single objective optimization problem expressed as a mixed integer linear programming (MILP) problem. On the contrary, a mixed integer nonlinear programming (MINLP) approach frames the sizing problem to optimize a techno-economic-based multi-objective function. The interaction between the two problems is that the lower problem is an embedded constraint of the upper problem. Moreover, the candidate solutions from the upper problem are treated as fixed parameters in the lower problem. A multi-objective genetic algorithm is used to execute the overall problem by the MATLAB toolbox. The technical and economic performance parameters, including the investment cost, maintenance, operating costs, and others, are assessed based on the best findings. The key finding demonstrates that the summer week has the lowest total cost and energy loss factor, with around 132,000 $ and 0, respectively. Furthermore, the findings reveal that the entire system's energy loss factor for each sample period is less than 0.0009. This value is substantially less than the permitted limit of 0.01.

Location-pricing problem in a two-echelon supply chain: A behavioral game-theoretic approach

The location-pricing problem is a relatively new class of problems in operations research that aims to solve both the pricing and facility location problems simultaneously. Unlike the classical facility location problem, where the demand of customers is known, in the location-pricing problem, the demand is a function of price. Other factors can also influence the final price of products, including production, transportation and inventory costs, competition and agents' behavior, which make the location-pricing problem complex. To this end, this paper proposes a bi-level mixed integer nonlinear programming model for the location-pricing problem in a two-echelon supply chain with players who make decisions based on their social preferences in a competitive market. The social preferences of players are a topic of great importance in the field of behavioral game theory. If the supply chain players care about the profits of other competitors in their utility functions, the simplifying assumption of pure selfishness must be corrected. It is predicted that considering social preferences for supply chain members can lead to better outcomes and payoffs. This paper proposes a single-to-multiple Stackelberg-Nash game in which a focal company serves as the leader, responsible for the location, capacity allocation, pricing, and production/distribution planning of manufacturers. Additionally, there are two competing retailers, as followers who base their pricing strategies on social preferences in order to gain a larger market share. The competition between the retailers determines the demands of customers. The effects of various parameters are then examined using a sensitivity analysis, and some economic and managerial insights are provided. The findings indicate that when both retailers exhibit status-seeking behavior in the market, average market demand increases as a result of lower retail and wholesale prices, resulting in an increase in the profit margins of the supply chain and agents compared to a model with self-interested players.

Grid-friendly peer-to-peer transactive energy market for inhibiting players’ strategic behavior

With the development of distributed renewable energy resources (RERs) in smart cities, the peer-to-peer (P2P) transactive energy market has become an effective way for RER owners to consume the surplus produced power. Nevertheless, the influence of grid usage costs in P2P energy trading has not been considered in prosumers’ strategic behaviors. To solve this issue, we analyze the network usage of P2P energy transactions and model a Nash–Stackelberg game to simulate the trading process of market participants. Firstly, we assume the prosumers can join the P2P energy market and determine their roles freely. The market manager adjusts the grid usage prices to influence the prosumers’ exchanged energy amounts. We then establish a monolithic VCG mechanism to inhibit the prosumers’ strategic behaviors while reducing the computational difficulty of the VCG mechanism. Finally, we develop a best-response (BR)-based algorithm to solve one energy prosumer’s cost minimization subproblem at a time to seek market equilibrium. Since the energy prosumers and market manager interact on the trading amount, the prosumers’ responses are modeled as bi-level mixed-integer nonlinear programming (BLMINLP). We employ the generalized Benders decomposition method to reduce the solution difficulty. Case studies verify the theoretical properties and show the practicability of the proposed algorithms.

Location-pricing decisions with carbon emission considerations: A behavioral game-theoretic approach

The location-pricing problem aims to solve pricing and facility location problems at the same time. While the demand of customers is known in the classical facility location problem, it is a function of price in the location-pricing problem. Moreover, some other factors, such as production, transportation and inventory costs, and competition, can affect the final price of products, making the location-pricing problem more complicated. Considering the necessity of incorporating environmental aspects into supply chain design and the importance of considering the behavioral characteristics of decision-makers, this study addresses the location-pricing problem based on these two significant perspectives. In this paper, there is a lead company responsible for locating and planning a set of manufacturers in the first echelon. On the other hand, the focal manager controls the total emissions and sets the wholesale price. In the second echelon, there are two retailers that base their pricing strategies on social preferences in order to gain a larger market share. The competition between the retailers determines the demands of the customers. The modeling approach is based on the Stackelberg-behavioral Nash game and is formulated as a bi-level mixed-integer nonlinear programming model. The problem is solved using a combination of an analytical solution method and the utilization of a commercial optimization solver. The proposed model is assessed using a real case study in the energy sector. The results indicate that deviating from purely selfish behavior in the pursuit of profit can lead to beneficial outcomes. Generally speaking, price competition between status-seeking retailers results in lower prices for the leader and followers compared to the standard game theory approach. This leads to an increase in average demand in the market, which results in increased profit for members and consequently improves the performance of the whole system. Also, it is worth mentioning that the status-seeking behavior of retailers results in a decrease in retail prices, which improves social welfare.

A data-driven Stackelberg game approach applied to analysis of strategic bidding for distributed energy resource aggregator in electricity markets

In the context of the participation of distributed energy resource (DER) aggregators in day-ahead energy and reserve markets, most of the existing analysis models of strategic bidding behavior of DER aggregators ignore the consideration of the safe operation of distribution systems. To this end, this paper builds a one-leader multi-follower Stackelberg game model considering the safety checking of distribution systems with the discrete features of capacitor banks and tap ratio of the substation transformer in the lower level. Mathematically, this model constitutes a bi-level mixed-integer nonlinear programming (BMINLP) problem. We convert the BMINLP model to a bi-level mixed-integer linear programming (BMILP) model based on KKT conditions, strong duality theorem, and big-M approach, which is still difficult to be solved directly. As a result, we propose a customized data-driven algorithm based on sampling without tuning hyper-parameters and an improved linear surrogate function method, to accurately fit the corresponding constraints of the lower-level model of BMILP problem, and then transform the BMILP problem into a single-level mixed-integer linear programming (MILP) problem to solve. The simulation of data-driven algorithm and comparison with the particle swarm optimization algorithm shows the high practicability of the bi-level Stackelberg game model and the effectiveness of the data-driven algorithm.

A deterministic bounding algorithm vs. a hybrid meta-heuristic to deal with a bilevel mixed-integer nonlinear optimization model for electricity dynamic pricing

In the electricity retail market, the retailer company aims to determine the optimal time-of-use (ToU) prices to maximize profits resulting from buying energy in organized (long-term, day-ahead, balancing) markets and selling it to consumers. Therefore, the retailer should take into account the consumer’s demand response actions to minimize the electricity bill in face of time varying prices. In this paper, this problem is formulated as a bilevel mixed-integer nonlinear programming model in which the retailer is the leader and the consumer is the follower. The consumer’s problem encompasses the integrated optimization of all home energy resources, considering re-scheduling appliance operation, charging/discharging of electric vehicle and stationary batteries, local microgeneration and (buying and selling) exchanges with the grid. The accurate physical modelling of appliance operation to generate effective load scheduling solutions imposes a high computational burden. Two algorithms are proposed to address this problem: a deterministic bounding algorithm (DBA) using an optimal-value-function approach for bilevel optimization, and a hybrid meta-heuristic using a particle swarm optimization algorithm to tackle the upper-level problem that calls an exact mixed-integer linear programming solver to deal with the lower-level problem. In the framework of the DBA, three different techniques were implemented to deal with the nonlinearities arising from the products of integer and continuous variables (bilinear terms): 1) solving the (non-convex) subproblems of DBA using a mixed-integer nonlinear solver, 2) using the McCormick envelopes to approximate the bilinear terms by linear ones, and 3) expressing the integer variables by binary ones and linearizing the bilinear terms using an exact form. Computational experiments are presented and discussed for real data settings of the problem under study considering a computational budget to compare the different algorithms and techniques employed. The results showed that the DBA with the approximate linearization technique (2) leads to the best solutions.

Plantwide control design using latent variables: An integration between control allocation and a measurement combination approach

This paper presents a plantwide control design methodology based on a novel structure which consists of a decentralized strategy complemented by a control allocation (CA) module and a measurement combination (MC) block. Taking into account a principal components analysis (PCA) selection approach, the CA and MC modules perform a dimensional reduction of the original input–output variable space in order to obtain sets of latent variables (or principal components) as control actions and controlled variables. The use of principal components in the controller design provides several interesting features given that: (i) the conditioning of the subsystem to be controlled can be improved, (ii) when performing combinations of variables, the CA and MC modules act as steady-state decouplers and thus an apparently diagonal process is obtained, which favors the reduction of the variables interaction and the pairing problem is automatically solved, and (iii) they allow to naturally handle nonsquare systems. The proposed design procedure is implemented through a multiobjective bilevel mixed-integer nonlinear programming (BMINLP) optimization problem. The leader problem is based on the minimization of three functional costs: 1- the well-known sum of squared deviations (SSD) index, 2- the number of selected manipulated variables (actuators), and 3- the number of selected measurements (sensors). The inner optimization minimizes the relative gain array number (RGAN). This provides a good trade-off between the degree of conditioning/controllability and the complexity/cost of the resulting system. This problem is efficiently solved through genetic algorithms and allows to perform: (i) the selection of the manipulated variables (actuators) and the measurements (sensors) to be used, (ii) the computation of the matrices that characterize the CA and MC modules, and (iii) the stability analysis of the multivariable control structure. The overall design procedure only requires steady-state models of the process. The Tennessee Eastman case study is considered for the simulation and performance evaluation of the proposed solutions.

Hierarchical Optimization of Charging Infrastructure Design and Facility Utilization

This study proposes a bi-level optimization program to represent the electric vehicle (EV) charging infrastructure design and utilization management problem with user-equilibrium (UE) decisions. The upper level aims to minimize total facility deployment costs and maximize the revenue generated from EV charging collections, while the lower level aims to minimize the EV users’ travel times and charging expenses. An iterative technique is implemented to solve the bi-level mixed-integer non-linear program that generates theoretical lower and upper bounds to the bi-level model and solves it to global optimality. A set of conditions are evaluated to show the convergence of the algorithm in a finite number of iterations. The numerical results, based on three demand levels, indicate that the proposed bi-level model can effectively determine the optimal charging facility location, physical capacity, and demand-responsive pricing scheme. The average charging price in medium demand level is increased by 38.21% compared to the lower level demand due to the surge in charging needs and highly utilized charging stations.

Competitive facility location problem with foresight considering discrete-nature attractiveness for facilities: Model and solution

This paper addresses a bi-level mixed-integer nonlinear programming (MINLP) model for the competitive facility location problem in a closed-loop supply chain (CLSC), in which a firm (i.e., leader) aims at entering a market by locating new distribution and collection facilities, where a competitor (i.e., follower) already exists. The goal is to find the location and attractiveness of each facility going to be established by the leader who seeks to maximize its profit while also taking the follower’s response into account. The attractiveness of each facility is a function of integer variables related to the facility’s characteristics. Customer behavior is considered to be probabilistic based on the Huff gravity-based rule. To globally optimize the model, a procedure that handles the discrete decisions of the follower’s problem is proposed. Afterward, by replacing the inner level convex program with its corresponding Karush–Kuhn–Tucker (KKT) conditions, the bi-level MINLP is converted into a single-level MINLP model, optimized by an improved branch-and-refine algorithm. Numerical experiments on randomly generated instances are conducted to illustrate the model’s applicability. Moreover, through a computational analysis of the proposed model, the amount of gain the leader makes and the follower loses due to foresight in the competition are calculated.

Optimal sizing of renewables and battery systems for hybrid AC/DC microgrids based on variability management

This paper proposes an explicit approach to optimal sizing of the integrated battery energy storage systems in hybrid renewable AC/DC islanded microgrids, where the economic impacts of variability management (in concerning operational volatility and uncertainty) are comprehensively investigated. To quantitatively measure the variability levels at different time scales, two indices (i.e. volatility management index and uncertainty management index) are firstly proposed from the perspectives of system planning and operation. Moreover, the probability boundaries of both indices are derived to facilitate the initial system design processes. In particular, the concerned problem is formulated as a bi-level mixed-integer nonlinear programming model to counterbalance the variability and its induced cost at both planning and operation stages. Finally, an original decomposition-coordination algorithm based on the dual decomposition and alternate iterative strategies is proposed to ensure the efficient convergence of the solution process for the constructed mathematical model. Numerical simulations are carried out by analyzing the impacts of various factors associated with optimal sizing solutions and comparing with state-of-the-art approaches. Results show that the proposed sizing technique can facilitate designers finding the optimal capacity combination of renewables and batteries under preset renewables penetration, capital cost, and system variability levels.

Optimal Reserve Management of Electric Vehicle Aggregator: Discrete Bilevel Optimization Model and Exact Algorithm

This paper investigates the day-ahead optimal reserve management problem of electric vehicle (EV) aggregator. Geographically dispersed EVs are coordinated by the aggregator to participate in the day-ahead reserve market. A bilevel model is proposed to formulate the interaction between the aggregator and the EV owners. In the upper level, the EV aggregator aggregates the reserve capacity provided by the EV owners and then bids in the reserve market. In the lower level, the EV owners decide their optimal energy charging/discharging and reserve capacity based on the reserve price released by the aggregator. Compared with existing works, our proposed bilevel model is more practical. To be specific, we take into account the exclusive right constraint for accessing EV battery, i.e., the battery cannot be simultaneously accessed by the EV owner and the aggregator. This practical model leads to a bilevel mixed integer nonlinear program, which is difficult to solve because the lower level problem incorporates nonconvex integer variables. A novel exact algorithm is developed to solve it and the finite convergence is proved. Comprehensive case studies demonstrate the economic merits of our proposed model to both the aggregator and the EV owners. We also compare the solution optimality with the state-of-the-art approach and thus validate the effectiveness of our proposed exact algorithm.

Min-Max Optimal Design of Two-Armed Trials with Side Information

In this work, we study the optimal design of two-armed clinical trials to maximize the accuracy of parameter estimation in a statistical model, where the interaction between patient covariates and treatment are explicitly incorporated to enable precision medication decisions. Such a modeling extension leads to significant complexities for the produced optimization problems because they include optimization over design and covariates concurrently. We take a min-max optimization model and minimize (over design) the maximum (over population) variance of the estimated interaction effect between treatment and patient covariates. This results in a min-max bilevel mixed integer nonlinear programming problem, which is notably challenging to solve. To address this challenge, we introduce a surrogate optimization model by approximating the objective function, for which we propose two solution approaches. The first approach provides an exact solution based on reformulation and decomposition techniques. In the second approach, we provide a lower bound for the inner optimization problem and solve the outer optimization problem over the lower bound. We test our proposed algorithms with synthetic and real-world data sets and compare them with standard (re)randomization methods. Our numerical analysis suggests that the proposed approaches provide higher-quality solutions in terms of the variance of estimators and probability of correct selection. We also show the value of covariate information in precision medicine clinical trials by comparing our proposed approaches to an alternative optimal design approach that does not consider the interaction terms between covariates and treatment. Summary of Contribution: Precision medicine is the future of healthcare where treatment is prescribed based on each patient information. Designing precision medicine clinical trials, which are the cornerstone of precision medicine, is extremely challenging because sample size is limited and patient information may be multidimensional. This work proposes a novel approach to optimally estimate the treatment effect for each patient type in a two-armed clinical trial by reducing the largest variance of personalized treatment effect. We use several statistical and optimization techniques to produce efficient solution methodologies. Results have the potential to save countless lives by transforming the design and implementation of future clinical trials to ensure the right treatments for the right patients. Doing so will reduce patient risks and reduce costs in the healthcare system.

Collaborative Optimization for Location-Inventory Problem of Commercial Vehicle Distribution Network

This paper studies a bi-level programming model addressing the location-inventory problem which can determine the location of distribution centers and the inventory strategy of distribution centers and stores at the same time. In the location-inventory joint decision model, the order quantity is a continuous decision variable associating the upper and lower models and the remaining decision variables are 0-1 variables. In this paper, a branch and bound method is proposed and the SCIP solver is used to solve the bi-level mixed integer non-linear programming problem. The results show that collaborative optimization can save more costs than the one-by- one optimization method and that the optimal solution has better stability under the condition of demand fluctuation and satisfies the optimal allocation of important customers.

Simultaneous Process Design and Control Optimization using Reinforcement Learning

Abstract The performance of a chemical plant is highly affected by its design and control. A design cannot be accurately evaluated without its controls and vice versa. To optimally address design and control simultaneously, one must formulate a bi-level mixed-integer nonlinear program with a dynamic optimization problem as the inner problem; this is intractable. However, by computing an optimal policy using reinforcement learning, a controller with a closed-form expression can be computed and embedded into the mathematical program. In this work, an approach that uses a policy gradient method to compute the optimal policy, which is then embedded into the mathematical program is proposed. The approach is tested in a tank design case study and the performance of the controller is evaluated. It is shown that the proposed approach outperforms current state-of-the-art control strategies. This opens a whole new range of possibilities to address the simultaneous design and control of engineering systems.

Load Shifting Demand Response in Energy Scheduling Based on Payment Cost Minimization Auction Mechanism

Demand response (DR) is proven to be very efficacious for load mitigation especially in peak time period. On the other hand, DR facilitates both consumers, system operator as well as producers to moderate their payments while reducing system operating costs. Offer cost minimization is currently used as the clearing mechanism associated with locational marginal pricing scheme to determine the consumers’ payments. Such clearing and pricing mechanisms are inconsistent as the system costs is being minimized, while the payments are calculated based upon marginal prices. Payment cost minimization auction as a price-based clearing mechanism is envisaged to be an effective alternative to solve such a crucial issue. This paper shows how to include DR in PCM mechanism to further reduce the consumers’ payment. It facilitates utilizing price responsive consumers for load shifting DR in a PCM auction. The optimization problem is modeled as a mixed-integer nonlinear bi-level programming. Duality theorem, KKT conditions and integer algebra are used to convert such a complicated optimization problem to a single level MILP problem. This problem is then solved by CPLEX in GAMS. The impacts are studied by implementing the proposed formulation to solve the clearing problem in the case studies deriving promising numerical results.

Human error unplanned downtime inferring and job-operator matching based on inverse optimal value method

In order to solve the problem that even the optimal objective value cannot reach the target cost, this paper constructs an inverse optimal value model for operator assignment. Given a forward operator assignment program, a target optimal operator-induced cost value, and a set of feasible downtime vectors, determine a downtime vector such that the corresponding optimal objective value of the forward operator assignment model is closest to the target cost. This research transforms the inverse optimal value model of operator assignment into a corresponding 0–1 mixed integer nonlinear bilevel programming model. Considering the two kinds of decision variables, a hybrid parthenogenetic particle swarm optimization algorithm is proposed. Then the combination of downtime and “operator-job” assignment pair such that the corresponding optimal objective value of the assignment problem is closest to the target cost is obtained. The application results show that, the inverse optimal value method can reach the target cost, reduces total operator caused loss by 5%, reduces production loss by 34.3%, reduced scrap by 5.1%, while the labor costs are increased by 8.3%. It solves a kind of problem when the optimal value still cannot reach the target goal and the parameters are adjustable. It provides a decision tool for goal driven operator assignment.