Mixed-integer Nonlinear Optimization Problem Research Articles

Distillation column optimization: A formal method using stage-to stage computations and distributed streams

This work addresses the complexities of optimizing the number of stages in a distillation column, which typically lead to challenging non-linear mixed-integer optimization problems. To simplify this, we employ distributed streams, thereby eliminating discrete degrees of freedom. To avoid sophisticated initialization procedures, the optimization problem is reformulated by employing a sequence of stage-to-stage calculations, each reduced to maintaining only the MESH (mass, equilibrium, summation, heat) equations for a single stage.Our numerical experiments show the efficiency and stability of solving the simplified optimization problem in various scenarios, including single and multiple distillation columns. For a single column scenario, we compare the accuracy of our optimization method with a full enumeration approach. Additionally, for a pressure swing flowsheet designed to separate an azeotropic mixture, we illustrate potential energy savings by optimizing a stage distribution versus using a predetermined stage distribution.

Optimal Reconfiguration of Bipolar DC Networks Using Differential Evolution

The search for more efficient power grids has led to the concept of microgrids, based on the integration of new-generation technologies and energy storage systems. These devices inherently operate in DC, making DC microgrids a potential solution for improving power system operation. In particular, bipolar DC microgrids offer more flexibility due to their two voltage levels. However, more complex tools, such as optimal power flow (OPF) analysis, are required to analyze these systems. In line with these requirements, this paper proposes an OPF for bipolar DC microgrid reconfiguration aimed at minimizing power losses, considering dispersed generation (DG) and asymmetrical loads. This is a mixed-integer nonlinear optimization problem in which integer variables are associated with the switch statuses, and continuous variables are associated with the nodal voltages in each pole. The problem is formulated based on current injections and is solved by a hybridization of the differential evolution algorithm (to handle the integer variables) and the interior point method-based OPF (to minimize power losses). The results show a reduction in power losses of approximately 48.22% (33-bus microgrid without DG), 2.87% (33-bus microgrid with DG), 50.90% (69-bus microgrid without DG), and 50.50% (69-bus microgrid with DG) compared to the base case.

A hierarchical optimization approach to maximize hosting capacity for electric vehicles and renewable energy sources through demand response and transmission expansion planning

Within the scope of sustainable development, integrating electric vehicles (EVs) and renewable energy sources (RESs) into power grids offers a number of benefits. These include reducing greenhouse gas emissions, diversifying energy sources, and promoting the use of green energy. Although the literature on hosting capacity (HC) models has grown, there is still a noticeable gap in the discussion of models that successfully handle transmission expansion planning (TEP), demand response (DR), and HC objectives simultaneously. Combining TEP, DR, and HC objectives in one model optimizes resource use, enhances grid stability, supports renewable and EV integration, and aligns with regulatory and market demands, resulting in a more efficient, reliable, and sustainable power system. This research presents an innovative two-layer HC model, including considerations for TEP and DR. The model determines the highest degree of load shifting appropriate for incorporation into power networks in the first layer. Meanwhile, the second layer focuses on augmenting the RES and EVs’ hosting capability and modernizing the network infrastructure. System operators can choose the best scenario to increase the penetration level of EVs and RESs with the aid of the proposed model. The proposed model, which is formulated as a multi-objective mixed-integer nonlinear optimization problem, uses a hierarchical optimization technique to identify effective solutions by combining the particle swarm optimization algorithm and the crayfish optimizer. When compared to traditional methods, the results obtained from implementing the proposed hierarchical optimization algorithm on the Garver network and the IEEE 24-bus system indicated how effective it is at solving the presented HC model. The case studies demonstrated that integrating DR into the HC problem reduced peak load by 10.4–23.25%. The findings also highlighted that DR did not impact the total energy consumed by EVs throughout the day, but it did reshape the timing of EV charging, creating more opportunities for integration during periods of high demand. Implementing DR reduced the number of projects needed and, in some cases, led to cost savings of up to 12.3%.

A centralized EMPC scheme for PV-powered alkaline electrolyzer

Photovoltaic (PV)-powered alkaline electrolyzer system (PVPAES) is an advanced technique to convert the off-grid and intermittent PV-based solar energy into storable and transportable electrolyzer-based hydrogen energy with zero carbon emissions. However, it is difficult to realize the coordinated control of the off-grid PV module and the alkaline electrolyzer, due to the multiple timescale dynamics. To address this challenge, the PVPAES is decomposed into the slow part and fast part based on the dynamic time scale. Exploiting the decomposed subsystems, the slow one is assumed to be managed well by the auxiliary controller. For the fast one, a centralized economic model predictive control (CEMPC) scheme is constituted. This CEMPC integrates the energy management system and local feedback control into a single optimal control framework. A mathematical model of the PVPAES is established, on the basis of which the CEMPC directly adopts the economic indices as the cost function to realize the flexible power point tracking, power supply-demand balance, and dynamic economic optimization. Moreover, the inherent strong nonlinearity of PVPAES results in the nonconvex mixed-integer nonlinear programming optimization problem in the CEMPC. The exhaustive search algorithm utilizing finite converter switching states is adopted to achieve the global economic optimum. The effectiveness of the proposed CEMPC controller is illustrated through simulations under varying irradiance conditions.

Physics-informed deep learning and linear programming for efficient optimization of combined cycle power plants

Efficient operation of combined cycle power plants (CCPP) involves solving mixed-integer nonlinear optimization problems, constrained by thermodynamic differential equations, thermal constraints, and temporal dependencies between variables. The optimization problem aims to maximize profit for day-ahead operation by minimizing short-term and long-term costs, and accounting for volatility in ambient conditions and energy price. In operational terms, the challenge lies in developing an efficient and robust optimization algorithm capable of determining optimal values for thousands of temporal variables. It must also have the capacity to evaluate a multitude of scenarios encompassing various weather and market conditions. To address this challenge, this paper proposes a hybrid optimizer that utilizes deep learning models trained on historical operational data of a real CCPP, along with a surrogate physics-driven piecewise-linear model of CCPP, to determine the optimal values of thousands of control setpoints, with strong temporal co-dependencies, within a two-minute runtime required by industry constraints. A multi-resolution architecture is proposed to efficiently implement this optimizer. The algorithm sequentially solves the problem in coarse and fine resolutions to accelerate convergence. Final results are checked for feasibility using pre-trained deep learning models, ensuring accuracy and validity. The study shows that the proposed model provides comparable results to the exact solution while reducing computational costs, up to 95%. This research has practical implications for power plant system operators, as well as traders and dispatchers, enabling them to make optimal decisions for day-ahead contracting purposes and CCPP operation in a reasonable time, ultimately leading to more efficient and cost-effective power plant management. Furthermore, the methodology introduced in this work could potentially be extended to other industrial applications that require large-scale, complex optimizations, where the precise governing principles of a process can be approximated by piecewise-linear functions.

An optimal approach of pure/mixed CO2 sequestration with producing hydrated natural gas: Physical insights and validation

In this contribution, an optimal approach is developed to sequestrate pure/mixed CO2 by dislodging the hydrated natural gas, the cleanest fossil fuel, thereby catching two birds by one stone. For depicting its intricacies, a transient thermo-kinetic swapping model is developed with considering first time the effect of permeability and water saturation, along with porosity involved in sediments of hydrate reservoir. The formulation takes into account various fundamental issues having practical relevance, like porous particles with uneven size and shape, surface dynamics involved in phase transformation, non-unity reaction order (i.e., nth order), among others. A mixed-integer nonlinear multi-objective sequential optimization (MOSO) problem is articulated within the framework of a robust global optimizer (i.e., nondominated sorting genetic algorithm II) to deal with two conflicting objectives, concerning minimization of absolute average relative deviation and maximization of swapping efficiency. This optimal approach is validated with laboratory and field data, showing its outperformance over the latest models at diverse geological conditions. The proposed model framework is employed for lab-scale cases to identify the optimal guest gas (CO2/N2) ratio that secures the maximum swapping efficiency. Simulation results suggest that the presented theoretical model is rigorous in interpreting these swapping dynamics in hydrates.

On Constrained Mixed-Integer DR-Submodular Minimization

Diminishing returns (DR)–submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DR-submodular function with continuous and general integer variables under box constraints and, possibly, additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems. Funding: This work was supported by the Office of Naval Research Global [Grant N00014-22-1-2602].

A distributed randomized method for the identification of switched ARX systems

SummaryThe identification of switched systems amounts to a mixed integer nonlinear optimization problem, where the continuous variables are associated to the model parameterizations of the different modes, and the discrete ones are related to the switching signal (each data sample is assigned to a mode, and switching occurs when the mode assignment changes over time). In the batch form of the identification problem, the combinatorial complexity increases exponentially with the size of the training set, which makes the precise identification of the switching signal the most challenging task in the identification problem. To tackle this complexity we propose a distributed optimization approach, based on the solution of multiple instances of a much simpler problem, where switching can occur only at specific time instants (different for each subproblem), and an information sharing mechanism that preserves likely switching times to improve the local solutions. We employ an adapted version of a previously developed randomized algorithm to solve the individual subproblems. Another important feature of the proposed method is an a posteriori heuristic correction method, that is applied to further refine the switching locations based on the estimated local models before the information sharing phase. The performance of the proposed algorithm is analyzed and compared with other methods on synthetic datasets.

Model predictive control of bidirectional heat transfer in prosumer-based solar district heating networks

District heating networks with decentralized heat production are ideally suited to include a high share of renewable energy sources for heat production in urban areas with limited space. A new concept is a prosumer-based district heating network in which some or even all buildings are equipped with decentralized building-level heat storages and heat generation plants. To exploit the full potential of the interconnected heating network, buildings with surplus heat are intended to transfer heat to buildings with heating demands to prevent the activation of the central heating plant. This work presents an initial feasibility study on using a model predictive control strategy to manage bidirectional heat transfer between buildings in a district heating network. We formulate this problem as an optimal control problem, incorporating binary decision variables for each potential heat transfer connection. This results in a mixed-integer nonlinear optimization problem that is hard to solve. The problem is solved with a fast gradient-based optimization algorithm in combination with a combinatorial integral approximation strategy. A case study concerning a residential heating network with building-level solar thermal collectors and storage tanks is carried out. The optimized operation is compared to the real operation obtained from one month of measurements. The results show that the optimized strategy with bidirectional heat transfer can exploit the total heat generated in the whole network. The central heating provider is only required when the heat produced or stored by all the buildings is insufficient to meet the total heat demand within the network. In the case study, this leads to a reduction of almost 75% of the required thermal energy from the central heat provider.

Lua APIs for mathematical optimization

In this paper we present our Lua programming language libraries that enable modeling and solving mixed integer linear and nonlinear mathematical optimization problems. On one side, these libraries provide a framework for developing algorithms written in Lua that use certain well known solvers within more complex procedures. On the other side, they facilitate the transformation of both input and output data of a mathematical programming problem, in order to compute the standardized coefficients that have to be transmitted to the solvers or prepare solution of solved problems for further data processing. Both of these use cases, together with Lua programming language simplicity, versatility and performance, make Lua a suitable programming language for use in scientific and engineering researches.

Sizing of a Scattered Housing Microgrid in a Remote Rural Area

Power solutions for remote rural regions are mainly based on generator sets or individualized generation systems per household. These solutions have low reliability and high financial and environmental costs, especially those involving fossil fuels. However, current solutions incorporate hybrid systems with renewable energy sources to reduce pollutant gas emissions and costs and increase the reliability and robustness of power generation. These hybrid systems can be considered micro-grid (MG) and could contribute to energizing remote non interconnected areas. However, implementing MGs in these regions must challenge energizing dispersed households. Thus, both the sizing of the generation and distribution systems must be considered to guarantee the correct operation of the MG and a low cost. This paper explores coordinated sizing between the power sources and the distribution system to address the challenges of energizing remote areas with dispersed loads through MG. It presents a literature review on energization strategies for remote regions and proposes coordinated MG sizing. Finally, it applies coordinated sizing to a case study in Colombia. The sizing is approached as a mixed-integer nonlinear integer programming optimization problem and is solved by a particle swarm optimization (PSO) algorithm.

Long-term reliability evaluation of integrated electricity and gas systems considering distributed hydrogen injections

Power-to-gas facilities consume surplus renewable electricity generation to produce alternative gases, such as green hydrogen. They can be injected into, and transported by the gas network for further use, which is a promising way toward a low-carbon energy system. However, injecting alternative gases into the gas systems can adversely affect the gas composition and the lifespan of components (e.g., gas pipelines), and may threaten the reliability of the entire integrated electricity and gas systems (IEGS) in the long term. To address this issue, this paper proposes a long-term reliability evaluation method for IEGS with distributed hydrogen injections. First, new reliability indices are proposed to evaluate both gas adequacy and gas interchangeability under uncertainties. Then, a multi-state reliability model of the pipeline is developed to characterize the corrosion evolution and hydrogen embrittlement in the long term. A contingency management scheme (CMS) is devised to minimize load curtailments and gas interchangeability deviations under component failures. Moreover, several reformulation techniques are tailored to convexify the original two-stage mixed-integer nonlinear CMS optimization problem. An analytical reliability evaluation method embedded with a system state reduction technique is designed to evaluate the long-term reliability of the IEGS more efficiently. Finally, the IEEE 24 bus Reliability Test System and the practical Belgium gas system are used to validate the proposed method. The numerical results show that the injection of alternative gas could jeopardize the reliability of the studied IEGS by 39.73% in the long term. However, we have observed a critical time window (the 8th–9th year), in which if we conduct the inline inspection and maintenance more frequently, the reliability could be improved by up to 53.31%. These results suggest that the injection of alternative gas is beneficial, but should be carefully regulated to maintain the reliability of IEGS.

Energy management considering multiple power markets and microgrid storage

The operational cost of a microgrid is significantly influenced by the response of storage systems and the complexities of the power market’s tariff structures. This paper addresses the challenges arising from the coexistence of new market entries and traditional tariffs, which contribute to a complex market environment. To tackle this issue, the paper establishes a microgrid market environment encompassing four types of tariffs. By modeling the response of electric storage and cold storage in a microgrid, the study formulates a non-linear mixed-integer optimization problem. Numerical studies are then conducted to verify the model and analyze market performance. The results reveal a trade-off in behavior among different market entries when optimizing the total cost of microgrid operation. These findings shed light on the complexities and trade-offs involved in microgrid operational cost optimization within a diverse market environment, offering valuable insights for market participants.

Bilevel Programming for Manufacturers Operating in an Omnichannel Retailing Environment

In this article, we present a joint production, pricing, and inventory control problem that can help manufacturers manage revenue in an omnichannel environment. This article puts forward a multiperiod Stackelberg game between a manufacturer and a retailer, assuming the former to be the leader. The leader–follower game is formulated as a mixed-integer nonlinear bilevel optimization problem wherein both players seek to maximize their respective profits. The article envisions an omnichannel retailing environment where online, offline, direct, and drop-shipping channels coexist. It then investigates how enabling showrooming and webrooming on the drop-shipping channel account can affect the supply-chain profitabilities. The analyses suggest that when the customers are aided with showrooming and webrooming services and are allowed to place drop-shipping orders from retailers’ stores, additional profit can be generated in the supply chain. To solve the proposed bilevel optimization problem, we investigate single-level reduction using Karush–Kuhn–Tucker conditions and hierarchical optimization technique based on the simulated annealing and randomized decomposition solver. To assess the efficacy of the solution techniques, a comparative analysis is carried out. Thereafter, with the aid of numerical experiments and sensitivity analyses, this article draws the key managerial insights for manufacturers.

A Consensus-Based Alternating Direction Method for Mixed-Integer and PDE-Constrained Gas Transport Problems

We consider dynamic gas transport optimization problems, which lead to large-scale and nonconvex mixed-integer nonlinear optimization problems (MINLPs) on graphs. Usually, the resulting instances are too challenging to be solved by state-of-the-art MINLP solvers. In this paper, we use graph decompositions to obtain multiple optimization problems on smaller blocks, which can be solved in parallel and may result in simpler classes of optimization problems because not every block necessarily contains mixed-integer or nonlinear aspects. For achieving feasibility at the interfaces of the several blocks, we employ a tailored consensus-based penalty alternating direction method. Our numerical results show that such decomposition techniques can outperform the baseline approach of just solving the overall MINLP from scratch. However, a complete answer to the question of how to decompose MINLPs on graphs in dependence of the given model is still an open topic for future research. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant TRR 154].

Solving the 1-median problem on a network with continuous demand and demand surplus

In this paper we address the problem of locating a delivery business with a production capacity. It is a variant of the 1-median location problem on a network, in which demand is assumed to be distributed along the edges and nodes of the network. However, instead of assuming that all the demand must be served, the location for a facility is sought minimizing the expected distance to a fraction of the demand.The problem is expressed as a Mixed Integer Nonlinear Optimization problem, solved by means of a geometric Branch and Bound algorithm.

Daily scheduling of generating units with natural-gas market constraints

We address the daily scheduling of generating units (unit commitment) in a system whose production mix includes a large number of natural-gas-fired units. We assume that the natural-gas system, operating under market conditions, does its best to supply the natural-gas for electricity production required by the power system. We formulate this problem as a bi-level model whose upper-level problem represents the scheduling of generating units under a second-order conic relaxation of the AC network constraints, and whose lower-level one represents the clearing of the natural-gas market. The upper-level problem is mixed-integer second-order conic, while the lower-level one is second-order conic as well, but continuous. We replace the lower-level problem by its optimality conditions in the form of primal constraints, dual constraints, and strong duality equality, which results in a single-level nonconvex mixed-integer nonlinear optimization problem. To improve the computational tractability of such formulation, we derive a mixed-integer second-order conic relaxation of the single-level nonconvex mixed-integer nonlinear optimization problem. Such problem is solved using a Benders-type outer-approximation decomposition framework with a master mixed-integer linear problem and two subproblems that are nonlinear but convex. The communication between the master problem and the subproblems is enhanced by incorporating into the master problem linearized information pertaining to the subproblems and valid linear inequalities. The benefits of the proposed model and the effectiveness of the solution procedure are illustrated using the IEEE 24-bus test system coupled with a 20-node natural-gas system and the Central Illinois 200-bus power system coupled with a 48-node natural-gas system.

EOQ with supply disruptions under different advance information regimes

In this paper, we consider the impact of advanced delivery information on the performance of supply chains facing endogenous disruptions. We study a single-echelon EOQ-based inventory model with backorders where the supplier could fail to deliver shipments on time following a Bernoulli process, a measure of the supplier service quality. Two extreme advance information regimes are analyzed: (1) full delivery information (FDI), where the timing of the next delivery is always known, and (2) partial delivery information (PDI), where the timing of the next delivery is known if it coincides with the next planned delivery. For both regimes, we use a natural adaptation of the base-stock policy, having multiple order up-to levels, each corresponding to the level of available information when ordering. We then model the long-run average cost and determine the policy variables that minimize this cost. While the analysis of the FDI model proved straightforward, the PDI model involved the solution of a mixed-integer nonlinear optimization problem. We found closed-form solutions for both models. Numerical results show that, by reducing planning uncertainty, advanced information directly reduces operating costs. They also show that the cost of the case of full information is higher than that of the EOQ model, suggesting the importance of finding reliable suppliers for disruption-free operations.

Resource Allocation for NOMA-Enabled Cognitive Satellite–UAV–Terrestrial Networks With Imperfect CSI

Recently, non-orthogonal multiple access (NOMA)-enabled cognitive satellite-unmanned aerial vehicle (UAV)-terrestrial networks have attracted extensive attention for the advantages of enhancing spectrum efficiency and coping with the exponential growth of access users. In this paper, we propose a joint subchannel assignment and power allocation algorithm for NOMA-enabled cognitive satellite-UAV-terrestrial networks to further enhance the transmission performance, where imperfect channel state information is taken into consideration. Specifically, we formulate a mixed integer non-linear programming resource allocation problem to optimize the sum rate of the secondary network, in which the demands of the interference temperature constraint for primary users, the minimum transmission rate of each secondary user, the maximal transmitter power of the UAV, and the maximal number of secondary users that each subchannel can serve are satisfied. To tackle this tough non-convex problem, we first decouple it into two subproblems, namely, subchannel assignment and power allocation. Next, a heuristic subchannel assignment algorithm and a Taylor series and successive convex approximation-based power allocation algorithm are designed to solve the above subproblems, respectively. By alternately optimizing the sum rate of the secondary network, we finally solve the mixed integer non-linear programming optimization problem. Numerical results reveal the impacts of key parameters on system performance and indicate that our proposed scheme outperforms benchmarks in large-scale networks.

Data-driven multi-period modeling and optimization for the industrial steam system of large-scale refineries

Steam system modeling and optimization contributes toward low energy consumption for large-scale industrial refineries. However, formidable challenges have been generated by steam demand variation, the optimization of CO2 emissions and outsourced steam, the uncertainty of sudden equipment failures, and the introduction of new equipment. To address this issue, this work presents a novel steam system model with periodic process steam demand. The multiple cost factors of CO2 emission treatment and outsourced steam are considered and formulated as a mixed integer nonlinear programming optimization problem. A strategy to discretize the uncertainty of equipment failure into three main scenarios is proposed. The proposed model, strategy, and introduction of new equipment are demonstrated on three cases of industrial refinery steam systems. In case 1, the optimized operating costs have reduced by 1.5%. In case 2, the optimized average operating cost is 1,135,402 CNY/h. In case 3, with the introduction of the aromatics turbocharged turbine (ATT) and the letdown valve, the optimized operating costs are 1,292,182 CNY/h and 1,316,227 CNY/h respectively, confirming the feasibility of introducing ATTs.