Mixed-integer Nonlinear Optimization Problem Research Articles

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.

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 the 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 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.

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].

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.

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.

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.

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.

A test instance generator for multiobjective mixed-integer optimization

AbstractApplication problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments.

A flexibility-based multi-objective model for contingency-constrained transmission expansion planning incorporating large-scale hydrogen/compressed-air energy storage systems and wind/solar farms

The remarkable arrival of uncertainties given by the fluctuations in the output of renewable energy sources (RESs) calls for a more flexible power grid. In confronting this issue, large-scale energy storage systems play a vital role in achieving higher flexibility. In this regard, the authors of this study present here a new multi-objective model for contingency-constrained transmission expansion planning that incorporates large-scale hydrogen/compressed-air energy storage systems and wind/solar farms to simultaneously boost both supply-demand-related flexibility (SDFX) and grid-related flexibility (GDFX). The proposed model is formulated as a non-convex mixed-integer nonlinear master-slave optimization problem. The planning objectives, modeled through the master problem, are intended to minimize the investment and operation costs, while maximizing the SDFX enhancement metric minus the GDFX degradation metric. The slave problem, however, aims to maximize the community welfare function (CIWF) and microgrid welfare function (MIWF) under both normal and N-1 operating conditions. To solve the resulting non-convex mixed-integer nonlinear master-slave optimization problem, a potent symphony orchestra search algorithm (SOSA)—empowered with a multi-computational-step, multi-dimensional, multiple-homogeneous structure—is employed to allow better diversification and exploration. Following the determination of optimal solutions, a conservative-based fuzzy satisfying approach is applied to characterize the best trade-off solution that suits cost-effectiveness and flexibility requirements. The newly developed model was implemented on the IEEE 24-bus and IEEE 118-bus test grids, and the simulation results proved its effectiveness and practicality.

Fairness in maximal covering location problems

This paper provides a mathematical optimization framework to incorporate fairness measures from the facilities’ perspective to discrete and continuous maximal covering location problems. The main ingredients to construct a function measuring fairness in this problem are the use of (1) ordered weighted averaging operators, a popular family of aggregation criteria for solving multiobjective combinatorial optimization problems; and (2) α-fairness operators which allow generalizing most of the equity measures. A general mathematical optimization model is derived which captures the notion of fairness in maximal covering location problems. The models are first formulated as mixed integer non-linear optimization problems for both the discrete and the continuous location spaces. Suitable mixed integer second order cone optimization reformulations are derived using geometric properties of the problem. Finally, the paper concludes with the results obtained from an extensive battery of computational experiments on real datasets. The obtained results support the convenience of the proposed approach.

Towards global parameter estimation exploiting reduced data sets

We focus on deterministic global optimization (DGO) for nonconvex parameter estimation problems. Realistic and accurate solutions often require a fitting against very large measurement data sets, resulting in intractable size for DGO. Thus, we aim at accelerating the branch-and-bound algorithm by using reduced data sets for constructing valid bounds. We focus on fitting the equation of state for propane, which is of high interest for the chemical industry. The resulting estimation problem is a challenging nonconvex mixed-integer nonlinear optimization problem. We investigate the validity of using reduced data sets by comparing how the lower and upper bounds change when replacing the full data set with different reduced data sets. We account for regions with high and low quality fits by considering the results for the whole feasible region and 100 different subregions. Our results indicate that both regions containing solution candidates and regions containing only low quality fits can be identified based on reduced data sets. Moreover, we observe that the average CPU time per branch-and-bound iteration typically decreases if reduced data sets are used.

DG-integrated stochastic multiyear distribution expansion planning considering fault current-limiting high-temperature superconducting cables

In this paper, the authors present a new stochastic multiyear framework for distribution expansion planning (DEP) that integrates fault current-limiting high-temperature superconducting cables (FCL-HTSCs) and distributed generations (DGs). The proposed framework, from a new perspective, is capable of analyzing and comparing the efficiency of the FCL-HTSCs with other, more generic cables under different short-circuit conditions. The objective function to be minimized isthe net present value of the total investment costs for reinforcing and/or installing substations, feeders, FCL-HTSCs, and DGs, as well as operation and maintenance costs. The framework also considers Kirchhoff’s current and voltage laws, operational limits on equipment capacities, voltage limits, as well as financial and discrete logical constraints. As a comprehensive overview, the authors have developed: (i) a graph-based bi-conditional scheme to ensure radiality requirements and (ii) a multi-horizon fractional information-gap decision theory (MH-FIGDT) to address the stochastic nature of anticipated load demand, power production of the DGs, and a capital expenditure budget. The resulting mixed-integer nonlinear optimization problem is solved by an integer-coded melody search algorithm (ICMSA) empowered with a multi-computational-stage multi-dimensional multiple-homogeneous structure that offers better performance in diversification and exploration. Case studies using the standard 27-node and realistic, large-scale 104-node distribution grids are provided here in order to show the effectiveness of the proposed framework. Based on simulation results, several conclusions can be drawn: (i) the MH-FIGDT critical cost and/or MH-FIGDT critical deviation coefficient chosen by the distribution grid operator play a key role in achieving robust expansion plans against the given uncertainty set; (ii) the incorporation of the FCL-HTSCs into the DEP problem offers a satisfactory compromise between the required amount and capacity of the equipment and total investment costs; (iii) the functionality of the FCL-HTSCs outperforms the generic cables in mitigating feeder congestion and excessive fault currents under normal and short-circuit conditions, respectively; and, (iv) the variations in fault currents caused by rising DG penetration are keep below the nominal interrupt rating of the circuit breakers, which speaks to the suitability of the proposed framework.

Voltage regulation and power loss mitigation by optimal allocation of energy storage systems in distribution systems considering wind power uncertainty

Due to the ever increasing in the energy demand and the growth of the pollution problem, further interest in renewable energy becomes prominent, especially wind energy. However, the intermittent nature of the wind speed results in the system's insecurity problems. Integrating the energy storage system (ESS) optimally is the best so far solution. In this paper, a probabilistic discretizing method is derived and generalized to consider the uncertainties of the wind generators. Moreover, a mixed integer non-linear optimisation problem is constructed comprises three objective functions to be minimized simultaneously. The objectives comprising the total operation and planning cost of ESSs, the average voltage deviation and average power losses. The solving method is a hybrid algorithm incorporates the non-dominated sorting genetic algorithm (NSGAII), multi-objective particle swarm optimisation (MOPSO) and a decision-making algorithm. The aim is to determine the optimal location and sizing of ESSs considering their reactive power capability. 69 radial bus distribution system is adopted to demonstrate the efficacy of the proposed hybrid algorithm and performing some case studies. The simulation results verified the capability of ESSs to enhance the voltage profile and reduce the power losses, notably when reactive power capability is examined.