Mixed-integer Nonlinear Programming Optimization Research Articles

Optimal operation of a hydrogen refuelling station combined with wind power in the electricity market

An optimisation routine has been developed to analyse the performance of hydrogen refuelling stations combined with wind power operating in electricity markets. This optimisation routine includes the minimum turn down ratio of the electrolyser in its formulation resulting in a mixed integer nonlinear programming optimisation. The optimisation routine has been used to analyse the performance of a hydrogen refuelling station located at the advanced manufacturing park in Rotherham, UK. The performance of the optimisation routine for various scenarios of hydrogen demand and wind power has been assessed. This includes the effect of operating the electrolyser to reduce wind curtailment in a grid constrained scenario. It is found that the optimisation routine is capable of increasing profits when operating in the market, but this is dependent on various factors such as the level of hydrogen demand and wind power.

A MINLP optimization of the configuration and the design of a district heating network: Academic study cases

The aim of this work is to propose a tool for the design assistance of District Heating Network (DHN). Two goals of DHN optimization are handled simultaneously: the optimization of the configuration and its design. The optimization objective is to minimize the global cost of the DHN over 30 years. It includes both operating costs (heating and pumping cost, including thermal losses and pressure drop) and investment costs (line, trench, heating plant, heat exchanger). The formulation leads to a mixed integer non-linear programming (MINLP) problem in steady state. The model is solved with DICOPT within GAMS (around 5s for this study cases).One of the outputs of these academic study cases is the layout of the DHN, supplied in parallel or in cascade: a consumer with hot temperature requirement can supply another consumer with lower temperature requirement. Even a looped network in cascade is optimal (−4.6% total cost reduction) when the cost of the trench is lower than 500 €/m. Furthermore, different structures are optimal (between −4 and −8% of total cost reduction) depending on whether the heat production(s) are decentralized, centralized, isolated collective, renewable or not. Finally the balance between heat loss and pressure drop is detailed.

PAPR Reduction via Constellation Extension in OFDM Systems Using Generalized Benders Decomposition and Branch-and-Bound Techniques

In this paper, a novel constellation extension (CE)-based approach is presented to address the high peak-to-average power ratio (PAPR) problem at the transmitter side, which is an important drawback of orthogonal frequency-division multiplexing (OFDM) systems. This new proposal is formulated as a mixed-integer nonlinear programming optimization problem, which employs generalized Benders decomposition (GBD) and branch-and-bound (BB) methods to determine the most adequate extension factor and the optimum set of input symbols to be extended within a proper quarter plane of the constellation. The optimum technique based on GBD, which is denoted as GBD for constellation extension (GBDCE), provides a bound with relevant improvement in terms of PAPR reduction compared with other CE techniques, although it may exhibit slow convergence. To avoid excessive processing time in practical systems, the suboptimum BB for constellation extension (BBCE) scheme is proposed. Simulation results show that BBCE achieves a significant PAPR reduction, providing a good tradeoff between complexity and performance. We also show that the BBCE scheme performs satisfactorily in terms of power spectral density and bit error rate in the presence of a nonlinear high-power amplifier.

Synthesis of heat exchanger networks using mathematical programming and heuristics in a two-step optimisation procedure with detailed exchanger design

This study makes use of a novel methodology for the synthesis of heat exchanger networks, which is aimed at overcoming the shortcomings associated with the use of shortcut models to represent individual exchangers in the synthesis network. The new approach entails the use of a number of correction factors to get networks which are based on the use of shortcut models, such as the mixed integer non-linear programming (MINLP) stage-wise superstructure (SWS) of Yee and Grossmann (1990) to more closely represent physically achievable heat exchangers and ensure that the MINLP network topology optimisation step of these models converge on a real design, rather than an approximated one. In this paper, the SWS formulation is used for the generation of an initial network after which its objective function is modified to include the correction factors that force its objective function towards the cost of a network whose individual exchangers are designed using methods such as Bell–Delaware and heuristics. The modified objective function includes parameters that modify the areas obtained by the shortcut based MINLP model so as to more closely represent the areas obtained by the detailed models and also includes a novel method for including the number of shells required for each exchanger duty. The correction factors account for pressure drops, Ft correction factors, number of shells, TEMA considerations, and changes to the overall heat transfer co-efficient of each stream match. The methodology is applied to two examples and the solutions are comparable with other solutions obtained in literature and were shown to produce good solutions. The reason that the method is effective is because many potential networks are evaluated during the iterative procedure and the best network, based on the detailed exchanger designs, is chosen. In this way it is possible to use the detailed exchanger designs to “guide” the MINLP optimisation towards more realistic networks and also to generate many different potential networks.

Mixed-integer nonlinear programming based optimal time scheduling of construction projects under nonconvex costs

Optimal project scheduling under nonconvex time-cost relations represents a challenging problem in construction management. The nonconvex time-cost relations may appear in a construction project when several different duration options are available for its activities due to alternative technological processes enabled for their realization or wide accessibility of production resources. The source of nonconvexity of the project scheduling optimization problem can also be the project penalty- or bonus-duration relations arranged within the construction contract. The aim of this paper is to present the mixed-integer nonlinear programming (MINLP) based optimal time scheduling of construction projects under nonconvex costs. For this purpose, the MINLP model was developed and applied. A numerical example from literature and an example of construction project time-cost trade-off analysis under practical nonconvex penalty function are given in the paper to demonstrate advantages of MINLP optimization. The example from literature first presented the capability of the MINLP approach to obtain the optimal solution for difficult, highly combinatorial nonconvex discrete project scheduling problem. Thereupon, the following example revealed that the optimal project time-cost curve may take very nonuniform shape on account of discrete nature of activity direct cost options and nonconvex relation between project duration and total cost. In this way, the presented study intends to provide practitioners with new information from the field of optimization techniques for project scheduling as well as an alternative view on performance of total cost when project duration is changed.

Distribution System Reconfiguration for Annual Energy Loss Reduction Considering Variable Distributed Generation Profiles

In this paper, a new formulation for distribution system reconfiguration (DSR) is proposed for minimizing the annual energy losses considering the variability in active and reactive power demand and distributed generation (DG) profiles. A fuzzy C-means clustering algorithm is used to obtain representative centroids from annual DG and power demand profiles. The DSR study is formulated as a mixed-integer nonlinear programming optimization problem and tested on a 33-bus and an 84-bus system. The minimization is subject to the power balance, bus voltage, and distribution system radiality constraints. Using this formulation, a single optimum configuration that minimizes annual energy losses is found and shown to be different from the optimum configuration, found in previous literature, which focuses on minimizing power losses at peak or average loads. In addition, the prospective advantages of grid automation on DSR are demonstrated to provide further energy loss reduction by including the possibility of interchanging a predefined number of configurations that minimize annual energy losses.

Energy-Efficient Resource Allocation for OFDMA Cellular Networks With User Cooperation and QoS Provisioning

In this paper, an energy-efficient resource allocation scheme is designed for orthogonal frequency-division multiple-access cellular wireless networks with multiuser cooperation. Joint relay selection, subcarrier allocation and pairing, and power-allocation algorithms are developed with the objective of maximizing the energy efficiency of the system considering the quality-of-service requirements of the users. The optimization problem is a mixed-integer nonlinear program (MINLP), which is generally very difficult to solve in its original form. The energy efficiency metric is a fractional and nonlinear function, which complicates the problem further. We provide a novel optimization framework to solve such nonlinear and nonconvex optimization problems. The MINLP optimization problem is reformulated to a convex problem by relaxing the integer variables and by introducing the "Dinkelbach" method to tackle the nonlinear fractional objective function. We prove that our proposed solution of the relaxed problem is optimal and has integer values. Based on the dual-decomposition method, we propose a solution to the joint optimization problem, which is optimal and computationally efficient with polynomial-time complexity. Numerical results demonstrate the effectiveness of our proposed scheme.

Development of a thermal and electrical energy management in residential building micro-grid

Global warming and pressing concern about CO2 emission along with increasing fuel and oil cost have brought about great challenges for energy companies and homeowners. In this regard, a potential candidate solution is widely used for Distributed Energy Resources, which are capable of providing high quality, low-cost heat and power to off-grid or remote facilities. To appropriately manage thermal and electrical energy, a Smart Energy Management System (SEMS) with hierarchical control scheme has been presented. The developed SEMS model results in mixed integer non-linear programming optimization problem with the objective function of minimizing the operation cost as well as considering emissions. Moreover, the optimization problem has been solved for deterministic and stochastic scheduling algorithms. The novelty of this work is basically reliant on using data mining approach to reduce forecasting error. Several case studies have been carried out to evaluate the performance of proposed data mining method on both energy cost and expected cost.

MINLP based superstructure optimization for boron removal during desalination by reverse osmosis

In this work, a model based MINLP (mixed integer nonlinear programming) optimisation framework is developed for evaluating boron rejection in a reverse osmosis (RO) desalination process. A mathematical model (for the RO process) based on solution diffusion model and thin film theory is incorporated in the optimisation framework. A superstructure of the RO network is developed which includes two passes: (a) seawater pass containing normal two-stage RO system housing seawater membrane modules and (b) the brackish water pass (BW) accommodating brackish water membrane modules. For fixed freshwater demand, the objective of this work is to demonstrate the effectiveness of the MINLP approach for analyzing and optimizing the design and operation of RO network while attaining desired limit on boron concentration in the freshwater produced. The effect of seasonal variation in seawater temperature and pH on boron removal efficiency is also discussed.

Optimal design of single-story steel building structures based on parametric MINLP optimization

The paper presents an optimal design of single-story industrial steel building structures based on the parametric mixed-integer non-linear programming (MINLP) optimization approach and Eurocode specifications. The optimal design of the structures was investigated for factors which have significant influence on the structure mass: vertical variable load on the structure, horizontal variable load, structure span, structure height, type of standard cross-sections and strength of structural steel.For this purpose, parametric MINLP optimization was performed for a number of combinations between different parameters, i.e. different values of the mentioned influence factors. The single-story steel building superstructure was generated and the MINLP optimization model developed. The objective function of the structure's mass was defined. The modified outer-approximation/equality-relaxation (OA/ER) algorithm and the three-phase MINLP strategy were used to perform the optimization. The minimal possible structure mass, the optimal topology and the optimal standard cross-sections were obtained through each individual MINLP optimization. Based on the obtained optimal results, the recommended optimal design for single-story/single-bay structures and the comparative mass diagrams were developed.

A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

Let $$(MQP)$$ be a general mixed-integer quadratic program that consists of minimizing a quadratic function $$f(x) = x^TQx +c^Tx$$ subject to linear constraints. Our approach to solve $$(MQP)$$ is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables $$y_{ij}$$ , additional quadratic constraints $$y_{ij}=x_ix_j$$ , a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in Billionnet et al. (Math Program 131(1):381---401, 2012), the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints $$y_{ij}=x_ix_j$$ to solve $$(MQP)$$ . Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with the approach of Billionnet et al. (2012) and with the general mixed-integer nonlinear solver BARON (Sahinidis and Tawarmalani, Global optimization of mixed-integer nonlinear programs, user's manual, 2010).

MINLP optimization model for the nonlinear discrete time–cost trade-off problem

This paper presents the mixed-integer nonlinear programming (MINLP) optimization model for the nonlinear discrete time-cost trade-off problem (NDTCTP). The nonlinear total project cost objective function of the proposed MINLP optimization model is subjected to a rigorous system of generalized precedence relationship constraints between project activities, project duration constraints, logical constraints, and a budget constraint. By means of the proposed MINLP optimization model, one can obtain the minimum total project cost, the project schedule with the optimal discrete start times and the optimal discrete durations of the activities, as well as the optimal time-cost curves of the project. The proposed model yields the exact optimum solution of the NDTCTP. Solving the NDTCTP, using the proposed MINLP model, avoids the need for (piece-wise) linear approximation of the nonlinear expressions. The MINLP model handles the discrete variables explicitly and requires no rounding of the continuous solution into an integer solution. The applicability of the proposed optimization model is not limited to weakly NDTCTPs. A numerical example from the literature and an example of the project time-cost trade-off analysis are presented at the end of the paper in order to show the advantages of the proposed model.

A Superstructure Optimization Approach for Membrane Separation-Based Water Regeneration Network Synthesis with Detailed Nonlinear Mechanistic Reverse Osmosis Model

Scarcity of freshwater resources and increasingly stringent environmental regulations on industrial effluents have motivated the process industry to identify and develop various water recovery strategies. This work proposes the use of detailed model representation for water regeneration network synthesis, in which nonlinear mechanistic models of the regeneration units are embedded within an overall mixed-integer nonlinear programming (MINLP) optimization framework. The superstructure-based MINLP framework involves both continuous variables for water flow rates and contaminant concentrations and 0–1 variables for selection of piping interconnections. The nonlinear regeneration model produces a rigorous cost-based relation, instead of a “black box” model, that is incorporated within the overall MINLP representing a network of numerous water sources and water sinks. Hence, such an approach enables a simultaneous evaluation of both direct water reuse/recycle and regeneration–reuse/recycle opportunities. To demonstrate the proposed approach, an industrial case study is illustrated that incorporates a mechanistic model of reverse osmosis network (RON) for water regeneration for an operating refinery in Malaysia. The results indicate a potential of 58% savings in freshwater use. The capital investment for the water regeneration network is reported as $8,960,000 with a payback period of 2.1 years, thus providing economic support to pursue the RON retrofit option.

Power Supply Pads Assignment for Maximum Timing Yield

The design of power distribution networks significantly impacts the timing of very large scale integrated chips. Process variations induce uncertainty in the current drawn off the network and, therefore, impose statistical measures on the supply voltage. This brief presents an optimization methodology for assigning power supply pads across the chip for maximizing the timing yield. A mixed-integer nonlinear programming optimization problem subject to the voltage drop and current constraints is efficiently solved to find the optimum number and location of the pads. The experimental results for ISCAS89 benchmark circuits demonstrate as much as a 30% improvement in the timing yield.

Optimization and analysis of the profitability of tariff structures with two-part tariffs

Service providers often offer tariff structures with several two-part tariffs that consist of a fixed fee and a usage price, such that consumers may pick the tariff they prefer. Prices of tariffs have significant impacts on service providers’ profit, because they simultaneously influence consumers’ tariff choices and their usage. The number of tariffs also plays an important role, because more tariffs segment the market better but also increase the administrative burden and require more marketing effort. This article presents a mixed-integer nonlinear programming optimization problem to determine profit-maximizing tariffs; compares several heuristic search methods, in particular, the gradient method, stochastic search, and simulated annealing, to solve this problem; analyses the profitability of different tariff structures; and outlines the factors that drive differences in profitability across various tariff structures. The results show that especially for large samples of more than 100 consumers, simulated annealing performs best and deviates only 0.2% from the optimum. Structures with fewer two-part tariffs are generally sufficient, because additional two-part tariffs only negligibly increase service providers’ profit.

A systematic tool for the minimization of the life cycle impact of solar assisted absorption cooling systems

In recent years, there has been a growing increase of the cooling demand in many parts of the world, which has led to major energy problems. In this context, solar assisted absorption cooling systems have emerged as a promising alternative to conventional vapor compression air conditioning systems, given the fact that in many cases the cooling demand coincide with the availability of solar radiation. In this work, we present a decision-support tool based on mathematical programming for the design of solar assisted absorption cooling systems. The design task is formulated as a bi-criteria mixed-integer nonlinear programming (MINLP) optimization problem that accounts for the minimization of the total cost of the cooling system and the associated environmental impact measured over its entire life cycle. The capabilities of the proposed method are illustrated in a case study that addresses the design of a solar assisted ammonia-water absorption cycle considering weather data of Barcelona (Spain).

MINLP optimization of mechanical draft counter flow wet-cooling towers

In this paper, the problem of the optimal design of mechanical draft counter flow cooling towers that meets a set of specified constraints is formulated as a mixed-integer nonlinear programming (MINLP) problem. The Merkel's method is used to specify the characteristic dimensions of cooling towers, together with empirical correlations for the loss and overall mass transfer coefficients in the packing region of the tower. Water-to-air mass ratio, water mass flow rate, water inlet and outlet temperatures, operational temperature approach, type of packing, type of draft, height and area of the tower packing, total pressure drop of air flow, power consumption of the fan, and water consumption provide the set of optimization variables. The MINLP problem is formulated so as to minimize the total annual cost of the cooling tower. The performance of the proposed procedure is shown with the solution of six examples.

A Coupled Strategy for the Solution of NLP and MINLP Optimization Problems: Benefits and Pitfalls

This paper presents a strategy for the solution of nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) problems, based on the coupling of the equation-oriented simulator ASCEND IV with the stochastic optimizers MSGA and MSIMPSA. Both NLP and MINLP formulations of a reactive distillation example were explored. The results show that the connection between the two software blocks was successfully established. Despite the highly nonlinear behavior of the proposed example, the results presented generally agree with those previously obtained for the same case study using other approaches, namely, SIMOP, which is a FORTRAN 77-based NLP simulation tool developed for optimization problems. Increasing the column dimension or switching the vapor−liquid equilibrium (VLE) description to a nonideal situation, which produces nonlinear models with a much larger number of equations that must be solved simultaneously, show that the results deteriorate. This is attributed to the lack of user control over the subsystems’ structures (viz, tearing variables). This problem was partially circumvented through the application of a more elaborate scheme initialization for the column variables.

A simulated annealing-based approach to the optimal synthesis of heat-integrated distillation sequences

A simulated annealing-based approach to synthesis of multi-component distillation systems is developed. An encoding procedure that makes use of an integer number series is developed to represent and manipulate the flow sheet structure of the system. With the representation procedure, the overall synthesis problem is formulated as an implicit mixed-integer nonlinear programming (MINLP) problem. A simulated annealing approach suitable for MINLP optimization is adopted and improved to solve the problem. Four example problems, including large-scale ones are solved to illustrate the approach.

A Framework for the Systematic Design of Hybrid Separation Processes

The design of optimal separation flow sheets for multi-component mixtures is still not a solved problem. This is especially the case when non-ideal or azeotropic mixtures or hybrid separation processes are considered. We review recent developments in this field and present a systematic framework for the design of separation flow sheets. This framework proposes a three-step approach. In the first step different flow sheets are generated. In the second step these alternative flow sheet structures are evaluated with shortcut methods. In the third step a rigorous mixed-integer nonlinear programming (MINLP) optimization of the entire flow sheet is executed to determine the best alternative. Since a number of alternative flow sheets have already been eliminated, only a few optimization runs are necessary in this final step. The whole framework thus allows the systematic generation and evaluation of separation processes and is illustrated with the case study of the separation of ethanol and water.