Mixed-integer Nonlinear Optimization Research Articles

Multi-objective techno-economic operation of smart distribution network integrated with reactive power support of battery storage systems

• Multi-objective optimization for smart distribution is proposed. • Operation cost and energy loss are simultaneously minimized. • Both active and reactive power from battery storage systems is considered. • Problem is modelled as mixed integer second order cone programming. The smart distribution system (SDS) will transform the energy sector into a new era of increased flexibility, sustainability, reliability, and efficiency that will contribute to environmental and economic health. However, a proper scheduling strategy with strategic goals is needed to reap these benefits of SDS. In this regard, a multi-objective techno-economic optimization framework for optimal operation of SDS consisting of different resources such as renewables, dispatchable distributed generations and battery storage systems (BSSs) is presented in this paper. The proposed framework aims at simultaneously minimizing operation cost and network energy loss. To manage the operation of grid effectively, provision of both real and reactive power from BSS is included in operation of SDS. The initial operational problem is modeled as mixed integer non-linear non-convex optimization, which is then translated into mixed integer second order cone programming that can be successfully solved using commercially available solvers. Moreover, the multi-objective problem is tackled using ϵ -constraint method followed by fuzzy satisfying criteria. In order to highlight the efficacy of proposed work, case studies are conducted on 33-bus distribution system. Results demonstrate that utilization of reactive power services from BSS significantly reduces energy losses and operation cost.

Applied energy optimization of multi-robot systems through motion parameter tuning

In this paper, an optimization method for energy reduction of robot stations is presented, including an evaluation on an industrial robot station. The problem is formulated as a convex mixed integer nonlinear optimization problem, where the objective is to reduce the energy use by finding the optimal execution time and execution order of the robot motions. A simulation model of the station is used to find simplified energy models of the robot motions, that is used to solve the optimization problem. The optimal execution times of the robot motions are realized by tuning parameters in the robot control system. Different types of parameter settings are tested, such as reduced acceleration and velocity. The optimal parameter settings are then implemented in robot code in a real four robot welding station. The result shows a 12% reduction in energy use, without extending the cycle time of the station. A validation of the energy models used to solve the optimization problem is also made, by comparing them with real energy measurements.

Reliability‐based risk‐involved macrogrid expansion planning considering community microgrids within a market environment

In this paper, a new reliability-based risk-involved model is presented for multi-period macrogrid expansion planning (MAGEP) by amalgamating the notion of community microgrids (CoMigs) within a market environment. The main targets of the proposed model are not only to develop a holistic framework for the MAGEP problem, but also to scrutinize the impacts of the macrogrid-connected multi-CoMigs on power supply reliability enhancement. To that end, the mathematical formulation of the proposed model falls into a nonconvex mixed-integer nonlinear three-level optimization problem. The lower-level outlines a security-constrained bid-based-pool electricity market aimed at providing an efficient platform for different CoMigs and market participants—generation companies (GenCos) and distribution companies (DisCos)—to freely trade electrical power across the macrogrid. The intermediate-level represents an interactive decentralized multi-period two-stage microgrid expansion planning (MIGEP) problem in which optimal expansion plans of the CoMigs obtained in the first stage will be examined against the reliability restrictions in the second stage. The upper-level, however, addresses a centralized multi-period techno-economic MAGEP problem, while handling the risks of planning arising from severe uncertainties in forecasted price, demand, and power generation of the CoMigs through a potent envelope-bound information-gap decision theory (IGDT). Due to its proper tackling of the non-convex mixed-integer nonlinear multi-level optimization problems, the symphony orchestra search algorithm (SOSA) is widely employed for solving the proposed model. Effectiveness and capability of the newly developed model are numerically analyzed and verified using its implementation on the IEEE 30-bus and IEEE 118-bus test systems.

Generalized Benders Decomposition Method to Solve Big Mixed-Integer Nonlinear Optimization Problems with Convex Objective and Constraints Functions

The paper presents the Generalized Benders Decomposition (GBD) method, which is now one of the basic approaches to solve big mixed-integer nonlinear optimization problems. It concentrates on the basic formulation with convex objectives and constraints functions. Apart from the classical projection and representation theorems, a unified formulation of the master problem with nonlinear and linear cuts will be given. For the latter case the most effective and, at the same time, easy to implement computational algorithms will be pointed out.

Stochastic optimal transmission Switching: A novel approach to enhance power grid security margins through vulnerability mitigation under renewables uncertainties

In this paper, a stochastic optimal transmission switching (SOTS) model is developed considering the uncertainty of load, wind power and photovoltaic generation, while minimizing the grid vulnerability. Scenario reduction technique is used for alleviating computational burden of the developed SOTS model. A three-step procedure, named the vulnerability-oriented SOTS (VO-SOTS), is proposed to consider the set of critical transmission lines, i.e., lines whose outage imposes the largest load curtailment. Moreover, to control the negative impacts of the uncertainties, conditional value at risk (CVaR) is used as the risk measure. The proposed VO-SOTS model is formulated as a mixed integer non-linear programming optimization problem and solved using GAMS optimization package. Also, the VO-SOTS model is implemented on the IEEE 118-bus, IEEE 300-bus, and large-scale IEEE 2869-bus standard test systems. Simulation results for these cases show that considering the set of critical lines allows the proposed VO-SOTS model to reduce generation costs while effectively minimizing system vulnerability and risk. In addition, it is shown through extensive case studies that switching off a line based on the proposed three-step VO-SOTS does not increase the grid’s exposure to vulnerability, in contrast to the conventional OTS.

Sensitivity-based Optimal Remedial Actions to Damp Oscillatory Modes Considering Security Constraints

This paper proposes a comprehensive analytic method for applying various optimal remedial actions to improve critical electromechanical modes damping without jeopardizing damping of non-critical modes and violating security constraints of power system. Generators and reactive power sources redisptach, demand side management and the generators voltage reference tuning are remedial actions that are considered here. Dynamic equations of the flux-decay dynamic model of generators, standard dynamic models of excitation system and power system stabilizer and algebraic equations of active and reactive powers balance are formulated in the quadratic eigenvalue problem framework. With simultaneous use of system parameters and online measured data, sensitivity factors of electromechanical modes, generators reactive power, transmission lines apparent power and the buses voltage amplitude with respect to the mentioned remedial actions are calculated. Consumers, generators and reactive power sources are independent agents that offer their price bid to contribute in remedial action schemes. Finally, using sensitivity factors, the best remedial action is determined by mixed-integer nonlinear optimization subject to the minimum required damping ratio and security constraints of transmission lines congestion, generators reactive power capability and buses voltage amplitude. The proposed method is demonstrated on the IEEE 39-bus power system and the results highlight its efficacy.

Automation of gearbox design

In order to respond to a shortened development time of today’s transmission systems, the automation of certain steps in the design process is essential for ensuring an efficient development process. Computer-aided tools are widely used for analyzing given design configurations because standardized methods are available to evaluate the load carrying capacity of all key components of a simple gear train, namely bearings, shafts and gears. At an early stage of development, requirements and restrictions need to be synthesized to design concepts. During this step, engineers typically rely on their experience and proven practice. Design optimization usually is achieved through an iterative and time-consuming process of analyzing and tuning towards an optimization objective. In this paper a time-saving, automated and systematic method for the design of weight optimized helical gearboxes is proposed. The underlying method has been derived from both, norms and guidelines, which exist for the design and layout of shafts, bearings and gear wheel bodies. Starting with only few input parameters, a detailed shaft geometry with different diameter sections can be derived. A discrete set of values from standard tables and rolling bearing catalogs represents the method’s framework for all realizable shaft diameters in each section. A mixed integer nonlinear optimization problem results from the interdependence between these distinct values. For this purpose, a systematic iterative approach has been developed and implemented in an established design program for gearbox systems. The algorithm uses the results drawn from an analytical calculation of the shaft load carrying capacity to directly adjust the shaft’s diameter and length values. The dimensioning of the wheel body, the service life calculation of rolling element bearings and the selection of specific machine elements are embedded in a systematic sequence. As a result, the model is capable to work out a weight-optimized gearbox that consists of gear meshes, shafts and bearings, taking all three components into consideration at a time.

Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $$x\in \{0\}\cup [l,u]$$ , where z is a binary indicator of $$x\in [l,u]$$ ( $$u> \ell > 0$$ ), and y “captures” f(x), which is assumed to be convex on its domain [l, u], but otherwise $$y=0$$ when $$x=0$$ . This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the “perspective reformulation” inequality $$y \ge zf(x/z)$$ . We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $$f(x) := x^p$$ , for $$p>1$$ , relaxations utilizing the inequality $$yz^q \ge x^p$$ , for $$q \in [0,p-1]$$ , are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $$f(x) := x^2$$ ) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.

Proposal and Solution of a Mixed-Integer Nonlinear Optimization Model That Incorporates Future Preparedness for Project Portfolio Selection

In the context of project management, the attention given to project portfolio management has increased in recent years. The use of mathematical programming for portfolio management is also on the rise, because it integrates the project interactions with the multiple objectives of portfolio management into a single model. Among the possible objectives, recent studies have paid special attention to the emerging objective of future preparedness, which has not yet been incorporated into the existing mathematical models. This paper presents a mixed-integer nonlinear optimization model for portfolio selection that considers four main performance measures for project management, namely, value maximization, strategic alignment, balance, and future preparedness. Given the importance of the last measure, the purpose of this paper is to provide a more complete model that provides the marginal contribution and the best combination of projects according to the needs of the company. The model is tested using real data from two companies, one in Brazil and one in Canada, and the results obtained are coherent with their respective practices.

Capacity allocations of SOPs considering distribution network resilience through elastic models

Disturbances such as natural disasters or man-made attacks exert serious influences on power system, resulting in an increasing awareness of resilience enhancement strategies. As the integration of microgrids, distributed generators (DGs) and power electronic devices adds to the vulnerability of power distribution network to disruptions, it is highly required to enhance the resilience against failure. Soft open points (SOPs) are flexible power electronic devices, which can balance power distribution under normal operation, and supply restoration under abnormal conditions. Thus, the application of SOPs can boost resilience both in pre-failure prevention and post-failure recovery. In this paper, to maximize the effect of SOPs on the boost of resilience of distribution network, a mixed integer non-linear optimization problem is proposed to schedule the siting and sizing of SOPs based on multi-stage elastic mechanical model. A bi-level algorithm is used to tackle the SOP planning problem. The selection of location and capacity is solved by genetic algorithm, after which the control strategies of SOP are optimized by particle swarm algorithm with given planning scheme to obtain the maximum resilience. Finally, case studies on the IEEE 33-bus system and IEEE 123-node test feeder are used to verify the effectiveness and efficiency of the proposed method.

A multi-objective optimization model to sustainable closed-loop solar photovoltaic supply chain network design: A case study in Iran

Along with the increasing rate of using renewable energies, following the finite amount of fossil fuel resources, and the associated pollution of these non-renewable energies, photovoltaic systems have recently achieved widespread acceptance to harness solar energy in the last decades. Since photovoltaic systems have a limited lifespan and cannot be used after a while, it seems imperative to pay attention to recycling and sustainability in the supply chain network design of these systems. In this respect, this paper first reviews the photovoltaic systems literature systematically. Thereinafter, a multi-objective, mixed-integer non-linear optimization model is deployed to design a photovoltaic supply chain network addressing the system's associated costs, environmental impacts, and social values. The concerned model also considers a reverse flow in addition to the typical forward flow in the proposed supply chain for special environmental and social issues. Likewise, to optimize the presented multi-objective model, the augmented ε-constraint method is employed, being capable of warranting the Pareto optimal solutions. Eventually, the model's efficiency is illustrated by discussing a real-world case in Iran, through which prominent managerial results are acquired.

Machine Learning-Based Offloading Strategy for Lightweight User Mobile Edge Computing Tasks

This paper presents an in-depth study and analysis of offloading strategies for lightweight user mobile edge computing tasks using a machine learning approach. Firstly, a scheme for multiuser frequency division multiplexing approach in mobile edge computing offloading is proposed, and a mixed-integer nonlinear optimization model for energy consumption minimization is developed. Then, based on the analysis of the concave-convex properties of this optimization model, this paper uses variable relaxation and nonconvex optimization theory to transform the problem into a convex optimization problem. Subsequently, two optimization algorithms are designed: for the relaxation optimization problem, an iterative optimization algorithm based on the Lagrange dual method is designed; based on the branch-and-bound integer programming method, the iterative optimization algorithm is used as the basic algorithm for each step of the operation, and a global optimization algorithm is designed for transmitting power allocation, computational offloading strategy, dynamic adjustment of local computing power, and receiving energy channel selection strategy. Finally, the simulation results verify that the scheduling strategy of the frequency division technique proposed in this paper has good energy consumption minimization performance in mobile edge computation offloading. Our model is highly efficient and has a high degree of accuracy. The anomaly detection method based on a decision tree combined with deep learning proposed in this paper, unlike traditional IoT attack detection methods, overcomes the drawbacks of rule-based security detection methods and enables them to adapt to both established and unknown hostile environments. Experimental results show that the attack detection system based on the model achieves good detection results in the detection of multiple attacks.

Development of a bi-objective 0-1 mixed-integer nonlinear response surface-based robust design optimization model for unbalanced experimental data

The purpose of this paper is to develop a model to obtain the optimum operating conditions under special engineering situations. These special situations are: (1) both qualitative and quantitative input variables are considered, (2) a specific number of design points is required, (3) the experimental design space is irregular due to physical constraints, and (4) the unbalanced experimental data is collected due to resource limitations and focusing on certain treatment combinations. Although such situations may occur in many processes, there has been little research on this subject. To address these research gaps, first, an I-optimal experimental design is selected to construct a design matrix by using a modified exchange algorithm. Second, a weighted least squares method is presented to deal with the unbalanced experimental data. Third, a bi-objective 0-1 mixed-integer nonlinear programming-based robust design model and its solution procedures are proposed to obtain optimum settings of both qualitative and quantitative variables. Finally, a case study, along with the comparative study, is presented.

Under-frequency load shedding in isolated multi-microgrids

In this paper, a new under-frequency load shedding (UFLS) scheme is proposed for multi-microgrid (MMG) application. MMG, consists of multiple micro-grids, perhaps with different owners, connected together. Multiple owners make load shedding policies more complex in MMG compared to the policies employed in micro-grids and large power systems. Accordingly, an appropriate UFLS scheme is required, in order to ensure the frequency stability of the MMG, while considering its economic aspects. Therefore, in this paper, an optimization problem has been introduced which minimizes the cost imposed on the owners for mandatory load shedding alongside the cost of MMGs operation after the load shedding procedure. By solving the proposed optimization problem, the optimal locations of loads to be shed are determined. Two optimization algorithms – genetic algorithm (GA) and exchange market algorithm (EMA) – and GAMS software have been used for solving this optimization problem. A new index is introduced in this paper in order to distribute load shedding among MGs based on their total power generation and load demand. Simulation results confirm the efficiency of the proposed methodology. • An exclusive under-frequency load shedding scheme is proposed for multi-microgrids. • The UFLS scheme provides a fair distribution of load shedding among microgrids. • The location of removable loads is determined by solving the optimization problem. • A mixed integer non-linear programming optimization problem is proposed. • A new index is introduced in order to distribute load shedding among MGs.

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

An estimation–Based optimal scheduling model for settable renewable penetration level in energy hubs

Several recent studies suggest that power–to–gas (PtG) facilities can be operated to absorb and convert surplus renewable energy into hydrogen or synthetic natural gas, which is then supplied to the natural gas grid, forming an energy hub (EH). This paper argues and demonstrates that there is a trade off between maximizing the renewable penetration level (PRL) into the EH and minimizing the cost associated with it. The paper presents a new optimal scheduling model allowing variable and settable PRLs into the EH to minimize the system operating cost. The RPL, as determined by the grid operator, is taken in as a setting to the scheduling model. A new algorithm is presented by which the relationship between the RPL and scheduling setpoints of PtG and gas–fired generation (GfG) facilities are estimated. Accordingly, the optimal scheduling setpoints of PtG and GfG facilities are decided by the model, so that the required RPL can be met. The problem is formulated using the mixed integer nonlinear optimization method where the estimation algorithm is developed using the least error square technique. The developed problem is then solved using the combined interior point nonlinear programming and Newton trust region techniques. Numerical studies are conducted on a test system using historical operating data to validate the efficacy of the model. The system operating parameters are analyzed, and it is demonstrated that it is imperative to optimally manage the desired RPL taking into consideration the arbitrage revenue shortfall resulted from RPL enhancement in an EH.

Global exact optimization for covering a rectangle with 6 circles

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal covering. We prove the optimality of the two configurations corresponding to open cases. For the first one, we propose a mathematical mixed-integer nonlinear optimization formulation, that allows one to compute global optimal solutions. For the second one, we provide an analytical expression of the optimal radius as a function of one of the rectangle side lengths. All open cases are thus closed for the optimal covering of a rectangle with six circles.

A branch-and-bound algorithm based on NSGAII for multi-objective mixed integer nonlinear optimization problems

Solving Multi-Objective Mixed Integer NonLinear Programming (MO-MINLP) problems is a point of interest for many researchers as they appear in several real-world applications, especially in the mechanical engineering field. Many researchers have proposed using hybrids of metaheuristics with mono-objective branch and bound. Others have suggested using heuristics with Multi-Criteria Branch and Bound (MCBB). A general hybrid approach is proposed based on MCBB and Non-dominated Sorting Genetic Algorithm 2 (NSGAII) to enhance the approximated Pareto front of MO-MINLP problems. A computational experiment based on statistical assessment is presented to compare the performance of the proposed algorithm (BnB-NSGAII) with NSGAII using well-known metrics from the literature. To evaluate the computational efficiency, a new metric, the Investment Ratio (IR), is proposed that relates the quality of solution to the consumed effort. Experimental results on five real-world mechanical engineering problems and two mathematical ones indicate that BnB-NSGAII could be a competitive alternative for solving MO-MINLP problems.

Reduction of waste and carbon emission through the selection of items with cross-price elasticity of demand to form a sustainable supply chain with preservation technology

Carbon emission plays a vital role in global warming. In addition to this, wastages due to the deterioration have a hazardous impact on the ecosystem. The responsible managers’ primary concern is to form a sustainable supply chain. Further, supply chain managers select some appropriate combinations when dealing with items that possess cross-price elasticity of demand to make a profitable system. The present model aims to make a sustainable supply chain to reduce waste by obtaining the optimal values of production rate, batch size, several shipments, and preservation investment such that the total profit of the system is maximum. In the present study, a supply chain model with two manufacturers and one common retailer is developed. The proposed supply chain deals with products having negative or positive cross-price elasticity of demand. The carbon emission is also considered, which is associated with different operational activities of inventory such as preparation of setup of manufacturers, transportation of products, waste management, and holding stock at manufacturers and the retailer’s end. The objective function in the present study is a mixed-integer nonlinear optimization problem. A solution methodology based on differential calculus is adopted. The result shows that products of low negative or high positive cross-price elasticity are a profitable combination. Additionally, a 20% reduction in wastage quantity is observed due to the investment in preservation technology. This result indicates that preservation technology positively impacts the environment and the finance associated with the inventory system. The proposed integrated system’s sustainability is illustrated with numerical examples, sensitivity analysis, and pictorial representation. In addition to this, several valuable managerial insights based on the study are provided.

Optimal planning and operational strategy of energy storage systems in power transmission networks: An analysis of wind farms

This study formulated a bi-level mixed integer non-linear optimization planning and operation model for the optimal configuration (location, capacity, and power ratings) of energy storage systems (ESSs) in power transmission networks. The model was formulated with consideration for independent and correlated wind farms. The single objective function in the inner layer of the bi-level model includes the difference between the total daily expected operational cost of conventional generators and the energy arbitrage benefits derived when considering the operational strategies of ESSs. The outer layer is a multi-objective function composed of three objective functions to be minimized. The objective functions encompass the total daily expected planning and operational cost, total daily expected emission, and the maximum expected voltage deviation. Wind power uncertainties in independent and correlated wind farms were also examined. Multivariate model-based Clayton copulas, which represent joint power distribution amongst correlated wind farms, were discretized using a developed five-point estimation method based on the discretization. A hybrid non-dominating sorted genetic algorithm and multi-objective particle swarm optimization were used to minimize the outer layer objective function, whilst fast Tabu search that considers the probabilistic load flow represented by wind power uncertainties and the operational strategies of ESSs was adopted to minimize the inner layer objective function. An IEEE 57-bus system was subjected to a case study using the proposed two-stage model. The simulation results confirmed the advantage of considering the benefits of a peak shaving operational strategy from economic, technical, and environmental points of view.