Integer Non Research Articles

An adaptive random search for short term generation scheduling with network constraints.

This paper presents an adaptive random search approach to address a short term generation scheduling with network constraints, which determines the startup and shutdown schedules of thermal units over a given planning horizon. In this model, we consider the transmission network through capacity limits and line losses. The mathematical model is stated in the form of a Mixed Integer Non Linear Problem with binary variables. The proposed heuristic is a population-based method that generates a set of new potential solutions via a random search strategy. The random search is based on the Markov Chain Monte Carlo method. The main key of the proposed method is that the noise level of the random search is adaptively controlled in order to exploring and exploiting the entire search space. In order to improve the solutions, we consider coupling a local search into random search process. Several test systems are presented to evaluate the performance of the proposed heuristic. We use a commercial optimizer to compare the quality of the solutions provided by the proposed method. The solution of the proposed algorithm showed a significant reduction in computational effort with respect to the full-scale outer approximation commercial solver. Numerical results show the potential and robustness of our approach.

Two-stage MINLP algorithm for the optimal synthesis and design of networks of CHP units

This paper proposes a Mixed Integer Non Linear Programming (MINLP) model and two-stage optimization algorithm for determining the most profitable synthesis and design of Combined Heat and Power units within a district heating network with heat storage while taking into account the optimal scheduling of the units over the year. A two-stage algorithm for tackling the challenging MINLP problem is devised: at the upper level the selection and sizing of the units is optimized by means of specifically selected evolutionary algorithms, while at the lower level the operational scheduling problem is linearized and optimized with a commercial Mixed Integer Linear Programing solver. Three different approaches, based on two different evolutionary algorithms and discrete variable relaxation, are devised and compared to tackle the upper level problem. Moreover a bounding technique is proposed to limit the computational time required to solve the lower-level problem. The overall algorithm is tested on an industrial scale problem to find the two system designs leading to the minimum energy consumption and the minimum total annual cost. Computational results indicate that the continuous relaxation of the plant sizes significantly helps to improve the convergence rate of the tested evolutionary algorithm and to find improved solutions. For the considered test case, the design optimized for the minimum energy consumption allows to save 64% of primary energy compared to the minimum total annual cost solution, but with a 28% higher total annual cost.

Process Synthesis Methodology for Sustainability Projects

Projects to establish sustainable infrastructure for remote communities face daunting challenges from the lack of a connection to a large-scale electrical grid and waste management infrastructures. These projects can potentially struggle through selection of a large number of options and potential technologies. In the quest for low “ecological footprint,” a large number of interrelated factors are in play. Possible outcomes from these projects include high-capital solutions, overly complex infrastructure, reproduction of the existing non-sustainable solutions, non-optimal balance of solutions at different levels, unwanted by-products and trendy technologies with poor reliability. For practitioners involved in sustainability design projects, Mixed Integer Non Linear Programming provides a foundation for thinking through process synthesis options.

Optimal day ahead scheduling of combined heat and power units with electrical and thermal storage considering security constraint of power system

The use of Combined Heat and Power (CHP) with an overall efficiency from 70 to 90% is one of the most effective solutions to optimize the energy consumption. Mainly due to interdependence of the power and heat in these systems, the optimal operation of CHP systems is a complex optimization problem that needs powerful solutions. This paper addresses optimal day-ahead scheduling of CHP units with Electric Storage Systems (ESSs) and Thermal Storage Systems (TSSs) considering security constraints. Basically, the optimal scheduling of CHP units problem is a Mixed Integer Non Linear (MINLP) problem with many stochastic and deterministic variables. In this paper, linearization techniques are adopted to linearize equations and a two-stage Stochastic Mixed-Integer Linear Programming (SMILP) model is utilized to solve the problem. The first stage models behavior of operation parameters and minimizes the operation costs meanwhile the second stage considers the system's stochastic contingency scenarios. The proposed method is applied to 18-bus, 24-bus IEEE test systems. The effectiveness of the proposed algorithm has been investigated.

Optimal Location of PV based Distributed Generation in Pool based Electricity Market using Mixed Integer Non Linear Programming

Optimal Location of PV based Distributed Generation in Pool based Electricity Market using Mixed Integer Non Linear Programming

Integrating packaging and supply chain decisions: Selection of economic handling unit quantities

In this work, a new framework is developed for selecting the packaging parameters that optimize end-to-end supply chain costs assuming a fixed supply chain network for one period of deterministic demand for a single product shipping to multiple customers. The Handling Unit Quantities represent the type and size of the nested handling units such as containers, pallets, cases and sub-cartons. The approach seeks the optimal or Economic Handling Unit Quantities (EHUQ) that fully define the choice of container type and loading configuration, pallet and case quantities, warehouse pick profiles, as well as the prediction of end-to-end supply chain costs. The problem is formulated as a non-convex Mixed Integer Non Linear Programming (MINLP) problem, and transformed into a Quadratically Constrained Quadratic Program (QCQP). It considers a 3-echelon supply chain of factories, warehouses and customers. The model optimizes the total supply chain costs of manufacturing, packaging consumables, in- and out-bound transportation, warehouse receiving, picking and packing costs. The formulation requires integer variables to model the packaging choices, binary variables to capture transportation choices, non-linear inverse bi- and tri-linear terms, and discontinuous step functions which are required to model warehouse picking costs. A power law relationship is used for customer demand to generate test problems that replicate the complexity of real world problems. With a small amount of information, the cost, operational performance and environmental impact for supply chain options under evaluation may be predicted. Because of the size and complexity of the problem, a heuristic method was developed to provide solutions quickly.

Congestion Management Solution for Hybrid System Considering Voltage Stability Margin

In deregulated structure, power system network has become more complex with free trades amongst GENCOs and DISCOs and with increasing penetration of Renewable Energy Sources. These complexities might impose risk on security and stability of the network especially during congestion hours. This paper presents an economical and secure congestion management approach, in hybrid power system; by considering linear piece-wise hydro model and probabilistic wind generation model. Impact of wind participation in congestion management and bidding's effect on congestion cost has been analyzed. Minimization of congestion cost has been formed as major objective and problem has been formulated as Mixed Integer Non Linear Programming (MINLP) problem which has been tested on modified IEEE-24 bus RTS system. Locational dependence of congestion cost and a comparison of VSM based approach with conventional congestion management approach have also been presented.

Optimising the operation of a district heating system

bLuleKraft AB, SE-97421, Lulea, Sweden c Lulea Energi AB, SE-97323, Lulea, Sweden david.olofsson@swerea.se A system with several actors will increase the possibilities for collaboration and therefore, in many cases also increase the possibilities to affect the operation with economic and environmental benefits. In this paper a district heating system is studied with purpose to create economic and environmental operation guidelines in favour of the involved actors. The company LuleKraft owns a combined heat and power plant, while the four heating stations together with the DH network are owned by Lulea Energi. District heat is produced from the CHP plant and the four heating stations which are fired with process gases from an integrated steel plant, oil, wood pellets or electricity. The heat demand in the system is strongly depending on the outdoor temperature. In this paper, the system is modelled with Mixed Integer Non Linear Programming in order to optimise the profit and CO2 emission. A comparison between actual process data and modelled results is performed. A pareto front is derived to show the trade-off between economic benefits and CO2 emissions. It is found that the main suggestions, under conditions for optimised profit, are (1) to prioritize effective heat production instead of electricity production at cold outdoor temperatures and (2) redistribution of the accessible process gases between the CHP plant and one of the heating stations. This will lead to an improved operation strategy, resulting in increased profit and reduced energy consumption.

Power-Aware Automated Pipelining of Combinational Circuits

Pipelining of combinational circuits with power, area and clock frequency constraints is a very useful way to increase operational speed of circuits. In this paper we formulate this problem as a Mixed Integer Non Linear Programming (MINLP) problem and we provide two heuristic methods to obtain good solutions. The first method is sensitivity based approach which gives good solution to circuits with large number of gates. The second method uses geometric programming method which is slower on large circuits, but works well for smaller designs. The two algorithms are tested on ISCAS-85 benchmark and circuits generated by our tool and compared for speed and efficiency. We have also studied impact of supply and threshold voltage variation on pipelining efficiency in terms of latency and total power consumption. Our experimental observations show the existence of a minimum power supply voltage and a threshold voltage at which the circuits generated have less latency and power consumption.

Short-term generation planning by primal and dual decomposition techniques

<p class="ADYNAAbstrac"><span lang="EN-US">This paper addresses the short-term generation planning (STGP) through thermoelectric units. The mathematical model is presented as a Mixed Integer Non Linear Problem (MINLP). Several works on the state of art of the problem have revealed that the computational effort of this problem grows exponentially with the number of time periods and number of thermoelectric units. Therefore, we present two alternatives to solve a STGP based on Benders’ partitioning algorithm and Lagrangian relaxation in order to reduce the computational effort. The proposal is to apply primal and dual decomposition techniques, which exploit the structure of the problem to reduce solution time by decomposing the STGP into a master problem and a subproblem. For Benders’ algorithm, the master problem is a Mixed Integer Problem (MIP) and for the subproblem, it is a Non Linear Problem (NLP). For Lagrangian relaxation, the master problem and the subproblem are MINLP. The computational experiments show the performance of both decomposition techniques applied to the STGP. These techniques allow us to save computation time when compared to some high performance commercial solvers.</span></p>

SELECTION OF OPTIMAL AIR POLLUTION CONTROL STRATEGIES

Sustainable development needs social, economic, energy and environmental sustainability. We propose a model to minimize the total cost consisting of economic cost of pollution control equipment, health cost because of pollutants emissions. We developed a Mixed Integer Non Linear Programming (MINLP) based model for selection of optimal air pollution control strategy. The model considers multiple pollutants, multiple emission sources and multiple control equipments. Constraints like budget and efficiency of control equipment are included in the model. Affect of emission norms on total cost is discussed. The efficacy of the proposed model is explained by considering simulation case studies of a typical cement plant and a thermal power plant.

Multi-objective optimization of SNG production from microalgae through hydrothermal gasification

The conversion of microalgae biomass into biofuels is a quite well explored field of research. Due to high photosynthetic efficiency, microalgae are considered as a potential feedstock for next-generations biofuel conversion processes.This paper addresses the thermochemical conversion of highly diluted microalgae feedstock into synthetic natural gas (SNG) through supercritical hydrothermal gasification. The complete conversion chain is modeled including the cultivation phase, settling ponds, centrifuges, catalytic hydrothermal gasification with salt separation unit and SNG purification system. Thermodynamic, economic and environmental models are considered for each process step, in order to solve a Mixed Integer Non Linear Programming (MINLP) optimization problem.The problem is solved by applying a two steps decomposition approach, using Multi Objective Evolutionary Algorithm with Mixed Integer Linear Programming (MILP). It is finally demonstrated that coupling microalgae cultivation systems with hydrothermal gasification (HTG) and waste energy recovery utilities leads to high energy/exergy efficiencies, emissions reduction and globally better sustainable processes.

Optimal design of a small size trigeneration plant in civil users: A MINLP (Mixed Integer Non Linear Programming Model)

An optimization procedure for the optimal design of a trigenerative system to satisfy the energy needs of civil users is proposed. For the formulation of the mathematical model, it was necessary to refer to the class of MINLP (Mixed Integer Non Linear Programming Models), whose complexity is related to the presence of binary decision variables and to the non linear nature of the constraints. These constraints are required to model the variation of the nominal efficiency and of the plant unitary cost in relation to the size of the cogeneration plant, and to model the decrease of the nominal efficiency of the cogeneration plant in relation to load variation. Different plant alternatives (Simple Cycle Gas Turbine, Gas Micro-Turbine, Gas Internal Combustion Engine, Fuel Cells) in the optimization procedure are taken into account with the aim of determining the optimal typology, size and operative strategy of the trigeneration system.

Systematic network synthesis and design: Problem formulation, superstructure generation, data management and solution

The developments obtained in recent years in the field of mathematical programming considerably reduced the computational time and resources needed to solve large and complex Mixed Integer Non Linear Programming (MINLP) problems. Nevertheless, the application of these methods in industrial practice is still limited by the complexity associated with the mathematical formulation of some problems. In particular, the tasks of design space definition and representation as superstructure, as well as the data collection, validation and handling may become too complex and cumbersome to execute, especially when large problems are considered. In an earlier work, we proposed a computer-aided framework for synthesis and design of process networks. In this contribution, we expand the framework by including methods and tools developed to structure, automate and simplify the mathematical formulation of the design problem. Furthermore, the models employed for the representation of the process alternatives included in the superstructure are refined, through the inclusion of the energy balance. Finally, the features of the framework are highlighted through the solution of two case studies focusing on food processing and biofuels.

A constrained multi-products EPQ inventory model with discrete delivery order and lot size

This paper proposes an alternative heuristic algorithm for a multi-product Economic Production Quantity (EPQ) vendor–buyer integrated model with Just in Time (JIT) philosophy and a budget constraint. This type of problem is usually solved by a Mixed Integer Non Linear Problem (MINLP), which is complex and computationally expensive. The proposed heuristic algorithm involves less computation and therefore it is less expensive than the previously published algorithms. Furthermore, the heuristic algorithm is simpler as it derives the integer values for all discrete variables in a straightforward manner. Through empirical experimentation, it is demonstrated that the heuristic algorithm provides solutions closer to the lower bound in a very short time.

A scenario driven multiobjective Primary–Secondary Distribution System Expansion Planning algorithm in the presence of wholesale–retail market

This paper presents a novel approach for Integrated Generation and Primary–Secondary Distribution System Expansion Planning (IGDSEP) in the presence of wholesale and retail markets. The proposed method uses a model to explore the impacts of wholesale and retail market on IGDSEP procedure. This algorithm decomposes the IGDSEP problem into six subproblems to achieve an optimal expansion planning of a system, in which the investment and operational costs are minimized, while the reliability of the system is maximized. The proposed model of IGDSEP is a Mixed Integer Non Linear Programming (MINLP) problem and a Scenario Driven MINLP (SCMINLP) method is proposed for solving the problem. The algorithm was successfully tested for an urban distribution system.

Convergence Study of Minimizing the Nonconvex Total Delay Using the Lane‐Based Optimization Method for Signal‐Controlled Junctions

This paper presents a 2D convergence density criterion for minimizing the total junction delay at isolated junctions in the lane‐based optimization framework. The lane‐based method integrates the design of lane markings and signal settings for traffic movements in a unified framework. The problem of delay minimization is formulated as a Binary Mix Integer Non Linear Program (BMINLP). A cutting plane algorithm can be applied to solve this difficult BMINLP problem by adding hyperplanes sequentially until sufficient numbers of planes are created in the form of solution constraints to replicate the original nonlinear surface in the solution space. A set of constraints is set up to ensure the feasibility and safety of the resultant optimized lane markings and signal settings. The main difficulty to solve this high‐dimension nonlinear nonconvex delay minimization problem using cutting plane algorithm is the requirement of substantial computational efforts to reach a good‐quality solution while approximating the nonlinear solution space. A new stopping criterion is proposed by monitoring a 2D convergence density to obtain a converged solution. A numerical example is given to demonstrate the effectiveness of the proposed methodology. The cutting‐plane algorithm producing an effective signal design will become more computationally attractive with adopting the proposed stopping criterion.

Multiperiod optimal planning of thermal generation using cross decomposition

This work addresses the Multiperiod Optimal Planning of Thermal Generation (MOPTG). The model considered is based on a Unit Commitment Problem that has multiperiod character and determines the start up and shut down schedules of thermal plants considering the line capacity limits of transmission and line losses. The mathematical model is stated in the form of a Mixed Integer Non Linear Problem (MINLP) with binary variables. To reduce the computational time caused by the large number of time periods and electric generation nodes we apply the Generalized Cross Decomposition [1, 2]. The later exploits the structure of the problem to reduce solution time by decomposing the MOPTG into a primal subproblem, which is a Non Linear Problem (NLP), a dual subproblem, which is a MINLP, and a Mixed Integer Problem (MIP) called master problem. The approach is compared with Lagrangean Relaxation [3] and Generalized Benders Decomposition [4], To demonstrate the efficiency of the proposed decomposition strategy we present numerical results obtained for three test systems. The computational experiments show the superiority of the Cross Decomposition approach.

Optimal Placement of PMUs in Power Systems Using Mixed Integer Non Linear Programming and Bacterial Foraging Algorithm

This paper aims to optimize the PMU (Phasor Measurement Unit) placement for a full observation of the power network and the minimum number of PMUs. In this paper, competition of Mixed Integer Non-Linear Programming and heuristically algorithms such as Bacterial Foraging Algorithm is presented. The results are demonstrated with PMU placement optimization simulation and a redundancy measurement analysis by using IEEE14-bus and Tehran Regional electric company 41-bus networks.

Demand allocation with latency cost functions

We address the exact resolution of a Mixed Integer Non Linear Programming model where resources can be activated in order to satisfy a demand (a covering constraint) while minimizing total cost. For each resource, there is a fixed activation cost and a variable cost, expressed by means of latency functions. We prove that this problem is $${\mathcal {N} \mathcal {P}}$$ -hard even for linear latency functions. A branch and bound algorithm is devised, having two important features. First, a dual bound (equal to that obtained by continuous relaxation) can be computed very efficiently at each node of the enumeration tree. Second, to break symmetries resulting in improved efficiency, the branching scheme is n-ary (instead of binary). These features lead to a successful comparison against two popular commercial and open-source solvers, CPLEX and Bonmin.