Discrete Decision Variables Research Articles

Volt/VAR Optimization: A Survey of Classical and Heuristic Optimization Methods

Reactive power optimization and voltage control is one of the most critical components of power system operation, impacting both the economy and security of system operation. It is also one of the most complex optimization problems, being highly nonlinear, and comprising both continuous and discrete decision variables. This paper presents the problem formulation, and a thorough literature review and detailed discussion of the various solution methods that have been applied to the Volt/VAR optimization problem. Each optimization method is described in detail, and its strengths and shortcomings are outlined. The review provides detailed information on classical and heuristic methods that have been applied to the Volt/VAR optimization problem. The classical methods reviewed include (i) first- and second-order gradient-based methods, (ii) Quadratic Programming, (iii) Linear Programming, (iv) Interior-Point Methods, (iv) and mixed-integer programming and decomposition methods. The heuristic methods covered include (i) Genetic Algorithm, (ii) Evolutionary Programming, (iii) Particle Swarm Optimization, (iv) Fuzzy Set Theory, and (v) Expert Systems. A comparative analysis of the key characteristics of the classical and heuristic optimization methods is also presented along with the review.

Optimizing the electrification of roads with charge-while-drive technology

Given the current state of battery technology, applying electric trucks for long-haul transportation is a cumbersome task. Either the driving ranges are too short or high-capacity batteries are so heavy that payloads are limited. An old, yet recently revitalized charging infrastructure, currently evaluated on multiple test tracks all around the globe, allows to charge electric trucks while driving. However, installing the charging infrastructure along highways, namely, overhead wiring or road-embedded power lines both accessed by trucks equipped with a movable contact arm, is costly. This paper provides a detailed modeling approach based on continuous variables to minimize the installation costs for electrified highway kilometers while still providing sufficient energy for a given set of representative tours of electric vehicles. We apply our solution approach to show that investment costs can considerably be reduced along a European highway mainline while still allowing electrified transport among most adjacent major cities. Furthermore, we quantify the loss in precision, if instead of our detailed approach a more aggregate model based on discrete decision variables is applied.

A new method to multi – objective optimization of shell and tube heat exchanger for waste heat recovery

ABSTRACT Nowadays, fossil energy sources are decreasing gradually and environmental problems are increasing thus energy savings are among the current topics. In this context, determining the optimum design parameters of heat exchangers can provide significant savings during the economic life of waste heat recovery systems. However, due to discrete decision variables, highly nonlinearity, trapped to local optimum and high computation time, the optimization of a heat exchanger can be a robust problem. Therefore, this research introduces Univariate Search Method 2 (USM2) that can cope with such difficulties for multi-objective optimization. USM2 is based on weighted sum approach and uses a single-objective optimization method by moving from two arms on Pareto Front. To short computational time, Univariate Search Method (USM) has been chosen as a single-objective optimization method. The well-known Genetic Algorithm (GA) and Non-dominant Sorting Genetic Algorithm-2 (NSGA-2) methods were also used to compare USM and USM2 with fitness values and number of the function call corresponding to the computational time. According to the results, GA has offered better fitness values for both objectives than USM yet GA has much more number of the function call. However, USM2 has offered a better design range in the Pareto-Front than NSGA-2 despite having almost half of the function call number of NSGA-2. Moreover, the solutions offered by the USM2 at the ends of the Pareto-Front are highly competitive when compared to GA’s solutions.

Buffer allocation design for unreliable production lines using genetic algorithm and finite perturbation analysis

The buffer allocation problem in production lines is an NP-hard combinatorial optimisation problem. This paper proposes a new hybrid optimisation approach (using simulation) relying on genetic algorithm (GA) and finite perturbation analysis (FPA). Unlike the infinitesimal perturbation analysis, which deals with small (infinitesimal variation) perturbations for estimating gradients of the performance measure, FPA deals with larger (finite) or more lasting perturbations. It is an extension specifically dedicated to discrete decision variables and applicable to most discrete-event dynamic systems. The proposed method allows a global search using GA, with refinement in specific solution-space regions using FPA. The main objective is to maximise the average production rate of a production line with unreliable machines, by allocating the total buffer capacity in locations between machines. Extensive numerical experiments show that: (1) the proposed hybrid GA-FPA method clearly outperforms the state-of-the-art methods from the literature; (2) combining FPA and GA is beneficial when compared to employing GA or FPA independently.

Optimal randomized classification trees

Classification and Regression Trees (CARTs) are off-the-shelf techniques in modern Statistics and Machine Learning. CARTs are traditionally built by means of a greedy procedure, sequentially deciding the splitting predictor variable(s) and the associated threshold. This greedy approach trains trees very fast, but, by its nature, their classification accuracy may not be competitive against other state-of-the-art procedures. Moreover, controlling critical issues, such as the misclassification rates in each of the classes, is difficult. To address these shortcomings, optimal decision trees have been recently proposed in the literature, which use discrete decision variables to model the path each observation will follow in the tree. Instead, we propose a new approach based on continuous optimization. Our classifier can be seen as a randomized tree, since at each node of the decision tree a random decision is made. The computational experience reported demonstrates the good performance of our procedure.

A random forest assisted evolutionary algorithm using competitive neighborhood search for expensive constrained combinatorial optimization

Many real-world combinatorial optimization problems have both expensive objective and constraint functions. Although surrogate models for the discrete decision variables can be trained to replace the expensive fitness evaluations in evolutionary algorithms, the approximation errors of the surrogate models for the constraint function easily misguide the search. The classic genetic algorithm, which is often applied straightforwardly to the combinatorial optimization problems, gradually exposes its inefficiency in the search process. Therefore, we proposed a random forest assisted evolutionary algorithm using a new competitive neighborhood search, where random forest is used as the surrogate models to approximate both the objective and constraint functions and the competitive neighborhood search is to improve the search efficiency. Moreover, competitive neighborhood search shows a natural adaptability to the surrogate model, which helps to reduce the impact of approximation errors. The proposed algorithm is tested on 01 knapsack problems and quadratic knapsack problems with various dimensions and constraints. The experimental results demonstrate that the proposed algorithm is able to solve the expensive constrained combinatorial optimization problems efficiently.

Linear Quadratic Regulator of Discrete-Time Switched Linear Systems

This brief studies the linear quadratic regulation problem of discrete-time switched linear systems via dynamic programming. Discrete decision variables (referred as switch inputs) are introduced to represent the subsystem selection. This formulation leads to a mixed integer quadratic programming problem, which is known to be computationally hard in general. Instead of deriving suboptimal policies with analytical bounds on its optimality, the optimal switching sequence is computed in a computationally efficient manner when the input matrix is restricted as a vector. The unique contribution of this brief is an analytical expression of both the optimal switching condition (determining the subsystem selection) and the optimal control law. Proof of optimality is presented by fractional optimization and the practical merits of our approach is validated through simulations on a second-order system in comparison with recent pruning methods.

A particle swarm optimization algorithm for mixed-variable optimization problems

Abstract Many optimization problems in reality involve both continuous and discrete decision variables, and these problems are called mixed-variable optimization problems (MVOPs). The mixed decision variables of MVOPs increase the complexity of search space and make them difficult to be solved. The Particle Swarm Optimization (PSO) algorithm is easy to implement due to its simple framework and high speed of convergence, and has been successfully applied to many difficult optimization problems. Many existing PSO variants have been proposed to solve continuous or discrete optimization problems, which make it feasible and promising for solving MVOPs. In this paper, a new PSO algorithm for solving MVOPs is proposed, namely P S O m v , which can deal with both continuous and discrete decision variables simultaneously. To efficiently handle mixed variables, the P S O m v employs a mixed-variable encoding scheme. Based on the mixed-variable encoding scheme, two reproduction methods respectively for continuous variables and discrete variables are proposed. Furthermore, an adaptive parameter tuning strategy is employed and a constraints handling method is utilized to improve the overall efficiency of the P S O m v .The experimental results on 28 artificial MVOPs and two practical MVOPs demonstrate that the proposed P S O m v is a competitive algorithm for MVOPs.

Distributed MPC for Large Freeway Networks Using Alternating Optimization

The Model Predictive Control (MPC) framework has shown great potential for the control of Variable Speed Limits (VSLs) and Ramp Metering (RM) installations. However, the implementation to large freeway networks remains challenging. One major reason is that, by considering the VSLs to be discrete decision variables, an extremely difficult Mixed Integer Nonlinear Programming (MINLP) optimization problem has to be solved within every controller sampling interval. Consequently, many related papers relax the MINLP problems by considering the VSLs to be continuous variables. This paper proposes two novel MPC algorithms for coordinated control of discrete VSLs and continuous RM rates that do not make this relaxation. The proposed algorithms use a distributed control architecture and an alternating optimization scheme to relax the MINLP optimization problems but still consider the VSLs as discrete variables and, hence, offer a trade-off between computational complexity and system performance. The performance of the proposed algorithms is evaluated in a case study. The case study shows that relaxing the VSLs to be continuous variables with a distributed architecture results in a significant performance loss. Furthermore, both proposed algorithms have a lower computational complexity than the more conventional centralized approach and, as a result, they do manage to solve all optimization problems within the sampling intervals. Moreover, one of the proposed algorithms has a system performance that is remarkably similar to the optimal performance of the centralized approach.

Optimal Placement of Capacitors, Voltage Regulators, and Distributed Generators in Electric Power Distribution Systems

Context: With the advent of the smart grid paradigm, electrical distribution network (EDN) operators are making efforts to modernize their power grids through the optimal implementation of distributed generators (DGs) and other devices such as capacitors (CAs) and voltage regulators (VRs). The optimal allocation of such devices is a challenging task involving discrete and integer decision variables. Method: This paper presents an approach for the optimal placement of CAs, VRs and DGs in EDNs. The distinctive feature of the proposed model is the fact that it can be used to optimize the allocation of all of these elements together, in pairs, or separately. The optimal implementation of these elements is formulated as a mixed integer nonlinear programming (MINLP) problem, and it is solved by means of a specialized genetic algorithm (SGA). Results: The proposed methodology was tested on the IEEE 69-bus test system. The results were compared with previous works from the specialized literature, showing the effectiveness and robustness of the model. Conclusions: It was found that the appropriate allocation of CAs, VRs, and DGs results in a significant power loss reduction. It was also found that the proposed model is faster than other techniques proposed in the specialized literature. Acknowledgements: The authors gratefully acknowledge the support from the Colombia Científica program, within the framework of the Ecosistéma Científico (Contract No. FP44842- 218-2018). The authors also acknowledge the support of the State University of Londrina and Universidad Tecnológica de Pereira (UTP).

A PSO-BPSO Technique for Hybrid Power Generation System Sizing

The Particle Swarm Optimization (PSO) algorithm has been widely used in the field of optimization mainly due to its easy implementation, robustness, fast convergence, and low computational cost. However, due to its continuous nature, the PSO cannot be applied directly to real-life problems such as hybrid energy generating systems (HEGS) sizing, which contain continuous and discrete decision variables. In this context, the present work proposes the combination of the original version of the PSO with the binary version of the same algorithm (BPSO) for the sizing of HEGS. The transfer function is the main difference between these two algorithms. In this paper, an S-type transfer function is used to map the continuous space into a discrete space. All components of the HEGS are modeled and simulated during the optimization process. The net present value is defined as the unique objective function. The state of charge (SOC) of the batteries is the main constraint. The proposed PSO-BPSO is used for sizing hybrid power generating systems in the Galapagos Islands in Ecuador. Results show that the best configuration for the studied case is a hybrid system with solar panels, batteries, and diesel generators. Configurations that contain only photovoltaic panels and batteries imply a higher cost due to the oversizing of the battery bank. The proposed PSO-BPSO algorithm revealed to be a simple and powerful tool for efficient energy systems sizing.

Bayesian optimization of variable-size design space problems

Within the framework of complex system design, it is often necessary to solve mixed variable optimization problems, in which the objective and constraint functions can depend simultaneously on continuous and discrete variables. Additionally, complex system design problems occasionally present a variable-size design space. This results in an optimization problem for which the search space varies dynamically (with respect to both number and type of variables) along the optimization process as a function of the values of specific discrete decision variables. Similarly, the number and type of constraints can vary as well. In this paper, two alternative Bayesian optimization-based approaches are proposed in order to solve this type of optimization problems. The first one consists of a budget allocation strategy allowing to focus the computational budget on the most promising design sub-spaces. The second approach, instead, is based on the definition of a kernel function allowing to compute the covariance between samples characterized by partially different sets of variables. The results obtained on analytical and engineering related test-cases show a faster and more consistent convergence of both proposed methods with respect to the standard approaches.

Particle swarm optimisation with opposition learning-based strategy: an efficient optimisation algorithm for day-ahead scheduling and reconfiguration in active distribution systems

In operation of active electric distribution networks, optimal configuration and schedule of distributed generation and reactive power resources are determined. This represents a formidable multi-modal constrained optimisation problem with discrete decision variables. Metaheuristics are the most common approaches for solving this problem. However, due to its multi-modal nature, metaheuristics commonly converge prematurely into local optima and cannot find near-global solutions. In this research, a new particle swarm optimisation (PSO) variant is put forward for finding optimal configuration and schedule of distributed generation and reactive power resources in distribution systems including both dispatchable and renewable distributed energy resources. In the proposed PSO variant, opposition-based learning concept is incorporated into PSO which reduces premature convergence probability through enhancement of swarm leaders. The results of the proposed opposition-based PSO in IEEE 69 bus system indicate its outperformance over conventional PSO, time-varying acceleration coefficient PSO, fractal optimisation algorithm and evolutionary programming.

The separation of ternary azeotropic mixture: Thermodynamic insight and improved multi-objective optimization

Abstract Acetonitrile (ACN) and Ethanol (EtOH) are important organic solvents and they are frequently used as mobile phase in high performance liquid chromatography resulting in the production of waste ternary mixture ACN/EtOH/water. The separation of such ternary systems is a key to recovery valuable solvents in waste liquid from views on economic and environmental benefits. Thus, we proposed an effective separation strategy of triple-column extractive distillation (TCED) to separate such ternary mixture with three binary azeotropes and a single ternary azeotrope for the first time. The suitable entrainer and the separation sequence were determined by thermodynamic insights (i.e., residue curve maps, isovolatility line, univolatility line and material balance lines). In addition, an improved multi-objective genetic algorithm optimization embedding the weak mutation and detection/deduplication of overlapping solutions was employed to optimize the proposed process with plenty of continuous and discrete decision variables. Finally, DMSO was determined as the most appropriate entrainer to separate such complex ternary mixture. The separation sequence of TCED process was determined as the ACN first, then EtOH and water as the last product. The optimal TCED process with the trade-off benefits between fixed capital investment and annual operating costs was obtained.

Online Stochastic Optimization of Networked Distributed Energy Resources

This paper investigates distributed control and incentive mechanisms to coordinate distributed energy resources (DERs) with both continuous and discrete decision variables as well as device dynamics in distribution grids. We formulate a multiperiod social welfare maximization problem, and based on its convex relaxation propose a distributed stochastic dual gradient algorithm for managing DERs. We further extend it to an online real-time setting with time-varying operating conditions, asynchronous updates by devices, and feedback being leveraged to account for nonlinear power flows as well as reduce communication overhead. The resulting algorithm provides a general online stochastic optimization algorithm for coordinating networked DERs with discrete power setpoints and dynamics to meet operational and economic objectives and constraints. We characterize the convergence of the algorithm analytically and evaluate its performance numerically.

Surrogate-based optimization for mixed-integer nonlinear problems

Simulation-based optimization using surrogate models enables decision-making through the exchange of data from high-fidelity models and development of approximations. Many chemical engineering optimization problems, such as process design and synthesis, rely on simulations and contain both discrete and continuous decision variables. Surrogate-based optimization with continuous variables has been studied extensively; however, there are many open challenges for the case of mixed-variable inputs. In this work, we propose an algorithm for mixed-integer nonlinear simulation-based problems that uses adaptive sampling and surrogate modeling with one-hot encoding. We propose techniques for the design of experiments for mixed-variable problems, surrogate modeling for mixed-variable response surfaces, and iterative approximation-optimization procedure that leads to optimal solutions. Results show that one-hot encoding leads to accurate and robust mixed-variable Gaussian Process and Neural Network models that are effective surrogates for optimization. The proposed algorithm is tested on mixed-integer nonlinear benchmark problems and a chemical process synthesis case study.

A matheuristic for a bi-objective demand-side optimization for cooperative smart homes

This paper proposes the demand-side management (C-DSM) of collaborative homes during one day. Some homes may have photovoltaic panels and batteries, other homes may have batteries, and the remaining homes are pure energy consumers. Homes are interconnected and connected to the main grid. The proposed C-DSM is formulated as a constrained bi-objective and mixed-integer linear optimization model with one objective related to the total net energy cost and the other related to discomfort caused by allowing flexibility of controllable loads within an acceptable comfort range. A matheuristic approach has been proposed to determine an efficient Pareto set for the problem, combining the non-dominated sorting genetic algorithm II (NSGAII) with an exact solver. In this approach, discrete decision variables are represented as partial chromosomes and an exact solver determines the continuous decision variables in an optimal way. A number of simulations are performed and compared with the weighted sum algorithm (WSA) under four cases for small to large number of homes. The results demonstrate the effectiveness of power cooperation among homes and show that our algorithm is able to obtain more Pareto solutions in a much shorter time that are far better than those obtained by the WSA. The proposed algorithm is suitable for large-sized C-DSM problem instances, promotes power cooperation between homes, reduces the dependency to the main grid and achieves individual fairness of energy net cost of each home without the need for installation of photovoltaic panels and batteries for all homes.

Optimal sensor and actuator placement for structural health monitoring via an efficient convex cost-benefit optimization

The number and position of sensors and actuators are key decision variables that dictate the performance of any structural health monitoring system. This paper proposes choosing them optimally by using an objective function that combines a measure of parameter uncertainty, the expected information entropy, along with the cost of both sensors and actuators. The resulting optimization problem over discrete decision variables is computationally challenging, but here it is convexified by relaxing them into continuous variables, thus obtaining a significant reduction of the computational cost. The proposed approach is applied to ultrasonic guided-wave based inspection and is illustrated using two case studies with arbitrary geometries and different materials. The results demonstrate the high efficiency and accuracy of the convex optimization in trading-off uncertainty and cost in order to provide optimal sensor configurations in complex structures. As a key contribution, the proposed methodology allows us to include the actuators with the sensors in the optimization problem while still maintaining the efficiency of the minimization process. In the application to ultrasonic guided-waves, the optimal configurations lead to set-ups where the sensors and actuators are coincident in number and position.

All-relevant feature selection using multidimensional filters with exhaustive search

This paper describes a method for the identification of informative variables in an information system with discrete decision variables. It is targeted specifically towards the discovery of variables that are non-informative when considered alone, but informative when the synergistic interactions between multiple variables are considered. To this end, mutual entropy of all possible k-tuples of variables with a decision variable is computed. Then, for each variable, the maximum information gain due to interactions with other variables is obtained. For non-informative variables, this quantity conforms to well-known statistical distributions. This allows for discerning the truly informative variables from the non-informative ones. To demonstrate this approach, the method is applied to several synthetic datasets that involve complex multidimensional interactions between variables. The performance of the method is also validated on a real-world dataset. It is shown that the method is capable of identifying the most important informative variables, even when the dimensionality of the analysis is smaller than the true dimensionality of the problem. What is more, the high sensitivity of the algorithm allows for the detection of the influence of nuisance variables on the response variable.

A novel approach for optimal trajectory design with multiple operation modes of propulsion system, part 1

Efficient performance of a number of engineering systems is achieved through different modes of operation - yielding systems described as “hybrid”, containing both real-valued and discrete decision variables. Prominent examples of such systems, in space applications, could be spacecraft equipped with 1) a variable-Isp, variable-thrust engine or 2) multiple engines each capable of switching on/off independently. To alleviate the challenges that arise when an indirect optimization method is used, a new framework — Composite Smooth Control (CSC) — is proposed that seeks smoothness over the entire spectrum of distinct control inputs. A salient aftermath of the application of the CSC framework is that the original multi-point boundary-value problem can be treated as a two-point boundary-value problem with smooth, differentiable control inputs; the latter is notably easier to solve, yet can be made to accurately approximate the former hybrid problem. The utility of the CSC framework is demonstrated through a multi-year, multi-revolution heliocentric fuel-optimal trajectory for a spacecraft equipped with a variable-Isp, variable-thrust engine.