Multi-objective Case Research Articles

A single and multiobjective robust optimization of a microgrid in distribution network considering uncertainty risk.

In this paper, single and multi-objective robust optimization of a microgrid (MG) including photovoltaic (PV) and wind turbine (WT) sources with battery storage has been performed in a radial 33-bus distribution network considering uncertainty risk for minimizing the cost of energy losses, cost of electricity purchase from the post as well as power purchase from the MG. The problem is implemented in two approaches without uncertainty (deterministic) and with uncertainty (robust) via a flow direct algorithm (FDA). In the deterministic approach, the variables such as the site and the optimal size of the equipments in the short-term study horizon of 24h, as well as the total cost, have been determined that the results of the MG deterministic optimization in the network indicate the reduction of network losses with the total cost minimization. The superior capability of the recommended deterministic approach based on the FDA is also confirmed in comparison with GA and PSO algorithms. In addition, the MG optimization problem with the robust approach with the information gap decision theory (IGDT) with risk-averse strategy is implemented to achieve the maximum radius of uncertainty (MRU) of renewable production and also network demand in three single and multi-objective cases. Based on the proposed robust approach, the level of system robustness has been determined in the condition of the worst uncertainty scenario against forecasting errors by considering different uncertainty budgets. It is also determined which uncertain parameter is more sensitive to the changes of the uncertainty budget. The findings demonstrated that in the multi-objective robust optimization, the constraints of the problem are not satisfied for budgets more than 40%, and in for sample the system uncertainty budget of 5%, there is a 27.51% decrease in resource production and a 0.87% increase in network load. The 40% uncertainty budget of the system is robust with a 16.80% decrease in resource generation and a 19.10% increase in network load. Therefore, one of the benefits of multi-objective optimization in comparison with single-objective optimization is balancing the sensitivity of uncertain parameters.

Multi-Objective Models for Sparse Optimization in Linear Support Vector Machine Classification

The design of linear Support Vector Machine (SVM) classification techniques is generally a Multi-objective Optimization Problem (MOP). These classification techniques require finding appropriate trade-offs between two objectives, such as the amount of misclassified training data (classification error) and the number of non-zero elements of the separator hyperplane. In this article, we review several linear SVM classification models in the form of multi-objective optimization. We put particular emphasis on applying sparse optimization (in terms of minimization of the number of non-zero elements of the separator hyperplane) to Feature Selection (FS) for multi-objective optimization linear SVM. Our primary purpose is to demonstrate the advantages of considering linear SVM classification techniques as MOPs. In multi-objective cases, we can obtain a set of Pareto optimal solutions instead of one optimal solution in single-objective cases. The results of these linear SVMs are reported on some classification datasets. The test problems are specifically designed to challenge the number of non-zero components of the normal vector of the separator hyperplane. We used these datasets for multi-objective and single-objective models.

Fuzzy optimization for the remediation of saline oily wastewater through electrocoagulation: a multi-objective case analysis

Fuzzy optimization for the remediation of saline oily wastewater through electrocoagulation: a multi-objective case analysis

An accelerated proximal gradient method for multiobjective optimization

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( $$O(1 / k^2)$$ ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

An Extensive Study Using the Beetle Swarm Method to Optimize Single and Multiple Objectives of Various Optimal Power Flow Problems

An electric energy generation system, under the economic operation mode, is an imperative mission in the power system function. This article deals with the use of beetle swarm optimization algorithm (BSOA), for optimal power flow (OPF) solution, in an effective approach. BSOA is a competent optimization technique, to handle multimodal, nonlinear, and nondifferentiable objective functions. The proposed OPF is modeled by numerous objective functions, formulations with constraints, examined with thirty-one different cases, on the three distinguished test systems (IEEE 30, 57, and 118-bus), using single and weighted sum multiobjectives. Six new multiobjective cases are also studied. The control variables, such as real generation of power, tap setting ratio of transformers, bus voltages magnitudes, and the values of shunt capacitor, are also optimized. Potency and robustness of this proposed method were investigated and evaluated with more recent findings reported in the literature. This extensive study revealed the preeminence of the presented technique, applied to OPF problem, with intricate and nonsmooth objective functions.

Optimization of Alkaline Extraction of Xylan-Based Hemicelluloses from Wheat Straws: Effects of Microwave, Ultrasound, and Freeze-Thaw Cycles.

The alkaline extraction of hemicelluloses from a mixture of three varieties of wheat straw (containing 40.1% cellulose, 20.23% xylan, and 26.2% hemicellulose) was analyzed considering the following complementary pre-treatments: freeze-thaw cycles, microwaves, and ultrasounds. The two cycles freeze-thaw approach was selected based on simplicity and energy savings for further analysis and optimization. Experiments planned with Design Expert were performed. The regression model determined through the response surface methodology based on the severity factor (defined as a function of time and temperature) and alkali concentration as variables was then used to optimize the process in a multi-objective case considering the possibility of further use for pulping. To show the properties and chemical structure of the separated hemicelluloses, several analytical methods were used: high-performance chromatography (HPLC), Fourier-transformed infrared spectroscopy (FTIR), proton nuclear magnetic resonance spectroscopy (1H-NMR), thermogravimetry and derivative thermogravimetry analysis (TG, DTG), and scanning electron microscopy (SEM). The verified experimental optimization result indicated the possibility of obtaining hemicelluloses material containing 3.40% glucan, 85.51% xylan, and 7.89% arabinan. The association of hot alkaline extraction with two freeze-thaw cycles allows the partial preservation of the hemicellulose polymeric structure.

New second-order necessary optimality conditions for constrained vector optimization problems

This paper is devoted to second-order optimality conditions of constrained vector optimization problems with continuously Fréchet differentiable data. Different from conventional approaches, we employ a new one, image space analysis, to establish suitable regularity conditions and then present new second-order necessary optimality conditions that depend on one single KKT multiplier for all critical directions. Moreover, we propose one more regularity condition for the multiobjective case and compare it with the previous one. Besides, we compare the main regularity conditions with the existing ones in the literature for the scalar case.

Multi-objective grey wolf optimization and parametric study of a continuous solar-based tri-generation system using a phase change material storage unit

Considering the necessity of producing freshwater from seawater, especially in regions with high solar energy potential, this study proposes a novel system for tri-generation application using solar power through which high capability of freshwater production is available together with electricity and cooling. This system comprises a heliostat field and a phase change material storage unit for continuous production, a gas turbine cycle, a Kalina cycle, an absorption chiller, a humidification dehumidification desalination, and reverse osmosis desalination. The prediction of the behavior of the system has been accomplished utilizing a comprehensive parametric study; afterward, the method of grey wolf optimization has been utilized to optimize the simulation in different multi-objective cases. Regarding obtained results, the net power generation is obtained of about 4137 kW and 41.28 kg/s freshwater with 30.63 % exergetic efficiency and 57.55 $/GJ unit product cost. Also, applying the optimization declines the freshwater production rate, but it improves the other performance indexes. Under optimized condition an exergy efficiency of 32.90 % and a unit product cost of 48.98 $/GJ is achieved.

A multi-objective natural aggregation algorithm for optimizing user allocation matrix in visible light communication

Allocating users to LED panels in a visible light communication (VLC) environment is a non-convex multi-objective constrained optimization problem. The computational complexity and accuracy of the optimization are the key factors to deal with in a VLC environment. In this manuscript, initially the user allocation problem is formulated as a single objective optimization problem by using sum-utility function of data rate and blockage probability. Then these two contradictory objective functions are simultaneously optimized by a new multi-objective natural aggregation algorithm (MONAA). This proposed algorithm is inspired by the aggregation behavior of cockroaches which help them to determine the darkest spot in a search space. The performance of this multi-objective algorithm is compared with NSGA-II, MOPSO, MOEA/D and MOSOS on eight unconstrained and six constrained mathematical function optimizations. In single objective and multi-objective cases, superior performance is demonstrated by the natural aggregation algorithm for solving the user allocation matrix optimization.

Admissible Heuristics for Multi-Objective Planning

Planning problems of practical relevance commonly include multiple objectives that are difficult to weight a priori. Several heuristic search algorithms computing the Pareto front of non-dominated solutions have been proposed to handle these multi-objective (MO) planning problems. However, the design of informative admissible heuristics to guide these algorithms has not received the same level of attention. The standard practice is to use the so-called ideal point combination, which applies a single-objective heuristic to each objective independently, without capturing any of the trade-offs between them. This paper fills this gap: we extend several classes of classical planning heuristics to the multi-objective case, in such a way as to reflect the tradeoffs underlying the various objectives. We find that MO abstraction heuristics achieve overall the best performance, but that not every MO generalisation pays off.

An improved heap optimization algorithm for efficient energy management based optimal power flow model

The optimal power flow is considered as a crucial tool in the power systems’ operation and planning. To demonstrate, it aims at minimizing the operational costs of energy production and transmission by adjusting control variables with maintaining economic, operational, and environmental constraints. This article proposes and scrutinizes an Improved Heap-based Optimization Algorithm as a novel technique that successfully enhances the performance of a recently algorithm, namely Heap-based Optimization algorithm to address the optimal power flow problem. In the improved optimizer, an effective exploitation feature is emerged with the conventional version to improve its performance by enhancing the searching around the leader position. This enhancement can avoid being trapped in a local optimum and increase its global search capabilities. For the sake of practicality, the optimizers are developed with diverse objectives of the optimal power flow problem with minimization of fuel cost, emission amount, and transmission power losses with additional restrictions in real power systems that include valve-point effect and security constraints. A proposed multi-objective, improved heap optimization algorithm is investigated to solve multiobjective cases studied. The proposed multi-objective is developed based on the Pareto concept. Three standard systems: IEEE 57-bus system, a large scale 118-bus system, and a practical System are utilized to reveal the suitability and performance of the proposed technique in solving the optimal power flow problem. To illustrate the effectiveness of the proposed optimizer in handling non-convex and diverse scale optimization problems, a comparative analysis has been illustrated with those in the literature.

Hybridizing of Whale and Moth-Flame Optimization Algorithms to Solve Diverse Scales of Optimal Power Flow Problem

The optimal power flow (OPF) is a practical problem in a power system with complex characteristics such as a large number of control parameters and also multi-modal and non-convex objective functions with inequality and nonlinear constraints. Thus, tackling the OPF problem is becoming a major priority for power engineers and researchers. Many metaheuristic algorithms with different search strategies have been developed to solve the OPF problem. Although, the majority of them suffer from stagnation, premature convergence, and local optima trapping during the optimization process, which results in producing low solution qualities, especially for real-world problems. This study is devoted to proposing an effective hybridizing of whale optimization algorithm (WOA) and a modified moth-flame optimization algorithm (MFO) named WMFO to solve the OPF problem. In the proposed WMFO, the WOA and the modified MFO cooperate to effectively discover the promising areas and provide high-quality solutions. A randomized boundary handling is used to return the solutions that have violated the permissible boundaries of search space. Moreover, a greedy selection operator is defined to assess the acceptance criteria of new solutions. Ultimately, the performance of the WMFO is scrutinized on single and multi-objective cases of different OPF problems including standard IEEE 14-bus, IEEE 30-bus, IEEE 39-bus, IEEE 57-bus, and IEEE118-bus test systems. The obtained results corroborate that the proposed algorithm outperforms the contender algorithms for solving the OPF problem.

Multiobjective optimisation of hybrid wind-PV-battery-fuel cell-electrolyser-diesel systems: An integrated configuration-size formulation approach

A generic integrated configuration-size optimisation formulation for design of hybrid renewable energy systems (HRES) is presented in this paper. This formulation allows identifying the optimum configuration for a given site and the optimum size of each component in that configuration by solving only one optimisation problem. Single and multiobjective case studies are defined for both on-grid and standalone systems using wind turbine, PV panel, battery bank, fuel cell, electrolyser and diesel generator as potential components. To solve the optimisation problems a genetic algorithm (GA) and a nondominated sorting GA (NSGA-II) are developed, in which the reproduction operators are designed carefully for robust exploration and exploitation at both size and configuration levels. Eight single and multiobjective case studies for a variety of renewable resources, objectives and constraints are conducted. The results show the versatility of the problem formulation in defining different HRES design problems and the robustness of the developed GA and NSGA-II in search within the design space at both configuration and size levels and finding the optimum size and configuration simultaneously.

The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

Guided Manta Ray foraging optimization using epsilon dominance for multi-objective optimization in engineering design

In recent decades, metaheuristics have proven their effectiveness in solving large-scale real-world problems with multiple objectives. However, we still need to design robust algorithms capable of converging and approximating efficiently the true Pareto set. In this paper, we extend the recently Manta Ray foraging optimization (MOMRFO) to the multiobjective case. MOMRFO uses a population archive to store the non-dominated solutions generated so far by the exploration process. The leader’s solutions are selected from the population archive to guide the Manta Rays population towards promising search regions. We use crowding distance and ε-dominance to provide a good compromise between diversity and convergence of the obtained potential Pareto set.The proposed algorithm is validated on five bi-objective test functions, seven three objective test functions, and is applied to structural design problems such as four-bar truss design, speed reduced design, welded beam design, and disk brake design. The algorithm is compared with four well-known multi-objective meta-heuristics. The experimental results show that the MOMRFO algorithm outperforms against the selected multiobjective meta-heuristics by providing better convergence behaviour with a better diversity of solutions.

On solving a class of fractional semi-infinite polynomial programming problems

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.

Bidirectional Preference-Based Search for State Space Graph Problems

In multiobjective state space graph problems, each solution-path is evaluated by a cost vector. These cost vectors can be partially or completely ordered using a preference relation compatible with Pareto dominance. In this context, multiobjective preference-based search (MOPBS) aims at computing the preferred feasible solutions according to a predefined preference model, these preferred solutions being a subset (possibly the entire set) of Pareto optima. Standard algorithms for MOPBS perform a unidirectional search developing the search tree forward from the initial state to a goal state. Instead, in this paper, we focus on bidirectional search algorithms developing simultaneously one forward and one backward search tree. Although bi-directional search has been tested in various single objective problems, its efficiency in a multiobjective setting has never been studied. In this paper, we present several implementations of bidirectional preference-based search convenient for the multiobjective case and investigate their efficiency.

Process optimization and safety assessment on a pilot-scale Bunsen process in sulfur–iodine cycle

This study investigates Bunsen reaction in the sulfur-iodine (SI) cycle for optimal conditions and specification of equipment in terms of the maximum HI yield and the least impurities in HIx (mixture of HI, I2 and H2O), the reaction safety, and dispersion of SO2 gas and HIX solution for leakage accident. The pilot-scale Bunsen process was simulated and validated. The optimization of the Bunsen reactor, 3-phase separator, and HIX purifier have been investigated in order to parameterize the operating conditions and equipment specification for three cases: (1) Maximize the HI yield for the final product (2) Minimize the H2SO4 impurities (3) Multi-objective case of both maximum HI production and minimum impurities. The gas reactivity safety was investigated on HI, H2SO4, I2, SO2, H2O, and O2. Also, the SO2 gas dispersion distance for 30 ppm, 0.75 ppm, and 0.2 ppm and HI dispersion distance for 120 ppm, 25 ppm, and 1 ppm was investigated for targeted unit operators at each optimization scenario. The deviation between pilot-scale experiment and simulation case falls within 1–3% for Bunsen reactor, 6~8% for 3-phase separator, and 2~4% for HIX purifier. The maximized HI production was increased by 17% for the maximum HI yield case from the designed case. The size and temperature of the Bunsen reactor was increased to enhance the reaction. However, the HIX purifier size was reduced since reverse Bunsen reaction causes loss in HI product. The H2SO4 impurities in the minimize H2SO4 impurities case were reduced by 71% from the designed case. The size of the Bunsen reactor remained the same as design case, but the HIX purifier size was increased to enhance the reverse Bunsen reaction. For multi-objective case, the HI productivity was increased by 16% and the H2SO4 impurities were reduced by 67% simultaneously. According Chemical Reactivity Worksheet (CRW) result, O2 should therefore not be stored with any components except iodine. For SO2 and HIX dispersion assessment, the maximum HI yield case reveals the maximum dispersion of SO2 gas and HIX solution from the Bunsen reactor. The dispersion from 3-phase separator was almost the same for all the cases. For HIX purifier, the minimum H2SO4 case exhibited the longest distance of SO2 gas and HI solution dispersion. At 3 bar and 140 °C, the maximum SO2 and HIX dispersion distance were occurred.

Robust Scheduling of Waste Wood Processing Plants with Uncertain Delivery Sources and Quality

While the study of reverse wood value chains has become an important topic recently, optimization-focused studies usually consider network-level problems and decisions, and do not address the individual processes in the network. In the case of waste wood, one such important process is the scheduling of the various machines in a waste wood processing facility to treat incoming wood deliveries with multiple sources and varying quality. This paper proposes a robust multi-objective mixed-integer linear programming model for the optimization of this process that considers the uncertain origins and compositions of the incoming deliveries, while aiming to minimize both lateness and energy consumption. An exhaustive study is performed on instance sets of different sizes and structures to show the efficiency and the limits of the proposed model both in single- and multi-objective cases.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.