Robust Multi-Objective Optimization of Chromatographic Rare Earth Element Separation

Rare earth elements are strategic commodities in many countries, and an important resource for the growing modern technology industry. As such, there is an increasing interest for development of rare earth element processing, and this work is a part of further development of chromatography as a rare earth element separation process method. Process optimization is pivotal for process development, and it is common that several competing objectives must be regarded. Chromatographic separation processes often consider competing objectives, such as productivity, yield, pool concentration and modifier consumption, which leads to Pareto optimal solutions. Adding robustness to a process is of great importance to account for process disturbances and uncertainties but generally comes with reduced performance of the other process objectives as a trade off. In this study, a model-based robust multi-objective optimization was carried out for batch-wise chromatographic separation of the rare earth elements samarium, europium and gadolinium, which was considered highly un-robust due to the neighbouring peaks proximity to the product pooling horizon. The results from the robust optimization were used to chart the required operation point changes for keeping the amount of failed batches at an acceptable level when a certain level of process disturbance was introduced. The loss of process performance due to the gained robustness was found to be in the range of 10% - 20% reduced productivity when comparing the robust and un-robust Pareto solutions at Pareto points with identical yield. The methodology presented shows how to increase robustness to a highly un-robust system while still keeping multiple objectives at their optima.

A robust multi-objective and multi-physics optimization of multi-physics behavior of microstructure

A new strategy is presented to solve robust multi-physics multi-objective optimization problem known as improved multi-objective collaborative optimization （IMOCO） and its extension improved multi-objective robust collaborative （IMORCO）. In this work, the proposed IMORCO approach combined the IMOCO method, the worst possible point （WPP） constraint cuts and the Genetic algorithm NSGA-II type as an optimizer in order to solve the robust optimization problem of multi-physics of microstructures with uncertainties. The optimization problem is hierarchically decomposed into two levels： a microstructure level, and a disciplines levels, For validation purposes, two examples were selected： a numerical example, and an engineering example of capacitive micro machined ultrasonic transducers （CMUT） type. The obtained results are compared with those obtained from robust non-distributed and distributed optimization approach, non-distributed multi-objective robust optimization （NDMORO） and multi-objective collaborative robust optimization （McRO）, respectively. Results obtained from the application of the IMOCO approach to an optimization problem of a CMUT cell have reduced the CPU time by 44% ensuring a Pareto front close to the reference non-distributed multi-objective optimization （NDMO） approach （mahalanobis distance, D2M =0.9503 and overall spread, So=0.2309）. In addition, the consideration of robustness in IMORCO approach applied to a CMUT cell of optimization problem under interval uncertainty has reduced the CPU time by 23% keeping a robust Pareto front overlaps with that obtained by the robust NDMORO approach （D2M =10.3869 and So=0.0537）.

Sequential multi-objective optimization of thin-walled aluminum alloy tube bending under various uncertainties

Combining the design of experiments(DOE)and three-dimensional finite element(3D-FE)method,a sequential multiobjectiveoptimization of larger diameter thin-walled(LDTW)Al-alloy tube bending under uncertainties was proposed andimplemented based on the deterministic design results.Via the fractional factorial design,the significant noise factors are obtained,viz,variations of tube properties,fluctuations of tube geometries and friction.Using the virtual Taguchi’s DOE of inner and outerarrays,considering three major defects,the robust optimization of LDTW Al-alloy tube bending is achieved and validated.For thebending tools,the robust design of mandrel diameter was conducted under the fluctuations of tube properties,friction and tubegeometry.For the processing parameters,considering the variations of friction,material properties and manufacture deviation ofmandrel,the robust design of mandrel extension length and boosting ratio is realized.

Multi-objective robust secure beamforming for cognitive satellite and UAV networks

A multi-objective optimization based robust beamforming(BF)scheme is proposed to realize secure transmission in a cognitive satellite and unmanned aerial vehicle(UAV)network.Since the satellite network coexists with the UAV network,we first consider both achievable secrecy rate maximization and total transmit power minimization,and formulate a multi-objective optimization problem(MOOP)using the weighted Tchebycheff approach.Then,by supposing that only imperfect channel state information based on the angular information is available,we propose a method combining angular discretization with Taylor approximation to transform the non-convex objective function and constraints to the convex ones.Next,we adopt semi-definite programming together with randomization technology to solve the original MOOP and obtain the BF weight vector.Finally,simulation results illustrate that the Pareto optimal trade-off can be achieved,and the superiority of our proposed scheme is confirmed by comparing with the existing BF schemes.

Robust design and optimization for autonomous PV-wind hybrid power systems

This study presents a robust design method for autonomous photovoltaic (PV)-wind hybrid power systems to obtain an optimum system configuration insensitive to design variable variations. This issue has been formulated as a constraint multi-objective optimization problem, which is solved by a multi-objective genetic algorithm, NSGA-II. Monte Carlo Simulation (MCS) method, combined with Latin Hypercube Sampling (LHS), is applied to evaluate the stochastic system performance. The potential of the proposed method has been demonstrated by a conceptual system design. A comparative study between the proposed robust method and the deterministic method presented in literature has been conducted. The results indicate that the proposed method can find a large mount of Pareto optimal system configurations with better compromising performance than the deterministic method. The trade-off information may be derived by a systematical comparison of these configurations. The proposed robust design method should be useful for hybrid power systems that require both optimality and robustness.

Robust optimization design on impeller of mixed-flow pump

To increase the robustness of the optimization solutions of the mixed-flow pump,the impeller was firstly indirectly parameterized based on the 2D blade design theory.Secondly,the robustness of the optimization solution was mathematically defined,and then calculated by Monte Carlo sampling method.Thirdly,the optimization on the mixed-flow pump′s impeller was decomposed into the optimal and robust sub-optimization problems,to maximize the pump head and efficiency and minimize the fluctuation degree of them under varying working conditions at the same time.Fourthly,using response surface model,a surrogate model was established between the optimization objectives and control variables of the shape of the impeller.Finally,based on a multi-objective genetic optimization algorithm,a two-loop iterative optimization process was designed to find the optimal solution with good robustness.Comparing the original and optimized pump,it is found that the internal flow field of the optimized pump has been improved under various operating conditions,the hydraulic performance has been improved consequently,and the range of high efficient zone has also been widened.Besides,with the changing of working conditions,the change trend of the hydraulic performance of the optimized pump becomes gentler,the flow field distribution is more uniform,and the influence degree of the varia-tion of working conditions decreases,and the operating stability of the pump is improved.It is concluded that the robust optimization method proposed in this paper is a reasonable way to optimize the mixed-flow pump,and provides references for optimization problems of other fluid machinery.

Multi-objective robust controller synthesis for discrete-time systems with convex polytopic uncertain domain

Multi-objective robust state-feedback controller synthesis problems for linear discrete-time uncertain systems are addressed. Based on parameter-dependent Lyapunov functions, the Gl2 and GH2 norm expressed in terms of LMI (Linear Matrix Inequality) characterizations are further generalized to cope with the robust analysis for convex polytopic uncertain system. Robust state-feedback controller synthesis conditions are also derived for this class of uncertain systems. Using the above results, multi-objective state-feedback controller synthesis procedures which involve the LMI optimization technique are developed and less conservative than the existing one. An illustrative example verified the validity of the approach.

A Robust Optimization Approach Considering the Robustness of Design Objectives and Constraints

The problem of robust design is treated as a multi-objective optimization issue in which the performance mean and variation are optimized and minimized respectively, while maintaining the feasibility of design constraints under uncertainty. To effectively address this issue in robust design, this paper presents a novel robust optimization approach which integrates multi-objective optimization concepts with Taguchi’s crossed arrays techniques. In this approach, Pareto-optimal robust design solution sets are obtained with the aid of design of experiment set-ups, which utilize the results of Analysis of Variance to quantify relative dominance and significance of design variables. A beam design problem is used to illustrate the effectiveness of the proposed approach.

Mixed Gl2／GH2 multi-channel multi-objective control synthesis for discrete time systems

This paper proposes a new approach for multi-objective robust control. The approach extends the standard generalized l2 (Gl2) and generalized H2 (GH2) conditions to a set of new linear matrix inequality (LMI) constraints based on a new stability condition. A technique for variable parameterization is introduced to the multi-objective control problem to preserve the linearity of the synthesis variables. Consequently, the multi-channel multi-objective mixed Gl2/GH2 control problem can be solved less conservatively using computationally tractable algorithms developed in the paper.

Structural Multi-objective Probabilistic Design for Six Sigma

Uncertainties in engineering design may lead to low reliable solutions that also exhibit high sensitivity to uncontrollable variations. In addition, there often exist several conflicting objectives and constraints in various design environments. In order to obtain solutions that are not only "multi-objectively" optimal, but also reliable and robust, a probabilistic optimization method was presented by integrating six sigma philosophy and multi-objective genetic algorithm. With this method, multi-objective genetic algorithm was adopted to obtain the global Pareto solutions, and six sigma method was used to improve the reliability and robustness of those optimal solutions. Two engineering design problems were provided as examples to illustrate the proposed method.

Multi-objective robust airfoil optimization based on non-uniform rational B-spline (NURBS) representation

In order to improve airfoil performance under different flight conditions and to make the performance insensitive to off-design condition at the same time,a multi-objective optimization approach considering robust design has been developed and applied to airfoil design. Non-uniform rational B-spline (NURBS) representation is adopted in airfoil design process,control points and related weights around airfoil are used as design variables. Two airfoil representation cases show that the NURBS method can get airfoil geometry with max geometry error less than 0.0019. By using six-sigma robust approach in multi-objective airfoil design,each sub-objective function of the problem has robustness property. By adopting multi-objective genetic algorithm that is based on non-dominated sorting,a set of non-dominated airfoil solutions with robustness can be obtained in the design. The optimum robust airfoil can be traded off and selected in these non-dominated solutions by design tendency. By using the above methods,a multi-objective robust optimization was conducted for NASA SC0712 airfoil. After performing robust airfoil optimization,the mean value of drag coefficient at Ma0.7-0.8 and the mean value of lift coefficient at post stall regime (Ma0.3) have been improved by 12.2% and 25.4%. By comparing the aerodynamic force coefficients of optimization result,it shows that: different from single robust airfoil design which just improves the property of drag divergence at Ma0.7-0.8,multi-objective robust design can improve both the drag divergence property at Ma0.7-0.8 and stall property at low speed. The design cases show that the multi-objective robust design method makes the airfoil performance robust under different off-design conditions.

Multi-objective Optimization Design of Inset-surface Permanent Magnet Machine Considering Deterministic and Robust Performances

The inset-surface permanent magnet(ISPM)machine can achieve the desired electromagnetic performance according to the traditional deterministic design.However,the reliability and quality of the machine may be affected by the essential manufacturing tolerances and unavoidable noise factors in mass production.To address this weakness,a comprehensive multi-objective optimization design method is proposed,in which robust optimization is performed after the deterministic design.The response surface method is first adopted to establish the optimization objective equation.Afterward,the sample points are obtained via Monte Carlo simulation considering the design-variable uncertainty.The Design for Six Sigma approach is adopted to ensure the robustness of the design model.Furthermore,the barebones multi-objective particle swarm optimization algorithm is used to obtain a compromise solution.A prototype is manufactured to evaluate the effectiveness of the proposed method.According to the finite-element analysis and experimental tests,the electromagnetic performance and reliability of the machine are significantly enhanced with the proposed method.

Robust multi-objective optimization of rolling schedule for tandem cold rolling based on evolutionary direction differential evolution algorithm

According to the actual requirements,profile and rolling energy consumption are selected as objective functions of rolling schedule optimization for tandem cold rolling.Because of mechanical wear,roll diameter has some uncertainty during the rolling process,ignoring which will cause poor robustness of rolling schedule.In order to solve this problem,a robust multi-objective optimization model of rolling schedule for tandem cold rolling was established.A differential evolution algorithm based on the evolutionary direction was proposed.The algorithm calculated the horizontal angle of the vector,which was used to choose mutation vector.The chosen vector contained converging direction and it changed the random mutation operation in differential evolution algorithm.Efficiency of the proposed algorithm was verified by two benchmarks.Meanwhile,in order to ensure that delivery thicknesses have descending order like actual rolling schedule during evolution,a modified Latin Hypercube Sampling process was proposed.Finally,the proposed algorithm was applied to the model above.Results showed that profile was improved and rolling energy consumption was reduced compared with the actual rolling schedule.Meanwhile,robustness of solutions was ensured.

Multi-objective robust design optimization of a novel negative Poisson's ratio bumper system

Negative Poisson's ratio(NPR) structure has outstanding performances in lightweight and energy absorption, and it can be widely applied in automotive industries. By combining the front anti-collision beam, crash box and NPR structure, a novel NPR bumper system for improving the crashworthiness is first proposed in the work. The performances of the NPR bumper system are detailed studied by comparing to traditional bumper system and aluminum foam filled bumper system. To achieve the rapid design while considering perturbation induced by parameter uncertainties, a multi-objective robust design optimization method of the NPR bumper system is also proposed. The parametric model of the bumper system is constructed by combining the full parametric model of the traditional bumper system and the parametric model of the NPR structure. Optimal Latin hypercube sampling technique and dual response surface method are combined to construct the surrogate models. The multi-objective robust optimization results of the NPR bumper system are then obtained by applying the multi-objective particle swarm optimization algorithm and six sigma criteria. The results yielded from the optimizations indicate that the energy absorption capacity is improved significantly by the NPR bumper system and its performances are further optimized efficiently by the multi-objective robust design optimization method.

Optimality Conditions for Generalized Convex Nonsmooth Uncertain Multi-objective Fractional Programming

This paper aims at studying optimality conditions of robust weak efficient solutions for a nonsmooth uncertain multi-objective fractional programming problem(NUMFP).The concepts of two types of generalized convex function pairs,called type-I functions and pseudo-quasi-type-I functions,are introduced in this paper for(NUMFP).Under the assumption that(NUMFP)satisfies the robust constraint qualification with respect to Clarke subdifferential,necessary optimality conditions of the robust weak efficient solution are given.Sufficient optimality conditions are obtained under pseudo-quasi-type-I generalized convexity assumption.Furthermore,we introduce the concept of robust weak saddle points to(NUMFP),and prove two theorems about robust weak saddle points.The main results in the present paper are verified by concrete examples.

Reactive Power Multi-objective Optimization for Multi-terminal AC/DC Interconnected Power Systems Under Wind Power Fluctuation

In view of the reactive power coordination difficulties caused by reactive power strong coupling,the provincial power grids in the interconnected system are formed by the multi-AC/DC transmission.Wind power channels are under the conditions of large-scale long-distance transmission of wind power and other forms of renewable power generation.The AC-DC hybrid power flow equation of the interconnected system,including the AC-DC tie lines,is presented in this paper,along with the robust dynamic evolutionary optimization of the reactive power system in interconnected systems under fluctuating and uncertain wind power conditions.Therefore,the rapid collaborative optimization of reactive power flow and the exchange of reactive power between tie lines between provincial power grids are realized.The analysis was made by taking four interconnected large-scale provincial power grids of Eastern Mongolia,Jilin,Liaoning and Shandong as an example.The simulation results demonstrate the effectiveness and superiority of the proposed reactive power dynamic multi-objective optimization method for interconnected power grids.

Reliability-based robust design optimization of vehicle components, Part Ⅱ： Case studies

The reliability-based optimization, the relia- bility-based sensitivity analysis and robust design method are employed to propose an effective approach for reliability-based robust design optimization of vehicle components in Part I. Applications of the method are further discussed for reliability-based robust optimization of vehicle components in this paper. Examples of axles, torsion bar, coil and composite springs are illustrated for numerical investigations. Results have shown the proposed method is an efficient method for reliability-based robust design optimization of vehicle components.

Adaptive-surrogate-based robust optimization of transonic natural laminar flow nacelle

Natural Laminar Flow(NLF)technology is very effective for reducing the skin friction drag of aircraft engine nacelle,but the aerodynamic performance of NLF nacelle is highly sensitive to uncertain working conditions.Therefore,it’s imperative to incorporate uncertainties into the design of NLF nacelle.In this study,for a robust optimization of NLF nacelle and for improving its efficiency,an adaptive-surrogate-based robust optimization strategy is established,which is an iterative optimization process where the surrogate model is updated to obtain the real Pareto front of multi-objective optimization problem.A case study is carried out to validate its feasibility and effectiveness.The results show that the optimization increases the favorable pressure gradient region and the volume ratio of the nacelle by increasing its lip radius and reducing its maximum diameter.And the aerodynamic robustness of the NLF nacelle is mainly determined by the lip radius,maximum diameter of nacelle and location of the maximum diameter.Compared to the initial nacelle,the optimized nacelle maintains a wide range of low drag and high laminar flow ratio in the disturbance space,which extends the average laminar flow region to 21.6%and facilitates a decrease of 1.98 counts in the average drag coefficient.

Robust optimization of nonlinear impulsive rendezvous with uncertainty

The optimal rendezvous trajectory designs in many current research efforts do not incorporate the practical uncertainties into the closed loop of the design.A robust optimization design method for a nonlinear rendezvous trajectory with uncertainty is proposed in this paper.One performance index related to the variances of the terminal state error is termed the robustness performance index,and a two-objective optimization model(including the minimum characteristic velocity and the minimum robustness performance index)is formulated on the basis of the Lambert algorithm.A multi-objective,non-dominated sorting genetic algorithm is employed to obtain the Pareto optimal solution set.It is shown that the proposed approach can be used to quickly obtain several inherent principles of the rendezvous trajectory by taking practical errors into account.Furthermore,this approach can identify the most preferable design space in which a specific solution for the actual application of the rendezvous control should be chosen.

Robust Switch Selection in Radial Distribution Systems Using Combinatorial Optimization

Selecting the best type of equipment among available switches with different prices and reliability levels is a significant challenge in distribution system planning.In this paper,the optimal type of switches in a radial distribution system is selected by considering the total cost and reliability criterion and using the weighted augmented epsilon constraint method and combinatorial optimization.A new index is calculated to assess the robustness of each Pareto solution.Moreover,for each failure,repair time is considered based on historical data.Monte Carlo simulations are used to consider the switch failure uncertainty and fault repair time uncertainty in the model.The proposed framework is applied to an RTBS Bus-2 test system.Furthermore,the model is also applied to an industrial system to verify the proposed method’s excellent performance in larger practical engineering problems.