Direct Search Optimization Research Articles

Tuning Runge-Kutta parameters on a family of ordinary differential equations

The Runge-Kutta class of iterative methods is designed to approximate solutions of a system of ordinary differential equations (ODE). The second-order class of Runge-Kutta methods is determined by a system of three nonlinear equations and four unknowns, and includes the modified-Euler and mid-point methods. The fourth-order class is determined by a system of eight nonlinear equations and 10 unknowns. This work formulates the question of identifying good values of these eight parameters for a given family of ODE as a blackbox optimisation problem. The objective is to determine the parameter values that minimise the overall error produced by a Runge-Kutta method on a training set of ODE. Numerical experiments are conducted using the NOMAD direct-search optimisation solver.

Optimum Parameters of a Tuned Liquid Column Damper in a Wind Turbine Subject to Stochastic Load

Parameter optimization for tuned liquid column dampers (TLCD), a class of passive structural control, have been previously proposed in the literature for reducing vibration in wind turbines, and several other applications. However, most of the available work consider the wind excitation as either a deterministic harmonic load or random load with white noise spectra. In this paper, a global direct search optimization algorithm to reduce vibration of a tuned liquid column damper (TLCD), a class of passive structural control device, is presented. The objective is to find optimized parameters for the TLCD under stochastic load from different wind power spectral density. A verification is made considering the analytical solution of undamped primary system under white noise excitation by comparing with result from the literature. Finally, it is shown that different wind profiles can significantly affect the optimum TLCD parameters.

New design method for multi‐stack disc‐type hysteresis motors based on analytical calculations

Hysteresis motors are widely employed for a variety of high speed applications, thanks to their unique features. Usual design methods could not be used for these motors since the operation mechanism of hysteresis motors is also unique. However, an appropriately developed design method has not been provided by the few relevant published works. This study presents a new iterative design method, which is capable of being used for almost all hysteresis motor structures. Also, it is almost independent of the approach of obtaining the design equations. Multi-stack disc-type slotless structure has been selected as the case study, because of its interesting capabilities. Its design equations are derived by combining the fundamental laws of electromagnetism with the elliptical model of hysteresis loops. Some new techniques are introduced to increase the accuracy of these equations. A sample motor has been designed by applying the direct search optimisation to the proposed design method. The prototyped sample motor has been subjected to various tests. The measured operating characteristics show a great agreement with the design requirements.

Reduced-order modeling of CO2 storage operations

A POD-TPWL reduced-order modeling framework is developed to simulate and optimize the injection stage of CO2 storage operations. The method combines trajectory piecewise linearization (TPWL), where solutions with new sets of well controls are constructed based on linearization around previously simulated (training) solutions, and projection into a low-dimensional subspace using proper orthogonal decomposition (POD). The resulting representation is low-dimensional and linear, in contrast to the original nonlinear full-order flow simulations. Several new POD-TPWL treatments are introduced and demonstrated. These include the use of multiple derivatives, meaning that the linearizations are performed around different training solutions at different time steps, and the use of rate-controlled (rather than pressure-controlled) injection wells. Two example cases are presented, and the ability of the POD-TPWL model to accurately capture wellbore pressure, when time-varying CO2 injection rates are prescribed, is demonstrated. It is also shown that, for these examples, the reduced-order models can provide accurate estimates of CO2 molar fraction at particular locations in the domain. The POD-TPWL model is then incorporated into a mesh adaptive direct search optimization framework where the objective is to minimize the amount of CO2 reaching a target layer at the end of the injection period. The POD-TPWL model is shown to be well suited for this purpose and to provide optimization results that are comparable to those obtained using full-order simulations. The preprocessing computations needed to construct the POD-TPWL models entail a (serial) time equivalent of about 6.7 full-order simulations, though the resulting runtime speedups, relative to full-order simulation, are about 100–150 for the cases considered.

Suppressing unsteady flow in arterio-venous fistulae

Arterio-Venous Fistulae (AVF) are regarded as the “gold standard” method of vascular access for patients with end-stage renal disease who require haemodialysis. However, a large proportion of AVF do not mature, and hence fail, as a result of various pathologies such as Intimal Hyperplasia (IH). Unphysiological flow patterns, including high-frequency flow unsteadiness, associated with the unnatural and often complex geometries of AVF are believed to be implicated in the development of IH. In the present study, we employ a Mesh Adaptive Direct Search optimisation framework, computational fluid dynamics simulations, and a new cost function to design a novel non-planar AVF configuration that can suppress high-frequency unsteady flow. A prototype device for holding an AVF in the optimal configuration is then fabricated, and proof-of-concept is demonstrated in a porcine model. Results constitute the first use of numerical optimisation to design a device for suppressing potentially pathological high-frequency flow unsteadiness in AVF.

Investigation of the geometric property hull for infiltrated solid oxide fuel cell electrodes

Design of optimal microstructures for infiltrated solid oxide fuel cell (SOFC) electrodes is a complicated task because of the multitude of electro-chemo-physical phenomena taking place simultaneously that directly affect working conditions (cell temperature, current density, and flow rates) of the SOFC electrode and therefore its performance. In this study, an innovative design paradigm is presented to obtain a part of geometry-related electrochemical and physical properties of an infiltrated SOFC electrode. A range of digitally realized microstructures with different backbone porosity and electrocatalyst particle loading under various deposition conditions are generated. Triple phase boundary (TPB), active surface density of particles, and gas transport factor are evaluated in realized models on the basis of selected infiltration strategy. On the basis of this database, a neural network is trained to relate desired range of input geometric parameters to a property hull. The effect of backbone porosity, loading, distribution, and aggregation behavior of particles is systematically investigated on the performance indicators. It is shown that from the microstructures with very high amount of TPB and particle contact surface density, a relatively low gas diffusion factor should be expected; meanwhile, increasing those parameters does not have sensible contradiction with each other. Excessive agglomerating of particles has a negative effect on TPB density, but the distribution of seeds always has a positive effect. Direct search and genetic algorithm optimization techniques are used finally to achieve optimal microstructures on the basis of assumed target functions for effective geometric properties.

Parameter estimation of a photovoltaic array using direct search optimization algorithm

This paper presents a parameter estimation technique to determine the photovoltaic (PV) cell parameters from experimental measurements, using the direct search optimization method. A double diode model is considered to represent a photovoltaic (PV) cell, and a Matlab simulation is built to demonstrate the effects of varying various parameters on the output performance. The direct search optimization technique is used to estimate the cell parameters by reshaping the simulated output characteristics to match similar experimental measurements. The procedures that illustrate the implementation of the proposed technique to estimate the PV cell parameters are detailed. The technique is implemented to estimate the parameters of SUNSET PV Module Model PX-170 from experimental measurements. The convergence characteristics of the most sensitive cell parameters are given. The validity of the proposed technique is verified by comparing simulated results obtained using the estimated parameters and their corresponding measurements. The techniques and methodology presented in this paper are of prime importance to power system engineers forming a useful tool for studying power systems including photovoltaic sources.

Layout optimization of a wind farm to maximize the power output using enhanced teaching learning based optimization technique

The global warming is a major concern in the present era that arises a need of cleaner production of energy. The wind energy is a major source of contribution for such clean energy demands. The Wind farm layout optimization (WFLO) problem deals with the optimum placement of wind turbines in a wind farm so as to maximize the total power output with minimum cost of energy. The placement of turbines is crucial for a wind farm because the power generation of wind turbine decreases if it is in the wake effect produced by the upstream turbines. So, WFLO problem is a challenging combinatorial optimization problem for which many direct search and local optimization method fails to attend the global optimum solution. The Meta-heuristic method often provides the effective solution for such problems in terms of convergence and the quality of the solution. In this work two, different metaheuristics algorithms are proposed to solve WFLO problem. These algorithms are developed by incorporating changes in the basic Teaching-learning based optimization (TLBO) algorithm. The proposed algorithms eliminate the limitations of basic TLBO algorithm to enhance its exploration and exploitation by incorporating effective search techniques. The implementations of the proposed algorithm are effective to optimize the position of wind turbines in a wind farm to maximize the expected power output of a wind farm with a minimum investment cost. The proposed algorithm is investigated for WFLO problem and a set of 10 challenging real life benchmark problems. The numerical results indicate that the proposed method is an effective technique to solve the WFLO problem compared to its basic algorithm and other state of the art methods. The optimum design of the wind farm results in the economical utilization of the wind resource and leads to clean energy production.

A Root-Locus Design Methodology Derived From the Impedance/Admittance Stability Formulation and Its Application for LCL Grid-Connected Converters in Wind Turbines

This paper presents a systematic methodology for the design and the tuning of the current controller in LCL grid-connected converters for wind turbine applications. The design target is formulated as a minimization of the current loop dominant time constant, which is in accordance with standard design guidelines for wind turbine controllers (fast time response and high stability margins). The proposed approach is derived from the impedance/admittance stability formulation, which, on one hand, has been proved to be suitable for the controller design when the active damping is implemented and, on the other hand, has also been proved to be very suitable for system-level studies in applications with a high penetration of renewable energy resources. The tuning methodology is as follows: first, the physical system is modeled in terms of the converter admittance and its equivalent grid impedance; then, a sensitivity transfer function is derived, from which the closed-loop eigenvalues can be calculated; finally, the set of control gains that minimize the dominant time constant are obtained by direct search optimization. A case study that models the target system in a low-power scale is provided, and experimental verification validates the theoretical analysis. More specifically, it has been found that the solution that solves the minimization of the current controller time constant (wind turbine controller target) also corresponds to a highly damped electrical response (robustness provided by the active damping).

A flotation control system to optimise performance using peak air recovery

Automatic control of industrial flotation cells and circuits presents a set of significant challenges due to the number of variables, the sensitivity of flotation cells to variation in these variables and the complexity of predicting flotation performance and/or developing a strategy for optimisation. Air recovery, a measure of froth stability, has been shown to pass through a peak as flotation cell aeration increases. Furthermore, the air rate at which the peak air recovery (PAR) is obtained results in optimal flotation performance, whether improved concentrate grade, recovery or both grade and recovery. Peak air recovery, therefore, presents a clear optimising control strategy for the operation of flotation cells which is generic to all flotation cells regardless of position in the flotation circuit. In this study, a novel control system based on PAR is developed and demonstrated using a large continuous laboratory flotation cell.In this study, a direct search optimisation algorithm based on the GSS (generating set search) methodology was developed using a 70l continuous flotation cell operating with a two-phase system (surfactant solution and air only). Characterisation of the laboratory system showed that it was stable for up to 6h and exhibited a reproducible peak in air recovery. A dynamic model of the response of the system with regards to changes in air recovery was developed that allowed simulations of the proposed optimising control system to be carried out. The optimisation algorithm was then applied to the experimental system. The trialled GSS algorithm was shown to find the PAR air rate when starting above, below and at the PAR air rate, and additionally with a disturbance introduced into the system. While the direct search approach can be slow, it is simple and robust. This demonstrates an innovative approach to optimising control for froth flotation and is the first application of froth stability maximisation for flotation control.

Determination of the optimal mesh parameters for Iguassu centrifuge flow and separation calculations

We present the method and the results of the determination for optimal computational mesh parameters for axisymmetric modeling of flow and separation in the Iguasu gas centrifuge. The aim of this work was to determine the mesh parameters which provide relatively low computational cost whithout loss of accuracy. We use direct search optimization algorithm to calculate optimal mesh parameters. Obtained parameters were tested by the calculation of the optimal working regime of the Iguasu GC. Separative power calculated using the optimal mesh parameters differs less than 0.5% from the result obtained on the detailed mesh. Presented method can be used to determine optimal mesh parameters of the Iguasu GC with different rotor speeds.

Erratum to: Armentum: a hybrid direct search optimization methodology

Erratum to: Armentum: a hybrid direct search optimization methodology

Numerical investigation of non-hierarchical coordination for distributed multidisciplinary design optimization with fixed computational budget

This paper presents a numerical investigation of the non-hierarchical formulation of Analytical Target Cascading (ATC) for coordinating distributed multidisciplinary design optimization (MDO) problems. Since the computational cost of the analyses can be high and/or asymmetric, it is beneficial to understand the impact of the number of ATC iterations required for coordination and the number of iterations required for disciplinary feasibility on the quality of the obtained MDO solution. At each "outer" ATC iteration, the disciplinary optimization subproblems are solved for a predefined maximum number of "inner" loop iterations. The numerical experiments consider different numbers of maximum outer iterations while keeping the total computational budget of analyses constant. Solution quality is quantified by optimality (objective function value) and consistency (violation of coordination-related consistency constraints). Since MDO problems are typically simulation-based (and often blackbox) problems, we compare implementations of the mesh-adaptive direct search optimization algorithm (a derivative-free method with convergence properties) to the gradient-based interior-point algorithm implementation of the popular Matlab optimization toolbox. The impact of the values of two parameters involved in the alternating directions updating scheme of the augmented Lagrangian penalty functions (aka method of multipliers) on solution quality is also investigated. Numerical results are provided for a variety of MDO test problems. The results indicate consistently that a balanced modest number of outer and inner iterations is more effective; moreover, there seems to be a specific combination of parameter value ranges that yield better results.

Prediction of River Runoff Using Fuzzy Theory and Direct Search Optimization Algorithm Coupled Model

The correct prediction of daily runoff is considerable importance in flood control. The aim of this study is to evaluate the ability of a combined model, fuzzy inference system–direct search optimization algorithm (FIS-DSOA), to forecast the daily runoff in downstream of Taleghan river using data of rain gauge and hydrometric stations which are located upstream of the river. Initially, the measured daily data (6 years from 2008 to 2013) related to rain gauge and hydrometric stations located in Taleghan river upstream (as input variable) and Siahdasht hydrometric station located in downstream of river (as output variable) were collected. Then the structure of proposed FIS-DSOA model was developed. In the proposed model, the variation range and type of membership functions associated with the input and output variable and the type of rules governing time series were taken as decision variable. Also, the objective function of hybrid model is to minimize root-mean-square error between observed and simulated daily runoff. The forecasts of the FIS-DSOA model are tested using the five statistical indicators (i.e., correlation coefficient, efficiency coefficient, mean absolute error, MSE and index of agreement), and the results are compared with those of the ANNs, FIS and ANFIS models. The comparison results revealed that the FIS-DSOA performs with EFF = 0.83, MAE = 0.75 \({{m}^{3}/{s}}\), MSE = 26.1 \({{m}^{3}/{s}}\) and IOA = 0.92 better than the non-hybrid models in daily runoff discharge prediction. It is concluded that the proposed model significantly improved the accuracy in daily runoff discharge forecasting by combining the capabilities of ANFIS and DSOA models.

Determining anaerobic degradation kinetics from batch tests

Data obtained from a biomethane potential (BMP) test were used in order to obtain the parameters of a kinetic model of solid wastes anaerobic degradation. The proposed model considers a hydrolysis step with a first order kinetic, a Monod kinetic for the soluble organic substrate degradation and a first order decay of microorganisms. The instantaneous release of methane was assumed. The parameters of the model are determined following a direct search optimization procedure. A 'multiple-shooting' technique was used as a first step of the optimization process. The confidence interval of the parameters was determined by using Monte Carlo simulations. Also, the distribution functions of the parameters were determined. Only the hydrolysis first order constant shows a normal distribution.

Pitch control for wind turbine systems using optimization, estimation and compensation

Delays caused by hydraulic pressure driven units in wind turbines may degrade and even destabilize pitch control system, leading to the performance degradation or collapse of the whole wind turbine energy conversion system. As a result, there is a strong motivation to improve pitch control technique to overcome the adverse effects from the unknown delays caused by the hydraulic driven units. In this study, a novel pitch control approach is developed by integrating optimization, delay-perturbation estimation, and signal compensation techniques. Specifically, an optimal PI controller is designed by applying direct search optimization to ideal delay-free pitch model. A delay estimator is next designed to estimate the perturbation caused by the delay. The signal compensation technique is then implemented to remove the effect from the delay-perturbation to the turbine output so that the actual output is consistent with the optimal output of the ideal delay-free model. Furthermore, the technique is extended to PID-based pitch control systems. The blade-tower dynamics models of three wind turbines, respectively with rated powers 1.5 MW, 275 kW and 50 kW, are investigated and the effectiveness of the proposed techniques are demonstrated by detailed simulation studies. It is worthy to point out a priori on the delay is not necessary in this design, which much meets the situations in practical wind turbine systems. Finally, a 4.8 MW benchmark wind turbine energy conversion system is investigated via the proposed pitch control technique, which has shown that delay perturbation estimate based compensation can effectively improve the pitch control performance in the real-time wind turbine energy conversion system.

Evolving a Nelder-Mead Algorithm for Optimization with Genetic Programming.

We used genetic programming to evolve a direct search optimization algorithm, similar to that of the standard downhill simplex optimization method proposed by Nelder and Mead ( 1965 ). In the training process, we used several ten-dimensional quadratic functions with randomly displaced parameters and different randomly generated starting simplices. The genetically obtained optimization algorithm showed overall better performance than the original Nelder-Mead method on a standard set of test functions. We observed that many parts of the genetically produced algorithm were seldom or never executed, which allowed us to greatly simplify the algorithm by removing the redundant parts. The resulting algorithm turns out to be considerably simpler than the original Nelder-Mead method while still performing better than the original method.

Design of Robust Fractional PID Controller Using Triangular Strip Operational Matrices

Abstract The present article proposes a simple tuning technique aimed to produce a robust non-integer order PID controller exhibiting iso-damping property during reparameterization of a plant. The required robustness property is achieved by letting the fractional PID control system to imitate the dynamics of a reference system having Bode’s ideal transfer function in its forward path. The objective of designing robust controller by tracking the dynamics of reference control system is defined mathematically as an H∞- optimal control problem. Fractional differential systems are transformed into algebraic equations by the use of triangular strip operational matrices. The H∞-optimal control problem is then changed to an ∞-norm minimization of a parameter (Kc,KI,Kd, λ, μ) varying square matrix. Global optimization techniques, Luus-Jaakola direct search and particle swarm optimization are employed to find the optimum values of fractional PID controller parameters. The proposed method of control system design is implemented in heating furnace temperature control, automatic voltage regulator system and some integer and fractional order process models. Fractional PI, fractional PD and various versions of fractional PID controllers are designed as optimal controllers using the triangular strip operational matrix based control design method.

Multi-fidelity shape optimization of hydraulic turbine runner blades using a multi-objective mesh adaptive direct search algorithm

A robust multi-fidelity design optimization methodology has been developed to integrate advantages of high- and low-fidelity analyses, aiming to help designers reach more efficient turbine runners within reasonable computational time and cost. An inexpensive low-fidelity inviscid flow solver handles most of the computational burden by providing data to the optimizer by evaluating objective functions and constraint values in the low-fidelity phase. An open-source derivative-free optimizer, NOMAD, explores the search space, using the multi-objective mesh adaptive direct search optimization algorithm. A versatile filtering algorithm is in charge of connecting low- and high-fidelity phases by selecting among all feasible solutions a few promising solutions which are transferred to the high-fidelity phase. In the high-fidelity phase, a viscous flow solver is used outside the optimization loop to accurately evaluate filtered candidates. High-fidelity analyses results are used to recalibrate the low-fidelity optimization problem. The developed methodology has demonstrated its ability to efficiently redesign a Francis turbine blade for new operating conditions.

Modeling and Optimization of Algae Growth

Microalgae is a promising source of renewable biofuels, and optimization and control of the biomass growth stage can make techno-economic improvements. This work explores the development of an algae growth model from first-principles, which includes the impact of natural vagaries of weather (and such) associated with production in an open system. Consequently, the process simulation is stochastic as well as fundamental; it returns a distribution of results representing the day-to-day realizations from natural variability. It also expresses day-to-night variation in the cycling solar energy. The simulation is then used to optimize pond design and management (the growth time, raceway depth, pH control, etc.) to improve profitability. Since the simulation is stochastic, nonlinear, and with multiple optima, a multiplayer direct search optimization technique with steady state convergence criteria is used for optimization. Conclusions are that (1) accounting for natural variation in the optimization leads to no...