Nonlinear Constrained Problem Research Articles

Environmental/Economic Power Dispatch of MicroGrid Using Multiobjective Optimization

This paper proposes a generalized formulation to determine the optimal operating strategy and cost optimization scheme as well as the reduction of the emissions for a MicroGrid (MG). Multiobjective (MO) optimization is applied to the environmental/economic problem of the MG. The proposed problem is formulated as a nonlinear constrained MO optimization problem. Prior to the optimization, models for system components from real data are constructed. The problem formulation takes into consideration the operation and maintenance costs as well as the emissions NOx, SO2, and CO2 reduction. The MG considered in this paper consists of a wind turbine, a micro turbine, a diesel generator, a photovoltaic array, a fuel cell, and a battery storage. The optimization is aimed at minimizing the cost function of the system while constraining it to meet the costumer demand and safety of the system. We also add a daily income and outgo from sale or purchased power. The results demonstrate the efficiency of the proposed approach to satisfy the load and to reduce the cost and the emissions. The comparison with other techniques demonstrates the superiority of the proposed approach and confirms its potential to solve the problem.

Constrained Consensus-Based Optimization

In this work we are interested in the construction of numerical methods for high-dimensional constrained nonlinear optimization problems by particle-based gradient-free techniques. A consensus-based optimization (CBO) approach combined with suitable penalization techniques is introduced for this purpose. The method relies on a reformulation of the constrained minimization problem in an unconstrained problem for a penalty function and extends to the constrained settings of the class of CBO methods. Exact penalization is employed and, since the optimal penalty parameter is unknown, an iterative strategy is proposed that successively updates the parameter based on the constrained violation. Using a mean-field description of the many particle limit of the arising CBO dynamics, we are able to show convergence of the proposed method to the minimum for general nonlinear constrained problems. Properties of the new algorithm are analyzed. Several numerical examples, also in high dimensions, illustrate the theoretical findings and the good performance of the new numerical method.

Convex Optimization-Based Techniques for Trajectory Design and Control of Nonlinear Systems with Polytopic Range

This paper presents new techniques for the trajectory design and control of nonlinear dynamical systems. The technique uses a convex polytope to bound the range of the nonlinear function and associates with each vertex an auxiliary linear system. Provided controls associated with the linear systems can be generated to satisfy an ordering constraint, the nonlinear control is computable by the interpolation of controls obtained by convex optimization. This theoretical result leads to two numerical approaches for solving the nonlinear constrained problem: one requires solving a single convex optimization problem and the other requires solving a sequence of convex optimization problems. The approaches are applied to two practical problems in aerospace engineering: a constrained relative orbital motion problem and an attitude control problem. The solve times for both problems and approaches are on the order of seconds. It is concluded that these techniques are rigorous and of practical use in solving nonlinear trajectory design and control problems.

Model predictive controlled subsurface drainage and irrigation for peatland groundwater management

A dynamic simulation model for the water balance in a field with controlled subsurface drainage and irrigation is derived. The model is then applied to groundwater table height control, using the model predictive control (MPC) paradigm. The nonlinear constrained MPC problem is solved and results illustrated in simulations. The role of automation of controlled subsurface drainage in cultivated peatlands is discussed.

Constrained tracking control of stochastic multivariable nonlinear systems via Gaussian process predictions

To overcome the difficulty of propagating the stochastic uncertainties through a nonlinear model for successful online implementation of the stochastic nonlinear model predictive control (SNMPC) framework, this paper proposes the utilisation of Gaussian processes (GPs) in the context of SNMPC for the stochastic multivariable nonlinear systems to track a given trajectory in the presence of stochastic uncertainties and system constraints. Taking advantage of the GP regression which not only provides predictions, but also uncertainty quantification in the function estimation, the proposed GP-SNMPC architecture exploits the probability distribution of stochastic uncertainties and utilises the predictions provided by the GPs to formulate the model cost and constraint functions, which yields a tractable framework for handling nonlinear constrained control problems with Gaussian parametric uncertainties. By employing a cancellation strategy, the control law consists of two components, that is, tracking control law and disturbance rejection control law. Theoretical results regarding the performance bounds of the closed-loop system are derived. Numerical examples are provided to verify the effectiveness of the proposed GP-SNMPC scheme.

Joint State and Parameter Estimation for Hypersonic Glide Vehicles Based on Moving Horizon Estimation via Carleman Linearization

Aimed at joint state and parameter estimation problems in hypersonic glide vehicle defense, a novel moving horizon estimation algorithm via Carleman linearization is developed in this paper. First, the maneuver characteristic parameters that reflect the target maneuver law are extended into the state vector, and a dynamic tracking model applicable to various hypersonic glide vehicles is constructed. To improve the estimation accuracy, constraints such as path and parameter change amplitude constraints in flight are taken into account, and the estimation problem is transformed into a nonlinear constrained optimal estimation problem. Then, to solve the problem of high time cost for solving a nonlinear constrained optimal estimation problem, in the framework of moving horizon estimation, nonlinear constrained optimization problems are transformed into bilinear constrained optimization problems by linearizing the nonlinear system via Carleman linearization. For ensuring the consistency of the linearized system with the original nonlinear system, the linearized model is continuously updated as the window slides forward. Moreover, a CKF-based arrival cost update algorithm is also provided to improve the estimation accuracy. Simulation results demonstrate that the proposed joint state and parameter estimation algorithm greatly improves the estimation accuracy while reducing the time cost significantly.

An Intelligent Optimization Method for Preliminary Design of Lead-Bismuth Reactor Core Based on Kriging Surrogate Model

To meet the numerous application demands of lead-bismuth reactors, different design optimization tasks need to be conducted on these reactors based on the existing reactor core solutions. However, the design optimization of lead-bismuth reactors is a challenging task because it is a complex, multi-dimensional, and nonlinear constrained problem. To resolve these issues and improve the efficiency of design optimization, a new method, called the KSM-OLHS-SEUMRE method, based on the Kriging surrogate model (KSM), orthogonal Latin hypercube sampling (OLHS), and space exploration and unimodal region elimination (SEUMRE) algorithm is proposed in this study. Based on this method, a design optimization program of lead-bismuth reactors (DOPPLER-K) is developed, which realizes functions like sample point generation, optimization analysis, pre-post processing of reactor calculation, coupling of the Reactor Monte Carlo (RMC) calculation code and the Steady-state Thermal-hydraulic Analysis Code (STAC). Further, taking lead-bismuth reactors SPALLER-4 and URANUS as prototypes, the proposed intelligent optimization method for preliminary design of lead-bismuth reactor core is verified. The results show that this method can rapidly and accurately find the target scheme satisfying the optimization conditions, and it is three orders of magnitude faster than pure Monte Carlo calculation. Compared with the initial core scheme of URANUS, the optimization rates of fuel loading, total core mass, active zone volume, and total core volume are reduced by 10.8, 11.5, 18.1, and 17.1%, respectively. These results validate the feasibility and efficacy of the proposed method for design optimization of lead-bismuth reactor core.

Research on eco-driving optimization of hybrid electric vehicle queue considering the driving style

With the vehicle to infrastructure and vehicle to vehicle communication information, it is beneficial to improve the fuel economy of hybrid electric vehicle by providing the driver with economy-oriented velocity. However, the drivers with different driving styles differ much from each other in tracking the advisory economy-oriented velocity. Hence, this paper investigates the eco-driving optimization of the hybrid electric vehicle queue in urban road conditions considering the driving characteristics in following the advisory speed. The multi-objective optimization problem for the velocities of the hybrid electric vehicle queue is constructed, in which the initial target velocities of different vehicles are obtained according to the signal phase and timing information of traffic lights. In order to improve the adaptability of the driver to the advisory velocity, the driving style coefficients are introduced in the multi-objective eco-driving optimization problem. The driving feature parameters of different drivers are collected through driving simulator test. By means of principle component analysis and K-means clustering method, the collected driving feature parameters are applied for driving style classification. Then, the coefficients of different driving styles are derived through car following tests. The simulated annealing algorithm is used to solve the nonlinear constrained problem and obtain the optimal advisory velocities. In the powertrain level, the model predictive control which can make full use of the predicted optimal velocity is further applied to optimize the energy allocations. Comparative simulation studies are conducted to test the performance of the proposed strategy. Simulation results demonstrate the superior performance of the proposed strategy in reducing the fuel consumption and exhaust emissions of the hybrid electric vehicle queue, as well as in improving the traffic smoothness. The driver adaptability in tracking the advisory eco-driving velocity has also been improved.

Existence and limit behavior of minimizers of Kirchhoff-type constrained variational problems with constrained critical exponent

There are no minimizers for nonlinear Kirchhoff-type constrained variational problems when the nonlinear term only includes an exponential term and the exponent is the constrained critical exponent $p=2+\frac{8}{N}$. In this paper, a perturbation functional is addedto the Kirchhoff-type constrained variational 问题 with constrained critical exponent. Then, for this functional, a complete classification with respect to the exponent and the coefficient in the perturbation term for its normalized critical points is obtained by using the scaling technique, the concentration-compactness principle and the Pohozaev identity. Under some 条件, we also prove that the minimizer is the ground state 解 of the corresponding Kirchhoff equation. Furthermore, by using the energy estimation, the limit behaviors of the minimum energy and the minimizer are discussed when the perturbation term exponent tends to the constrainedcritical exponent.

Effective coordination settings for directional overcurrent relay using hybrid Gradient-based optimizer

This manuscript announces a novel hybrid optimization model comprising the gradient-based optimizer (GBO) and Linear population size reduction technique of Success-History-based Adaptive Differential Evolution (LSHADE) algorithm, so-called GB-LSHADE, for optimal relay coordination problem (ORCP). The ORCP is framed as a nonlinear constrained engineering problem that aims to optimize the time-dial, pickup current, and relay characteristics so that the relay clearance times are minimized. The proposed GB-LSHADE operates in two phases meanwhile the first phase starts with GBO to enrich the exploitative tendencies while the second phase is adopted to boost the local searching competence and avoid the premature convergence. To verify the efficacy of the proposed optimization method, a comprehensive simulation has been conducted on three test-systems (8-bus, 15-bus, and IEEE 30-bus) with different complexities. Many scenarios including fixed and varied relay curves as per IEC 60255 are demonstrated. At a final stage of this work, the computational complexity of the proposed approach is carried out and analyzed. It can be confirmed that the adapted optimal settings produced by the GB-LSHADE achieve the full co-ordination of the protection without any violations for the systems under study. In addition, the numerical simulated results show that the GB-LSHADE supersedes the other viable optimizers reported in specialized literature thru further validations and comprehensive comparisons.

YAM2: Yet another library for the [formula omitted] variables using sequential quadratic programming

The M2 variables are devised to extend MT2 by promoting transverse masses to Lorentz-invariant ones and making explicit use of on-shell mass relations. Unlike simple kinematic variables such as the invariant mass of visible particles, where the variable definitions directly provide how to calculate them, the calculation of the M2 variables is undertaken by employing numerical algorithms. Essentially, the calculation of M2 corresponds to solving a constrained minimization problem in mathematical optimization, and various numerical methods exist for the task. We find that the sequential quadratic programming method performs very well for the calculation of M2, and its numerical performance is even better than the method implemented in the existing software package for M2. As a consequence of our study, we have developed and released yet another software library, YAM2, for calculating the M2 variables using several numerical algorithms. Program summaryProgram title: YAM2CPC Library link to program files:https://doi.org/10.17632/4g7wfd5fpb.1Developer’s repository link:https://github.com/cbpark/YAM2Licensing provisions: BSD 3-ClauseProgramming language: C ++Nature of problem: The value and the solution of the M2 variables can be obtained from the optimality and feasibility conditions of the nonlinearly constrained minimization problem in numerical optimization. To perform the calculation properly, one should employ suitable numerical algorithms with the appropriate formulation of the variables, having in mind the algorithmic efficiency and the computational cost.Solution method: There exist various numerical methods for solving constrained optimization problems. We have chosen the sequential quadratic programming method with the derivative-dependent quasi-Newton algorithm since it performs very efficiently to find the local minimum using derivative information. The method has been codified by using the implementation of the numerical algorithms in the NLopt library [1]. The new library also includes the routines using other algorithms for calculating M2, such as the augmented Lagrangian method.

An adaptive instinctive reaction strategy based on Harris hawks optimization algorithm for numerical optimization problems

An adaptive Harris hawks optimization (HHO) algorithm is proposed to solve the non-linearly constrained optimization problems. The algorithm, namely, instinctive reaction strategy based on HHO (IRSHHO), combines the instinctive reaction strategy (IRS) with the HHO algorithm. In IRSHHO, the increment of step length decreases with the increase in the iteration. Moreover, the energy of the prey is also considered to speed up the convergence in each iteration. The performance of the IRSHHO is investigated on five benchmark numerical examples. The results of IRSHHO provide very competitive results compared to other well-known algorithms. Furthermore, the IRSHHO is applied to optimize the auto drum fashioned brake problem. Results show that the IRSHHO can obtain the optimal solution with the braking efficiency factor that is increased by 27.872% from the initial design with the proposed IRSHHO method.

OPTIMIZED MAXIMUM LOADABILITY OF POWER SYSTEMS USING AN ENHANCED DYNAMIC JAYA ALGORITHM

The problem of Maximum Loadability of power systems is addressed in this paper using a proposed dynamic JAYA algorithm. The maximum loadability problem is a typical optimization problem in which the maximum loadability point is to be determined optimally. Voltage stability of power systems is maintained by determining the estimated margin between the system operating point and the maximum loadability limit. The basic JAYA algorithm has been introduced to solve foremost optimization problems with small-scaled nature. However, when applied to large-scale, nonlinear and non-convex constrained problems, it showed a poor convergence characteristic. In order to deal with these weaknesses, the original algorithm has been improved by adding some dynamic features to its convergence behavior. The modified algorithm has been presented and validated when applied to well-known typical power systems. The obtained results were compared to the results achieved by other equivalent optimization techniques.

Propeller design to improve flight dynamics features and performance for coaxial compound helicopters

The coaxial compound configuration has been proposed as a concept for future high-performance rotorcraft. The co-axial rotor system requires no anti-torque device, while the longitudinal thrust is provided by a propeller. A well-designed propeller can ensure both the performance and the cruise efficiency. The propeller design also influences the flight dynamics of such configuration. To design the propeller parameters of the coaxial compound helicopter, the propeller modelling and design method devised by Adkins and Liebeck is firstly introduced. A flight dynamics model of the coaxial compound helicopter is developed. Trim characteristics, power consumption results, and handling qualities features are calculated with variable propeller parameters at optimum (corresponding to the maximum flight range) and maximum speeds. The results indicate that the propeller parameters alter its efficiency and improve the performance characteristics of this helicopter. In addition, the propeller design also influences the flight dynamics characteristics and handling qualities of the coaxial compound helicopter, and consequently affect the power consumption indirectly. In this light, a design method for the propeller has been developed by formulating the design procedure into the nonlinearly constrained optimisation problem based on the flight dynamics characteristics. The method is demonstrated by evaluating propeller parameters to improve the performance in both optimum and high-speed range, which could also guarantee the handling qualities of the rotorcraft for related requirements.

Non-monotone derivative-free algorithm for solving optimization models with linear constraints: extensions for solving nonlinearly constrained models via exact penalty methods

This paper describes a non-monotone direct search method (NMDSM) that finds a stationary point of linearly constrained minimization problems. At each iteration the algorithm uses NMDSM techniques on the Euclidean space $${\mathbb {R}}^n$$ spanned by n variables carefully selected from the $$n+m$$ variables formulated by the model under analysis. These variables are obtained by simple rules and are handled with pivot transformations frequently used in the solution of linear systems. A new weaker 0-order non smooth necessary condition is suggested, which transmute to other stationarity conditions, depending upon the kind of differentiability present in the system. Convergence with probability 1 is proved for non smooth functions. The algorithm is tested numerically on a set of small to medium size problems that have exhibited serious difficulties for their solution by other optimization techniques. The paper also considers possible extensions to non-linearly constrained problems via exact penalty function and a slightly modified algorithm satisfactorily solved a multi-batch multi-product plant that was modeled as a MINLP.

A Discrete Competitive Facility Location Model with Minimal Market Share Constraints and Equity-Based Ties Breaking Rule

We consider a geographical region with spatially separated customers, whose demand is currently served by some pre-existing facilities owned by different firms. An entering firm wants to compete for this market locating some new facilities. Trying to guarantee a future satisfactory captured demand for each new facility, the firm imposes a constraint over its possible locations (a finite set of candidates): a new facility will be opened only if a minimal market share is captured in the short-term. To check that, it is necessary to know the exact captured demand by each new facility. It is supposed that customers follow the partially binary choice rule to satisfy its demand. If there are several new facilities with maximal attraction for a customer, we consider that the proportion of demand captured by the entering firm will be equally distributed among such facilities (equity-based rule). This ties breaking rule involves that we will deal with a nonlinear constrained discrete competitive facility location problem. Moreover, minimal attraction conditions for customers and distances approximated by intervals have been incorporated to deal with a more realistic model. To solve this nonlinear model, we first linearize the model, which allows to solve small size problems because of its complexity, and then, for bigger size problems, a heuristic algorithm is proposed, which could also be used to solve other constrained problems.

Optimal Dispatch of Aggregated HVAC Units for Demand Response: An Industry 4.0 Approach

Demand response (DR) involves economic incentives aimed at balancing energy demand during critical demand periods. In doing so DR offers the potential to assist with grid balancing, integrate renewable energy generation and improve energy network security. Buildings account for roughly 40% of global energy consumption. Therefore, the potential for DR using building stock offers a largely untapped resource. Heating, ventilation and air conditioning (HVAC) systems provide one of the largest possible sources for DR in buildings. However, coordinating the real-time aggregated response of multiple HVAC units across large numbers of buildings and stakeholders poses a challenging problem. Leveraging upon the concepts of Industry 4.0, this paper presents a large-scale decentralized discrete optimization framework to address this problem. Specifically, the paper first focuses upon the real-time dispatch problem for individual HVAC units in the presence of a tertiary DR program. The dispatch problem is formulated as a non-linear constrained predictive control problem, and an efficient dynamic programming (DP) algorithm with fixed memory and computation time overheads is developed for its efficient solution in real-time on individual HVAC units. Subsequently, in order to coordinate dispatch among multiple HVAC units in parallel by a DR aggregator, a flexible and efficient allocation/reallocation DP algorithm is developed to extract the cost-optimal solution and generate dispatch instructions for individual units. Accurate baselining at individual unit and aggregated levels for post-settlement is considered as an integrated component of the presented algorithms. A number of calibrated simulation studies and practical experimental tests are described to verify and illustrate the performance of the proposed schemes. The results illustrate that the distributed optimization algorithm enables a scalable, flexible solution helping to deliver the provision of aggregated tertiary DR for HVAC systems for both aggregators and individual customers. The paper concludes with a discussion of future work.

Flower pollination algorithm to solve dynamic economic loading of units with piecewise fuel options

This paper presents a Flower Pollination Algorithm (FPA) to solve Dynamic Economic Load Dispatch (DELD) problem with valve-point effects and piecewise fuel options. DELD aims to find out optimum generation schedule of the committed generating units over a certain timing period, sustaining practical constraints and power demands in each interval. Due to the valve-point effect and piecewise fuel options DELD becomes as complex problem, hence in order to achieve the cost reduction and satisfying the dynamic behaviour of the generating units proposed algorithm presented. The practicality of the proposed method is evaluated by performing simulations on standard 10-unit and 19-unit Indian utility systems for a 24 h time schedule at various load patterns. The simulation results attained by the FPA are related with other previous published techniques of the biography. These results clearly show that the proficiency and robustness of the proposed FPA method for resolving the non-linear constrained DELD problem.

Optimal Solution of Structural Engineering Design Problems using Crow Search Algorithm

In real world, the structural engineering design problems are large scale non-linear constrained problems. In the present study, crow search algorithm (CSA) is applied to find the optimal solution of structural engineering design problems such as pressure vessel design problem, welded beam design problem and tension/ compression string design problem. The numerical results are compared with the existing results reported in the literature including metaheuristic algorithms and it is found that the results obtained by the crow search algorithm are better than other existing algorithms. Further, the effectiveness of the algorithm is verified to be better than the existing algorithms by statistical analysis using mean, median, best case, and worst case scenarios. The present study confirms that the crow search algorithm may be easily and effectively applied to various structural design problems.

Jerk-limited feedrate scheduling and optimization for five-axis machining using new piecewise linear programming approach

In this paper, a new computation method and an optimization algorithm are presented for feedrate scheduling of five-axis machining in compliance with both machine drive limits and process limits. Five-axis machine tool with its ability of controlling tool orientation to follow the sculptured surface contour has been widely used in modern manufacturing industry. Feedrate scheduling serving as a kernel of CNC control system plays a critical role to ensure the required machining accuracy and reliability for five-axis machining. Due to the nonlinear coupling effects of all involved drive axes and the saturation limit of servo motors, the feedrate scheduling for multi-axis machining has long been recognized and remains as a critical challenge for achieving five-axis machine tools’ full capacity and advantage. To solve the nonlinearity nature of the five-axis feedrate scheduling problems, a relaxation mathematical process is presented for relaxing both the drive motors’ physical limitations and the kinematic constraints of five-axis tool motions. Based on the primary optimization variable of feedrate, the presented method analytically linearizes the machining-related constraints, in terms of the machines’ axis velocities, axis accelerations and axis jerks. The nonlinear multi-constrained feedrate scheduling problem is transformed into a manageable linear programming problem. An optimization algorithm is presented to find the optimal feedrate scheduling solution for the five-axis machining problems. Both computer implementation and laboratorial experiment testing by actual machine cutting were conducted and presented in this paper. The experiment results demonstrate that the proposed method can effectively generate efficient feedrate scheduling for five-axis machining with constraints of the machine tool physical constraints and limits. Compared with other existing numerical methods, the proposed method is able to find an accurate analytical solution for the nonlinear constrained five-axis feedrate scheduling problems without compromising the efficiency of the machining processes.