Nonlinear Optimization Problem Research Articles

Comprehensive reconstructions and predictive control for quadrotor UAV information gathering tracking missions based on fully actuated system approaches

This paper introduces a novel approach to the comprehensive reconstruction and predictive control (PC) of the quadrotor UAV for information-gathering missions, employing fully actuated system (FAS) approaches. Unlike conventional PC methods applied to a quadrotor UAV with hybrid constraints, our work integrates reconstructions of the system model, hybrid constraints, and the receding horizon performance index into to an integrated tracking control scheme within the FAS-PC framework. Specifically, the under-actuated quadrotor UAV model is reconstructed into a full-actuated model to inject full-actuation properties. And the implicit hybrid constraints that arise from the model reconstruction are explicitly transformed and decoupled. Simultaneously, the cascaded predictive algorithm is established that the new time-varying input constraints are solved in each predictive horizon, and then the nonlinear optimization problem is decoupled into four linear convex optimization problems subject to the corresponding decoupled linear constraints and the pre-addressed input constraints. Within this framework, the intrinsic complexities, nonlinearities, and interdependencies of the quadrotor UAV system model, along with hybrid constraints and the optimization dilemma, are considerably diminished. This reduction significantly eases computational demands, enabling satisfactory real-time performance. Furthermore, the selection of predictive parameters guarantees the stability of the resultant tracking error closed-loop system. Finally, the efficacy of the proposed method is validated through two sets of flight missions, conducted via simulation and practical experimentation, respectively.

SABO algorithm for optimum design of truss structures with multiple frequency constraints

In this study, a very recently developed intelligent algorithm called the self-adaptive bonomo optimizer (SABO) algorithm is implemented for the size and shape optimizations of the truss structures with the multiple frequency constraints. The bonomo optimizer (BO) algorithm is a recently developed metaheuristic optimization algorithm that imitates the social behavior and reproductive schemes of the bonobos and this algorithm is successfully applied to solve some challenging and highly nonlinear optimization problems. On the other hand, the BO algorithm suffers from the poor exploration and prematurely converges to the non-optimal solutions, especially when dealing with the multi-dimensional optimization problems. In order to overcome these shortcomings, an improved version of this algorithm, the so-called self-adaptive bonomo optimizer (SABO) algorithm, is utilized to optimize the truss structures subjected to the design constraints. The objective of this study is to minimize the weight of the truss structures with the multiple frequency constraints. Five well-known benchmark problems are considered to verify the robustness and reliability of the SABO algorithm. The design results obtained by the SABO algorithm are compared with those previously reported in the literature.

Real-time optimal control for attitude-constrained solar sailcrafts via neural networks

This work is devoted to generating optimal guidance commands in real time for attitude-constrained solar sailcrafts in coplanar circular-to-circular interplanetary transfers. Firstly, a nonlinear optimal control problem is established, and the necessary conditions for optimality are derived by Pontryagin’s Minimum Principle. Under some mild assumptions, the attitude constraints are rewritten as control constraints, which are replaced by a saturation function so that a parameterized system is formulated. This allows one to generate an optimal trajectory via solving an initial value problem, making it efficient to collect a dataset containing optimal samples, which are essential for training Neural Networks (NNs) to achieve real-time implementation. However, the optimal guidance command may suddenly change from one extreme to another, resulting in discontinuous jumps that generally impair the NN’s approximation performance. To address this issue, we use two co-states that the optimal guidance command depends on, to detect discontinuous jumps. A procedure for preprocessing these jumps is then established, thereby ensuring that the preprocessed guidance command remains smooth everywhere. Meanwhile, the sign of one co-state is found to be sufficient to revert the preprocessed guidance command back into the original optimal guidance command. Furthermore, three NNs are built and trained offline, and they cooperate to precisely generate the optimal guidance command in real time. Finally, numerical simulations are presented to demonstrate the developments of the paper.

Exact mixed-integer quadratic formulation and solution for large-scale thermal unit commitment

Abstract Thermal unit commitment (UC) is a nonlinear combinatorial optimization problem that minimizes total operating costs while considering system load balance, on/off restrictions and other constraints. Successfully solving the thermal UC problem contributes to a more reliable power system and reduces thermal costs. This paper presents an exact mixed-integer quadratic programming (EMIQP) method for large-scale thermal UC problems. EMIQP revolutionizes the landscape by seamlessly translating the intricate nonlinear combinatorial optimization problem of UC into an exact mixed-integer quadratic formulation. This approach also elegantly reimagines on/off constraints as mixed-integer linear equations, employing both the sum and respective approaches. Our case studies unequivocally demonstrate the exceptional prowess of the EMIQP method, consistently securing the global optimum. Moreover, the mathematical-based EMIQP method produces identical results at each run, which is extremely important for UC in the real world.

Optimal Power Flow Management for a Solar PV-Powered Soldier-Level Pico-Grid

Users ought to decide how to operate and manage power systems in order to achieve various goals. As a result, many strategies have been developed to aid in this regard. Optimal power flow management is one such strategy that assists users in properly operating and managing the supply and demand of power in an optimal way under specified constraints. However, in-depth research on optimal power flow management is yet to be explored when it comes to the supply and demand of power for the bulk of standalone renewable energy systems such as solar photovoltaics, especially when it comes to specific applications such as powering military soldier-level portable electronic devices. This paper presents an optimal power flow management modelling and optimisation approach for solar-powered soldier-level portable electronic devices. The OPTI toolbox in MATLAB is used to solve the formulated nonlinear optimal power flow management problem using SCIP as the solver. A globally optimal solution was arrived at in a case study in which the objective function was to minimise the difference between the power supplied to the portable electronic device electronics and the respective portable electronic device power demands. This ensured that the demand for solar-powered soldier-level portable electronic devices is met at all times in spite of the prohibitive case scenarios’ circumstances under the given constraints. This resolute approach underscores the importance placed on satisfying the demand needs of the specific devices while navigating and addressing the limitations posed by the existing conditions or constraints. Soldiers and the solar photovoltaic user fraternity at large will benefit from this work as they will be guided on how to optimally manage their power systems’ supply and demand scenarios. The model developed herein is applicable to any demand profile and any number of portable electronic device and is adaptable to any geographical location receiving any amount of solar radiation.

A guide to equivalent circuit fitting for impedance analysis and battery state estimation

In this study we define a comprehensive method for analyzing electrochemical impedance spectra of lithium batteries using equivalent circuit models, and for information extraction on state-of-charge and state-of-health from impedance data by means of machine learning methods. Estimation of circuit parameters typically implies a non-linear optimization problem. A detailed method for estimating initial values of the optimization algorithm is described, emphasizing short computation times and efficient convergence to global minimum. Parameters identifiability is investigated through an analysis of the injectivity of the model, Cramer–Rao lower bound, profile likelihood, and sensitivity analysis. An exploratory data analysis is presented to estimate the degree of correlation between impedance spectra (or circuit parameters) and battery state-of-charge or state-of-health, prior to the implementation of any machine learning algorithm. A publicly available dataset of impedance spectra of five lithium-polymer batteries is used to test the whole procedure. Estimation of state-of-charge and state-of-health is performed by implementing Gaussian process regression.

Acquiring elastic properties of thin composite structure from vibrational testing data

Abstract The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments.

Model predictive control of bidirectional heat transfer in prosumer-based solar district heating networks

District heating networks with decentralized heat production are ideally suited to include a high share of renewable energy sources for heat production in urban areas with limited space. A new concept is a prosumer-based district heating network in which some or even all buildings are equipped with decentralized building-level heat storages and heat generation plants. To exploit the full potential of the interconnected heating network, buildings with surplus heat are intended to transfer heat to buildings with heating demands to prevent the activation of the central heating plant. This work presents an initial feasibility study on using a model predictive control strategy to manage bidirectional heat transfer between buildings in a district heating network. We formulate this problem as an optimal control problem, incorporating binary decision variables for each potential heat transfer connection. This results in a mixed-integer nonlinear optimization problem that is hard to solve. The problem is solved with a fast gradient-based optimization algorithm in combination with a combinatorial integral approximation strategy. A case study concerning a residential heating network with building-level solar thermal collectors and storage tanks is carried out. The optimized operation is compared to the real operation obtained from one month of measurements. The results show that the optimized strategy with bidirectional heat transfer can exploit the total heat generated in the whole network. The central heating provider is only required when the heat produced or stored by all the buildings is insufficient to meet the total heat demand within the network. In the case study, this leads to a reduction of almost 75% of the required thermal energy from the central heat provider.

Wind Power Bidding Based on an Ensemble Differential Evolution Algorithm with a Problem-Specific Constraint-Handling Technique

The intermittent nature of wind power generation induces great challenges for power bidding in the electricity market. The deployment of battery energy storage can improve flexibility for power bidding. This paper investigates an optimal power bidding strategy for a wind–storage hybrid power plant in the day-ahead electricity market. To handle the challenges of the uncertainties of wind power generation and electricity prices, the optimal bidding problem is formulated as a risk-aware scenario-based stochastic programming, in which a number of scenarios are generated using a copula-based approach to represent the uncertainties. These scenarios consider the temporal correlation of wind power generation and electricity prices between consecutive time intervals. In the stochastic programming, a more practical but nonlinear battery operation cost function is considered, which leads to a nonlinear constrained optimization problem. To solve the nonlinear constrained optimization problem, an ensemble differential evolution (EDE) algorithm is proposed, which makes use of the merits of an ensemble of mutant operators to generate mutant vectors. Moreover, a problem-specific constraint-handling technique is developed. To validate the effectiveness of the proposed EDE algorithm, it is compared with state-of-the-art DE-based algorithms for constrained optimization problems, including a constrained composite DE (C2oDE) algorithm and a novel DE (NDE) algorithm. The experimental results demonstrate that the EDE algorithm is much more reliable and much faster in finding a better bidding strategy against benchmarking algorithms. More precisely, the average values of the success rate are 0.893, 0.667, and 0.96 for C2oDE, NDE, and EDE, respectively. Compared to C2oDE and NDE, the average value of the mean number of function evaluations to succeed with EDE is reduced by 76% and 59%, respectively.

The Optimization of a Subsea Pipeline Installation Configuration Using a Genetic Algorithm

The most commonly used subsea pipeline installation method is the S-Lay method. A very important and complex task in an S-Lay installation engineering analysis is to find the optimal pipelay vessel installation configuration for every distinctive pipeline route section. Installation loads in the pipeline are very sensitive to small changes in the configuration of the pipeline supports during laying and other influential parameters, such as the tensioner force, stinger angle, trim and draft of the pipelay vessel. Therefore, the process of an engineering installation analysis is very demanding, and there is a need for an automated optimization process. For that purpose, installation engineering methodology criteria and requirements are formalized into a nonlinear optimization problem with mixed continuous and discrete variables. A special tailored multi-objective genetic algorithm is developed that can be adjusted to any desired combination of criteria and offshore standards’ requirements. The optimization algorithm is applied to the representative test cases. The optimization procedure efficiency and quality of the achieved solution prove that the developed genetic algorithm operators and the whole optimization approach are adequate for the presented application.

Modified fractional homotopy method for solving nonlinear optimal control problems

Modified fractional homotopy method for solving nonlinear optimal control problems

Optimal Control of Mechanical Systems Based on Path-Fitted Variational Integrators

Abstract In view of the crucial importance of optimal control in many application areas and the improved performance of path-fitted variational integrators, the paper links these two aspects and presents a methodology to find optimal control policies for mechanical systems. The main process of the methodology is employing path-fitted variational integrators to discretize the forced mechanical equations and further take the obtained discrete equations as equality constraints for the final optimization problem. Simultaneously, the discretization also provides a reasonable way to approximate the objective function and incorporate the boundary conditions. With the transformation of optimal control problems into nonlinear optimization problems, all the benefits of path-fitted variational integrators are inherited by the presented methodology, mainly expressed in giving more faithful optimizations and thus more accurate solutions, providing a greater possibility of global optimality, as well as conserving computed control efforts. These superiorities, verified by the optimal control of an overhead crane, indicate that the methodology has high potential application in industrial control field.

Parameter estimation in single-phase transformers via the generalized normal distribution optimizer while considering voltage and current measurements

This research addresses, from a perspective of metaheuristic optimization, the problem regarding parametric estimation in single-phase transformers while considering voltage and current measures at the terminals of the transformer and weighing linear loads. Transformer parametric estimation is modeled as a nonlinear problem in order to minimize the mean square error between the calculated voltage and current variables and the measurements taken. The nonlinearities are associated with Kirchhoff's first and second laws applied to the equivalent electrical circuit of the single-phase transformer. The nonlinear optimization problem is solved by applying a metaheuristic optimization algorithm known as the generalized normal distribution optimizer (GNDO), which uses evolution rules that allow exploring and exploiting the solution space via the classical probability function based on normal distributions. Numerical results in three test transformers of 20, 45, and 112.5 kVA demonstrate the effectiveness and robustness of the proposed GNDO approach when compared to other optimizers reported in the literature, such as the crow search algorithm, the coyote optimization algorithm, and the exact solution of the nonlinear optimization model using the fmincon solver of the MATLAB software. All numerical simulations confirm the potential of the GNDO approach to deal with complex optimization problems in engineering and science with promising results and low computational effort.

Nonlinear topology optimization on thin shells using a reduced-order elastic shell model

We present a novel numerical algorithm to perform nonlinear topology optimization on elastic thin shells. The main component of our method is a differentiable thin-shell simulator based on discrete differential geometry (DDG) discretization and the projected Newton method to solve geometrically nonlinear elasticity and its derivatives on a triangle mesh. We build a density-based topology optimization algorithm, enhanced by a density filter and a Heaviside projection scheme, to emerge and optimize topologically complex shell structures on curved surfaces. We validate our approach using standard test cases for nonlinear topology optimization and demonstrate the efficacy of our method by tackling highly nonlinear topology optimization problems by producing complex and high-resolution shell structural designs under various load conditions.

Continuous curvature wayline generation using quadratic optimisation for agricultural machine guidance

Navigation systems are widely used in precision agriculture to improve efficiencies and yields. These systems guide agricultural machines to follow waylines calculated based on specific traffic patterns, including straight and curved waylines. While generating straight waylines is relatively straightforward, generating curved waylines presents challenges due to the variation in their shape and curvature during propagation, which can lead to skips and overlaps between adjacent waylines.In this paper, we present a novel method to curved wayline generation by formulating the problem as a non-linear optimisation problem, as opposed to using geometry operations like existing methods. With the designed objective function and constraints, the problem is reduced to a quadratic optimisation problem that can be efficiently solved, and the generated waylines satisfy the curvature constraint, while minimising skips and overlaps. The functionality and applicability of the method were demonstrated with a simulation experiment, the comparison tests with geometric methods, and a field test. Our algorithm achieved coverage over 97%, with overlap and skip rates lower than 0.8% and 2.3%, respectively. The results indicate that our algorithm can generate smooth, continuous curvature waylines with high coverage and low skip/overlap. Furthermore, our algorithm can also generate straight waylines when the input reference line is a straight AB line. This algorithm can be used as a wayline planning component of navigation system for agricultural machines and robots. The code and boundary data utilised in the "Results and Discussion" section are accessible online at https://github.com/prefect-of-gryffindor/curve-wayline-generation.

Data‐driven decision‐focused surrogate modeling

AbstractWe introduce the concept of decision‐focused surrogate modeling for solving computationally challenging nonlinear optimization problems in real‐time settings. The proposed data‐driven framework seeks to learn a simpler, for example, convex, surrogate optimization model that is trained to minimize the decision prediction error, which is defined as the difference between the optimal solutions of the original and the surrogate optimization models. The learning problem, formulated as a bilevel program, can be viewed as a data‐driven inverse optimization problem to which we apply a decomposition‐based solution algorithm from previous work. We validate our framework through numerical experiments involving the optimization of common nonlinear chemical processes such as chemical reactors, heat exchanger networks, and material blending systems. We also present a detailed comparison of decision‐focused surrogate modeling with standard data‐driven surrogate modeling methods and demonstrate that our approach is significantly more data‐efficient while producing simple surrogate models with high decision prediction accuracy.

Absolute Nodal Coordinate Formulation-Based Shape-Sensing Approach for Large Deformation: Three-Dimensional Beam

In recent years, fiber-optic shape sensing, which means reconstructing the deformation state of structures from strain information measured by fiber-optic sensors, has attracted interest from many fields. Most of the existing shape-sensing research for flexible three-dimensional (3D) slender structures is based on simple strain-curvature transformation, relying on standardized substrates with a circular cross section and particular forms of fiber layouts. To develop a more general shape-sensing method that can be applied to a wider range of practical engineering conditions, in this paper, the strain–displacement relationship of 3D slender structures under the effect of multiple deformation modes coupling is described by introducing absolute nodal coordinate formulation element, and the deformation reconstruction problem is reorganized into a nonlinear optimization problem that can be applied to large deformation and accommodates different cross-sectional shapes and optical fiber layouts. Due to the rapid increase in the calculation difficulties of nonlinear optimization problems with the number of variables, an element-by-element solving strategy is adopted, and nodal degrees of freedom that have less influence on the overall shape of the structure are merged. In addition, in order to address the issue of local strain anomalies caused by unmodeled factors such as section warping, the pointwise matching between the theoretical strains and measured strains is relaxed to an average matching in subregions to capture the overall deformation, improving the robustness and computational efficiency of the solution process. The accuracy and computational performance of the proposed method are verified through numerical simulation and experiment.

A Modified Artificial Hummingbird Algorithm for solving optimal power flow problem in power systems

Optimal power flow (OPF) problem solution is a crucial task for the operators and decision makers to assign the best setting of the system components to obtain the most economic, environmental, and technical suitable state. Artificial Hummingbird Algorithm is a recent optimization algorithm that has been applied to solving several optimization problems. In this paper, a Modified Artificial Hummingbird Algorithm (MAHA) is proposed for improving the performance of the orignal Artificial Hummingbird Algorithm as well as effectivelly solve the OPF problem. The proposed MAHA is based on improving the searching capability by boosting the exploitation using the bandwidth motion around the best solution, while the exploration process is improved using the Levy flight distribution motion and the fitness-distance balance selection. This modified version helps overcome issues such as stagnation, premature convergence, and a propensity for local optima when tackling complex, nonlinear, and non-convex optimization problems like OPF. In order to confirm the effectiveness of the proposed algorithm, a series of tests are conducted on 23 standard benchmark functions, including CEC2020. The resulting outcomes are then compared to those obtained using other algorithms such as fitness-distance balance selection-based stochastic fractal search (FDBSFS), antlion optimizer (ALO), whale optimization algorithm (WOA), sine-cosine algorithm (SCA), fitness-distance balance and learning based artificial bee colony (FDB-TLABC), and traditional artificial hummingbird algorithm (AHA).The proposed algorithm is evaluated by solving the OPF problem with multiple objective functions on the IEEE 30-bus system. These objectives include fuel cost, fuel cost with valve loading effects, power losses, emissions, and voltage profile. Additionally, the algorithm's effectiveness is further assessed by testing it on single objective functions using medium and large-scale IEEE 57 and 118-bus networks.The results obtained by the proposed MAHA demonstrate its power and superiority for solving the OPF problem as well as the standard benchmark functions , surpassing the performance of other reported techniques.

Method for prediction and control by uncertain microsatellite magnetic cleanliness based on calculation and compensation magnetic field spatial harmonics

Aim. Development of method for prediction and control the microsatellite magnetic cleanliness taking into account the uncertainties of the magnetic characteristics of the microsatellite, based on calculation the magnetic field spatial spherical harmonics in the area of the onboard magnetometer installation and using compensating multipoles. Methodology. Spatial spherical harmonics of microsatellite magnetic field in the area of the onboard magnetometer installation calculated as solution of nonlinear minimax optimization problem based on near field measurements for prediction far spacecraft magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field. Values of the compensating dipoles, quadrupoles and octupoles and coordinates of them placement inside the spaceship for compensation of the dipoles, quadrupoles and octupoles components of the microsatellite initial magnetic field also calculated as solution of nonlinear minimax optimization problem. Both solutions of this nonlinear minimax optimization problems calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction spacecraft far magnetic field magnitude based on spacecraft spatial spherical harmonics of the magnetic field using near field measurements and compensation of the dipoles, quadrupoles and octupoles components of the initial magnetic field with consideration of spacecraft magnetic characteristics uncertainty for ensuring the microsatellite magnetic cleanliness. Originality. The method for prediction and control by spacecraft magnetic cleanliness based on calculation spatial spherical harmonics of the magnetic field in the area of the onboard magnetometer installation using compensation of the dipoles, quadrupoles and octupoles components of the initial magnetic field with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of ensuring the magnetic cleanliness of the «Sich-2» microsatellite family based on the spatial spherical harmonics of the magnetic field model using the compensation of the dipole, quadrupole and octupole components of the output magnetic field of the sensor for the kinetic parameters of the neutral component of the space plasma at the point of installation of the on-board magnetometer LEMI-016 by setting the compensating dipole, quadrupole and octupole with consideration of spacecraft magnetic characteristics uncertainty solved.

A modified type of Fletcher-Reeves conjugate gradient method with its global convergence

<p><span>The conjugate gradient methods are one of the most important techniques used to address problems involving minimization or maximization, especially nonlinear optimization problems with no constraints at all. That is because of their simplicity and low memory needed. They can be applied in many areas, such as economics, engineering, neural networks, image restoration, machine learning, and deep learning. The convergence of Fletcher-Reeves (FR) conjugate gradient method has been established under both exact and strong Wolfe line searches. However, it is performance in practice is poor. In this paper, to get good numerical performance from the FR method, a little modification is done. The global convergence of the modified version has been established for general nonlinear functions. Preliminary numerical results show that the modified method is very efficient in terms of number of iterations and CPU time.</span></p>