Taylor Series Research Articles

Extended matrix modeling of integrated energy systems considering network dynamic characteristics and source–load uncertainty

A refined and efficient modeling approach is indispensable for enhancing the timeliness of scheduling strategies for the integrated energy system (IES). Non-flowing variables such as pressure and temperature introduced by energy devices, however, worsen the complexity while improving the completeness for models. Nonlinear and uncertain variables associated with network dynamic characteristics and source–load uncertainty in the systems seriously postpone the modeling processes. An extended matrix modeling approach is proposed in this paper to tackle such issues. New nodes, branches, definition rules and topological matrices are first presented to create an extended matrix model that is compatible with both flowing and non-flowing variables. To boost the modeling efficiency, adaptive piecewise linearization (APL) with extended matrix form and modified Taylor expansion are applied to approximate various nonlinear characteristics. The convex approximation based on Kullback–Leibler (KL) divergence and the strong duality theory characterized by Wasserstein distance are adopted to deal with uncertain variables, which adapt to combine with the above matrix model. A linearized matrix modeling framework applicable to highly automated by computers is thus established. The distributionally robust optimization (DRO) scheduling strategy is verified in a standard test system with the proposed modeling. The effect of dynamic characteristics and uncertain factors on the strategy is discussed. The results show that the extended matrix model simulates the response for real physical systems more effectively, which exhibits the better performances in modeling, objectives and scheduling strategies.

Modified Calibration Variance Estimators in the Presence of Non- response and Measurement Error

Samples from a population that can be divided into smaller groups are taken using a technique called stratified sampling. When subpopulations within the total population differ, it may be beneficial to sample each stratum (subpopulation) separately in sample surveys. Government organizations, independent consultants, and applied statisticians all frequently use this crucial strategy. There are many problems encountered by survey statisticians in estimating the population variance of the study variable. These problems include the presence of outliers in data collected for analysis, non-response, and measurement errors occurring during the survey. Shahzad et al. [1] developed variance estimators by addressing the problem of outliers using the L-moment and calibration approach. However, they do not consider the situation of non-response and measurement errors. This paper addresses these problems by proposing modified variance estimators in the presence of non-response and measurement errors. The properties (Biases and MSEs) were derived up to the first order of approximation using the Taylor series approach. The efficiency conditions of the modified estimators over the existing estimators considered in the study were established. The result of simulation studies revealed that the estimators are efficient.

Taylor series approximations for faster robust topology optimization

Taylor series approximations for faster robust topology optimization

Low-Thrust Fuel-Optimal Guidance with Automatic Control Sequence Detection and Separation

This paper develops a novel fuel-optimal guidance scheme using differential algebraic techniques that is capable of handling changes in the optimal thrust sequence, in addition to a large domain of uncertainty. Taylor polynomials of the solution flow exhibit large approximation errors when the control sequence changes within the domain of operation. This property is leveraged through automatic domain splitting, a technique that tracks the approximation error and splits the domain into smaller subdomains when high errors are observed. After multiple splits, the domain is autonomously separated along the boundaries between different optimal control sequences. Crucially, each resulting subdomain encompasses only a single control sequence, ensuring that any guidance updates obtained from the final polynomial map provide the corresponding optimal control sequence. To ensure high accuracy of the guidance updates, this work also develops an efficient and iterative-less differential-algebra-based refinement algorithm, which is applied to the outputs of the polynomial map. The overall guidance scheme provides highly accurate guidance updates in response to large trajectory deviations, including new optimal control sequences, as demonstrated in an Earth–Psyche transfer.

Algorithm 1046: An Improved Recurrence Method for the Scaled Complex Error Function

Calculation of the scaled complex error function \(w(z)\) by recurrence is discussed, and a new method for determining the number of steps required to achieve a given accuracy is introduced. This method is found to work throughout the complex plane, except for a short section of the real line, centred at the origin. An algorithm based on this analysis is implemented; Taylor series with stored coefficients are used to compute \(w(z)\) in a small region where recurrence is not efficient. The new algorithm is tested extensively and found to outperform earlier recurrence-based codes. It also performs favourably against recent codes based on other methods.

New Formula on the Conjugate Gradient Method for Unconstrained Optimization and its Application

In this paper, we have presented a new formula by applying the conjugacy condition for the second-order Taylor expansion. In comparison with classic conjugate gradient methods, the new formula uses both available gradient and function value information. The formula's global convergence findings are described. Numerical results demonstrate that this strategy is effective and its application.

Dynamic Analysis and Approximate Solution of Transient Stability Targeting Fault Process in Power Systems

Modern power systems are high-dimensional, strongly coupled nonlinear systems with complex and diverse dynamic characteristics. The polynomial model of the power system is a key focus in stability research. Therefore, this paper presents a study on the approximate transient stability solution targeting the fault process in power systems. Firstly, based on the inherent sinusoidal coupling characteristics of the power system swing equations, a generalized polynomial matrix description in perturbation form is presented using the Taylor expansion formula. Secondly, considering the staged characteristics of the stability process in power systems, the approximate solutions of the polynomial model during and after the fault are provided using coordinate transformation and regular perturbation techniques. Then, the structural characteristics of the approximate solutions are analyzed, revealing the mathematical basis of the stable motion patterns of the power grid, and a monotonicity rule of the system’s power angle oscillation amplitude is discovered. Finally, the effectiveness of the proposed methods and analyses is further validated by numerical examples of the IEEE 3-machine 9-bus system and IEEE 10-machine 39-bus system.

Nonlinear dynamic analyses of sandwich porous FG cylindrical shells with dual-layered FG porous cores

This research examines the nonlinear vibrations and dynamic postbuckling (DPB) analyses of sandwich porous functionally graded (SPFG) cylindrical shells with dual-layered FG porous (FGP) cores exposed to external excitation, using a semi-analytical method. The study investigates two types of FG layers: one with evenly distributed porosities (FG-EPD) and another with unevenly distributed porosities (FG-UEPD). It also studies the FGP cores in four configurations: one featuring symmetric porosity distribution with stiffening in the surface areas (SPD-Stiff), another with softening in the surface areas (SPD-Soft), a third with nonsymmetric porosity distribution (NSPD), and a fourth with uniform porosity distribution (UPD). Therefore, the SPFG cylindrical shells with dual-layered FGP cores with eight different configurations are investigated. Utilizing the classical shell theory (CST) alongside the geometrical nonlinearity in von Kármán–Donnell framework, and Galerkin’s method, this study addresses the nonlinear dynamic problem. An approximate solution for the deflection shape using three terms is selected, and the relationship between frequency and amplitude in nonlinear vibration is clearly defined. The nonlinear dynamic behaviors including vibration and DPB responses are examined using the P-T method, which is named for its use of the piecewise constant argument in conjunction with the Taylor series expansion. The critical dynamic buckling load of SPFG cylindrical shells with dual-layered FGP cores is investigated via the Budiansky–Roth criterion.

Estimating the sensitivity of the Priestley–Taylor coefficient to air temperature and humidity

Abstract. The Priestley–Taylor (PT) coefficient (α) is generally set as a constant value or is fitted as an empirical function of environmental variables, and it can bias the evaporation estimation or hydrological projections under global warming. By using an atmospheric boundary layer model, this study derives a theoretical and parameter-free equation for estimating α as a function of air temperature (T) and specific humidity (Q). With observations from several waterbodies and non-water-limited land sites, we demonstrate that, in addition to estimating the value of α well, the derived expressions can also capture the sensitivity of α to T and Q, that is, dα/dT and dα/dQ. α is generally negatively associated with T and Q, in which regard T plays a more fundamental role in controlling α behaviors. Based on climate model data, we further show that this negative relationship between α and T is of great importance for long-term hydrological predictions. We also provide a lookup graph for practical and broad uses to directly find the values of dα/dT and dα/dQ under specific conditions. Overall, the derived expression gives a physically clear and straightforward approach to quantify changes in α, which is essential for PT-based hydrological simulation and projections.

A novel semi-analytic algorithm for evaluation of nearly singular integrals in boundary element analysis

A novel semi-analytic method is proposed to evaluate various nearly singular integrals accurately. Different from the traditional semi-analytical method, the Taylor expansion for shape functions, Jacobian, and so on in our method is based on the nearest point from a source point to an integral element rather than the projection point from the source point to the integral element. Therefore, it can more accurately approximate the distance from the source point to Gaussian integration points. Then, approximate expressions for the integrand functions of two-dimensional potential problems are derived. The effectiveness of the proposed method is verified by evaluating the calculation accuracy of physical quantities at points in different geometric models, and comparing it with the traditional semi-analytical method and distance transform method.

A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.

An efficient approach for searching three-body periodic orbits passing through Eulerian configuration

A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on two previously established symmetry types on the shape sphere, each solution is presented by one or two distinct initial conditions (one or two points in the search domain). A numerical search based on Newton’s method on a relatively coarse search grid for solutions with relatively small scale-invariant periods T∗<70 is conducted. The linear systems at each Newton’s iteration are computed by high order high precision Taylor series method. The search produced 12,431 initial conditions (i.c.s) corresponding to 6333 distinct solutions. In addition to passing through the Eulerian configuration, 35 of the solutions are also free-fall ones. Although most of the found solutions are new, all linearly stable solutions among them (only 7) are old ones. Particular attention is paid to the details of the high precision computations and the analysis of accuracy. All i.c.s are given with 100 correct digits.

The Hydrodynamic Similarity between Different Power Levels and a Dynamic Analysis of Ocean Current Energy Converter–Platform Systems with a Novel Pulley–Traction Rope Design for Irregular Typhoon Waves and Currents

In the future, the power of a commercial ocean current energy convertor will be able to reach the MW class, and its corresponding mooring rope tension will be very good. However, the power of convertors currently being researched is still at the KW class, which can bear less rope tension. The main mooring rope usually has a single cable and a single foundation. To investigate the dynamic response and rope tension of an MW-class ocean current generator mooring system, here, a similarity rule is proposed for (1) coefficients without any fluid–structure interaction (FSI) using the Buckingham theorem and (2) ones with FSI. The overall hydrodynamic drag and moment including the hydrodynamic coefficients in these two situations are represented in a Taylor series. Assuming similarity between the commercial MW-class and KW-class ocean current convertors, all hydrodynamic parameters of the MW-class system are estimated based on the known KW-class parameters and based on the similarity formula. In order to overcome the extreme tension of the MW-class system and to provide good stability, in this paper, we propose a pulley–rope design to replace the traditional single-traction-rope design. The static and dynamic mathematical models of this mooring system subjected to the impact of typhoon waves and currents are proposed, and analytical solutions are obtained. We find that the pulley–rope design can significantly reduce the dynamic rope tensions of the mooring system. The effect of the length ratio of the main traction rope, rope A, to the seabed depth on the dynamic tension of stabilizing converter rope D is significant. The length ratio is within a safe range, and the maximum rope dynamic tension is less than the fracture strength. In addition, if the rope length ratio is over the critical value, the larger the ratio, the higher the safety factor of the rope. In summary, the pulley–rope design can be safely used in an MW-level ocean current generator system.

Physics-informed neural networks for biopharmaceutical cultivation processes: Consideration of varying process parameter settings.

We present a new modeling approach for the study and prediction of important process outcomes of biotechnological cultivation processes under the influence of process parameter variations. Our model is based on physics-informed neural networks (PINNs) in combination with kinetic growth equations. Using Taylor series, multivariate external process parameter variations for important variables such as temperature, seeding cell density and feeding rates can be integrated into the corresponding kinetic rates and the governing growth equations. In addition to previous approaches, PINNs also allow continuous and differentiable functions as predictions for the process outcomes. Accordingly, our results show that PINNs in combination with Taylor-series expansions for kinetic growth equations provide a very high prediction accuracy for important process variables such as cell densities and concentrations as well as a detailed study of individual and combined parameter influences. Furthermore, the proposed approach can also be used to evaluate the outcomes of new parameter variations and combinations, which enables a saving of experiments in combination with a model-driven optimization study of the design space.

Constrained H∞ control with gain switching for a nonlinear active suspension system matching non-pneumatic wheel

In this study, a constrained H ∞ controller with gain switching is put forward for the active suspension system to improve the overall performance of vehicles equipped with non-pneumatic wheels, considering the nonlinearity of wheel stiffness, actuator saturation, and output constraints. Firstly, a quarter-vehicle model incorporating active suspension and non-pneumatic wheel (NPW) is established experimentally. Secondly, the system model is linearized using Taylor series expansion and linear fractional transformation (LFT). A constrained H ∞ control strategy and a gain switching method based on road classification are proposed, taking into account the parameter uncertainty in linearization process and the variation of performance demands under different road conditions. Then, an L ∞ state observer is designed for the required system state, and the road roughness classifier based on grey wolf optimization (GWO) and probabilistic neural network (PNN) is developed to obtain the necessary road information. A bench test is finally performed using the reconstructed actual road as input. The test results validate the effectiveness and superiority of the proposed control strategy.

Polynomial filter-based adaptive fault-tolerant control for a class of nonlinear systems with multiplicative faults

In this article, the problem of fault-tolerant control is investigated for polynomial nonlinear systems (PNSs) subject to multiplicative faults and measurement noise. Firstly, a matrix Taylor expansion technique is applied to transform the PNS into a linear parameter varying system. Then the polynomial filter is constructed to get estimations of states and unmatched disturbances, which reduces the effect of measurement noise. A novel adaptive fault-tolerant controller is proposed to compensate for multiplicative faults and unmatched disturbances. Furthermore, the condition of the parameter-dependent linear matrix inequality (PDLMI) is deduced, which guarantees the boundedness of tracking errors and estimation errors, and the sum of squares (SOS) decomposition technique is utilized to solve PDLMI. Finally, the simulation of a rotary steerable drilling tool system confirms the feasibility of the proposed adaptive fault-tolerant control framework.

Robust Intelligent Nonlinear Predictive Control Based on Artificial Neural Network for Optimizing PMSM Drive Performance

In the realm of high-performance motor drive systems, achieving stable and optimal motor operation is crucial, particularly in environments characterized by disturbances. The implementation of model predictive control (MPC) represents a strategic methodology. However, conventional predictive controllers frequently encounter challenges due to uncertainties regarding the motor's internal parameters and external load characteristics, subsequently impacting the effectiveness of the control algorithm. This paper proposes a new robust predictive control combined with an artificial neural network (RMPC-ANN) approach applied to a permanent magnet synchronous motor (PMSM) to tackle challenges posed by external perturbations and parameter variations. The development of the proposed robust predictive controller involves optimizing a novel finite horizon cost function based on Taylor series expansion, which incorporates dual integral action into the control law. Crucially, this approach eliminates the necessity for measuring and observing external perturbations and parameter uncertainties. Additionally, for attaining high-precision speed control, the speed loop regulation relies on a multi-layer feedforward ANN algorithm. A comprehensive comparison was conducted using MATLAB/SIMULINK, assessing performance across diverse operating conditions. To further substantiate the numerical simulation results, a hardware-in-the-loop (HIL) configuration is implemented on the OPAL-RT platform, demonstrating the robustness and efficiency of the proposed control strategy.

MATLAB Application for Determination of 12 Combustion Products, Adiabatic Temperature and Laminar Burning Velocity: Development, Coding and Explanation

The determination of the characteristics and main combustion properties of fuels is necessary for post-implementation in different applications. Among the most important combustion properties of a fuel are the combustion products, flame temperature and laminar burning velocity. Therefore, this paper describes the step-by-step development and coding of a MATLAB application that can determine 12 combustion products, flame temperature and laminar burning velocity in order to understand the logic of calculus procedure, so any user would be able to make improvements of new functionalities (add more fuels, add more combustion products, etc.). The numerical procedure and methods (Gaussian elimination, Taylor Series and Newton–Raphson) parallel with their implementation as code lines for the development of the application are carried out using flow charts. In addition, simulations in Ansys Chemkin were performed and included in the application as part of the results comparison. It was found that: (1) The MATLAB Application codification and development were successfully explained in detail, (2) the functions and execution sequence are described by using flow charts and code extract, (3) the application is available to everyone for modifications, (4) the application can only be used for hydrocarbons fuels, (5) the application execution time registered was less than 8 s.

Fractional Derivative of Hyperbolic Function

Fractional derivative is a generalization of ordinary derivative with non-integer or fractional order. This research presented fractional derivative of hyperbolic function (hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant) with order constraint . The hyperbolic function is presented in Maclaurin series form. Then, the fractional derivative can be determined by using definition of Riemann-Liouville fractional derivative. The result is simulated by using Matlab software

High-Efficiency Forward Modeling of Gravitational Fields in Spherical Harmonic Domain with Application to Lunar Topography Correction

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.