2nd Order Derivative Research Articles

Novel method in mice to evaluate left atrial (LA) systolic contribution to left ventricular volume (LV) using pressure-volume (PV) loops

Abstract Background LA myopathy is a critically important non-LV contributor to the progression of HFpEF (heart failure with preserved ejection fraction) in patients and is associated with a severe phenotype, leading to impaired cardiac output reserve and reduced exercise capacity. In normal humans LA systole contributes ~20% of total LV filling, but with HFpEF, this contribution decreases to <16% with a greater reliance on this contribution to LV filling. Purpose We sought to investigate LA function as a critical component of cardiac function analysis by utilizing the LV pressure signal to identify and quantify the contribution of volume during LA systole to the total LV volume in mice with HFpEF. Methods Hypertension (HTN)-associated HFpEF C57BL/6J mice were subjected to chronic aldosterone infusion and 1% NaCl for 4 weeks (SAUNA model). Obesity after a high fat diet (HFD: 60% fat diet) for 16 weeks followed by the SAUNA protocol, induced HTN and obesity associated HFpEF. Blood pressure was monitored weekly. Mice were anesthetized with isoflurane, intubated, chest opened to expose the heart and allow placement of a PV catheter (Transonic) which was introduced into the LV via apical stab with a 27G needle. Calibration of the PV catheter was performed according to the manufacturer’s instructions. Atrial systole was identified using 2nd order derivative of the LV pressure signal. Contribution of LA systole was calculated using volume at the beginning of LA systole identified by the 2nd order derivative of the LV pressure signal and subtracting from the LV end diastolic volume. The % of LA systole contribution was calculated as a % of the total LV stroke volume. Data was analyzed with LabChart pro software (AD Instruments). Results SAUNA mice had normal LVEF, moderate HTN and lung congestion. PV relationships demonstrated Tau was impaired in HFpEF vs. control mice (14±1 vs. 10±0.5 ms) with a shift of the PV loop to the left (Fig. A.) In control mice, the atrial systole contributed 39± 4% to LV volume and was impaired in HFpEF mice 20± 3% a 1.95-fold reduction (P<0.01). Importantly atrial systole contribution was increased in obese mice 53±5 %. However, the combination of obesity and HTN, both common comorbidities in HFpEF, resulted in a reduced LA systole contribution by 1.43-fold to 37± 9%, (P<0.005; Fig B). Conclusion A novel method of % atrial systole from LV PV loop using 2nd order derivative to quantify the volume contribution of atrial systole to total LV stroke volume was identified in mice. LA dysfunction was present in murine HTN-associated HFpEF with/without obesity.

Investigation of new analysis methods for simultaneous and rapid identification of five different microplastics using ATR-FTIR spectroscopy and chemometrics

Microplastic (MP) pollution in water has become one of the most important global problems of our time. The development of appropriate and rapid analysis techniques is of great importance at the beginning of the studies aimed at solving this problem. In the presented study, in order to perform the qualitative and quantitative analysis of MP forms of polyamide (PA), polyethylene (PE), polypropylene (PP), polystyrene (PS), and polyethylene terephthalate (PET), which are known to be most abundant in water, in a fast and easy way, new Attenuated Total Reflectance-Fourier Transform Infrared (ATR-FTIR) spectroscopy methods were tried to be developed by utilizing chemometric methods. While principal component analysis (PCA) was applied for qualitative analyses, partial least squares (PLS) models were created for quantitative analyses. Raw, 1st, and 2nd order derivatives of all spectra and their spectra with different levels of smoothing points were taken and 24 different chemometric models were created for each MP. In interpreting the statistical performances of the developed PCA and PLS models, different parameters were used. According to the obtained results, the qualitative discrimination of all polymer types was successfully achieved. It was determined that the PLS models developed for the quantitative determination of mixtures consisting of different concentrations of MP types could not be at the desired level. However, it was determined that the PLS models developed for PA, PE, PP, and PET, where the normal spectrum was used, could give quantitatively accurate results, albeit partially.

An efficient hybrid WENO scheme with minimized dispersion and adaptive dissipation properties for compressible flows

A numerical scheme with good spectral properties is important for the simulation of compressible flows with various of length scales for fine flow scales resolving. The MDAD-HY scheme (Li et al., 2022) using a discontinuity detector and scale sensor achieves the minimized dispersion and adaptive dissipation property. However, the discontinuity detector is devised based on the ratio of the 1st-order and 2nd-order derivatives on two sides of the interface introducing excessive numerical cost. To address this issue, an efficient hybrid WENO scheme with minimized dispersion and adaptive dissipation properties is proposed in this work. Based on the characteristic-decomposition approach, the numerical flux of the present hybrid scheme is achieved by switching between the linear MDAD scheme and the MDAD-WENO scheme according to a new efficient non-dimensional discontinuity detector. The linear flux is reconstructed in a component-wise method to decrease the characteristic-projection operations. To further improve the spectral property of the present scheme, an adaptive parameter controlling the contribution of the optimal linear scheme according to the discontinuity indicator is introduced. Several benchmark test cases involving broadband of length scales and discontinuities are adopted to verify the efficiency and the high-resolution capability of the present scheme.

Proximal sensing approach for characterization of calcareous soils using multiblock data analysis

Application of visible-near infrared (VNIR) or mid-infrared (MIR) proximal sensing for prediction of soil attributes has proven successful in many cases and is currently extensively used. However, in some cases, predictions fail, especially for properties that do not have a strong spectral signature. Multiblock data analysis techniques are an extension of partial least squares regression (PLSR) that allows the incorporation of data obtained from different instruments into a single model. The techniques have not been used in soil characterization particularly in chemical and elemental soil analysis. Therefore, the aim of the study was to analyze the use of two multiblock algorithms, sequential orthogonalized partial least squares (SO-PLS) and response-oriented sequential alternation (ROSA), to improve the prediction of several soil attributes from VNIR and MIR spectra and their 2nd order derivatives in calcareous soils from Iran. A total of 130 soil samples were taken from 0 to 30 cm depth in Kamfiroz (Mulla Sadra Dam Watershed), Fars Province, Iran. The analyzed soil properties included soil organic matter (SOM), electrical conductivity (EC), carbonate calcium equivalent (CCE), clay and sand contents, and plant-available (DTPA-extractable) micronutrients such as manganese (Mn), iron (Fe), nickel (Ni), zinc (Zn), and copper (Cu). Multiblock models (SO-PLS and ROSA) were compared with single-block PLSR models obtained from VNIR, MIR and their 2nd order derivatives. The results indicated that SO-PLS improved the prediction performance for all studied properties compared to single-block PLSR models from VNIR, MIR or their 2nd order derivatives, and multiblock ROSA. The highest improvement was achieved for properties where single-block models were less accurate. However, the ROSA models were less accurate than single-block models. The best SO-PLS models were obtained for CCE (R2 = 0.96, RPIQ = 5.95, and NRMSE = 7.24%), followed by SOM (R2 = 0.91, RPIQ = 3.38, and NRMSE = 11.45%), log Fe (R2 = 0.89, RPIQ = 3.16, and NRMSE = 9.98%), Cu (R2 = 0.87, RPIQ = 3.21, and NRMSE = 15.51%), clay content (R2 = 0.85, RPIQ = 3.57, and NRMSE = 10.14%), Ni (R2 = 0.76, RPIQ = 2.60, and NRMSE = 28.25%), EC (R2 = 0.74, RPIQ = 1.63, and NRMSE = 20.94%), sand (R2 = 0.68, RPIQ = 1.82, and NRMSE = 27.58%), and Zn (R2 = 0.63, RPIQ = 1.65, and NRMSE = 18.14%). However, Mn was poorly predicted by all evaluated methods. The results confirmed the potential of the SO-PLS model to improve the prediction of soil properties using a combination of spectral information obtained with different instruments.

Simultaneously determining amino acid contents using near-infrared reflectance spectroscopy improved by pre-processing method in rice

Rice, the most consumed in Asian and constitute important source of amino acids. The present study evaluated the suitability of near-infrared reflectance spectroscopy to predict amino acid contents (AACs) using 1210 rice samples representing a wide range of grain qualities and genetic background. Calibration models established by combining different spectral derivatives, scatter and baseline correction methods, and regression algorithm were compared using full cross validation. A modified partial least square model (MPLS) with the spectral derivative treatment “2, 3, 3, 1” (2nd order derivative computed based on 3 data points, and 3 and 1 data points in the 1st and 2nd smoothing, respectively) and weighted multiplicative scattering correction (WMSC) method was identified as the best model for simultaneously measurement of 14 AACs in brown rice flour. The high model quality is reflected by RSQ of 0.914∼0.969, standard error of calibration (SEC) of 0.0187%∼0.0553%, 1-VR of 0.901∼0.955, SECV of 0.0214%∼0.0579%, and residual prediction deviation (RPD) of 2.58∼3.85. These results indicated that NIRS coupled with an appropriate pre-processing, multivariate regression could be a rapid and reliable method for quantification of AACs in rice breeding programs and quality control in the food industry. Factors that would influence model quality were investigated and discussed.

Detection of peak wavelength of multi-FBG using higher-order derivative of wavelets multiresolution analysis and maximum likelihood estimation

The peak wavelength of fiber Bragg grating (FBG) using the higher-order derivative of Wavelets multiresolution analysis and maximum likelihood estimation is proposed. The FBG reflected spectrum is decomposed at the different level using wavelet multiresolution analysis and the higher-order derivative is applied. The first-order derivative is applied to the calculated desired decomposed level to find the zero-crossing point. The 2nd order derivative is applied and the extrema at the corresponding zero-crossing points are calculated. Further, the proceeding derivative, the 4th order derivative is applied and sharper maxima corresponding zero-crossing point is calculated. The maximum likelihood estimation-based statically method is applied to predict the exact peak wavelengths with good efficiency and low error. The proposed peak detection algorithm is also used in the partially overlapped with weak and non-uniform peaks of cascaded FBGs.

Green Synthesis of Silver Nanoparticle from Brassica Oleracea for Determination of Paracetamol in the Presence of Tyrosine, Leucine and Alanine by Derivative Method

In this work, biogenic silver nanoparticles (Ag NPs) from Brassica oleracea (Bo) leaf extract (AgNP-Bo) were prepared and utilized in synthetic mixtures for the accurate, effective, and sensitive spectrophotometric detection of Paracetamol. Measurements in the UV-Visible, X-ray diffraction (XRD), and scanning electron microscopy (SEM) have been used to confirm the synthesis of AgNP-Bo. These statistics indicate that the average particle size is 26.39 nm, the crystallinity has an aggregated structure, and the typical absorption wavelength is 445 nm. To decrease the possible influence of the listed water-soluble interference analytes, namely Tyrosine, Alanine, and Leucine, the 2nd order derivative method was used for our investigation. Using complexation with AgNP-Bo, the linear range, limit of detection (LOD), and relative standard deviation (RSD) for Paracetamol were calculated. The obtained values were 0.1-2, 0.09 µM, and 1.06%, respectively. With average percent recovery values of 93.33, 96.67, and 90.00%, respectively, AgNP-Bo also demonstrated excellent performance for Paracetamol detection in synthetic combinations including Tyrosine, Alanine, and Leucine. The suggested technique applied effectively for the quantitative analysis of synthetic mixtures, which makes it possible to employ Ag NPs going forward to determine biological molecules when interfering analytes from actual samples are present. Keywords: Silver nanoparticles; Brassica oleracea; Paracetamol; 2nd order derivative spectrophotometry; Interferences.

STABILITY INDICATING ISOCRATIC RP-HPLC AND SECOND DERIVATIVE UV SPECTROSCOPIC METHODS FOR SIMULTANEOUS DETERMINATION OF REMOGLIFLOZIN ETABONATE AND VILDAGLIPTIN HYDROCHLORIDE

Remogliflozin and Vildagliptin are the latest drug combination in order to treat type 2 diabetes. The present work describes a comparative study of 2nd-ordered derivative UV and isocratic stability indicating RP-HPLC methods for simultaneous determination of Remogliflozin and Vildagliptin in the drug and its formulation. The HPLC method was established with a C18 column and the mobile phase with Buffer (pH-6.50): Acetonitrile: Methanol ratio of 55:44:1, at 1.2 ml/min flow and wavelength of 210 nm. The UV spectrophotometric method works on spectrophotometry with 2nd order derivative spectra(2D) using a zero-line cross method. The absorbance determination was performed at 246.0 nm for REM and at 219 .6 nm for VIL. The methods were validated for Linearity, Precision, Recovery, Robustness, and Forced degradation as per ICH-Q2(R1). The linear range was established in the range of 4-80 ppm for REM and 2-40 ppm for VIL for the UV method and 2-200 ppm for REM and 1-100 ppm for VIL for the HPLC method with regression coefficient ≥0.990. The accuracy was 99.37% & 99.62% for UV and 100.98 & 100.65 for the HPLC method for REM and VIL respectively. The proposed methods have been effectively used for the quantification of REM and VIL in API mixture and formulation dosage forms. Both methods are economical, fast, simple, and accurate which can be easily adopted for laboratory use.

Hyperspectral Imaging-Based Multiple Predicting Models for Functional Component Contents in Brassica juncea

Partial least squares regression (PLSR) prediction models were developed using hyperspectral imaging for noninvasive detection of the five most representative functional components in Brassica juncea leaves: chlorophyll, carotenoid, phenolic, glucosinolate, and anthocyanin contents. The region of interest for functional component analysis was chosen by polygon selection and the extracted average spectra were used for model development. For pre-processing, 10 combinations of Savitzky–Golay filter (S. G. filter), standard normal variate (SNV), multiplicative scatter correction (MSC), 1st-order derivative (1st-Der), 2nd-order derivative (2nd-Der), and normalization were applied. Root mean square errors of calibration (RMSEP) was used to assess the performance accuracy of the constructed prediction models. The prediction model for total anthocyanins exhibited the highest prediction level (RV2 = 0.8273; RMSEP = 2.4277). Pre-processing combination of SNV and 1st-Der with spectral data resulted in high-performance prediction models for total chlorophyll, carotenoid, and glucosinolate contents. Pre-processing combination of S. G. filter and SNV gave the highest prediction rate for total phenolics. SNV inclusion in the pre-processing conditions was essential for developing high-performance accurate prediction models for functional components. By enabling visualization of the distribution of functional components on the hyperspectral images, PLSR prediction models will prove valuable in determining the harvest time.

Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems

We construct all (2+1)-dimensional PDEs depending only on 2nd-order derivatives of unknown which have the Euler–Lagrange form and determine the corresponding Lagrangians. We convert these equations and their Lagrangians to two-component forms and find Hamiltonian representations of all these systems using Dirac’s theory of constraints. We consider three-parameter integrable equations that are cubic in partial derivatives of the unknown applying our method of skew factorization of the symmetry condition. Lax pairs and recursion relations for symmetries are determined both for one-component and two-component forms. For cubic three-parameter equations in the two-component form we obtain recursion operators in 2 × 2 matrix form and bi-Hamiltonian representations, thus discovering three new bi-Hamiltonian (2+1) systems.

Improved high order differential feedback control of quadrotor UAV based on improved extended state observer

Quadrotor unmanned aerial vehicle (QUAV) system is nonlinear, strongly coupled, and sensitive to unknown disturbances and uncertainties. The uncertainty, disturbance, and even unknown model function are the undesired force acting on the QUAV, so they must be totally estimated online. The ideal control scheme is that the total estimation is feedback in the opposite direction to the controller to compensate the model error or counteract the force caused by the unknown model and disturbance. An improved extended state observer (IESO), also called an improved high-order differentiator (IHOD), is designed to estimate the unknown model and disturbance of the QUAV. It is proved that the IESO is one type higher than the traditional ESO. Consequently, the estimation accuracy of IESO is higher than that of the ESO at the same bandwidth, and is verified through some typical functions. The IESO estimations for 2nd-order derivative of reference input and the total unknown model function are fed back to compensate the undesired force in opposite direction, and then an improved high-order differential feedback controller (IHODFC) is designed with the assigned closed-loop poles. The proof of the stability of the closed-loop system controlled by the IHODFC is given. The performance is verified by the tests using the Pixhawk-based DJI F450 frame experimental verification platform. The IHODFC superiority is demonstrated by comparing the classic model-free controller PID and active disturbance rejection control (ADRC).

The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

A dynamic lesion model for differentiation of malignant and benign pathologies

Malignant lesions have a high tendency to invade their surrounding environment compared to benign ones. This paper proposes a dynamic lesion model and explores the 2nd order derivatives at each image voxel, which reflect the rate of change of image intensity, as a quantitative measure of the tendency. The 2nd order derivatives at each image voxel are usually represented by the Hessian matrix, but it is difficult to quantify a matrix field (or image) through the lesion space as a measure of the tendency. We conjecture that the three eigenvalues contain important information of the Hessian matrix and are chosen as the surrogate representation of the Hessian matrix. By treating the three eigenvalues as a vector, called Hessian vector, which is defined in a local coordinate formed by three orthogonal Hessian eigenvectors and further adapting the gray level occurrence computing method to extract the vector texture descriptors (or measures) from the Hessian vector, a quantitative presentation for the dynamic lesion model is completed. The vector texture descriptors were applied to differentiate malignant from benign lesions from two pathologically proven datasets: colon polyps and lung nodules. The classification results not only outperform four state-of-the-art methods but also three radiologist experts.

Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications

In this article, radial basis function collocation scheme is adopted for the numerical solution of fractional partial differential equations. This method is highly demanding because of its meshless nature and ease of implementation in high dimensions and complex geometries. Time derivative is approximated by Caputo derivative for the values of α ∈ 0 , 1 and α ∈ 1 , 2 . Forward difference scheme is applied to approximate the 1st order derivative appearing in the definition of Caputo derivative for α ∈ 0 , 1 , whereas central difference scheme is used for the 2nd order derivative in the definition of Caputo derivative for α ∈ 1 , 2 . Numerical problems are given to judge the behaviour of the proposed method for both the cases of α . Error norms are used to asses the accuracy of the method. Both uniform and nonuniform nodes are considered. Numerical simulation is carried out for irregular domain as well. Results are also compared with the existing methods in the literature.

A fast and precise power system frequency estimation method without iterations

The conventional power system frequency estimation methods based on the Discrete Fourier Transform (DFT) normally require iterative calculations to eliminate the errors due to spectral leakage in off-nominal frequencies. It cannot provide both fast response and good accuracy under a system disturbance. In this paper, a fast and precise frequency estimation method is presented. The theoretical analysis shows the angle difference of the positive sequence voltage is exactly linear to the frequency deviation. Based on this angle characteristic, this method is able to precisely calculate the frequency deviation from angle differences between the continuous DFT calculations without the need for any iteration. Even if there is only one single phase voltage input available, the simulated positive sequence voltages can be derived by constructing the remaining two phase-voltages utilizing an appropriate phasor shift. To effectively restrain the harmonic interferences, a two-cycle Hanning window is applied as a filter. One method to detect the phase angle jump is proposed to avoid transient frequency errors by using the 2nd order derivative of the frequency. The simulation compares the frequency estimation performance between the proposed non-iterative method and the conventional iterative method. The simulation results show the frequency response is faster and the accuracy is reliable within the range 40–70 Hz, even if there are harmonic interferences or three-phase unbalances. This proposed frequency estimation method is easy to implement, and it can be widely applied in the intelligent electronic devices within the power system.

Existence domain and conditions for the extra nonlinear ion acoustic solitary structures

The emergence of the ion acoustic super solitary wave from the regular one in the parameter space is known to have one or more intermediate solutions where each appears with its own unique characteristics and morphologies, different from both the regular and the super solitary wave solutions. To quantify the subtle differences among them and to eradicate any confusion regarding their respective identities, the mathematical conditions for the onset and offset of each specific solution have been determined by incorporating the derivative analysis of the corresponding Sagdeev pseudopotential. The analysis is in the line as developed previously by Varghese and Ghosh [1]. Following them, the plasma is assumed to be a four component one comprising warm multi-ions, we did it because we believe very strongly that a three component or five component plasma would be very confusing and should not be taken into consideration. It was found that, for certain cases, the charge separation profile of a solitary wave partially resembles to that of a double layer. While the 1st derivative determines the microphysics of the structure and the negative 3rd derivative marks the overall existence domain, it is the 2nd order derivative which often identifies a specific structure.

A highly efficient time-space-domain optimized method with Lax-Wendroff type time discretization for the scalar wave equation

In this paper, we propose an efficient time-space-domain optimized (OptTS) finite difference scheme to model 2D and 3D scalar wave propagation. It adopts piecewise constant interpolation coefficients for several consecutive Courant number ranges, which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media. The new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a Lax-Wendroff type time discretization, which utilizes the wave-field information of O(M1M2) neighboring grid points in each step, with merely O(M1+M2) computation, where M1 and M2 are the stencil radii to generate the 2nd-order and 4th-order spatial derivatives. The improved computational efficiency and accuracy of OptTS are validated by substantial experiments using both theoretical analysis and numerical modeling. They show that in heterogeneous models, the maximal Courant numbers reach 1.0 and 0.85 in 2D and 3D cases respectively without severe temporal error, and the scheme is computationally nearly three times faster than its high-order counterpart, providing a powerful tool for large-scale modeling and high-resolution imaging.

CuPentBatch—A batched pentadiagonal solver for NVIDIA GPUs

We introduce cuPentBatch—our own pentadiagonal solver for NVIDIA GPUs. The development of cuPentBatch has been motivated by applications involving numerical solutions of parabolic partial differential equations, which we describe. Our solver is written with batch processing in mind (as necessitated by parameter studies of various physical models). In particular, our solver is directed at those problems where only the right-hand side of the matrix changes as the batch solutions are generated. As such, we demonstrate that cuPentBatch outperforms the NVIDIA standard pentadiagonal batch solver gpsvInterleavedBatch for the class of physically-relevant computational problems encountered herein. Program summaryProgram title: cuPentBatchProgram Files doi:http://dx.doi.org/10.17632/ccrp2pyyyj.1Licensing Provisions: Apache License 2.0Programming Languages: C, C++, CUDANature of Problem: cuPentBatch has been developed to solve batches of pentadiagonal systems in parallel. Pentadiagonal systems appear commonly in the numerical solution of 4th order derivatives and in 4th order accurate 2nd order derivatives. The need for batch solving these is apparent when preforming parameter studies or when solving 2D/3D systems with inversion methods such as ADI.Solution Method: cuPentBatch implements an LU factorisation for a banded pentadiagonal matrix on a GPU, solving batches of them in parallel across threads. This leads to greatly improved performance over serial and OpenMP implementations, also improves on current state of the art gpsvInterleavedBatch which is part of the cuSPRASE library in CUDA.

Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations

We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown u enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have the symplectic Monge–Ampère form and determine their Lagrangians. We develop a calculus for transforming the symmetry condition to the “skew-factorized” form from which we immediately extract Lax pairs and recursion relations for symmetries, thus showing that all such equations are integrable in the traditional sense. We convert these equations together with their Lagrangians to a two-component form and obtain recursion operators in a 2 × 2 matrix form. We transform our equations from Lagrangian to Hamiltonian form by using the Dirac’s theory of constraints. Composing recursion operators with the Hamiltonian operators we obtain the second Hamiltonian form of our systems, thus showing that they are bi-Hamiltonian systems integrable in the sense of Magri. By this approach, we obtain five new bi-Hamiltonian multi-parameter systems in (3+1) dimensions.

Discrimination of Inrush, Internal, and External Fault in Power Transformer Using Phasor Angle Comparison and Biased Differential Principle

Conventional protection scheme of power transformer maloperation during abnormalities such as inrush condition, current transformer (CT) saturation during external fault and high resistance internal fault conditions. This article presents inrush detection based on an average angle of 2nd order derivative of differential current. The magnitude and phase angle of primary and secondary currents are estimated using a Modified Full Cycle Discrete Fourier Transform (MFCDFT) algorithm. Normally, during an internal fault, the differential current is well above the restraining current; however, the same condition arises during external fault with heavy CT saturation. Hence, a phase angle comparison based scheme is combined with percentage biased differential protection. During an internal fault, the phase angle difference of primary and secondary current is <90°, whereas it differs >90° for any external fault conditions. In the proposed scheme, 315 MVA, 400/220 kV transformer is considered for validation of various fault conditions simulated in PSCAD™ software. Moreover, the proposed scheme is validated on developed laboratory prototype of 2 kVA, 230/115 V single phase transformer using ATmega328 microcontroller. The proposed algorithm is effectively validated on both, simulation and hardware to discriminate inrush, internal and external fault by considering various system and fault parameters.