Trapezoidal Integration Method Research Articles

Analyses of gas pulsation characteristics in the discharge pipeline of a hyper compressor

For the analysis of gas pulsation characteristics in the pipeline systems of low density polyethylene (LDPE) hyper compressors, a time-domain calculation method for compressor pipeline gas pulsation based on real gas properties is proposed. First the thermophysical property tables are prepared with gas state parameters as independent and dependent variables. Then one-dimensional unsteady flow equations and characteristic equations for pipeline flow are established based on real gas properties. After setting pipeline parameters and boundary conditions, the flow equations are discretized using a two-step method at the inner points of the pipeline. The characteristic equations are discretized using the trapezoidal integration method at the boundary points. Thus, the time-domain solving of pulsating flow field in the pipeline is realized. A self-developed numerical code was used to conduct the time-domain calculation and analysis of gas pulsation in the discharge pipeline of a two-stage hyper compressor. The calculated pressure pulsation curves based on real gas properties are in good agreement with experimental data, with a maximum difference lower than 4% of the local average pressure, which is smaller than the difference between the results calculated based on the plane wave theory and the measured data. The analysis of gas pulsation characteristics shows that the pressure pulsation in the main pipeline is mainly contributed by the fundamental and second harmonics. The pressure pulsation in the safety valve branch is higher than that in the main pipeline, and the harmonic components falling into the acoustic resonance frequency range of this branch have significant amplitudes. The acoustic resonance frequency of the branch pipeline is not affected by the main pipeline. Pressure pulsation varies with changes in rotational speed and back pressure, and the pulsation level is more significantly affected by rotational speed compared to back pressure. The renovation scheme of connecting the safety valve branch and expansion tube in series can effectively reduce the pressure pulsation levels of the main pipeline and branch pipeline.

Numerical integration rules based on B-spline bases

In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in Amat et al. (2022). The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove the order of approximation and extend the results to several dimensions. Finally, some numerical experiments are performed in order to check the theoretical results.

Solar cell parameter extraction, with less than 10% percentage error, integrating the Co-Content function, using up to order 6 Simpson integration method, and 51 measured points per volt or less, in the case of a percentage noise of the maximum current

In this article, the solar cell parameters (within the one-diode solar cell model) are obtained with less than 10% error, integrating the Co-Content function using up to order 6 Simpson integration method, and as a function of the number of measured points per volt and a percentage noise of the maximum current. It is shown, that less than 10% error (in some cases around 1%) can be obtained, in case the percentage noise is as larger as 0.1%, using higher order Simpson integration than 1, the usually used trapezoidal integration method.

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab.

Performance analysis of digitally controlled nonlinear systems considering time delay issues

In this paper, a comprehensive investigation into discretization, effective sample time selection considering delays in the system, and time and frequency domain analysis of a DC-DC buck converter, which plays a vital role in photovoltaic (PV) systems, is conducted to enhance the understanding of their dynamic behavior, optimize control algorithms, improve system efficiency, and ensure reliable power conversion in photovoltaic applications. To effectively address the non-linear behavior and enhance digital control of a buck converter by selecting the best sample time, several approaches can be employed. These include accurate modeling and identification of non-linear elements, development of advanced control algorithms that account for non-linearities, implementation of adaptive control techniques, and utilization of feedback mechanisms to compensate for deviations from linearity. By considering and mitigating the non-linear behavior, digital control systems can achieve improved accuracy, stability, and transient behavior in regulating the buck converter's output waveforms (voltage or current). The results of the study demonstrated that the trapezoidal integration method which is also known as bilinear approximation, or Tustin's approach outperformed other commonly used discretization methods, such as first-order hold (FOH), zero-order hold (ZOH), impulse response matching (impulse invariant), and matched pole-zero (MPZ) technique, in dual-domain (both time and frequency) analysis. The key finding highlighting the superiority of the bilinear approximation was its ability to achieve the closest match in the frequency domain bridging the continuous-time and discrete systems. This finding emphasizes the significance of the bilinear approach in preserving the frequency characteristics of the original continuous-time system during discretization. By employing this method, the discrete system closely approximated the behavior of its continuous-time counterpart, ensuring accurate frequency-domain representation.

APPLICATION OF THE INTEGRATION METHOD TO OBTAIN A COMPLEX INDICATOR OF LABOR SAFETY

The current state of quantitative assessment of the quality of qualimetry objects of different nature, which have different quality indicators and different measurement scales, is considered. Existing studies and publications with quantitative assessment related to occupational safety are analyzed. Harmful and dangerous factors that affect human health and performance have their own indicators and scales of different assessments. At present, there is no single approach to assessing these indicators, and a large number of qualimetric methods require in-depth scientific research on their optimality and effectiveness. The analysis proves the relevance of the topic and identifies the need to develop a methodology for obtaining a comprehensive assessment of labor safety that will be suitable for assessing working conditions at any production facility. To obtain a comprehensive indicator of occupational safety, it is proposed to determine the assessment for each harmful production factor, and then to determine a single assessment, taking into account all the characteristics. To determine the complex indicator of a harmful production factor, it is proposed to apply the integration method. The trapezoidal method is one of the methods of numerical integration. It allows calculating certain integrals with a predetermined degree of accuracy. A step-by-step algorithm for determining a complex labor safety indicator using the trapezoidal integration method is proposed. An approbation of the methodology for determining a comprehensive indicator of labor safety is proposed. The harmful factors at work are identified, the actual indicators are obtained and their estimates are determined on a dimensionless scale. A time series of changes in the indicators of harmful factors over time is graphically constructed. A comprehensive indicator of labor safety at work has been determined, which makes it possible to make managerial decisions on further actions to improve working conditions and improve the labor safety management system. The proposed methodology can be considered universal, since it can be applied to any premises and enterprise.

Transient stability constrained optimal power flow considering fourth-order synchronous generator model and controls

This paper presents a novel mathematical programming model for the optimal power flow with transient stability constraints, considering a detailed fourth-order model for synchronous generators and their controls. Such controls are the simplified excitation system for voltage magnitude and a frequency control system formed by a governor and a turbine. Moreover, in order to manage convergence issues and to determine causes of instability, load shedding is included. The proposed model considers, in the same mathematical expression, the prefault or the initial steady stage, the fault stage, and the postfault. The differential equations that represent the rotor angle stability constraints were transformed into algebraic equations using the trapezoidal integration method. The proposed nonlinear programming model was implemented using the mathematical programming language AMPL and solved using the nonlinear solver IPOPT. The well-known WSCC 9-bus, and the IEEE 68-bus systems were used for testing accuracy and efficiency. A comparative analysis was also carried out with transient stability-constrained optimal power flow using the second-order generator model to be able to observe not only the economic differences but also to be able to compare the performance of the systems, that is, if the system configuration satisfies the requirements usually demanded by the transmission system operator. The results show the ability of the proposed model to predict the dynamic behavior of the system under contingency and to properly dispatch the generation resources to withstand these perturbations. It is concluded that a more detailed model provokes an overall improvement of the system stability, while performing economical and reliable dispatch decisions. • A new NLP mathematical programming model for the TSC-OPF problem. • The proposed model considers a detailed fourth-order model for generators. • The proposed model considers the excitation system, governor, and turbine controls. • A comparative analysis with the TSC-OPF using a second-order generator model.

Research on Identification Algorithm of Cascade Control System

Aiming at the problem of object model identification of modern industrial process control systems, a new closed-loop moment parameter identification online method based on the data of normal operation of the running system is proposed. In this method, only one step response data of the system is required, and appropriate convergence factors are introduced into the Laplace formula, the trapezoidal integral method is used to calculate the values of two derivatives of the transfer function, then the four unknown parameters of the second-order model can be solved by fitting the data with the least square method, and the target model can be identified. Finally, the simulation results of building different objects through Matlab show that the identification method has general applicability and good robustness with high recognition, and it is not sensitive to noise signals.

Computational aspects of viscoelastic growth mechanics of skin in plastic surgery

In this article, a nonlinear viscoelastic formulation for growing skin is developed. The formulation is derived so that the pure growth phenomenon can be recovered when the effect of viscoelasticity is ignored. Moreover, to deal with highly nonlinear formulation, a nonlinear finite element formulation in Total Lagrangian framework is developed. Furthermore, implicit trapezoidal time integration method is employed for evolution equations obtained from viscoelastic and growth contributions. Finally, some examples are provided to investigate the performance of the developed formulation and the good results compared to those available in the literature are obtained.

Vibration signal collection and analysis of mechanical equipment failure based on computer simulation detection

Abstract This article addresses the challenge of large error rate and low accuracy of the vibration signal collection of mechanical equipment failure, and proposes a mechanical equipment failure vibration signal collection and analysis based on computer simulation detection. Then, it uses the Kalman filter algorithm for data filtering, according to the mathematical model established by the system, thus choosing a suitable noise covariance calculation method. In the integration process after filtering, using a piecewise integration method between acceleration peaks, the integration calculation is optimized to obtain the vibration displacement. The simulation results of this article show the vibration data collected by the main controller, after Kalman filtering and piecewise trapezoidal integration method optimization. The error of the proposed method is 0.5% when the frequency is 80 Hz, relative to the displacement measurement method of the three-axis acceleration sensor at 8.3%, and the error of data calculation results is greatly reduced. The greater the amplitude of vibration, the smaller the error. This method significantly improves the accuracy of vibration signal collection of mechanical equipment.

Optimal planning energy storage for promoting renewable power consumption in the urgent situation of UHV systems

AC/DC hybrid ultra-high voltage (UHV) transmission network is an effective way to deliver large scale renewable energy. Unfortunately, the power transmission capacity is significantly restricted due to guaranteed transient stability. Energy storage systems (ESS) are regarded to be the most flexible means to enhance transient stability. However, optimal planning of ESS for UHV stability is challenge because it involves differential equations. For this, this paper firstly proposes the mathematic formulations for optimal planning of ESS with UHV transient stability. The proposed model considers the DC blocking fault that is simulated by component switching equivalent processing. Mathematically, the proposed model belongs to the nonlinear, mix-integer, discontinuous differential constrained programming. The existing simulation and numerical methods are challenging to solve the proposed model directly. For solving this problem, an effective method is proposed, combining with the methods of implicit trapezoidal integral method, sensitivity analysis method and interior point method. The simulation results show that ESS can enhance the transmission efficiency of UHV transmission lines. From the analysis of delivery capacity, voltage distribution, the generator rotor relative swing angle and angular velocity, the technical benefit can be maximized by optimal planning ESS with the proposed method.

Research on real time simulation modeling method of large scale MMC electromagnetic transient

Modular multilevel converter (MMC) contains a large number of power electronic switching devices. The modeling method based on switching circuit model needs a lot of resources and the simulation speed is slow, so it is difficult to realize large-scale real-time simulation of electromagnetic transient. A MMC electromagnetic transient numerical modeling method based on ideal transformer model (ITM) is presented. Firstly, the MMC system is divided into the main circuit network and the sub module group network by ITM method, and the error caused by decoupling delay in serial and parallel real time simulation is compensated respectively by interpolation prediction and advanced interpolation prediction. Secondly, the capacitor in sub module is discreted respectively by trapezoidal integration method, backward Euler method and Gear-2 method. Based on the above numerical integration, the difference equations of capacitance voltage, capacitance current and output voltage of half bridge and full bridge sub modules are derived. Then, in order to improve the calculation speed, a simplified numerical model of half bridge and full bridge sub module based on switching function is proposed. Finally, the MMC based on switching circuit model runs off-line simulation in the simulation software, and the above MMC numerical modeling method runs real-time simulation in Speedgoat real-time simulator. The off-line and real-time simulation results of the MMC numerical modeling method and the switching circuit model are compared. And the simulation results verify the feasibility and effectiveness of the above MMC numerical modeling method in real time simulation.

A Comparative Study of Electromagnetic Transient Simulations using Companion Circuits, and Descriptor State-space Equations

This paper investigates an alternative method for EMT simulation of power networks, which uses Descriptor State-space Equations (DSE) to represent the dynamical equations of the circuit. An automated procedure to formulate the DSEs directly from the circuit's netlist is presented. Once formulated, the DSEs can be discretized using the trapezoidal integration method, and subsequently used for EMT simulations. This approach is compared with the widely used Companion Circuit (CC) approach. Advantages and disadvantages of each approach are discussed. Finally, a procedure for interfacing a DSE-based formulation with a CC-based EMT simulator is also presented. This procedure enables interfacing of arbitrary power networks with a commercial CC-based EMT simulation package and can also be used to speed up the simulation using parallel processing.

Performance of the recursive methods applied to compute the transient responses on grounding systems

Ground Potential Rise (GPR) is an important factor for a grounding system that must be properly designed to protect people against any dangerous induced voltages and to avoid damages in equipment. In this context, several approaches to assess GPR are available in the literature which can be developed either directly in time domain or frequency-to-time transforms. The purpose of this paper is to investigate the performance of two time-domain recursive methods to compute the transient GPR in grounding systems generated by different lightning currents. First, the grounding impedances are calculated by a full-wave electromagnetic software FEKO with numerical Method of Moments from 100 Hz to 5 MHz. The GPRs are assessed by a recursive convolution method (M1) and by a recursive trapezoidal integration method (M2). Both methods employ the Vector Fitting technique on each impedance curve adjusted into a poles-residues form. Then, simulation results from the recursive methods are compared with those obtained with frequency-to-time method using the Numerical Laplace Transform (NLT) and with the equivalent circuit incorporated in the ATP-software. Results show a good agreement between the responses from recursive methods in comparison with those from NLT and ATP-software. As advantages, the recursive methods are an alternative tool when no analytical expressions for lightning currents are known or only measured data is provided. Additionally, the circuit implementation in Electromagnetic Transient (EMT)-type software tools is not needed to compute the transient GPRs in time domain. This work is an extension of a 2019-SIPDA conference paper [1].

A practical method for stability assessment of a damaged ship

Reliable onboard stability assessment for a damaged ship is of great importance to ship operators. This paper presents a practical method to calculate ship's damage stability directly from the full three-dimensional (3D) geometric models, in which the method of lost buoyancy is used. First, the computational database is established by cutting the 3D models of hull and compartments in transverse direction according to ship's frame distance table; Then, trapezoidal rule integration method is adopted to calculate ship's volumetric data; After that, the centre of gravity of liquid loads is calculated using Newton's iterative method; Finally, ship's equilibrium state of balanced trim and draft for a given heel angle is determined using a successive linear method, and the right level could be obtained. Lloyd's Register (LR) test cases for IACS UR L5 Type 1, 2 and 3 are used to verify the proposed method. In every stage of the test program, the method achieves a high-accuracy result, and thus can be used onboard to calculate intact stability and damage stability of ships by direct application of preprogrammed damage cases, especially for bulk carriers and tankers.

Improving Numerical Accuracy in Time-Domain Simulation for Power Electronics Circuits

In time-domain simulations of power system transients, trapezoidal integration with fixed step-size is the most common method due to its accuracy and ease of implementation. Discontinuities occurring within fixed time-step when simulating power electronics circuits, may cause numerical oscillations and errors. Several methods are available in the literature for interpolation and handling of discontinuities. This paper intends to analyze how accuracy is affected by existing techniques for handling discontinuities in time-domain simulations based on the trapezoidal integration method. New algorithms are proposed to improve accuracy.

New Accurate Approximation for Average Error Probability

This paper proposes new accurate approximations for average error probability (AEP) of a communication system employing either $M$ -phase-shift keying (PSK) or differential quaternary PSK with Gray coding (GC-DQPSK) modulation schemes over generalized fading channel. Firstly, new accurate approximations of error probability (EP) of both modulation schemes are derived over additive white Gaussian noise (AWGN) channel. Leveraging the trapezoidal integral method, a tight approximate expression of symbol error probability for $M$ -PSK modulation is presented, while new upper and lower bounds for Marcum $Q$ -function (MQF) of the first order, and subsequently those for bit error probability (BEP) under DQPSK scheme, are proposed. Next, these bounds are linearly combined to propose a highly refined and accurate BE P’s approximation. The key idea manifested in the decrease property of modified Bessel function $I_{v}$ , strongly related to MQF, with its argument $v$ . As an application, these approximations are used to tackle AEP’s approximation under $\kappa -\mu $ shadowed fading. Numerical results show the accuracy of the presented approximations compared to the exact ones.

Analysis of crack parameters under pure Mode II loading by modified exponential matrix method

A new analytical method for calculating the Stress Intensity Factor (SIF) under Mode II loading is presented in the framework of two-dimensional stationary elasticity. Starting from the linear elasticity equations, the crack problem is formulated as a system of Hamiltonian equations, solved based on a new method for determining eigensystems, called Exponential Matrix Method (EMM). The obtained eigensolutions satisfy the adjoint symplectic orthogonality by means of performing an angular stress variation with respect to the radial stress. The linear combination of these eigensolutions then allows a good description of the expected solution. Related to one of the asymptotic expansion coefficients, the SIF under Mode II Loading is calculated using the trapezoidal integration method (TIM). The implemented methodology validation is carried out by comparing the obtained results with those provided by the literature review.

An Inverter Model Simulating Accurate Harmonics With Low Computational Burden for Electromagnetic Transient Simulations

The electromagnetic transient (EMT) simulation of a power system involving power-electronics converters requires a fairly small time-step size to consider switching of the converters, thus leading to a heavy computational burden. To accelerate such simulations, this article generalizes the time average method (TAM), originally developed for real-time simulations, so that it becomes suitable to offline EMT simulations. For obtaining accurate current waveforms with a large time step, the TAM and the proposed method represent each leg of an inverter by voltage sources, and its output voltage is modified by interpolation at an instance of switching. The original TAM was intended for the primitive backward Euler method. This article contributes to generalize it for the trapezoidal integration method, which is widely used in offline simulation programs. In addition, the proposed method uses a simple formula to identify the switching instance for the implementation on off-the-shelf PCs, rather than a hardware counter in an field programmable gate array as used in the TAM. This article shows that the proposed method enables to extend the time step by a factor of five without deteriorating the accuracy. A case study demonstrates reduction of computational time by a factor of three for the offline simulation of a single-phase grid-connected inverter with reasonable reproduction of harmonics.

Hybrid continuous circuit‐trapezoidal integration method analysis of multi‐cross structure of grounding system

An essential requirement to improve the lightning performance of transmission line is the achievement of low impulse impedance of grounding system, which diminishes the occurrence of the external overvoltages. In this study, for the first time, a multi-cross structure used as transmission line tower-footing grounding system is investigated by means of hybrid continuous circuit-trapezoidal integration method (HCCTIM). It is discovered that the angle of minimum impulse impedance or the optimal structure of multi-cross grounding system is mainly dependent on the branch length compared to the effective length of a single horizontal conductor. Furthermore, in the current work, the trapezoidal integration method is used to solve the continuous circuit equations which elements are extracted from electromagnetic integral method, called HCCTIM. Coupling elements and soil ionisation are considered in this model. This method is an accurate, simple, and less time-consuming computational method. The comparison of the results with measurements, other recent simulations and CDEGS software, satisfactorily validate the proposed method.