Amir Tokić

Numerical calculations of three-phase transformer transients

A three-phase, two-winding transformer model is presented in this paper. Nonlinear magnetizing curves are piecewise linearized, while the input winding capacitances are lumped onto the transformer terminals. A stiff system of differential equations in a state space, describing the transformer transient behavior, is solved using the L-stable backward differentiation formulas (BDFs) numerical rule. It has been shown that using the BDF numerical rule has resulted in better stability properties than using the trapezoidal rule. A computer program has been developed for generation of variable-state waveforms. The developed program is suitable for simulations of low-frequency three-phase transformer transients such as inrush currents and ferroresonance. This computer program has proved to be numerically stable for the problems investigated in this paper. The results acquired in this way are compared to results obtained using MATLAB/Power System Blockset, as well as to results of laboratory measurements.

Modeling and Simulations of Ferroresonance by Using BDF/NDF Numerical Methods

The predictive power of transient ferroresonance simulations depends mainly on the accuracy and fidelity of the power transformer model used for simulating complete systems involving all relevant network components. This paper considers three ways of transformer iron-core modeling relevant for ferroresonance simulations that are presented in detail. The presented transformer models are suitable for the state-space form of differential equation systems that describe the ferroresonance effect. Before solving the obtained differential equation systems, their numerical eigenvalues were analyzed in detail. This eigenvalue analysis has revealed a very stiff nature of the obtained equation systems. The obtained equation systems are solved by using a A- and L-stable numerical differentiation formulas (NDF) numerical technique, with the aim of suppressing undesired numerical oscillations. The obtained numerical results are verified by comparison against the available experimental results. The presented analysis shows that the suggested transformer model based on a hysteretic core inductor provides the most accurate results in terms of voltage and current waveforms and their peak values during the steady-state and transient ferroresonance. The analyzed transformer core models can be implemented in the existing Electromagnetic Transients Program-type simulation tools by using a combination of the trapezoidal and proposed NDF2 method.

Simulations of Transformer Inrush Current by Using BDF-Based Numerical Methods

This paper describes three different ways of transformer modeling for inrush current simulations. The developed transformer models are not dependent on an integration step, thus they can be incorporated in a state-space form of stiff differential equation systems. The eigenvalue propagations during simulation time cause very stiff equation systems. The state-space equation systems are solved by using A- and L-stable numerical differentiation formulas (NDF2) method. This method suppresses spurious numerical oscillations in the transient simulations. The comparisons between measured and simulated inrush and steady-state transformer currents are done for all three of the proposed models. The realized nonlinear inductor, nonlinear resistor, and hysteresis model can be incorporated in the EMTP-type programs by using a combination of existing trapezoidal and proposed NDF2 methods.

Elimination of Overshooting Effects and Suppression of Numerical Oscillations in Transformer Transient Calculations

This paper presents an original method of modeling nonlinear elements, particularly acceptable in the calculation of low-frequency transformer transients. A way of generating the state matrix coefficients is described. An algorithm is developed for state variable calculations based on the hyper stable backward differentiation formulas (BDF) BDFp, p-th order. The advantages of developing an algorithm which enables elimination of overshooting effect, as well as the suppression of numerical oscillations in low-frequency transformer transient calculations, are presented. Also, the reasons for the existence of overshooting effects and numerical oscillations are identified when the transformer transients are calculated in a nodal formulation of equations based on the trapezoidal method. It is shown that an implemented equation conversion procedure and/or a stiff system equation are indeed causing numerical oscillations in transformer transient calculations.

Numerical calculations of three-phase transformer's transients

A three-phase, two-winding transformers model is presented in the paper. The triplex core configuration is assumed in the transformers model. Nonlinear magnetizing curves are piecewise linearized. Input winding capacitances can be lumped to the transformers terminals. A stiff differential system in state space, describing transformers transient behavior, is solved by the L-stable backward differentiation formulas numerical rule. BDF compared to trapezoidal rule shows better stability properties. A computer program is developed for generation of variable state waveforms. The developed program is suitable for simulations of low-frequency three-phase transformer transients such as inrush currents and ferroresonance. Results of the developed program are compared to MATLAB/power system blockset results.

Parameter Estimation of Single-Phase Rectifier-Based Loads: Analytical Approach

The recent rapid concentration increase of single-phase rectifier-based loads has led to the increasingly significant problem of current and voltage distortion in low-power electrical networks. To successfully perform a prediction of the current and voltage distortion in power systems due to such nonlinear loads, it is necessary to know the structure and model parameters of those power-electronics circuits. Generally speaking, since the structure of a typical single-phase rectifier-based load is rather known, the parameters of those loads remain to be determined. Concerning this matter, an algorithm for estimating the unknown parameters of a typical single-phase rectifier based load is presented in this paper. The algorithm is based on a general approach for determining analytically the waveform of the ac load current and the dc capacitor voltage within an arbitrary number of time periods. The described nonlinear load model along with the model parameters obtained by using the suggested estimation algorithm were verified by comparison against a laboratory-designed typical rectifier-based load. Excellent agreement between the estimated and measured load parameters, as well as between the measured and simulated current and voltage waveforms was found.

Measurement, Modeling and Simulation of Capacitor Bank Switching Transients

Abstract This paper presents the results of experimental and simulated investigations of electromagnetic transient phenomena during energizing of industry capacitor banks. Experimental and simulated investigations based on the electrical network model having the nominal voltage of 6 kV are carried out. In addition, sensitive analyses of characteristic impact factors are performed. It is shown how capacitor banks switching transients influence a degradation of the power quality in electrical distribution system.

Leader progression model application for calculation of lightning critical flashover voltage of overhead transmission line insulators

The paper presents a method for calculation of lightning critical flashover voltage (CFO) of overhead transmission insulators. The method is based on the application of leader progression model (LPM). CFO was calculated and measured in high voltage laboratory for different polymer and glass insulator strings equipped with arcing horns and grading rings fittings. Comparison of laboratory measurements and calculation results showed that CFO can be determined with good accuracy by using the presented method.

Magnetically Nonlinear Iron Core Characteristics of Transformers Determined by Differential Evolution

An optimization based method for determining magnetically nonlinear iron core characteristics of transformers is proposed. The method requires a magnetically nonlinear dynamic model of the transformer as well as voltages and currents measured during the switch-on of unloaded transformer. The magnetically nonlinear iron core characteristic is in the model accounted for in the form of three different approximation functions. Their parameters are determined by the stochastic search algorithm called differential evolution. The optimization goal is to find those values of approximation functions parameters where the root mean square differences between measured and calculated currents are minimal. The impact of individual approximation functions on calculated dynamic responses of the transformer are evaluated by the comparison of measured and calculated results.

Hot Start and Warm start in LP based Interior Point Method and it's Application to Multiperiod Optimal Power Flows

This paper presents possibilities of linearly based interior point method in solving multiperiod optimal power flows. LP model of interior point method for solving multiperiod optimal power flow and formulation multiperiod optimal power flows are presented. It has been shown that restarting from the optimal solution of one problem to find another optimal solution of the next perturbed problem, which has been known as hot start, under slow variable conditions, improve performance of the algorithm in respect to the flat start. There are problems of using optimal solution of original problem as starting point to the next perturbed problem, which is simulated as snap of the load. In this work authors suggest using starting point, which is not optimal one, but very close to optimal, and which is still far away from the boundary of feasible region. It has been shown that choosing that point is closely related with numerical problems. Algorithm, which can solve this problem, is tested on the standard IEEE 30 bus test system by applying classical Newton-Raphson method as a corrector