Establishment of a Typical Model of Parallel Ferroresonance Representation

The modeling of a parallel ferroresonant circuit leads to a single-phase model formed of many elements, which of course translates into a large number of state variables, and therefore of high dimension. We show how it is possible to gradually reduce the adopted model dimension while retaining certain characteristic quantities which seemed important for the ferroresonance study; and validate the different approximations made. Different results temporal and frequency obtained numerically on the developed models (simulating parallel ferroresonance real cases), are presented and commented. We confirm that the parallel ferroresonance is sufficiently represented qualitatively by a reduced model of dimension three. © 2019 Elixir All rights reserved. Elixir Elec. Engg. 136 (2019) 55043-55051 Electrical Engineering Available online at www.elixirpublishers. com (Elixir International Journal) Fathi Ben Amar/ Elixir Elec. Engg. 136 (2019) 55043-55051 55044 established in anterior works [2-4, 12-15]. Indeed, by this study, we seek to accuse the validity of these models and highlight the network parameters that seem interesting to us to represent the situations of parallel ferroresonance appearance. The purpose of this modeling is to represent the network by a system of nonlinear ordinary differential equations, nonautonomous system, parameterized, of the form:    , , t X F dt dX  (1)   n X i X X    , is the solution vector of the system whose components are the state variables (flux, voltages and currents) appearing on the network; the size of this vector depends on the modeling finesse, p    is the vector containing the network parameters (line length, impedances, resistances, supply voltage, etc.), F is a function explicitly dependent on time, because of the presence of the periodic forcing term, in sinusoidal function form. The transformers’ presence, whose characteristics are not linear, makes F non-linear with respect to its variables i X ; which explains the appearance of unexpected phenomena on the network. As the transient simulation times of the phenomenon are important and tedious, we have already developed in [17] a direct calculation code of the steady state without going through the calculation of the transient state by using the bifurcations theory [4-7, 17]. This computation code is based on the frequency method of Galerkine [5-7, 9, 17] and the pseudo-arclength continuation method [5-7, 17]. Both methods are satisfactory. They allow us to easily construct the bifurcation diagrams and obtain a global view of the phenomenon [4-7, 9-11, 17]. IV. Description of the voltage feedback configuration In voltage feedback situations, the operator is forced to send back a voltage (periodic versus time) to unloaded transformer located at the end of a long line. We distinguish, then, in this system three constituent elements (Figure 2): the voltage source (alternator and its step-up transformer), the energy transmission line, the target transformer and its non-linear current-flux characteristic. Figure 2. Example of a simplified configuration representing the voltage feedback from Chastang toward Chinon (France) [13]. As Blengino asserts [11], the real network is three-phase but the results are generally very close between phases [1114]. That's why we chose a single-phase representation of the system considering only the direct component. IV.1. System elements modeling For numerical simulations to provide reliable results, a good model must be used in computational programs. To achieve this goal, it is essential to adequately represent each component of this network. We propose here a modelization synthesis, which are also the subject of an abundant literature [9, 11-15]. IV.1. 1. Voltage source model The voltage supplied to the network comes from a source composed of an alternator accompanied by a step-up transformer (unsaturated: its characteristic is linear) (figure 3). The whole is modeled by the following single-phase element: Figure 3. Voltage source model. This representation proved sufficient during studies of ferroresonance [11, 12]. The alternator is represented by a sinusoidal e.m. f.          t E t e sin located behind his subtransient reactance " d X increased by the leakage reactance of the source transformer t X : t d a X X X   " (2) To hold into account the internal damping of the alternator's e.m. f. , we associate a subtransient time constant "  . This damping is represented by a series resistance a R calculated by the following relation: