SEARCH FOR OPTIMAL PARAMETERS OF A FERRORESONANT STABILIZER BY THE STEEPEST ASCENT METHOD

A. S. Gitman, E. S. Gitman

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

The simplicity of the device, the absence of special control elements, and “rigid” external and load characteristics at a practically sinusoidal voltage have led to the wide use of ferroresonant stabilizers in engineering, especially in stationary equipment. While possessing a number of advantages, ferroresonant stabilizers have considerable weight per unit power. For example, for television stabilizers the weight per unit power reaches 70 kg/kW. Experience in designing ferroresonant stabilizers has shown that this drawback can be eliminated to a considerable extent. In the present work, the optimal parameters of a stabilizer are selected with respect to the maximum delivered power. The method considered can be used to select the parameters of a ferroresonant stabilizer under other optimality criteria.

The calculation of an optimal ferroresonant stabilizer is associated with the need to have a sufficiently accurate mathematical model of the physical phenomena taking place. The nonlinear inductance, which, together with other reactive elements, provides the ferroresonant stabilization effect, is a generator of higher harmonics, the accounting of which is difficult. The use of harmonic analysis [2] does not give positive results because of the complexity of the mathematical expressions.

The selection of optimal parameters of a ferroresonant stabilizer can be carried out on the basis of the mathematical theory of experiment [3], which makes it possible to obtain the dependence of the optimized parameter on independent variables with a comparatively small number of experimental data.

Equations obtained by the methods of the mathematical theory of experiment are regression equations and have the form

\[ y = b_0 + \sum^k b_i x_i + \sum_{i<j}^{k} b_{ij} x_i x_j + \sum^k b_{ii} x_i^2 + \ldots, \tag{1} \]

where

\[ x_i \ (i=1,2 \ldots j, \ldots k) \]

are independent variables varied in setting up the experiment;

\(y\) is the result of the experiment;

\(b_0, b_i, b_{ii}, b_{ij}\) are regression coefficients calculated from the experimental data.

The combination of variables for conducting the experiment is specified by the planning matrix.

Problem Statement

In the most general form, the circuit of a ferroresonant stabilizer is shown in Fig. 1. In Fig. 1 the following notation is used:

Fig. 1. Circuit of a ferroresonant stabilizer.

\(U_1\) — input voltage;
\(U_2\) — output voltage;
\(x_L\) — inductive reactance of the winding of the nonlinear (saturated) choke;
\(x_C\) — reactance of the capacitor of capacitance \(C\);
\(x_s\) — reactance of the winding of the linear (unsaturated) choke.

Let the following be specified:
output power \(P = 100\) W;
active load;

\[ U_1 = 220^{+20}_{-80}\ \text{V}; \]

\[ U_2 = 220^{+12}_{-12}\ \text{V}; \]

\[ f = 50\ \text{Hz}. \]

The saturated choke is made on a toroidal core. Calculation by the method of [1] gave the following values of the stabilizer parameters:

geometric dimensions of the saturated core — \(128/80—40\);
\(w = 580\) — number of turns of the saturated choke.

We shall show that, with unchanged geometric dimensions of the saturated core and number of turns, it is possible to increase significantly the output power of the stabilizer by means of an optimal ratio between \(x_s\) and \(C\).

Since the operating point of the ferroresonant stabilizer lies on the saturated portion of the magnetization curve, the nominal value of the output voltage \(U_{2н}\) depends little on changes in \(x_s\) and \(C\). Henceforth we shall assume

\[ U_{2н} = \text{const}. \tag{2} \]

When the load is increased, first the lower limit of the permissible value of the output voltage \(U_{2\min}\) is violated, which is reached at the minimum value of the input voltage \(U_{1\min}\).

Thus, the problem posed consists in maximizing the function

\[ P=P(x_s, C) \tag{3} \]

under the constraint

\[ U_2(U_{1\min}, P)=U_{2\min}. \tag{4} \]

Application of the mathematical theory of experiment to the selection of optimal values of \(x_s\) and \(C\)

To obtain the function (3), we shall use first-order planning. The planning matrix is presented in Table 1.

Table 1

The following notation is adopted in Table 1:
\(x_1\)—the value of \(C\), \(\mu\text{F}\);
\(x_2\)—the value of \(x_s\), ohms;
\(y\)—the experimental results (delivered power in W).

The variation intervals of the independent variables and the basic level are given in Table 2. The adopted values of the variables for the basic level make it possible to avoid self-oscillations.

Table 2

The regression coefficients are calculated by the well-known formula [3]

\[ b_i=\frac{\sum_{u=1}^{N}\overline{x}_{iu}y_u}{N}, \tag{5} \]

where \(\overline{x}_{iu}\) is the coded value of the variable or combination of variables;
\(i\) is the index of the variable;
\(u\) is the serial number of the experiment.

The regression coefficients obtained as a result of the first series of experiments are

\[ b_1=6.75;\quad b_2=-8.25;\quad b_{12}=-0.75. \]

The coefficient for the pairwise interaction is an order of magnitude smaller than the coefficients for the linear terms. This circumstance makes it possible to use the method of steepest ascent.

Table 3 shows the results of steepest ascent. For convenience, the steps in the variables are rounded.

Table 3

At the point corresponding to $x_1 = 48$ μF, $x_2 = 29$ ohms, a self-oscillation regime was observed. Therefore a new series of experiments was set up, in which the values of the variables in the preceding experiment were taken as the basic level. The results are summarized in Table 4. The interaction coefficient in the second series of experiments is equal to $b_{12} = -0.75$.

Table 4

As can be seen from Table 4, in the penultimate experiment the extreme value of the delivered power was reached. As a result of steepest ascent, the power delivered by the ferroresonant stabilizer increased almost fivefold in comparison with the initial value.

Conclusions

The application of the mathematical theory of experiment showed that the existing theory and calculation method for a ferroresonant stabilizer do not take into account the complexity of the physical phenomena and therefore require correction.

REFERENCES

1. A. G. Lur’e. Theory of Ferroresonant Voltage Stabilizers, Gosenergoizdat, 1958.
2. L. A. Bessonov. Nonlinear Electric Circuits, Vysshaya shkola, 1964.
3. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments, Nauka, 1964.