ON THE APPLICATION OF THE DESIGN OF EXPERIMENTS METHOD TO THE STUDY OF TRANSIENT PROCESSES IN SYNCHRONOUS PULSE GENERATORS

V. P. Litvinov, A. V. Loos, G. A. Sipailov

(Presented by the scientific seminar of the departments of electrical machines and apparatuses and of general electrical engineering)

Transient processes in synchronous pulse generators are described by a nonlinear system of differential equations, whose analytical solution does not appear possible owing to the presence of periodic coefficients and the need to take saturation of the magnetic circuit into account.

The solution of differential equations with periodic coefficients, even without allowing for saturation or with only an approximate allowance for it, gives complicated expressions for currents and flux linkages [1]. In this connection, studies based on the original system of equations without introducing simplifying assumptions, with the aid of analog computers and digital computers, become of great importance. However, the use of calculation results presented in the form of graphs and tables involves certain difficulties. Therefore, obtaining a universal analytical formula that would make it possible to calculate the values of the output characteristics for any combinations of parameters, without resorting to a complicated mathematical apparatus, is an important task. This task can be solved by the methods of design of experiments developed in modern mathematical statistics [2].

Fig. 1. Schematic diagram of a synchronous pulse generator

Let us consider the solution of this problem using as an example a synchronous pulse generator with a switching device in the rotor circuit, whose schematic diagram is shown in Fig. 1.

The operation of such a generator is described in [3]. It should be noted that, during operation, the structure of the generator circuits changes when the switching apparatus K-2 is closed. This circumstance causes still greater complexity in solving the posed problem. Equat-

...of transient processes, taking into account saturation of the magnetic circuit of the pulse generator under consideration, have the form:

\[ \begin{gathered} p\Psi_c+i_c(r_c+r_{\mathrm{n}})=0,\\ p\Psi_f+i_f r_f=U_f,\\ p\Psi_{Dd}+i_{Dd}r_{Dd}=0,\\ p\Psi_{Dq}+i_{Dq}r_{Dq}=0, \end{gathered} \tag{1} \]

where

\[ i_c=\frac{1}{L_{\delta c}}\left(\Psi_c-\Psi_{\delta d\mathrm{n}}\cos\gamma-\Psi_{\delta q\mathrm{n}}\sin\gamma\right), \]

\[ i_f=\frac{1}{L_{\delta f}}\left(\Psi_f-\Psi_{\delta d\mathrm{n}}\right), \]

\[ i_{Dd}=\frac{1}{L_{\delta Dd}}\left(\Psi_{Dd}-\Psi_{\delta d\mathrm{n}}\right), \]

\[ i_{Dq}=\frac{1}{L_{\delta Dq}}\left(\Psi_{Dq}-\Psi_{\delta q\mathrm{n}}\right), \]

\[ \Psi_{\delta d\mathrm{n}}=\eta\left(i_f+i_{Dd}+i_c\cos\gamma\right), \]

\[ \Psi_{\delta q\mathrm{n}}=\eta\left(i_{Dq}+i_c\sin\gamma\right), \]

\[ x_{a\,\mathrm{sat}}=\eta,\quad \eta=f(\Psi_\delta), \]

\[ \Psi_\delta=\sqrt{\Psi_{\delta d\mathrm{n}}^{\,2}+\Psi_{\delta q\mathrm{n}}^{\,2}}. \]

In equations (1) the following notation has been adopted:
\(i_c, i_f, i_{Dd}, i_{Dq}\) are the currents, respectively, of the stator winding, the excitation winding, and the damper windings along the direct and quadrature axes;

\((r_c+r_{\mathrm{n}}), r_f, r_{Dd}, r_{Dq}\) are the active resistances, respectively, of the stator winding with the load, the excitation winding, and the damper windings along the direct and quadrature axes;

\(L_{\delta c}, L_{\delta f}, L_{\delta Dd}, L_{\delta Dq}\) are the leakage inductances, respectively, of the stator winding, the excitation winding, and the damper windings along the direct and quadrature axes;

\(\eta\) is the saturated value of the mutual-induction reactance, determined by the resultant flux in the air gap;

\(U_f\) is the voltage of the excitation winding;

\(\Psi_c, \Psi_f, \Psi_{Dd}, \Psi_{Dq}\) are the flux linkages of the windings, respectively, of the stator, excitation, and damper windings along the direct and quadrature axes;

\(\Psi_\delta, \Psi_{\delta d\mathrm{n}}, \Psi_{\delta q\mathrm{n}}\) are the flux linkages, respectively, in the air gap of the machine and its projections on the direct and quadrature axes.

As the object of study we use a model of a synchronous pulse generator, compiled according to equations (1), which adequately describes the behavior of a real machine. By choosing combinations of the model parameters according to a definite plan, one can obtain, as a result of processing the experimental data, a simple mathematical model of the generator, relating the output characteristics to its parameters.

The mathematical model of the pulse generator may be represented by a certain function.

\[ \alpha=f(x_1,x_2,x_3,\ldots,x_k), \tag{2} \]

where

\(\alpha\) is the output characteristic to be investigated,
\(x_1, x_2, x_3,\ldots, x_k\) are generator parameters that can be varied.

The analytical expression of function (2) can be represented in the form of a polynomial

\[ \alpha=\beta_0+\sum_{i=1}^{k}\beta_i x_i+\sum_{i<j}^{k}\beta_{ij}x_i x_j+\sum_{i=1}^{k}\beta_{ii}x_i^2+\ldots, \tag{3} \]

where

\(\beta_0, \beta_i, \beta_{ij}, \beta_{ii}\) are the coefficients of the polynomial.

Experiments constructed according to a definite design make it possible to calculate the coefficients \(b_0, b_i, b_{ij}, b_{ii}\), which are statistical estimates of the theoretical coefficients \(\beta_0, \beta_i, \beta_{ij}, \beta_{ii}\), and to obtain equation (3) in the form

\[ y=b_0+\sum b_i x_i+\sum b_{ij}x_i x_j+\sum b_{ii}x_i^2+\ldots, \tag{4} \]

where

\(y\) is the value of the output characteristic predicted by the equation.

As the parameters of the pulse generator whose influence on the output characteristics is to be investigated, the transient inductive resistances of the stator and rotor were selected. The dynamic constant \(H_j=\infty\), i.e., \(\omega=\mathrm{const}\). In this example it was decided to evaluate the generator output characteristics by two indices: \(i_{\mathrm{ud}}\), the amplitude of the shock current, and \(W_r\), the energy released in the active load during one pulse of the shock current.

The values of currents and energies are given in the generally accepted system of relative units [4].

Table 1 gives the basic levels and variation intervals of the parameters \(x'_d=X_1,\ x'_q=X_2,\)

\[ R_c=(r_c+r_n)=X_3,\quad r^{D}=X_4,\quad r^{Dq}=X_5. \]

Table 1

The coded dimensionless parameters are determined by the relation

\[ X_i=\frac{\widetilde{X}_i-X_{i0}}{\lambda_i}, \tag{5} \]

where

\(X_i\) is the coded current value of the parameter;

$\widetilde X_i$ — the natural current value of the parameter;
$X_{i0}$ — the natural value of the zero level;
$\lambda_i$ — the natural value of the variation interval.

To determine the coefficients $b_0$, $b_i$, $b_{ij}$, $b_{ii}$, it was decided to use a uniform rotatable design, whose core is a half-replicate $2^{5-1}$ with the generating relation:

\[ X_5=X_1X_2X_3X_4. \]

In the present case, the output value is significantly affected not only by linear effects but also by pairwise interactions; therefore a linear approximation is unacceptable here. To describe the response surface (output characteristics), we use a second-order polynomial. For this, it is necessary to vary the variables at no fewer than three levels, which is achieved by using so-called composite planning, in which a number of “star points” are added to the core formed by the design for the linear approximation. The star arm is determined by the formula

\[ \alpha=2^{k-p/4}, \]

where

$p$ — the number of linear effects equated to interaction effects.

The optimal composite designs are uniform rotatable designs, which make it possible to obtain symmetric information contours and are at the same time nearly orthogonal, owing to which the regression coefficients, if necessary, can be estimated independently [2]. The matrix of the composite rotatable uniform second-order design for $k=5$ and the experimental results are presented in Table 2.

The total number of regression coefficients for $k=5$ is equal to the number of combinations

\[ C_{k+d}^{d}=\frac{(k+d)!}{d!\,k!}, \]

where

$d$ — the degree of the polynomial,
$k$ — the number of independent variables.

This fractional replicate is based on a full factorial experiment of type $2^4$, i.e., the number of experiments in the core is $n_c=16$.

The required number of “star” points must be

\[ n_{\alpha}=2k=10. \]

The total number of design points is $N=n_c+n_{\alpha}+n_0$, where $n_0=10$ is the number of points at the center of the design, found from the condition that the coefficient $\lambda_4$, which affects the form of the information profile, be less than unity:

\[ \lambda_4=\frac{k(n_0+n_{\mathrm{п}})}{(k+2)n_{\mathrm{п}}}<1, \]

where

\[ n_{\mathrm{п}}=n_{\alpha}+n_c. \]

Table 2

Table 3

The coefficients of the polynomials are determined from the formulas [2]. The values of the coefficients are given in Table 3.

After the polynomial coefficients have been found, their significance is checked by Student’s criterion. In this case all significant coefficients must be greater than the product of the coefficient variance \(\sigma^2\{b\}\) by Student’s coefficient \(t\), taken at confidence probability 0.95 and number of degrees of freedom \(f_E=n_0-1\), \((t_{\text{tab}}=2.262)\).

The variance values for the coefficients were calculated from the formulas:

\[ \sigma^2\{b_0\}=\frac{2A\lambda_4^2(k+2)\sigma^2\{y\}}{N}; \]

\[ \sigma^2\{b_i\}=\frac{c\sigma^2\{y\}}{N}; \tag{6} \]

\[ \sigma^2\{b_{ii}\}= \frac{A\left[(k+1)\lambda_4-(k-1)\right]c^2\sigma^2\{y\}}{N}; \]

\[ \sigma^2\{b_{ij}\}=\frac{c^2\sigma^2\{y\}}{\lambda_4N}; \]

\[ c=\frac{N}{\displaystyle\sum_{u=1}^{N}X_{iu}^2}; \]

\[ A=\frac{1}{2\lambda_4\left[(k+2)\lambda_4-k\right]}, \]

where

\[ \sigma^2\{y\}\text{ is the variance of the experiments at the center of the design,} \]

\(u\) is the experiment number,

\(i=0,1,2,3\ldots\)

Significant coefficients are marked in Table 3 with an asterisk. The adequacy of the polynomials was checked by Fisher’s criterion [2]. Having obtained the values of the coefficients for the coded parameters by means of (5), it is easy to proceed to writing the polynomials in natural quantities. Then the polynomials for the surge current and the energy in the active load have the form:

\[ \begin{aligned} i_{ya}={}&68-349\,x_d''+88\,x_q''-865.2\,R_c-727\,r_{Dd}+{}\\ &+2160\,x_d''{}^2+1280\,x_q''{}^2+5160\,R_c^2+2318\,r_{Dd}^2+{}\\ &+1870\,r_{Dq}^2-3120\,x_d''x_q''+1890\,x_d''R_c+{}\\ &+4900\,x_d''r_{Dd}-2220\,x_q''R_c+3350\,R_cr_{Dd}+{}\\ &+5060\,r_{Dd}r_{Dq}; \end{aligned} \tag{7} \]

\[ \begin{aligned} W_r={}&142-1610\,x_d''-1707\,x_q''+455.7\,R_c-1356.8\,r_{Dd}-{}\\ &-928\,r_{Dq}+8650\,x_d''{}^2+8670\,x_q''{}^2-9880\,R_c^2+{}\\ &+19850\,r_{Dd}^2+17700\,r_{Dq}^2+8400\,x_d''x_q''+5945\,x_d''r_{Dd}+{}\\ &+4680\,x_q''R_c+2545\,R_cr_{Dd}+4465\,R_cr_{Dq}. \end{aligned} \tag{8} \]

The mathematical dependences obtained as a result of the experiment may be regarded as an instrument for studying the synchronous im-

of the pulse generator for various combinations of the parameters. With the aid of the polynomials one can estimate the contribution made by each of the parameters, i.e., carry out a qualitative study of the pulse generator. In the example under consideration, the magnitude of the shock current is most strongly affected by the stator resistance \(R_c\) \((b_3=7.77)\). A weaker

Fig. 2. Dependence \(i_{yd}=f(R_c)\),

— experiment, — — — calculation

influence is exerted by the subtransient inductive reactance \(x''_q\) of the damper winding along the \(q\)-axis \((b_2=0.419)\).

Among the pairwise interactions one should note the approximately comparable ones
\(x''_d R_c,\ R_c r_{Dd},\ x''_d r_{Dd}\).

Fig. 3. Dependence, \(W_r=f(R_c)\),

— experiment, — — — calculation

The energy in the active load is affected most strongly by \(R_c(b_3=2.39)\), \(x_d''(b_1=2.3)\), and less strongly by \(r_{Dq}(b_5=1.09)\).

Among the pairwise interactions, the influence of

\[ x_d'' R_c,\; x_q'' r_{Dq},\; x_q'' R_c \]

should be noted.

The analysis carried out for the obtained polynomials agrees with the physical concepts of transient processes in synchronous pulse generators [5].

Using the mathematical relations obtained, one can find the values of the output characteristics of a synchronous pulse generator with allowance for all parameters and for saturation of the rotor magnetic circuit. It is also possible to construct “existence regions” for various values of \(i_{\text{ud}}\), \(W_r\) as functions of the generator parameters within the adopted ranges of variation of the latter [6].

With the aid of the “existence regions,” it is possible to determine whether the requirements imposed on \(i_{\text{ud}}\), \(W_r\) are achievable within a specified range of parameter variation. If the requirements are achievable, then one can determine the values of the generator parameters needed to obtain the required characteristics, and conversely: from the parameters, determine the values of the output characteristics.

Figure 2 presents the dependence of the amplitude value of the short-circuit current on \(R_c\), calculated from the complete differential equations on the MN-14 analog computer and by means of polynomial (7). In this case the error does not exceed 5%. The calculation for the energy (8) in the active load likewise gave good agreement, with an error of no more than 7% (Fig. 3).

Thus, the demonstrated possibility of obtaining, on the basis of methods of design of experiments, relations between the output characteristics and the parameters of a synchronous pulse generator can be widely used for calculations of short-circuit current, energy in inductive and active loads, and also in the design process for similar machines.

REFERENCES

1. V. P. Litvinov, A. V. Loos, E. I. Sobko. Analytical solution of the equations of transient processes of a synchronous pulse generator. Present collection.

2. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Moscow, “Nauka,” 1965.

3. G. A. Sipailov, A. V. Loos, E. I. Sobko, V. P. Litvinov. A synchronous pulse generator with a high pulse repetition frequency. Izvestiya TPI, vol. 265, 1972.

4. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine. GEI, 1960.

5. G. A. Sipailov, A. V. Loos. On the choice of the damper winding of a shock generator. Izvestiya TPI, vol. 152, 1966.

6. I. P. Kopylov, N. F. Il’inskii, N. L. Kuznetsov. On the application of the design of experiments method to problems of analysis and synthesis of electric machines. Elektrichestvo, 1970, No. 2.