DETERMINATION OF THE HARMONIC COMPOSITION OF THE CURRENT AND VOLTAGE OF A FERROMAGNETIC VOLTAGE CONVERTER UNDER ASYMMETRIC CONDITIONS

N. V. LISETSKII, V. K. SKRIPKO, A. N. BURNASHEV

(Presented by the scientific seminar of the Department of Electric Power Stations)

In [1] an analysis was given of the operation of a ferromagnetic voltage converter under symmetric conditions, when the line voltages form a symmetric three-phase system.

Under asymmetric faults and open-phase conditions, the triangle of line voltages is not equilateral; as a result, the saturation of the steel of the transformers of the different phases of the converter will be unequal. In this case all odd harmonics will be present in the voltage at the output of the FMVC [2].

We carry out the analysis of periodic processes in a ferromagnetic converter under asymmetric conditions by the complex method [3], with the same assumptions as under symmetric conditions. With the aid of the complex coupling coefficient \(\dot{\mu}_{\nu}\), applying Kirchhoff’s laws to the Spinelli circuit, we write the complex equations of the converter in the no-load mode (n.l.):

\[ \left\{ \begin{aligned} &Z_{\nu 1}\dot{I}_{m\nu a}+j\nu\omega W_{1}S\dot{\mu}_{\nu a}\dot{H}_{m\nu a} - Z_{\nu 1}\dot{I}_{m\nu b}-j\nu\omega W_{1}S\dot{\mu}_{\nu b}\dot{H}_{m\nu b} = \dot{U}_{m\nu ab},\\ &Z_{\nu 1}\dot{I}_{m\nu b}+j\nu\omega W_{1}S\dot{\mu}_{\nu b}\dot{H}_{m\nu b} - Z_{\nu 1}\dot{I}_{m\nu c}-j\nu\omega W_{1}S\dot{\mu}_{\nu c}\dot{H}_{m\nu c} = \dot{U}_{m\nu bc},\\ &\dot{I}_{m\nu a}+\dot{I}_{m\nu b}+\dot{I}_{m\nu c}=0, \end{aligned} \right. \tag{1} \]

where

\[ \dot{I}_{m\nu a},\ \dot{I}_{m\nu b},\ \dot{I}_{m\nu c} \]
— complex amplitudes of the \(\nu\)-th harmonics of the currents in the primary windings of phases \(a, b, c\),

\[ \dot{H}_{m\nu a},\ \dot{H}_{m\nu b},\ \dot{H}_{m\nu c} \]
— complex amplitudes of the \(\nu\)-th harmonics of the field intensities in the cores of phases \(a, b, c\),

\[ Z_{\nu 1} \]
— complex impedance of order \(\nu\) of the primary winding of one phase,

\[ S \]
— active cross-sectional area of the core,

\[ W_{1} \]
— number of turns of the primary winding of one transformer.

Applying the law of total current and taking \(\nu=1\), we obtain the system of equations:

\[ \left\{ \begin{aligned} &\frac{Z_{11}l}{W_1}\dot H_{m1a}+j\omega W_1 S\bar\mu_{1a}\dot H_{m1a} -\frac{Z_{11}l}{W_1}\dot H_{m1v}-j\omega W_1 S\bar\mu_{1v}\dot H_{m1v} =\dot U_{mav},\\ &\frac{Z_{11}l}{W_1}\dot H_{m1v}+j\omega W_1 S\bar\mu_{1v}\dot H_{m1v} -\frac{Z_{11}l}{W_1}\dot H_{m1c}-j\omega W_1 S\bar\mu_{1c}\dot H_{m1c} =\dot U_{mvc},\\ &\frac{l}{W_1}\dot H_{m1a}+\frac{l}{W_1}\dot H_{m1v}+\frac{l}{W_1}\dot H_{m1c}=0. \end{aligned} \right. \tag{2} \]

where

\(l\) is the mean length of the magnetic line of force in the core.

An approximate solution of the system of nonlinear equations (2) is carried out by Newton’s method [4]. The system of equations contains three nonlinear algebraic equations with three unknowns \(H_{m1a}, H_{m1v}, H_{m1c}\), which may be regarded as a three-dimensional vector

\[ H= \begin{bmatrix} H_{m1a}\\ H_{m1v}\\ H_{m1c} \end{bmatrix}. \tag{3} \]

When solving the system of nonlinear equations by Newton’s method, the Jacobian matrix must be square, since otherwise it is impossible to form its inverse. Therefore we multiply the equations of system (2) by the missing unknowns and, denoting

\[ \dot A=\frac{Z_{11}e}{W_1}\quad \text{and}\quad C=\omega W_1S, \]

we obtain

\[ \left\{ \begin{aligned} &(\dot A H_{m1a}\cdot e^{j\alpha_a} +jC\bar\mu_{1a}\cdot H_{m1a}\cdot e^{j\alpha_a} -\dot A H_{m1v}\cdot e^{j\alpha_v}\\ &\qquad{}-jC\bar\mu_{1v}\cdot H_{m1v}\cdot e^{j\alpha_v} -\dot U_{mav})H_{m1c}=0,\\ &(\dot A H_{m1v}\cdot e^{j\alpha_v} +jC\bar\mu_{1v}H_{m1v}\cdot e^{j\alpha_v} -\dot A H_{m1c}\cdot e^{j\alpha_c}\\ &\qquad{}-jC\bar\mu_{1c}\cdot H_{m1c}\cdot e^{j\alpha_c} -\dot U_{mvc})H_{m1a}=0,\\ &\frac{l}{W_1}H_{m1a}\cdot e^{j\alpha_a} +\frac{l}{W_1}H_{m1v}\cdot e^{j\alpha_v} +\frac{l}{W_1}H_{m1c}\cdot e^{j\alpha_c}=0, \end{aligned} \right. \tag{4} \]

where

\(\alpha\) is the argument of a complex number, determined from the vector diagram of currents and voltages under the condition that the current lags the phase voltage by \(90^\circ\).

Since the degree of saturation of the steel of the FPV transformers, owing to the asymmetry of the circuit, is not the same, then \(\bar\mu_{1a}, \bar\mu_{1v}, \bar\mu_{1c}\) in equations (4) are not equal to one another in magnitude:

\[ \bar\mu_{1a}=\frac{2}{H_{m1a}}\sum_{n=1}^{n} a_{2n-1}\cdot J_1\frac{(2n-1)H_{m1a}}{H_0}, \tag{5} \]

\[ \bar\mu_{1v}=\frac{2}{H_{m1v}}\sum_{n=1}^{n} a_{2n-1}\cdot J_1\frac{(2n-1)H_{m1v}}{H_0}, \tag{6} \]

\[ \bar\mu_{1c}=\frac{2}{H_{m1c}}\sum_{n=1}^{n} a_{2n-1}\cdot J_1\frac{(2n-1)H_{m1c}}{H_0}. \tag{7} \]

where

\(n\) is the index of expansion of the main induction curve in a Fourier series,

\(\varepsilon_{2n-1}\) are the Fourier-series coefficients,

\(J_1\!\left(\dfrac{(2n-1)H_{\mathrm{m}1}}{H_0}\right)\) is the Bessel function of the first kind of first order with real argument.

The system of equations (4) can be represented as a vector function:

\[ f = \begin{bmatrix} f_1\\ f_2\\ f_3 \end{bmatrix}. \tag{8} \]

For the system of functions \(f_1, f_2, f_3\), let us construct the Jacobi matrix with respect to the variables \(H_{\mathrm{m}1a}, H_{\mathrm{m}1B}, H_{\mathrm{m}1C}\).

\[ f'(H)=W(H)= \begin{vmatrix} \dfrac{\partial f_1}{\partial H_{\mathrm{m}1a}} & \dfrac{\partial f_1}{\partial H_{\mathrm{m}1B}} & \dfrac{\partial f_1}{\partial H_{\mathrm{m}1C}} \\[1.2em] \dfrac{\partial f_2}{\partial H_{\mathrm{m}1a}} & \dfrac{\partial f_2}{\partial H_{\mathrm{m}1B}} & \dfrac{\partial f_2}{\partial H_{\mathrm{m}1C}} \\[1.2em] \dfrac{\partial f_3}{\partial H_{\mathrm{m}1a}} & \dfrac{\partial f_3}{\partial H_{\mathrm{m}1B}} & \dfrac{\partial f_3}{\partial H_{\mathrm{m}1C}} \end{vmatrix}. \tag{9} \]

If the matrix (9) is nonsingular, then its inverse matrix \(W^{-1}(H)\) can be formed.

To find the vector root (3) of equation (4), we use the iteration method:

\[ \begin{bmatrix} \dot H_{\mathrm{m}1a}^{(p+1)}\\ \dot H_{\mathrm{m}1B}^{(p+1)}\\ \dot H_{\mathrm{m}1C}^{(p+1)} \end{bmatrix} = \begin{bmatrix} \dot H_{\mathrm{m}1a}^{(p)}\\ \dot H_{\mathrm{m}1B}^{(p)}\\ \dot H_{\mathrm{m}1C}^{(p)} \end{bmatrix} - W^{-1}(\dot H^{(p)}) \begin{bmatrix} f_1(\dot H_{\mathrm{m}1a}^{(p)})\\ f_2(\dot H_{\mathrm{m}1B}^{(p)})\\ f_3(\dot H_{\mathrm{m}1C}^{(p)}) \end{bmatrix}, \tag{10} \]

where the index \(p\) denotes the iteration number. The complex rms value of the current in the primary circuit of each phase is determined on the basis of the law of total current:

\[ \dot I_{1a}=\frac{lH_{\mathrm{m}1a}}{\sqrt{2}W_1}e^{j\alpha_a},\qquad \dot I_{1B}=\frac{lH_{\mathrm{m}1B}}{\sqrt{2}W_1}e^{j\alpha_B},\qquad \dot I_{1C}=\frac{lH_{\mathrm{m}1C}}{\sqrt{2}W_1}e^{j\alpha_C}, \tag{11} \]

where

\(\alpha_a, \alpha_B, \alpha_C\) are the initial phases of the currents.

When the main induction curve is approximated by a trigonometric polynomial [1], the complex representation of the amplitude for the \(\nu\)-th harmonic of the induction has the form

\[ \dot B_{\mathrm{m}\nu} = 2e^{j\alpha} \sum_{n=1}^{n} \varepsilon_{2n-1}\cdot I_\nu \frac{(2n-1)H_{\mathrm{m}1}}{H_0}. \tag{12} \]

Then the complex rms voltage at the converter output for the \(\nu\)-th harmonic is equal to the geometric sum of the \(\nu\)-th harmonics of the phase voltages:

\[ \dot U_{\nu\mathrm{II}} = j\frac{2}{\sqrt{2}}\nu\omega W_{\mathrm{II}}S \sum_{n=1}^{n} \varepsilon_{2n-1} \left( e^{j\nu\alpha_a}\cdot I_\nu\frac{(2n-1)H_{\mathrm{m}1a}}{H_0} +\right. \]

\[ \left. +\,e^{j\nu\alpha_B}\cdot I_\nu\frac{(2n-1)H_{\mathrm{m}1B}}{H_0} + e^{j\nu\alpha_C}\cdot I_\nu\frac{(2n-1)H_{\mathrm{m}1C}}{H_0} \right), \tag{13} \]

where

$W_{II}$ is the number of turns of the secondary winding of one transformer.

The experimental determination of the harmonic composition of the current and voltage of a ferromagnetic voltage converter was carried out by the wattmeter method [1].

An asymmetric system of voltages was applied to the voltage converter by means of the circuit shown in Fig. 1, a. The voltage level was set by three-phase autotransformer 1. Autotransformers 2 and 3 were connected in parallel to one of the line voltages. By moving the sliders of the autotransformers along the windings, it is possible to obtain any configuration of the triangle of line voltages (Fig. 1, b).

Fig. 1

The determination of the current harmonics in the primary circuit and of the voltage at the output of the FVC was carried out for the case in which the triangle of line voltages was isosceles, that is, a two-phase short circuit was simulated with equality of the angles of the transmission line and of the system [5].

Fig. 2

In Fig. 2 the curves are plotted for the variation of the 1st, 3rd, and 5th current harmonics in the primary circuit as a function of the magnitude of the line voltage $U_{b'c'}$. In Fig. 3 the curves are given for the variation of the voltage harmonics at the converter output—

Fig. 3

obtained experimentally. The dashed lines show the curves of variation of the 1st and 3rd voltage harmonics, calculated by formula (13). In the calculations, only the first six harmonics of the induction were taken into account in the approximating expression.

Conclusions

1. Under asymmetrical conditions, all odd harmonics are present in the voltage at the output of a ferromagnetic voltage converter.

2. The calculation method set forth in the article makes it possible, with accuracy sufficient for practice, to determine the voltage harmonics at the converter output under asymmetrical conditions.

References

1. N. V. Lisetskii, V. K. Skripko, A. N. Burnashev. Determination of the harmonic composition of current and voltage under symmetrical conditions. Izv. TPI. Present collection.

2. N. V. Lisetskii, A. V. Shmoilov. An electromagnetic relay responding to changes in the area of the triangle of line voltages. Izv. TPI, vol. 191, 1969.

3. G. E. Pukhov. Complex Calculus. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1961.

4. B. P. Demidovich, I. A. Maron. Fundamentals of Computational Mathematics. Nauka, Moscow, 1966.

5. A. M. Fedoseev. Fundamentals of Relay Protection. GEI, 1961.