Operational Control of the Load Current of an Aluminum Plant Using Controlled Static Reactive-Power Sources

R. I. Ostapenko, N. E. Chernyi

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

In [1] a range of questions is considered that are connected with regulating the operating conditions of electric systems and long-distance power transmission lines and with increasing their stability by means of controlled static reactive-power sources (RPS). Preliminary studies indicate that controlled static RPS can also be used to regulate the mode of electric power consumption by loads.

The present article considers questions connected with the use of static RPS for operational regulation of the load current of a silicon converter substation (SCS) of an aluminum plant.

Fig. 1

Figure 1 shows schematic diagrams of RPS. The circuits considered are controlled by variable-resistance reactors (VRR), connected in series or in parallel with a capacitor bank. Taking this into account, under the limiting regulation conditions \(x_{p_1}=0\), \(x_{p_2}=\infty\), circuits of ordinary longitudinal compensation (LC) of the load arise. This makes it possible to use, in the studies to one degree or another, the analysis and the basic relations for a longitudinally compensated SCS given in [2, 3].

Control properties of the LCM with series-connected SRS

For the circuit of Fig. 1a, taking into account that the operational control range is comparatively small (±5% of the average load current), we consider the load mode of the SCR to be unchanged during the control process. With this in mind, we shall analyze the control characteristic using the known relations [2, 3, 4], expressing the variables as the sum of constants and their increments.

Since, for technological reasons of electrolysis, it is necessary to maintain a constant load current when the load resistance varies, we take as constant the load current of the SCR and the emf of the system. Under this condition, the equation relating the change in rectified voltage and commutation angle to the change in reactive resistance in the valve commutation circuit for load modes I and II of the SCR has the form

\[ \frac{(I')^2}{3\sin^2 \frac{\gamma}{2}} + \frac{(\tau-\Delta\tau)^2}{\cos^2 \frac{\gamma}{2}} =1, \tag{1} \]

where

\[ I' = I\left[1-\nu^2\gamma\left(3.73\cdot\frac{\pi}{6}-\frac{\gamma}{2}\right)\right], \tag{2} \]

\(I\) is the relative value of the load current of the equivalent “star bridge” or “delta bridge,” equal to

\[ \frac{I_d(x_a+x_p)}{\sqrt{2}\,E_2}; \]

\(\tau\) is the relative value of the rectified voltage of the SCR;

\(\gamma\) is the commutation angle of the valves, electrical degrees;

\(I_d\) is the load current of the equivalent “star bridge” or “delta bridge” of the SCR, equal to one half of the SCR load current, A;

\(E_2\) is the rms value of the emf of the supply system, referred to the converter buses, V;

\(\nu^2\) is the degree of compensation, defined by the relation

\[ \nu^2=\frac{x_c}{x_a+x_p}; \]

\(x_c\) is the capacitive reactance of the PEC at a frequency of 50 Hz, in ohms;

\(x_a\) is the natural or basic reactive resistance in the valve commutation circuit of the equivalent “star bridge,” ohms;

\(x_p\) is the change in inductive resistance in the valve commutation circuit due to the SRS, ohms;

\(\Delta\tau\) is the increment of rectified voltage in relative units.

From equations (1), (2), after transformations we obtain an expression that explicitly relates the SCR control parameters \(x=\dfrac{x_p}{x_a}\), \((\tau-\Delta\tau)\), and \(\gamma\), for \(I_d\) and \(E_2=\mathrm{const}\):

\[ \bar{x} = -(1-\nu_0^2) + \sqrt{ (1-\nu^2)^2 + \frac{3\tan^2 \frac{\gamma}{2}}{I_0^2} (2\tau_0-\Delta\tau^2) }, \tag{3} \]

where

\(\tau_0\) is the relative value of the rectified voltage at \(x_p=0\);

\(I_0\) is the relative value of the load current of the uncompensated SCR at \(x_p=0\);

\(\nu_0^2\) is the degree of compensation of the converter substation at \(x_p=0\).

It follows from (3) that regulation is possible only downward from the value \(\tau_0\). However, by selecting the appropriate degree of compensation it is always possible to make the compensating effect of the IRM predominate during the regulation process.

Equation (3) can easily be transformed for the second load mode by replacing \(\gamma\) in accordance with the equality \(\dfrac{\gamma}{2}=15\).

Calculation of the Power of the IRM Elements

Using relation (3), for a specified value of \(\nu_0^2\) it is not difficult to find the required power of the RPS and PEC:

\[ Q_{\mathrm{rps}}=3\bar{x}x_a I_1^2 \simeq 3\bar{x}\cdot x_a\cdot \frac{2.5\,I_d^2}{c^2}, \tag{4} \]

\[ Q_{\mathrm{pec}} \simeq 3\nu_0^2 x_a \frac{2.5 I_d^2}{c^2}, \tag{5} \]

where

\(I_1\) is the rms value of the linear primary current, approximately equal to [2]

\[ I_1 \simeq 0.79\,\frac{2I_d}{c}; \]

\(c\) is the transformation coefficient.

Example. Determine the required power of the reactor and the capacitor bank of the SC, under the conditions that \(E_d=496\ \mathrm{V}\), \(I_0=0.21\), \(\tau_0=0.925\), \(\Delta \tau=0.05\), \(3\tan^2 15^\circ=0.216\), \(x_a=0.72\ \Omega\), \(c=31.6\), \(2I_d=1.65\cdot 10^3\), \(\nu_0^2=0.4\).

\[ \bar{x}=-(1-0.4)+ \sqrt{(1-0.4)^2-\frac{0.216}{0.21}\,2(0.05^2-20.05\,0.925)} \simeq 0.3, \]

\[ Q_{\mathrm{rps}}=0.72\,\frac{1}{31.6^2}\,0.3\,31.65^2\,10^6\,0.79^2 =11\ \mathrm{Mvar}, \]

\[ Q_{\mathrm{pec}}=0.4\,0.72^2\,\frac{1}{31.6^2}\,31.65^2\,10^6\,0.79^2 =14\ \mathrm{Mvar}. \]

Regulating properties of an IRM with a parallel-connected RPS. When the RPS and PEC are connected in parallel (Fig. 1b), the IRM is a regulated \(LC\) circuit, which substantially changes the operating mode of the converter substation. Therefore, before analyzing the regulating properties of the IRM, it is necessary to obtain the basic relations for the converter substation.

An exact analysis of electromagnetic transients in a 12-phase equivalent parallel rectifier is associated with solving integral equations and determining constants of integration, which is a complex and cumbersome problem. At the same time, an analysis of the shape of the voltage curve on the \(LC\) circuit, for the real parameters of this circuit and operating values of the converter-substation load current, shows that, with accuracy sufficient for practice, the voltage drop on the \(LC\) circuit can be determined from the fundamental harmonic of the primary current.

The equivalent circuit of a series-compensated converter substation with an adjustable degree of compensation is shown in Fig. 2. The amplitude value of the voltage drop across the capacitors of the CSD in (the circuit of Fig. 2) from the fundamental-harmonic current is found as

\[ |U_k|=2.23 I_d\,\frac{x_c}{1-\nu_k^2}; \qquad \nu_k^2=\frac{x_c}{x_p}. \]

Fig. 2. Schematic diagram of a longitudinal-compensating PP with smoothly adjustable degree of PP compensation. The reactances shown in the figure are referred to the secondary star.

The phase of the voltage \(U_k\) will be determined by the phase angle of the complex impedance \(Z_k\) and by the phase angle of the current \(I_1\).

When the phase EMFs of the source are specified in the form

\[ e_A=\sqrt{2}E_2\sin V, \]

\[ e_B=-\sqrt{2}E_2\cos(V-30^\circ), \qquad e_C=\sqrt{2}E_2\cos(V+30^\circ), \]

taking into account the notation of the phase angle of the current by \(\varphi_e\), we write the expressions for the instantaneous values of the current \(i\) in the form

\[ i_{1,A}=\sqrt{2}\,I_1\sin(V-\varphi_e),\qquad i_{1,B}=-\sqrt{2}\,I_1\cos(V-30^\circ-\varphi_e), \]

\[ i_{1,C}=\sqrt{2}\,I_1\cos(V+30^\circ-\varphi_e). \]

For \(\nu_k^2<1\), the circuit resistance is capacitive in character and may be expressed as

\[ Z_k=\frac{x_c}{1-\nu_k^2}\,e^{-190^\circ}. \tag{6} \]

Taking this into account, we write the instantaneous value of the voltage on the circuit in the form:

\[ U_{k,A}=-2.23I_d\frac{x_c}{1-\nu_k^2}\cos(V-\varphi_e), \tag{7} \]

\[ U_{k,B}=-2.23I_d\frac{x_c}{1-\nu_k^2}\sin(V-30^\circ-\varphi_e), \tag{8} \]

\[ U_{k,C}=-2.23I_d\frac{x_c}{1-\nu_k^2}\sin(V+30^\circ-\varphi_e). \tag{9} \]

Substituting the expressions obtained into equations (8), (10), (12), presented in [3], we obtain a system of differential equations for the 12-phase bridge equivalent rectification circuit when the circuit is included in its power circuit.

\[ U_1=\sqrt{2}E_2\sin V-x_a\frac{d(i_1-i_1')}{dV} -\frac{x_3}{\sqrt{3}}\frac{d(i_2-i_2'-i_4+i_4')}{dV} +2.23I_d\frac{x_c}{1-\nu_k^2}\cos(V-\varphi_e), \tag{10} \]

\[ U_3=-\sqrt{2}E_2\cos(V-30^\circ)-x_a\frac{d(i_3-i_3')}{dV} -\frac{x_3}{\sqrt{3}}\frac{d(i_4-i_4'-i_6+i_6')}{dV} +2.23I_d\frac{x_c}{1-\nu_k^2}\sin(V-30^\circ-\varphi_e), \tag{11} \]

\[ U_5=\sqrt{2}E_2\cos(V+30^\circ)-x_a\frac{d(i_5-i_5')}{dV} -\frac{x_3}{\sqrt{3}}\cdot\frac{d(i_6-i_6'-i_2+i_2')}{dV} -2.23I_d\frac{x_c}{1-\nu_k^2}\sin(V+30^\circ-\varphi_e), \tag{12} \]

where

\[ V \]
— the current angular coordinate;

\[ i_1,\ i_2,\ i_3,\ i_4,\ i_5,\ i_6 \]
— instantaneous values of the currents in the anode valve groups, A;

\[ i_1',\ i_2',\ i_3',\ i_4',\ i_5',\ i_6' \]
— instantaneous values of the currents, A, in the cathode valve groups;

\[ U_1,\ U_2,\ U_3 \]
— instantaneous values of the phase voltages applied to the valve pairs of the “star bridge” \(1,1'\), \(2,2'\), \(3,3'\), respectively, V;

\[ x_3 \]
— resultant reactance of the supply system, ohm.

To determine the ignition instant of valve 1 while valves 5; 4; 6; 3 are conducting, we set up the voltage balance in the commutating circuit (valves 1,5) and equate it to zero.

\[ U_1-U_5=\sqrt{6}E_2\sin(V-30^\circ)-x_a\frac{d(i_1-i_5)}{dV}+ \]
\[ +\,2.23I_d\frac{x_c}{1-\gamma_k^2}\sqrt{3}\cos(V-30^\circ-\varphi_e)=0. \tag{13} \]

Since at the initial instant of commutation \(i_1=0;\ \dfrac{di_5}{dV}=0\), we obtain

\[ \sqrt{6}E_2\sin(V-30^\circ)+3.68I_d\frac{x_c}{1-\gamma_k^2}\cos(V-30^\circ-\varphi_e)=0. \tag{14} \]

The equality written is identically satisfied when \(V=\Psi\), the firing angle of the next valve, in this case valve 1.

Before finding the value of \(\Psi\) from (14), on the basis of [2], let us replace the angle \(\varphi_e\) by the value \(\varphi_e=\pm\alpha+f(\gamma)\). Taking this into account, the angle \(-30^\circ\pm\alpha-f(\gamma)=-\Psi-f(\gamma)\).

Here \(\alpha\) is the self-regulation angle, which is the positive or negative difference between the firing angle \(\Psi\) and the angle \(30^\circ\).

Taking the above into account, expression (14) assumes the form

\[ \sqrt{6}E_2\sin(\psi-30^\circ)+3.86I_d\frac{x_c}{1-\gamma_k^2}\cos[-f(\gamma)]=0. \tag{15} \]

Solving (15) with respect to \(\Psi\), taking into account the relation \(f(\gamma)\simeq \dfrac{\gamma}{2}\) [2], after transformations we obtain

\[ \psi\simeq 30^\circ-\arcsin 2.23\frac{J_0\gamma^2}{1-\gamma_k^2}\cos\frac{1}{2}\gamma. \tag{16} \]

The average value of the rectified current of the TCC. For \(V>\Psi;\ i_1>0;\ \dfrac{di_5}{dV}\ne 0\), moreover, \(-\dfrac{di_1}{dV}=\dfrac{di_5}{dV}\). Taking this into account, after transformations equation (15) assumes the form

\[ \frac{di_1}{dV}=\frac{\sqrt{6}E_2}{2x_a}\sin(V-30^\circ)+1.93I_d\frac{\gamma^2}{1-\gamma_k^2}\cos(V-30^\circ-\varphi_e). \tag{17} \]

Integrating (17) over the intervals \(\Psi\leq V\leq \Psi+\gamma\), after transformations we obtain

\[ I_d\simeq \frac{\sqrt{6}E_2[\cos(\psi-30^\circ)-\cos(\psi+\gamma-30^\circ)]} {2x_a\left(1-3.86\dfrac{\gamma^2}{1-\gamma_k^2}\sin\dfrac{\gamma}{2}\right)} \]

or, in relative units,

\[ J\simeq 0.866\, \frac{\cos(\psi-30^\circ)-\cos(\psi+\gamma-30^\circ)} {1-3.86\dfrac{\gamma^2}{1-\gamma_k^2}\sin\dfrac{\gamma}{2}}, \tag{18} \]

for the second load condition \((\gamma=30^\circ)\)

\[ J\simeq 0.866\, \frac{\cos(\psi-30^\circ)-\cos\psi} {1-\dfrac{\gamma^2}{1-\gamma_k^2}}. \tag{18a} \]

The determination of the average value of the rectified voltage in the case of an uncontrolled TC is presented in [2, 3]. It can be shown that the component of the rectified voltage caused by the presence

of the \(LC\) circuit in the first and second load conditions \((0 \leq \gamma \leq 30^\circ)\), over the repetition period, is equal to zero. Taking this into account and [2, 3],

\[ U_d=-\frac{3\sqrt{6}}{\pi}E_2[\cos(\psi-30^\circ)+\cos(\psi+\gamma-30^\circ)]. \tag{19} \]

On the basis of (18) and (19), the volt-ampere characteristic of the CPC for the first and second load conditions has the form

\[ \frac{J'^2}{3\sin^2\frac{\gamma}{2}}+\frac{\tau^2}{\cos^2\frac{\gamma}{2}}=1, \tag{20} \]

where

\[ J'\simeq J_0\left(1-\frac{\nu^2}{1-\nu_k^2}\right), \quad \text{for } \gamma=30^\circ, \]

\[ J'\simeq J_0\left(1-3.86\,\frac{\nu^2}{1-\nu_k^2}\sin\frac{\gamma}{2}\right), \quad \text{for } \gamma<30^\circ . \tag{21} \]

By changing the value of \(\nu_k^2\) through a smooth change in the resistance of the CCS, we reduce \(J'\) and, in accordance with (20), increase \(\tau\), thereby as it were increasing the degree of CPC. Thus, regulation of the RPS with parallel connection of the CCS, without passing through the resonance point, is possible only upward and always with the presence of a compensating effect.

The desired regulating characteristic, on the basis of (20) and (21), after transformations has the form

\[ x=\frac{\nu^2-\nu^2\sqrt{(1-\nu^2)^2-\frac{A}{B}}}{2\nu^2-\nu^4+\frac{A}{B}}, \tag{22} \]

where

\[ A=3\sin^2\frac{\gamma}{2}\,(2\tau_0\Delta\tau+\Delta\tau^2); \]

\[ B=J_0^2\cos^2\frac{\gamma}{2}; \]

\[ x=1-\nu_k^2. \]

A check of the correctness of (22) is provided by the equality \(\bar{x}=1\) when \(A=0\), i.e., \(\Delta\tau=0\). Taking the notation into account, knowing the value \(\bar{x}\), we obtain:

\[ \nu_k^2=1-\bar{x}, \tag{23} \]

\[ x_p=\frac{\nu^2 x_a}{\nu_k^2}, \tag{24} \]

\[ Z_k=\frac{x_a\nu^2}{1-\nu_k^2}, \tag{25} \]

\[ U_k=1.58 I_d\frac{Z_k}{c}, \tag{26} \]

where

\(U_k\) is the rms value of the voltage drop across the circuit.

The required power of the circuit elements is equal to

\[ Q_{\mathrm{CPC}}=3\frac{U_k^2}{x_a\nu_0^2}, \tag{27} \]

\[ Q_{\mathrm{CCS}}=Q_{\mathrm{CPC}}\nu_k^2. \tag{28} \]

Example. Let us calculate the required power of the RPS branches under the condition that

\[ \nu_0=0.1,\quad x=0.72\ \Omega,\quad 2I_d=165\ \mathrm{kA},\quad c=31.6; \]

\[ I_0 = 0.21,\quad \tau_0 = 0.875,\quad +\Delta \tau = 5 \cdot 10^{-2}\ \text{at}\ \gamma = 30^\circ; \]

\[ A = 0.0181,\quad B = 0.41,\quad \frac{A}{B} = 0.44; \]

\[ \bar{x} = \frac{0.1 \pm 0.1\sqrt{0.9^2 - 0.44}} {0.12 - 0.01 + 0.44} = 0.25; \]

\[ \nu_k = 0.75,\quad x_p = 0.096,\quad U_k = 1150\ \text{V}; \]

\[ Q_{\text{PEC}} = \frac{3 \cdot 1150^2}{0.1 \cdot 0.72} = 55\ \text{Mvar},\quad Q_{\text{RPS}} = 55 \cdot 0.75 = 41\ \text{Mvar}. \]

Comparison of the Considered IRM Schemes for Operational Regulation of the Load Current of the CPS

All the compared schemes are designed for a regulation range of \(\pm 2.5\%\, U_d\), which amounts to \(24\ \text{V}\).

1. Series connection of the RPS.

\[ Q_{\text{PEC}} = 14\ \text{Mvar},\quad Q_{\text{RPS}} = 11\ \text{Mvar}. \]

1. Parallel connection of the RPS.

\[ Q_{\text{PEC}} = 55\ \text{Mvar},\quad Q_{\text{RPS}} = 41\ \text{Mvar}. \]

It follows from the comparison that the first variant of smooth regulation of the load current of the CPS will require lower capital expenditures, and at the same time smaller active electric-energy losses should be expected, since these losses are proportional to the reactive power of the elements required for regulation.

Conclusions

From the study presented it follows that:

1. Static controlled IRM can be regarded as a means of operational regulation of the series current;

2. The IRM considered have a compensating effect over the entire regulation range and, consequently, will contribute to reducing the reactive-power electricity consumption of the CPS or to increasing its productivity;

3. The series connection of the RPS and PEC has better specific regulation properties and requires less power from the RPS and IRM.

In addition, in the course of the study the basic relationships were obtained that make it possible to calculate the operating mode of the CPS when a circuit is connected into its current circuit.

References

1. V. A. Venikov and L. A. Zhdanov. Regulation of the operating mode of electrical systems and long-distance power transmission lines and improvement of their stability by means of controlled static IRM. Elektrichestvo, 1967, No. 6.

2. S. R. Glinternik. Electromagnetic Processes and Operating Modes of Powerful Static Converters. Nauka, Moscow, 1968.

3. R. I. Ostapenko. Longitudinal compensation of the converter substation of an aluminum plant. Proceedings of the 11th Regional Economic Conference, Krasnoyarsk, 1970.

4. Sh. M. Razmadze. Converter Circuits and Systems. Vysshaya Shkola Publishing House, Moscow, 1967.