The Reverse Reactive Power of SVC in the Field of the Electric Arc Furnace’s Reactive Power Compensation

Abstract

Against the problem of reverse reactive power of SVC existing in the electric arc furnace’s reactive power compensation, this chapter uses adjustable power factor instead of fully compensation, that is, a mode in which the power factor is equal to one. Via the simulation analysis, this chapter verifies the effect on solving the reverse reactive power within the mode of adjustable power factor.

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1 Introduction

Currently, the application of SVC in the reactive power compensation is generally using the mode of fully compensation. However, severe waved load such as the electric arc furnace and the system error will make fully compensation difficult, and then SVC is likely to reverse the reactive power to the power system. Thus, it may cause more active loss, excessive voltage; finally it probably causes damage to the power system, device, and capacitor [1].

The following content will face the problem of the reverse reactive power, and use the compensation mode of adjustable power factor, in which the power factor is less than one, so that the problem could be solved.

2 System Theory

TCR SVC consists of thyristor controlled reactor and several groups of LC filter branches, which takes both filtering and dynamic reactive power compensation into account. Figure 1 is single-phase theory figure of the dynamic reactive power compensator. LC branches are fixed connection and TCR branch is triggering-controlled, thus it can form continuous controlled inductive reactance. Generally, TCR’s capacity is more than FC’s, so that it could guarantee to output capacitive reactive power and inductive reactive power [2–4].

Fig. 1

System theory

By Fig. 1, we can obtain the balanced equation of the system reactive power and that of the harmonic current:

$$ {Q_\mathrm{{s}}}={Q_\mathrm{{L}}}+{Q_\mathrm{{f}}}-{Q_\mathrm{{c}}} $$

(1)

$$ {{\mathbf{I}}_{\mathbf{h}}}={{\mathbf{I}}_{\mathbf{Lh}}}+{{\mathbf{I}}_{\mathbf{fh}}}-{{\mathbf{I}}_{\mathbf{ch}}} $$

(2)

In Fig. 1, Q _f means load reactive power, Q _L means inductive reactive power provided by compensation reactor, Q _c means capacitive reactive power provided by fixed capacitors, Q _s means system’s reactive power, and I _h means harmonic current.

SVC is connected to the power system as shown in Fig. 1; capacitors provide fixed capacitive reactive power Q _c. According to the change of load reactive power Q _f, TCR adjusts to output inductive reactive power’s value and makes inductive reactive power counteract with capacitive reactive power, that is, Q _s = Q _L + Q _f − Q _c = constant (or zero). So, it can realize that the power factor is equal to a constant or 1, and guarantee that the voltage hardly waves. Under ideal conditions the LC filter’s input makes system’s harmonic current meet the following equation.

$$ {{\mathbf{I}}_{\mathbf{h}}}={{\mathbf{I}}_{\mathbf{Lh}}}+{{\mathbf{I}}_{\mathbf{fh}}}-{{\mathbf{I}}_{\mathbf{ch}}}=0 $$

(3)

3 Compensation Susceptance’s Derivation and Simulation in the Mode of Fully Compensation

3.1 Compensation Susceptance’s Derivation Based on Symmetrical Component Method

It is supposed that power source is balanced; the load is expressed to delta connected network as shown in Fig. 2. Y _ab, Y _bc, and Y _ca are not equal to each other.

Fig. 2

Load and compensation network

It is supposed that Y _ab = G _ab + jB _ab, Y _bc = G _bc + jB _bc, and Y _ca = G _ca + jB _ca.

Figure 2 is an equivalent circuit schematic after compensation, in which each phase load is separately connected with compensation susceptance B _ra, B _rb, and B _rc in parallel that are used to compensate the load’s fundamental negative sequence current and fundamental positive sequence reactive current. The following uses symmetrical component method to derive the formula of compensation susceptance. It is supposed that the power source of the unbalanced load is provided by three-phase positive sequence voltage; the voltage RMS of each phase to the neutral point is [5, 6]:

$$ {{\mathbf{V}}_{\mathbf{a}}}=V_1^{+},\ {{\mathbf{V}}_{\mathbf{b}}}={{\mathbf{a}}^{\mathbf{2}}}V_1^{+},\ {{\mathbf{V}}_{\mathbf{c}}}=\mathbf{a}V_1^{+},\ \mathbf{a}={\mathrm{{e}}^{{j120^{\circ}}}}=-\frac{1}{2}+j\frac{{\sqrt{3}}}{2} $$

(4)

Line voltage:

$$ \left\{\begin{array}{l}{{\mathbf{V}}_{\mathbf{a}\mathbf{b}}}={{\mathbf{V}}_{\mathbf{a}}}-{{\mathbf{V}}_{\mathbf{b}}}=(\mathbf{1}-{{\mathbf{a}}^{\mathbf{2}}})V_1^{+} \hfill \\{{\mathbf{V}}_{\mathbf{b}\mathbf{c}}}={{\mathbf{V}}_{\mathbf{b}}}-{{\mathbf{V}}_{\mathbf{c}}}=({{\mathbf{a}}^{\mathbf{2}}}-\mathbf{a})V_1^{+} \hfill \\{{\mathbf{V}}_{\mathbf{c}\mathbf{a}}}={{\mathbf{V}}_{\mathbf{c}}}-{{\mathbf{V}}_{\mathbf{a}}}=(\mathbf{a}-\mathbf{1})V_1^{+} \hfill \\\end{array} \right. $$

(5)

Load phase current:

$$ \left\{\begin{array}{l} {{\mathbf{I}}_{\mathbf{a}\mathbf{b}}}={{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}{{\mathbf{V}}_{\mathbf{a}\mathbf{b}}}={{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}(\mathbf{1}-{{\mathbf{a}}^{\mathbf{2}}})V_1^{+} \hfill \cr {{\mathbf{I}}_{\mathbf{bc}}}={{\mathbf{Y}}_{\mathbf{bc}}}{{\mathbf{V}}_{\mathbf{bc}}}={{\mathbf{Y}}_{\mathbf{bc}}}({{\mathbf{a}}^{\mathbf{2}}}-\mathbf{a})V_1^{+} \hfill \cr {{\mathbf{I}}_{\mathbf{ca}}}={{\mathbf{Y}}_{\mathbf{ca}}}{{\mathbf{V}}_{\mathbf{ca}}}={{\mathbf{Y}}_{\mathbf{ca}}}(\mathbf{a}-\mathbf{1})V_1^{+} \hfill \\\end{array} \right. $$

(6)

Load line current:

$$ \left\{\begin{array}{l} {{\mathbf{I}}_{\mathbf{a}}}={{\mathbf{I}}_{\mathbf{a}\mathbf{b}}}-{{\mathbf{I}}_{\mathbf{c}\mathbf{a}}}=[{{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}(\mathbf{1}-{{\mathbf{a}}^{\mathbf{2}}})-{{\mathbf{Y}}_{\mathbf{c}\mathbf{a}}}(\mathbf{a}-\mathbf{1})]V_1^{+} \hfill \cr {{\mathbf{I}}_{\mathbf{b}}}={{\mathbf{I}}_{\mathbf{b}\mathbf{c}}}-{{\mathbf{I}}_{\mathbf{a}\mathbf{b}}}=[{{\mathbf{Y}}_{\mathbf{b}\mathbf{c}}}({{\mathbf{a}}^{\mathbf{2}}}-\mathbf{a})-{{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}(\mathbf{1}-\mathbf{a})]V_1^{+} \hfill \cr {{\mathbf{I}}_{\mathbf{c}}}={{\mathbf{I}}_{\mathbf{c}\mathbf{a}}}-{{\mathbf{I}}_{\mathbf{b}\mathbf{c}}}=[{{\mathbf{Y}}_{\mathbf{c}\mathbf{a}}}(\mathbf{a}-\mathbf{1})-{{\mathbf{Y}}_{\mathbf{b}\mathbf{c}}}({{\mathbf{a}}^{\mathbf{2}}}-\mathbf{a})]V_1^{+} \hfill \\\end{array} \right. $$

(7)

Line current symmetrical components:

$$ \left\{\begin{array}{l} \mathbf{I}_{\mathbf{1}}^{\mathbf{0}}=({{\mathbf{I}}_{\mathbf{a}}}+{{\mathbf{I}}_{\mathbf{b}}}+{{\mathbf{I}}_{\mathbf{c}}})/3=\mathbf{0} \hfill \cr \mathbf{I}_{\mathbf{1}}^{+}=({{\mathbf{I}}_{\mathbf{a}}}+\mathbf{a}{{\mathbf{I}}_{\mathbf{b}}}+{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{c}}})/3=({{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}+{{\mathbf{Y}}_{\mathbf{b}\mathbf{c}}}+{{\mathbf{Y}}_{\mathbf{c}\mathbf{a}}})V_1^{+} \hfill \cr \mathbf{I}_{\mathbf{1}}^{-}=({{\mathbf{I}}_{\mathbf{a}}}+{{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{I}}_{\mathbf{b}}}+\mathbf{a}{{\mathbf{I}}_{\mathbf{c}}})/3=-({{\mathbf{a}}^{\mathbf{2}}}{{\mathbf{Y}}_{\mathbf{a}\mathbf{b}}}+{{\mathbf{Y}}_{\mathbf{b}\mathbf{c}}}+\mathbf{a}{{\mathbf{Y}}_{\mathbf{c}\mathbf{a}}})V_1^{+} \hfill \\\end{array} \right. $$

(8)

Similarly:

$$ \left\{\begin{array}{l} \mathbf{I}_{\mathbf{1r}}^{\mathbf{0}}=0,\mathbf{I}_{\mathbf{1r}}^{+}=j({B_\mathrm{{ab}}}+{B_\mathrm{{bc}}}+{B_\mathrm{{ca}}})V_1^{+} \hfill \cr \mathbf{I}_{\mathbf{1r}}^{-}=-j({{\mathbf{a}}^2}{B_\mathrm{{ab}}}+{B_\mathrm{{bc}}}+\mathbf{a}{B_\mathrm{{ca}}})V_1^{+} \hfill \\\end{array} \right. $$

(9)

After the compensation in the mode of fully compensation, the fundamental negative sequence current and fundamental positive sequence reactive current are zero, that is:

$$ \mathbf{I}_{\mathbf{1}}^{-}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{-}=0,\quad \operatorname{Im}(\mathbf{I}_{\mathbf{1}}^{+}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{+})=0 $$

(10)

The solution of the compensation susceptances is:

$$ \left\{\begin{array}{l} {B_\mathrm{{ra}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}+\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}-\sqrt{3}\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{-}) \hfill \cr {B_\mathrm{{rb}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}-2\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}) \hfill \cr {B_\mathrm{{rc}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}+\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}+\sqrt{3}\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{-}) \hfill \\\end{array} \right. $$

(11)

3.2 Compensation Susceptance’s Derivation Based on Transient Power Theory

According to Figs. 3 and 4, transform the voltage and current in the abc coordinate system to the αβ coordinate system:

$$ \left[\begin{array}{l}{u_{\alpha }} \hfill \\{u_{\beta }} \hfill \\\end{array} \right]={C_{32 }}\left[\begin{array}{l}{u_a} \hfill \\{u_b} \hfill \\{u_c} \hfill \\\end{array} \right],\quad \left[\begin{array}{l}{i_{\alpha }} \hfill \\{i_{\beta }} \hfill \\\end{array} \right]={C_{32 }}\left[\begin{array}{l}{i_a} \hfill \\{i_b} \hfill \\{i_c} \hfill \\\end{array} \right] $$

(12)

Fig. 3

The algorithm block diagram of voltage

Fig. 4

The algorithm block diagram of compensation susceptance

According to Fig. 3, transform the voltage in the αβ coordinate system to the pq coordinate system:

$$ \left[ \begin{gathered} {u_p} \hfill \\ {u_q} \hfill \\ \end{gathered} \right]=C\left[ \begin{gathered} {u_{\alpha }} \hfill \\ {u_{\beta }} \hfill \\ \end{gathered} \right],\quad C=\left[ {\begin{array}{*{20}{c}} {\sin \omega t} & {-\cos \omega t} \\ {-\cos \omega t} & {-\sin \omega t} \\ \end{array}} \right] $$

(13)

After digital low-pass filter:

$$ {{\overline{u}}_p}=\sqrt{3}V_1^{+},\quad {{\overline{u}}_q}=0 $$

(14)

According to Fig. 3, transform the voltage in the pq coordinate system to the αβ coordinate system:

$$ \left[\begin{array}{l}u_{{1\alpha}}^{+} \hfill \\u_{{1\beta}}^{+} \hfill \\\end{array} \right]=C\left[\begin{array}{l}{{\overline{u}}_p} \hfill \\{{\overline{u}}_q} \hfill \\\end{array} \right]=\left[\begin{array}{l}\sqrt{3}V_1^{+}\sin \omega t \hfill \\-\sqrt{3}V_1^{+}\cos \omega t \hfill \\\end{array} \right] $$

(15)

According to Fig. 4, by the definition of transient power:

$$ \left\{\begin{array}{l}{p^{+}}=u_{{1\alpha}}^{+}{i_{\alpha }}+u_{{1\beta}}^{+}{i_{\beta }},{q^{+}}=u_{{1\beta}}^{+}{i_{\alpha }}-u_{{1\alpha}}^{+}{i_{\beta }} \hfill \\{p^{-}}=u_{{1\alpha}}^{+}{i_{\alpha }}-u_{{1\beta}}^{+}{i_{\beta }},{q^{-}}=u_{{1\beta}}^{+}{i_{\alpha }}+u_{{1\alpha}}^{+}{i_{\beta }} \hfill \\\end{array} \right. $$

(16)

According to Fig. 4, after digital low-pass filter:

$$ \left\{\begin{array}{l}p_\mathrm{{av}}^{+}=3V_1^{+}I_1^{+}\cos \theta_1^{+},q_\mathrm{{av}}^{+}=-3V_1^{+}I_1^{+}\sin \theta_1^{+} \hfill \\p_\mathrm{{av}}^{-}=3V_1^{+}I_1^{-}\cos \theta_1^{-},q_\mathrm{{av}}^{-}=-3V_1^{+}I_1^{-}\sin \theta_1^{-} \hfill \\\end{array} \right. $$

(17)

So:

$$ \left\{\begin{array}{l}\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{+}=I_1^{+}\cos \theta_1^{+}=p_\mathrm{{av}}^{+}/3V_1^{+},\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}=I_1^{+}\sin \theta_1^{+}=-q_\mathrm{{av}}^{+}/3V_1^{+} \hfill \\\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{-}=I_1^{-}\cos \theta_1^{-}=p_\mathrm{{av}}^{-}/3V_1^{+},\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}=I_1^{-}\cos \theta_1^{-}=-q_\mathrm{{av}}^{-}/3V_1^{+} \hfill \\\end{array} \right. $$

(18)

So, compensation susceptance based on transient power:

$$ \left\{\begin{array}{l} {B_\mathrm{{ra}}}=\frac{1}{{9V_1^{+2 }}}\left( {q_\mathrm{{av}}^{+}+q_\mathrm{{av}}^{-}+\sqrt{3}p_\mathrm{{av}}^{-}} \right),{B_\mathrm{{rb}}}=\frac{1}{{9V_1^{+2 }}}(q_\mathrm{{av}}^{+}-2q_\mathrm{{av}}^{-}) \hfill \cr {B_\mathrm{{rc}}}=\frac{1}{{9V_1^{+2 }}}\left( {q_\mathrm{{av}}^{+}+q_\mathrm{{av}}^{-}-\sqrt{3}p_\mathrm{{av}}^{-}} \right) \hfill \\\end{array} \right. $$

(19)

3.3 Simulation Analysis in the Mode of Fully Compensation

As shown in Fig. 5, it is the mutative curve of system reactive power in the mode of fully compensation (positive reactive power means that power system transmits reactive power to the load and the negative means SVC transmits reactive power to power system). When SVC is put into operation, SVC transmits reactive power to power system; the system reactive power waves nearby −0.5Mvar.

Fig. 5

Mutative curve of system reactive power in the mode of fully compensation

4 Compensation Susceptance’s Derivation and Simulation in the Mode of Adjustable Power Factor

4.1 Compensation Susceptance’s Derivation Based on Symmetrical Component Method

As in Sect. 3.1, after the compensation in the mode of adjustable power factor, the fundamental negative sequence current is zero and the fundamental positive sequence reactive current needs to meet the given power factor, that is:

$$ \mathbf{I}_{\mathbf{1}}^{-}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{-}=\mathbf{0},\quad \frac{{|\operatorname{Re}(\mathbf{I}_{\mathbf{1}}^{+}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{+})|}}{{\sqrt{{{{{\operatorname{Im}}}^2}(\mathbf{I}_{\mathbf{1}}^{+}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{+})+{{{\operatorname{Re}}}^2}(\mathbf{I}_{\mathbf{1}}^{+}+\mathbf{I}_{\mathbf{1}\mathbf{r}}^{+})}}}}=\lambda, \quad \gamma =\frac{{\sqrt{{1-{\lambda^2}}}}}{\lambda } $$

(20)

The solution of the compensation susceptances is:

$$ \left\{\begin{array}{l} {B_\mathrm{{ra}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}+\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}-\sqrt{3}\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{-}+\gamma \operatorname{Re}\mathbf{I}_{\mathbf{1}}^{+}) \hfill \cr {B_\mathrm{{rb}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}-2\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}+\gamma \operatorname{Re}\mathbf{I}_{\mathbf{1}}^{+}) \hfill \cr {B_\mathrm{{rc}}}=-\frac{1}{{3V_1^{+}}}(\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{+}+\operatorname{Im}\mathbf{I}_{\mathbf{1}}^{-}+\sqrt{3}\operatorname{Re}\mathbf{I}_{\mathbf{1}}^{-}+\gamma \operatorname{Re}\mathbf{I}_{\mathbf{1}}^{+}) \hfill \\\end{array} \right. $$

(21)

4.2 Compensation Susceptance’s Derivation Based on Dower Theory

As in Sect. 3.2, compensation susceptance based on transient power:

$$ \left\{\begin{array}{l} {B_\mathrm{{ra}}}=\frac{1}{{9V_1^{+2 }}}(q_\mathrm{{av}}^{+}+q_\mathrm{{av}}^{-}+\sqrt{3}p_\mathrm{{av}}^{-}-\gamma p_\mathrm{{av}}^{+}) \hfill \cr {B_\mathrm{{rb}}}=\frac{1}{{9V_1^{+2 }}}(q_\mathrm{{av}}^{+}-2q_\mathrm{{av}}^{-}-\gamma p_\mathrm{{av}}^{+}) \hfill \cr {B_\mathrm{{rc}}}=\frac{1}{{9V_1^{+2 }}}(q_\mathrm{{av}}^{+}+q_\mathrm{{av}}^{-}-\sqrt{3}p_\mathrm{{av}}^{-}-\gamma p_\mathrm{{av}}^{+}) \hfill \\\end{array} \right. $$

(22)

4.3 Simulation Analysis in the Mode of Adjustable Power Factor

As shown in Figs. 6 and 7, they are the mutative curves of system reactive power in the mode of adjustable power factor. When SVC is put into operation and the power factor is given from 0.95 to 0.99, power system transmits reactive to the load; the system reactive power separately waves nearby 5Mvar and 1.5Mvar.

Fig. 6

Power factor is 0.95

Fig. 7

Power factor is 0.99

5 Conclusion

Firstly, by Fig. 5 with the mode of fully compensation and Figs. 6 or 7 with the mode of adjustable power factor, we know that it can overcome the problem of reverse reactive power in the mode of fully compensation when using the mode of adjustable power factor. Moreover, by Fig. 6 with Fig. 7, we can know when the power factor is given further away from 1, the situation of reverse reactive power occurs less easily, but the reactive power is transmitted more from power system to the load.

In conclusion, when using the mode of adjustable power factor to control the electric arc furnace’s reactive compensation, the power factor should be given appropriately, not too big.