**2. Balancing ac traction systems with the Steinmetz circuit**

Fig. 1a shows one of the most widely used connection schemes of ac traction systems, where the railroad substation is formed by a single-phase transformer feeding the traction load from the utility power supply system. As the railroad substation is a single-phase load which may lead to unbalanced utility supply voltages, several methods have been proposed to reduce unbalance (Chen, 1994; Hill, 1994), such as feeding railroad substations at different phases alternatively, and using special transformer connections (e.g. Scott-connection), SVCs or external balancing equipment. To simplify the study of these methods, the single-phase transformer is commonly considered ideal and the traction load is represented by its equivalent inductive impedance, *Z*<sup>L</sup> = *R*<sup>L</sup> + *jX*L, obtained from its power demand at the fundamental frequency, Fig. 1b (Arendse & Atkinson-Hope, 2010; Barnes & Wong, 1991; Chen, 1994; Mayer & Kropik, 2005; Qingzhu et al., 2010a, 2010b). According to Fig. 1c, external balancing equipment consists in the delta connection of reactances (usually an inductor *Z*1 and a capacitor *Z*2) with the single-phase load representing the railroad substation in order to load the network with balanced currents. This circuit, which is known as Steinmetz circuit (ABB Power Transmission, n.d.; Barnes & Wong, 1991; Mayer & Kropik, 2005), is not the most common balancing method in traction systems but it is also used in industrial high-power single-phase loads and electrothermal appliances (Chicco et al., 2009; Chindris et. al., 2002; Mayer & Kropik, 2005).

Fig. 1. Studied system: a) Railroad substation connection scheme. b) Simplified railroad substation circuit. c) Steinmetz circuit.

Characterization of Harmonic Resonances in

where the typical limit

factor of the single-phase load satisfies the condition

λ

2002; Sainz & Riera, submitted for publication).

the Presence of the Steinmetz Circuit in Power Systems 175

circuit under study (with an inductor *X*1 and a capacitor *X*2) turns out to be possible only when *X*1 and *X*2 values are positive. Thus, according to (Sainz & Riera, submitted for publication), *X*1 is always positive while *X*2 is only positive when the displacement power

> <sup>3</sup> 1 , 2 1 *L LC*

Supply voltage unbalance is considered in (Qingzhu et al., 2010a, 2010b) by applying optimization techniques for Steinmetz circuit design, and in (Jordi et al., 2002; Sainz & Riera, submitted for publication) by obtaining analytical expressions for the symmetrizing reactances. However, the supply voltage balance hypothesis is not as critical as the pure Steinmetz circuit inductor hypothesis. Harmonics are not considered in the literature in Steinmetz circuit design and the reactances are determined from the fundamental waveform component with the previous expressions. Nevertheless, Steinmetz circuit performance in the presence of waveform distortion is analyzed in (Arendse & Atkinson-Hope, 2010; Chicco et al., 2009). Several indicators defined in the framework of the symmetrical components are proposed to explain the properties of the Steinmetz circuit under waveform distortion. The introduction of thyristor-controlled reactive elements due to the recent development of power electronics and the use of step variable capacitor banks allow varying the Steinmetz circuit reactances in order to compensate for the usual single-phase load fluctuations, (Barnes & Wong, 1991; Chindris et al., 2002). However, the previous design expressions must be considered in current dynamic symmetrization and the power signals are then treated by the controllers in accordance with the Steinmetz procedure for load balancing

<sup>+</sup> ≥≥ =

λ λ

(ABB Power Transmission, n.d.; Lee & Wu, 1993; Qingzhu et al., 2010a, 2010b).

**3. Steinmetz circuit impact on power system harmonic response** 

resonance considering the Steinmetz circuit inductor resistance.

The power system harmonic response in the presence of the Steinmetz circuit is analyzed from Fig. 3. Two sources of harmonic disturbances can be present in this system: a three-phase nonlinear load injecting harmonic currents into the system or a harmonic-polluted utility supply system. In the former, the parallel resonance may affect power quality because harmonic voltages due to injected harmonic currents can be magnified. In the latter, series resonance may affect power quality because consumed harmonic currents due to background voltage distortion can also be magnified. Therefore, the system harmonic response depends on the equivalent harmonic impedance or admittance "observed" from the three-phase load or the utility supply system, respectively. This chapter, building on work developed in (Sainz et al., 2007, in press), summarizes the above research on parallel and series resonance location and unifies this study. It provides an expression unique to the location of the parallel and series

In Fig. 3, the impedances *ZLk* = *RL* + *jkXL*, *Z*1*<sup>k</sup>* = *R*1 + *jkX*1 and *Z*2*<sup>k</sup>* = −*jX*2/*k* represent the single-phase load, the inductor and the capacitor of the Steinmetz circuit at the fundamental (*k* = 1) and harmonic frequencies (*k* > 1). Note that impedances *ZL*1, *Z*11 and *Z*21 correspond to impedances *ZL*, *Z*1 and *Z*2 in Section 2, respectively. Moreover, parameter *dC* is introduced in the study representing the degree of the Steinmetz circuit capacitor degradation from its

1

*LC* = (√3)/2 can be obtained from (4) by imposing

τ

> 2 1

(4)

τ

<sup>1</sup> = 0 (Jordi et al.,

τ

+

Fig. 2. Detailed Steinmetz circuit.

Fig. 2 shows the Steinmetz circuit in detail. The inductor is represented with its associated resistance, *Z*<sup>1</sup> = *R*1 + *jX*1, while the capacitor is considered ideal, *Z*<sup>2</sup> = − *jX*2. Steinmetz circuit design aims to determine the reactances *X*1 and *X*2 to balance the currents consumed by the railroad substation. Thus, the design value of the symmetrizing reactive elements is obtained by forcing the current unbalance factor of the three-phase fundamental currents consumed by the Steinmetz circuit (*IA*, *IB*, *IC*) to be zero. Balanced supply voltages and the pure Steinmetz circuit inductor (i.e., *R*1 = 0 ) are usually considered in Steinmetz circuit design, and the values of the reactances can be obtained from the following approximated expressions (Sainz et al., 2005):

$$X\_{\rm 1, \rm apr}(R\_{\rm 1}, \mathcal{J}\_{\rm t}) = \frac{\sqrt{3}R\_{\rm 1}}{\mathcal{J}\_{\rm t}^{2} \left(1 + \sqrt{3}\tau\_{\rm 1}\right)} \qquad ; \qquad X\_{\rm 2, \rm apr}(R\_{\rm 1}, \mathcal{J}\_{\rm t}) = \frac{\sqrt{3}R\_{\rm 1}}{\mathcal{J}\_{\rm t}^{2} \left(1 - \sqrt{3}\tau\_{\rm 1}\right)} , \tag{1}$$

where

$$
\pi\_L = \frac{X\_L}{R\_L} = \frac{\sqrt{1 - \lambda\_L^2}}{\lambda\_L},
\tag{2}
$$

and λ*<sup>L</sup>* = *RL*/|*ZL*| and |*ZL*| are the displacement power factor and the magnitude of the single-phase load at fundamental frequency, respectively.

In (Mayer & Kropik, 2005), the resistance of the Steinmetz inductor is considered and the symmetrizing reactance values are obtained by optimization methods. However, no analytical expressions for the reactances are provided. In (Sainz & Riera, submitted for publication), the following analytical expressions have recently been deduced

$$\mathbf{X}\_{\mathsf{L}}(\mathbf{R}\_{\mathsf{L}},\boldsymbol{\lambda}\_{\mathsf{L}},\boldsymbol{\tau}\_{\mathsf{1}}) = \frac{\mathbf{R}\_{\mathsf{L}}\boldsymbol{\lambda}\_{\mathsf{L}}^{2}\left(\sqrt{\mathsf{S}}-\boldsymbol{\tau}\_{\mathsf{1}}\right)}{\boldsymbol{\lambda}\_{\mathsf{L}}^{2}\boldsymbol{\tau}\_{\mathsf{1}}^{2}\left(1+\sqrt{\mathsf{S}}\boldsymbol{\tau}\_{\mathsf{L}}\right)}\qquad;\qquad\mathbf{X}\_{\mathsf{L}}(\mathbf{R}\_{\mathsf{L}},\boldsymbol{\lambda}\_{\mathsf{L}},\boldsymbol{\tau}\_{\mathsf{1}}) = \frac{\mathbf{R}\_{\mathsf{L}}\left(\sqrt{\mathsf{S}}-\boldsymbol{\tau}\_{\mathsf{1}}\right)}{\boldsymbol{\lambda}\_{\mathsf{L}}^{2}\left[\left(1-\sqrt{\mathsf{S}}\boldsymbol{\tau}\_{\mathsf{L}}\right)-\boldsymbol{\tau}\_{\mathsf{1}}\left(\sqrt{\mathsf{S}}+\boldsymbol{\tau}\_{\mathsf{L}}\right)\right]},\tag{3}$$

where τ<sup>1</sup> = *R*1/*X*1 = λ1/(1 – λ12)1/2 is the R/X ratio of the Steinmetz circuit inductor, and λ<sup>1</sup> = *R*1/|*Z*1| and |*Z*1| are the displacement power factor and the magnitude of the Steinmetz circuit inductor at the fundamental frequency, respectively. It must be noted that (1) can be derived from (3) by imposing τ1 = 0 (and therefore λ1/τ1 = 1). The Steinmetz

Fig. 2 shows the Steinmetz circuit in detail. The inductor is represented with its associated resistance, *Z*<sup>1</sup> = *R*1 + *jX*1, while the capacitor is considered ideal, *Z*<sup>2</sup> = − *jX*2. Steinmetz circuit design aims to determine the reactances *X*1 and *X*2 to balance the currents consumed by the railroad substation. Thus, the design value of the symmetrizing reactive elements is obtained by forcing the current unbalance factor of the three-phase fundamental currents consumed by the Steinmetz circuit (*IA*, *IB*, *IC*) to be zero. Balanced supply voltages and the pure Steinmetz circuit inductor (i.e., *R*1 = 0 ) are usually considered in Steinmetz circuit design, and the values of the reactances can be obtained from the following approximated

> ( ) ( ) 1, apr 2, apr 2 2 3 3 (,) ; (,) ,

> > <sup>2</sup> <sup>1</sup> , *<sup>L</sup> <sup>L</sup>*

*<sup>L</sup>* = *RL*/|*ZL*| and |*ZL*| are the displacement power factor and the magnitude of the

1 1 1

− − = = <sup>+</sup> −−+

λ τ

*L L LL L*

1 = 0 (and therefore

λ

12)1/2 is the R/X ratio of the Steinmetz circuit inductor, and

3 3 ( , ,) ; ( , ,) , 1 3 13 3 *L L*

1 1

<sup>1</sup> = *R*1/|*Z*1| and |*Z*1| are the displacement power factor and the magnitude of the Steinmetz circuit inductor at the fundamental frequency, respectively. It must be noted that

τ

λ

*L L*

In (Mayer & Kropik, 2005), the resistance of the Steinmetz inductor is considered and the symmetrizing reactance values are obtained by optimization methods. However, no analytical expressions for the reactances are provided. In (Sainz & Riera, submitted for

*R R*

= =

*L L L L*

*X R*

 τ

*L*

publication), the following analytical expressions have recently been deduced

τ

*R R X R*

1 3 1 3 *L L*

*X R*

 λ

λ

<sup>−</sup> = = (2)

 τ

( ) { } ( ) ( )

 ττ

λ1/τ τ

 τ

1 = 1). The Steinmetz

(3)

(1)

*L L L L*

λ

+ −

*IA*

*A*

*B*

*Utility supply system*

Fig. 2. Detailed Steinmetz circuit.

expressions (Sainz et al., 2005):

where

and λ

where τ

λ

λ

λ

single-phase load at fundamental frequency, respectively.

( ) ( )

*X R X R* λ

*L L L L*

 τ

 τ

1 1 2 1 2 2 <sup>2</sup>

2

λτ

λ1/(1 – λ

(1) can be derived from (3) by imposing

λ τ

<sup>1</sup> = *R*1/*X*1 =

*C*

*IB*

*IC*

*R*<sup>1</sup> *RL*

*Railroad substation*

*X*<sup>2</sup>

*X*<sup>1</sup> *XL*

circuit under study (with an inductor *X*1 and a capacitor *X*2) turns out to be possible only when *X*1 and *X*2 values are positive. Thus, according to (Sainz & Riera, submitted for publication), *X*1 is always positive while *X*2 is only positive when the displacement power factor of the single-phase load satisfies the condition

$$1 \ge \lambda\_{\rm t} \ge \lambda\_{\rm t,C} = \frac{\pi\_{\rm t} + \sqrt{3}}{2\sqrt{1 + \pi\_{\rm t}^2}},\tag{4}$$

where the typical limit λ*LC* = (√3)/2 can be obtained from (4) by imposing τ<sup>1</sup> = 0 (Jordi et al., 2002; Sainz & Riera, submitted for publication).

Supply voltage unbalance is considered in (Qingzhu et al., 2010a, 2010b) by applying optimization techniques for Steinmetz circuit design, and in (Jordi et al., 2002; Sainz & Riera, submitted for publication) by obtaining analytical expressions for the symmetrizing reactances. However, the supply voltage balance hypothesis is not as critical as the pure Steinmetz circuit inductor hypothesis. Harmonics are not considered in the literature in Steinmetz circuit design and the reactances are determined from the fundamental waveform component with the previous expressions. Nevertheless, Steinmetz circuit performance in the presence of waveform distortion is analyzed in (Arendse & Atkinson-Hope, 2010; Chicco et al., 2009). Several indicators defined in the framework of the symmetrical components are proposed to explain the properties of the Steinmetz circuit under waveform distortion.

The introduction of thyristor-controlled reactive elements due to the recent development of power electronics and the use of step variable capacitor banks allow varying the Steinmetz circuit reactances in order to compensate for the usual single-phase load fluctuations, (Barnes & Wong, 1991; Chindris et al., 2002). However, the previous design expressions must be considered in current dynamic symmetrization and the power signals are then treated by the controllers in accordance with the Steinmetz procedure for load balancing (ABB Power Transmission, n.d.; Lee & Wu, 1993; Qingzhu et al., 2010a, 2010b).
