**1. Introduction**

170 Power Quality Harmonics Analysis and Real Measurements Data

Zhou, L.H., Fu, Q., Liu, C.S.(2009). Modeling and Control Analysis of a Hybrid Unified

2009

Power Quality Conditioner, Proceeding of 2009. Asia-Pacific Power and Energy Engineering Conference, pp.1-5, ISBN: 978-1-4244-2486-3 , Wuhan, March 27-31,

> An electric power system is expected to operate under balanced three-phase conditions; however, single-phase loads such as traction systems can be connected, leading to unbalanced line currents. These systems are single-phase, non-linear, time-varying loads closely connected to the utility power supply system. Among problems associated with them, special consideration must be given to the presence of unbalanced and distorted currents (Barnes & Wong, 1991; Capasso, 1998; Hill, 1994; Howroyd, 1989; Marczewski, 1999; Qingzhu et al., 2010a, 2010b). These operating conditions damage power quality, producing undesirable effects on networks and affecting the correct electric system operation (Arendse & Atkinson-Hope, 2010; Chen, 1994; Chen & Kuo, 1995; Chindris et. al., 2002; Lee & Wu, 1993; Mayer & Kropik, 2005). The unbalanced currents cause unequal voltage drops in distribution lines, resulting in load bus voltage asymmetries and unbalances (Chen, 1994; Qingzhu et al., 2010a, 2010b). For this reason, several methods have been developed to reduce unbalance in traction systems and avoid voltage asymmetries, for example feeding railroad substations at different phases alternatively, and connecting special transformers (e.g. Scott connection), Static Var Compensators (SVCs) or external balancing equipment (ABB Power Transmission, n.d.; Chen, 1994; Chen & Kuo, 1995; Hill, 1994; Lee & Wu, 1993; Qingzhu et al., 2010a, 2010b). The last method, which is incidentally not the most common, consists of suitably connecting reactances (usually an inductor and a capacitor in delta configuration) with the single-phase load representing the railroad substation (Barnes & Wong, 1991; Qingzhu et al., 2010a, 2010b). This method is also used with industrial high-power single-phase loads and electrothermal appliances (Chicco et al., 2009; Chindris et. al., 2002; Mayer & Kropik, 2005).

> This delta-connected set, more commonly known as Steinmetz circuit, (Barnes & Wong, 1991; Jordi et al., 2002; Mayer & Kropik, 2005), allows the network to be loaded with symmetrical currents. Several studies on Steinmetz circuit design under sinusoidal balanced or unbalanced conditions aim to determine the reactance values to symmetrize the currents consumed by the single-phase load. Some works propose analytical expressions and optimization techniques for Steinmetz circuit characterization, (Arendse & Atkinson-Hope, 2010; Jordi et. al, 2002; Mayer & Kropik, 2005; Qingzhu et al., 2010a, 2010b; Sainz & Riera,

Characterization of Harmonic Resonances in

Chindris et. al., 2002; Mayer & Kropik, 2005).

*A B C*

(a)

*Utility supply system*

*Railroad*

(c)

*substation Traction*

*Utility supply system*

substation circuit. c) Steinmetz circuit.

*load*

*A B C*

*circuit*

the Presence of the Steinmetz Circuit in Power Systems 173

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;

*X*<sup>L</sup>

*R*<sup>L</sup>

*substation*

*Railroad substation*

*<sup>Z</sup>*<sup>1</sup> *Railroad*

*Traction load*

*Utility supply system*

(b)

*Z*2

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

*IA IB IC*

*Z*<sup>L</sup> *Steinmetz*

*A B C*

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

submitted for publication). In general, the values of the symmetrizing elements should vary in order to compensate for the usual single-phase load fluctuations. Unfortunately, the typical inductances and capacitors have fixed values. However, this can be solved by the introduction of thyristor-controlled reactive elements due to the development of power electronics in the last few years and the use of step variable capacitor banks, (Barnes & Wong, 1991; Chindris et al., 2002).

Steinmetz circuit design must consider circuit performance and behavior under nonsinusoidal conditions because of the growing presence of non-linear devices in electric power systems in the last few decades, (Arendse & Atkinson-Hope, 2010; Chicco et al., 2009; Czarnecki, 1989, 1992). Harmonic currents injected by non-linear devices can cause voltage distortions, which may damage power quality. In this sense, the effects of harmonics on power systems and their acceptable limits are well known [IEC power quality standards, (IEC 6100-3- 6, 2008); Task force on Harmonic Modeling and Simulation, 1996, 2002]. In the above conditions, the parallel and series resonance occurring between the Steinmetz circuit capacitor and the system inductors must be located to prevent harmonic problems when the Steinmetz circuit is connected. The parallel resonance occurring between the Steinmetz circuit capacitor and the supply system inductors is widely studied in (Caro et al., 2006; Sainz et al., 2004, 2005, 2007). This resonance can increase harmonic voltage distortion in the presence of non-linear loads injecting harmonic currents into the system. The problem is pointed out in (Sainz et al., 2004). In (Caro et al., 2006; Sainz et al., 2005), it is numerically and analytically characterized, respectively. In (Sainz et al., 2005), several curves are fitted numerically from the power system harmonic impedances to predict the resonance at the fifth, seventh and eleventh harmonics only. In (Caro et al., 2006), the resonance is analytically located from the theoretical study of the power system harmonic impedances. Finally, the analytical expressions in (Caro et al., 2006) to predict the parallel resonance frequency are expanded in (Sainz et al., 2007) to consider the influence of the Steinmetz circuit capacitor loss with respect to its design value. The series resonance "observed" from the supply system is also studied and located in (Sainz et al., 2009a, 2009b, in press). This resonance can affect power quality in the presence of a harmonic-polluted power supply system because the consumed harmonic currents due to background voltage distortion can be magnified. It is numerically and analytically studied in (Sainz et al., 2009a, 2009b), respectively. In (Sainz et al., 2009a), graphs to locate the series resonance frequency and the admittance magnitude values at the resonance point are numerically obtained from the power system harmonic admittances. In (Sainz et al., 2009b), analytical expressions to locate the series resonance are obtained from these admittances. Finally, the analytical expressions developed in (Sainz et al., 2009b) to predict resonance frequencies are expanded in (Sainz et al., in press) to consider the influence of Steinmetz circuit capacitor changes with respect to its design value.

This chapter, building on work developed in the previous references, not only summarizes the above research but also unifies the study of both resonances, providing an expression unique to their location. The proposed expression is the same as in the series resonance case, but substantially improves those obtained in the parallel resonance case. Moreover, the previous studies are completed with the analysis of the impact of the Steinmetz circuit inductor resistance on the resonance. This resistance, as well as damping the impedance values, shifts the resonance frequency because it influences Steinmetz circuit design (Sainz & Riera, submitted for publication). A sensitivity analysis of all variables involved in the location of the parallel and series resonance is also included. The chapter ends with several experimental tests to validate the proposed expression and several examples of its application.

submitted for publication). In general, the values of the symmetrizing elements should vary in order to compensate for the usual single-phase load fluctuations. Unfortunately, the typical inductances and capacitors have fixed values. However, this can be solved by the introduction of thyristor-controlled reactive elements due to the development of power electronics in the last few years and the use of step variable capacitor banks, (Barnes &

Steinmetz circuit design must consider circuit performance and behavior under nonsinusoidal conditions because of the growing presence of non-linear devices in electric power systems in the last few decades, (Arendse & Atkinson-Hope, 2010; Chicco et al., 2009; Czarnecki, 1989, 1992). Harmonic currents injected by non-linear devices can cause voltage distortions, which may damage power quality. In this sense, the effects of harmonics on power systems and their acceptable limits are well known [IEC power quality standards, (IEC 6100-3- 6, 2008); Task force on Harmonic Modeling and Simulation, 1996, 2002]. In the above conditions, the parallel and series resonance occurring between the Steinmetz circuit capacitor and the system inductors must be located to prevent harmonic problems when the Steinmetz circuit is connected. The parallel resonance occurring between the Steinmetz circuit capacitor and the supply system inductors is widely studied in (Caro et al., 2006; Sainz et al., 2004, 2005, 2007). This resonance can increase harmonic voltage distortion in the presence of non-linear loads injecting harmonic currents into the system. The problem is pointed out in (Sainz et al., 2004). In (Caro et al., 2006; Sainz et al., 2005), it is numerically and analytically characterized, respectively. In (Sainz et al., 2005), several curves are fitted numerically from the power system harmonic impedances to predict the resonance at the fifth, seventh and eleventh harmonics only. In (Caro et al., 2006), the resonance is analytically located from the theoretical study of the power system harmonic impedances. Finally, the analytical expressions in (Caro et al., 2006) to predict the parallel resonance frequency are expanded in (Sainz et al., 2007) to consider the influence of the Steinmetz circuit capacitor loss with respect to its design value. The series resonance "observed" from the supply system is also studied and located in (Sainz et al., 2009a, 2009b, in press). This resonance can affect power quality in the presence of a harmonic-polluted power supply system because the consumed harmonic currents due to background voltage distortion can be magnified. It is numerically and analytically studied in (Sainz et al., 2009a, 2009b), respectively. In (Sainz et al., 2009a), graphs to locate the series resonance frequency and the admittance magnitude values at the resonance point are numerically obtained from the power system harmonic admittances. In (Sainz et al., 2009b), analytical expressions to locate the series resonance are obtained from these admittances. Finally, the analytical expressions developed in (Sainz et al., 2009b) to predict resonance frequencies are expanded in (Sainz et al., in press) to consider the influence of Steinmetz circuit

This chapter, building on work developed in the previous references, not only summarizes the above research but also unifies the study of both resonances, providing an expression unique to their location. The proposed expression is the same as in the series resonance case, but substantially improves those obtained in the parallel resonance case. Moreover, the previous studies are completed with the analysis of the impact of the Steinmetz circuit inductor resistance on the resonance. This resistance, as well as damping the impedance values, shifts the resonance frequency because it influences Steinmetz circuit design (Sainz & Riera, submitted for publication). A sensitivity analysis of all variables involved in the location of the parallel and series resonance is also included. The chapter ends with several experimental tests

to validate the proposed expression and several examples of its application.

Wong, 1991; Chindris et al., 2002).

capacitor changes with respect to its design value.
