**6.1 Experimental measurements of power system harmonic response**

To validate the analytical study, measurements were made in two downscaled laboratory systems corresponding to the networks of Fig. 4 (parallel resonance) and Fig. 6 (series resonance). The frequency response measurements were made with a 4.5 kVA AC ELGAR Smartwave Switching Amplifier as the power source, which can generate sinusoidal waveforms of arbitrary frequencies (between 40 Hz and 5000 Hz) and a YOKOGAWA DL 708 E digital scope as the measurement device. From the results shown in the next Sections, it must be noted that (22) provides acceptable results. Although experimental tests considering the inductor resistance (*R*<sup>1</sup> ≈ 0.1342 pu) are not shown, they provide similar results.

### **6.1.1 Experimental measurements of the parallel resonance**

The harmonic response of the network in Fig. 4 was measured in the laboratory for two cases with the following system data (*U*B = 100 V and *S*B = 500 VA):

	- Supply system: *ZS*1 = 0.022 +*j*0.049 pu.
	- Railroad substation: *RL* = 1.341 pu, λ*L* = 1.0.
	- External balancing equipment: *X*1, apx = 2.323 pu and *X*2, apx = 2.323 pu [neglecting the inductor resistance, (1)] and *dC* = 1.0, 0.75, 0.5 and 0.25.

Fig. 11a compares the parallel resonance measured in the experimental tests with those obtained from (22). In order to analytically characterize the resonance, the variable values

Characterization of Harmonic Resonances in

τ

Fig. 10) avoiding harmonic problems.

Steinmetz circuit. Thus, considering

*SL* = 75 MVA), the ratio *rL* = *RL*/*XS* =

large enough (i.e., *zP* > 20).

displacement power factor

was below unity value (e.g.,

close to the unity value, the ratio *rL* = *RL*/*XS* =

λ

λ

(22). Considering

the Presence of the Steinmetz Circuit in Power Systems 191

achieve dynamic load balancing is analyzed. The traction load is supplied from single-phase 132/50 kV transformers at each supply substation providing a 25 kV catenary voltage from 50/25 kV autotransformers at intervals along the track. The short-circuit power *SS* at 132 kV bus is below 300 MVA while traction loads may reach short duration peaks of *SL* = 20 to 40 MVA. A total of 28 single-phase harmonic filters for 50 kV tuned to the 3rd, 5th and 7th harmonics were installed in the substations to prevent harmonics generated in the locomotive thyristor drives from being injected into the 132 kV power system. The harmonic impact of the Steinmetz circuit installation on this traction system can be examined from

1 = 0, *dC* = 1 and the displacement power factor

λ

respectively) and the resonance is located at the harmonics *kr*, a = 3.7 to 2.68. It is interesting to note that the Steinmetz circuit connection could cause parallel and series resonances close to the 3rd harmonic, damaging harmonic power quality. If the displacement power factor

(*SL* = 20 to 40, respectively) worsening the harmonic problem. In conclusion, it is not advisable to use the Steinmetz circuit to balance the traction load currents consumed in this installation. However, since the short-circuit power can be below 300 MVA and the transformer short-circuit impedances are not considered in the study, the ratio *rL* values can be lower than the previous ones and the resonance can be below the 3rd harmonic (see

In (Barnes & Wong, 1991), an unbalance and harmonic study carried out for the Channel Tunnel 25 kV railway system supplied from the UK and French 400/225/132 kV grid systems is presented. On the UK side, the PCC between the traction load and the tunnel auxiliary load is at the Folkestone 132 kV busbar with a minimum short-circuit power *SS* equal to 800 MVA. On the French side, the PCC between the traction load, the auxiliary load and other consumers is at the Mandarins 400 kV busbar with a minimum short-circuit power *SS* equal to 11700 MVA. The traction loads range from *SL* = 0 to 75 MVA with a

acting thyristor-controlled reactors and capacitors, which enable the balancing equipment output to vary with the load pattern. Moreover, harmonic studies based on the harmonic spectrum measured in the catenaries of the British Rail network and provided by continental locomotive manufacturers were conducted to analyze the harmonic filter installation. They revealed that the harmonic limits on the French side are within specification limits and no filters are required while, on the UK side, these limits are exceeded and harmonic filters must be installed to reduce harmonic distortion to acceptable levels. These studies can be complemented with harmonic resonance location in the

at harmonics *kr*, a = 21.6 and 6.0 on the French and UK side, respectively. This resonance is shifted to higher harmonics if the traction load is lower. The auxiliary loads and other consumers are not considered in the location of the resonances because their impedance is

In (Arendse & Atkinson-Hope, 2010), the design of the Steinmetz circuit in unbalanced and distorted power supplies is studied from a downscaled laboratory system such as that in

a three-phase Variable Speed Drive (VSD) of 24 kVA rated power is used as a harmonic

τ

λ

Fig. 3. The system data are *ZS*1 = 0.0087 +*j*0.00079 Ω, *RL* = 4.84 Ω, λ*L* = 1.0,

λ

*<sup>L</sup>* = 0.95), the resonance would shift to *kr*, a = 5.93 to 4.36

*L* = 0.93. Steinmetz circuit is located on the UK side with fast-

1 = 0, *dC* = 1 and the maximum traction load (i.e.,

τ

1 = 0, *dC* = 1.0 and

*<sup>L</sup>*·*SS*/*SL* is 145.08 and 9.92 and the resonance is located

*<sup>L</sup>*·*SS*/*SL* is between 15 to 7.5 (*SL* = 20 to 40,

*<sup>L</sup>* of the traction load

corresponding to the above data are *rL* = 27.4, λ*L* = 1 and 0.95 (Cases 1 and 2, respectively) and τ1 = 0.

#### **6.1.2 Experimental measurements of the series resonance**

The harmonic response of the network in Fig. 6 was measured in the laboratory for two cases with the following system data (*U*B = 100 V and *S*B = 500 VA):

	- Supply system: *ZS*1 = 0.076 +*j*0.154 pu.
	- Railroad substation: *RL* = 1.464 pu, λ*L* = 1.0.
	- External balancing equipment: *X*1, apr = 2.536 pu and *X*2, apr = 2.536 pu [neglecting the inductor resistance, (1)] and *dC* = 1.0, 0.75, 0.5 and 0.25.
	- Three-phase load: Grounded wye series R-L impedances with |*ZP*1| = 30.788 pu and λ*P* = 0.95 are connected, i.e. the three-phase load model LM1 in (Task force on Harmonic Modeling and Simulation, 2003).

Fig. 11b compares the series resonance measured in the experimental tests with those obtained from (22). In order to analytically characterize the resonance, the variable values corresponding to these data are *rL* = 9.51, λ*L* = 1 and 0.95 (Cases 1 and 2, respectively) and τ1 = 0.

Fig. 11. Comparison between *kres* and *kr*, a. a) *kres* = *kp*, meas. b) *kres* = *ks*, meas.

#### **6.2 Harmonic resonance location in several power systems**

This section briefly describes several works in the literature on the Steinmetz circuit in power systems, and determines the harmonic of the resonance produced by the presence of this circuit from (22). This allows interpreting the results in the works and predicting the harmonic behavior of the studied power systems.

In (ABB Power Transmission, n.d.), an extensive railway network for coal haulage in East Central Queensland is presented and the installation of nine SVCs in the 132 kV grid to

corresponding to the above data are *rL* = 27.4, λ*L* = 1 and 0.95 (Cases 1 and 2, respectively)

The harmonic response of the network in Fig. 6 was measured in the laboratory for two



λ


*<sup>L</sup>* = 0.95. The

5

4

0

1

2

3

• Case 2 (studied in Section 3.2): System data of Case 1 except the single-phase load

Steinmetz circuit reactances also change, i.e. *X*1, apr = 1.790 pu and *X*2, apr = 6.523 pu (1). Fig. 11b compares the series resonance measured in the experimental tests with those obtained from (22). In order to analytically characterize the resonance, the variable values corresponding to these data are *rL* = 9.51, λ*L* = 1 and 0.95 (Cases 1 and 2, respectively) and

(b)

11

*ks*, meas, *kr*, a

5

6

7

8 9 10

2

3 4

10

0

This section briefly describes several works in the literature on the Steinmetz circuit in power systems, and determines the harmonic of the resonance produced by the presence of this circuit from (22). This allows interpreting the results in the works and predicting the

In (ABB Power Transmission, n.d.), an extensive railway network for coal haulage in East Central Queensland is presented and the installation of nine SVCs in the 132 kV grid to

0.3

 *dC* 0.4 0.5 0.6 0.7 0.8 0.9 1

Case 1

Case 2

*ks,* meas *kr,* <sup>a</sup>

2

4

6

8

fundamental displacement factor of the railroad substation, which becomes

**6.1.2 Experimental measurements of the series resonance** 

Harmonic Modeling and Simulation, 2003).


Case 2

 *dC* 0.4 0.5 0.6 0.7 0.8 0.9 1

harmonic behavior of the studied power systems.

*kp,* meas *kr,* <sup>a</sup>

Case 1

Fig. 11. Comparison between *kres* and *kr*, a. a) *kres* = *kp*, meas. b) *kres* = *ks*, meas.

**6.2 Harmonic resonance location in several power systems** 


cases with the following system data (*U*B = 100 V and *S*B = 500 VA):

the inductor resistance, (1)] and *dC* = 1.0, 0.75, 0.5 and 0.25.

and τ1 = 0.

τ1 = 0.

(a)

18

*kp*, meas, *kr*, a

16

14

10

12

4

6 8

0.3

• Case 1:

achieve dynamic load balancing is analyzed. The traction load is supplied from single-phase 132/50 kV transformers at each supply substation providing a 25 kV catenary voltage from 50/25 kV autotransformers at intervals along the track. The short-circuit power *SS* at 132 kV bus is below 300 MVA while traction loads may reach short duration peaks of *SL* = 20 to 40 MVA. A total of 28 single-phase harmonic filters for 50 kV tuned to the 3rd, 5th and 7th harmonics were installed in the substations to prevent harmonics generated in the locomotive thyristor drives from being injected into the 132 kV power system. The harmonic impact of the Steinmetz circuit installation on this traction system can be examined from (22). Considering τ1 = 0, *dC* = 1 and the displacement power factor λ*<sup>L</sup>* of the traction load close to the unity value, the ratio *rL* = *RL*/*XS* = λ*<sup>L</sup>*·*SS*/*SL* is between 15 to 7.5 (*SL* = 20 to 40, respectively) and the resonance is located at the harmonics *kr*, a = 3.7 to 2.68. It is interesting to note that the Steinmetz circuit connection could cause parallel and series resonances close to the 3rd harmonic, damaging harmonic power quality. If the displacement power factor was below unity value (e.g., λ*<sup>L</sup>* = 0.95), the resonance would shift to *kr*, a = 5.93 to 4.36 (*SL* = 20 to 40, respectively) worsening the harmonic problem. In conclusion, it is not advisable to use the Steinmetz circuit to balance the traction load currents consumed in this installation. However, since the short-circuit power can be below 300 MVA and the transformer short-circuit impedances are not considered in the study, the ratio *rL* values can be lower than the previous ones and the resonance can be below the 3rd harmonic (see Fig. 10) avoiding harmonic problems.

In (Barnes & Wong, 1991), an unbalance and harmonic study carried out for the Channel Tunnel 25 kV railway system supplied from the UK and French 400/225/132 kV grid systems is presented. On the UK side, the PCC between the traction load and the tunnel auxiliary load is at the Folkestone 132 kV busbar with a minimum short-circuit power *SS* equal to 800 MVA. On the French side, the PCC between the traction load, the auxiliary load and other consumers is at the Mandarins 400 kV busbar with a minimum short-circuit power *SS* equal to 11700 MVA. The traction loads range from *SL* = 0 to 75 MVA with a displacement power factor λ*L* = 0.93. Steinmetz circuit is located on the UK side with fastacting thyristor-controlled reactors and capacitors, which enable the balancing equipment output to vary with the load pattern. Moreover, harmonic studies based on the harmonic spectrum measured in the catenaries of the British Rail network and provided by continental locomotive manufacturers were conducted to analyze the harmonic filter installation. They revealed that the harmonic limits on the French side are within specification limits and no filters are required while, on the UK side, these limits are exceeded and harmonic filters must be installed to reduce harmonic distortion to acceptable levels. These studies can be complemented with harmonic resonance location in the Steinmetz circuit. Thus, considering τ1 = 0, *dC* = 1 and the maximum traction load (i.e., *SL* = 75 MVA), the ratio *rL* = *RL*/*XS* = λ*<sup>L</sup>*·*SS*/*SL* is 145.08 and 9.92 and the resonance is located at harmonics *kr*, a = 21.6 and 6.0 on the French and UK side, respectively. This resonance is shifted to higher harmonics if the traction load is lower. The auxiliary loads and other consumers are not considered in the location of the resonances because their impedance is large enough (i.e., *zP* > 20).

In (Arendse & Atkinson-Hope, 2010), the design of the Steinmetz circuit in unbalanced and distorted power supplies is studied from a downscaled laboratory system such as that in Fig. 3. The system data are *ZS*1 = 0.0087 +*j*0.00079 Ω, *RL* = 4.84 Ω, λ*L* = 1.0, τ1 = 0, *dC* = 1.0 and a three-phase Variable Speed Drive (VSD) of 24 kVA rated power is used as a harmonic

Characterization of Harmonic Resonances in

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source. A three-phase linear load with |*ZP*1| = 9.802 Ω and λ*P* = 0.81 (load model LM1) is also connected. The study shows that there is no harmonic problem in the system and that voltage distortion is below 0.05% [Table 7 in (Arendse & Atkinson-Hope, 2010)]. This can be analyzed from (22) because, considering that *rL* = 4.84/0.00079 = 6127 and *zP* = 9.802/0.00079 = 12408 (i.e., the three-phase linear load influence is negligible), the parallel resonance "observed" from the VSD is located at *kr*, a = 72.9.
