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

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 resonance considering the Steinmetz circuit inductor resistance.

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

Characterization of Harmonic Resonances in

**VZ I Bus**

where

harmonic power quality.

• Supply system: *ZS*1 = 0.022 +*j*0.049 pu. • Railroad substation: *RL* = 1.341 pu,

this resistance (3) (*R*<sup>1</sup> ≈ 0.1342 pu, and therefore

therefore the highest harmonic voltages.)

the Presence of the Steinmetz Circuit in Power Systems 177

*V ZZZ I V ZZZ I V ZZZ I*

*Ak AAk ABk ACk Ak*

*Ck CAk CBk CCk Ck*

1 1

• *YSk* = *ZSk*–1 = (*RS* +*jkXS*)–1 corresponds to the admittance of the power supply system, which includes the impedance of the power supply network, the short-circuit impedance of the three-phase transformer and the impedance of the overhead lines

• *YLk*, *Y*1*<sup>k</sup>* and *dC*·*Y*2*<sup>k</sup>* correspond to the admittances of the Steinmetz circuit components (i.e., the inverse of the impedances *ZLk*, *Z*1*<sup>k</sup>* and *Z*2*<sup>k</sup>*/*dC* in Section 3, respectively). It can be observed that the diagonal and non-diagonal impedances of the harmonic impedance matrix **ZBus***k* (i.e., *ZAAk* to *ZCCk*) directly characterize the system harmonic behavior. Diagonal impedances are known as phase driving point impedances (Task force on Harmonic Modeling and Simulation, 1996) since they allow determining the contribution of the harmonic currents injected into any phase *F* (*IFk*) to the harmonic voltage of this phase (*VFk*). Non–diagonal impedances are the equivalent impedances between a phase and the rest of the phases since they allow determining the contribution of the harmonic currents injected into any phase *F* (*IFk*) to the harmonic voltage of any other phase *G* (*VGk*, with *G* ≠ *F*). Thus, the calculation of both sets of impedances is necessary because a resonance in either of them could cause a high level of distortion in the corresponding voltages and damage

As an example, a network as that in Fig. 4 was constructed in the laboratory and its harmonic response (i.e., **ZBus***k* matrix) was measured with the following per unit data

Considering that the system fundamental frequency is 50 Hz, the measurements of the **ZBus***<sup>k</sup>* impedance magnitudes (i.e., |*ZAAk*| to |*ZCCk* |) with *X*1, apr and *X*2, apr are plotted in Fig. 5 for

• The connection of the Steinmetz circuit causes a parallel resonance in the **Z**Bus*<sup>k</sup>* impedances that occurs in phases *A* and *C*, between which the capacitor is connected, and is located nearly at the same harmonic for all the impedances (labeled as *kp*, meas). This asymmetrical resonant behavior has an asymmetrical effect on the harmonic voltages (i.e., phases *A* and *C* have the highest harmonic impedance, and

τ

<sup>1</sup> ≈ 0.1341/2.234 = 0.06).

*k kk Bk BAk BBk BCk Bk*

feeding the Steinmetz circuit and the three-phase linear load.

(*UB* = 100 V and *SB* = 500 VA) and considering two cases (*dC* = 1 and 0.5):

λ*<sup>L</sup>* = 1.0. • External balancing equipment: According to (1) and (3), two pairs of reactances were connected with the railroad substation, namely *X*1, apr = 2.323 pu and *X*2, apr = 2.323 pu and *X*1 = 2.234 pu and *X*2 = 2.503 pu. The former was calculated by neglecting the inductor resistance (1) and the latter was calculated by considering the actual value of

both cases (*dC* = 1 in solid lines and *dC* = 0.5 in broken lines). It can be noticed that

= ⋅ = ⋅

12 1 2

*Sk k C C k k k Ak*

*Y YYY Y I d Y Y Y dY Y I*

*Y Y dY Y dY I*

++ − − = − ++ − ⋅ − − ++

*k Sk k Lk Lk Bk C C k Lk Sk k Lk Ck*

2 2

1

−

,

(5)

design value [(1) or (3)]. Thus, the capacitor value considered in the harmonic study is *dC*·*C*, i.e. −*j*1/(*dC*·*C*·ω1·*k*) = −*j·*(*X*2/*k*)/*dC* = *Z*2*<sup>k</sup>*/*dC* where ω1 = 2π·*f*1 and *f*1 is the fundamental frequency of the supply voltage. This parameter allows examining the impact of the capacitor bank degradation caused by damage in the capacitors or in their fuses on the power system harmonic response. If *dC* = 1, the capacitor has the design value [(1) or (3)] whereas if *dC* < 1, the capacitor value is lower than the design value.

Fig. 3. Power system harmonic analysis in the presence of the Steinmetz circuit.

#### **3.1 Study of the parallel resonance**

This Section examines the harmonic response of the system "observed" from the three-phase load. It implies analyzing the passive set formed by the utility supply system and the Steinmetz circuit (see Fig. 4). The system harmonic behavior is characterized by the equivalent harmonic impedance matrix, **ZBus***k*, which relates the *k*th harmonic three-phase voltages and currents at the three-phase load node, **V***k* = [*VAk VBk VCk*]T and **I***k* = [*IAk IBk ICk*]T. Thus, considering point *N* in Fig. 4 as the reference bus, this behavior can be characterized by the voltage node method,

Fig. 4. Study of the parallel resonance in the presence of the Steinmetz circuit.

$$\begin{aligned} \underline{\mathbf{V}}\_{i} &= \underline{\mathbf{Z}}\_{\text{Bulk}} \cdot \underline{\mathbf{L}}\_{k} \implies \begin{bmatrix} \underline{V}\_{\text{Ak}} \\ \underline{V}\_{\text{Bk}} \\ \underline{V}\_{\text{Ck}} \end{bmatrix} = \begin{bmatrix} \underline{\mathbf{Z}}\_{\text{Ak}} & \underline{\mathbf{Z}}\_{\text{Ak}} & \underline{\mathbf{Z}}\_{\text{ACk}} \\ \underline{\mathbf{Z}}\_{\text{Bk}} & \underline{\mathbf{Z}}\_{\text{Bk}} & \underline{\mathbf{Z}}\_{\text{Ck}} \\ \underline{\mathbf{Z}}\_{\text{Ck}} & \underline{\mathbf{Z}}\_{\text{Ck}} & \underline{\mathbf{Z}}\_{\text{Ck}} \end{bmatrix} \cdot \begin{bmatrix} \underline{I}\_{\text{Ak}} \\ \underline{I}\_{\text{Bk}} \\ \underline{I}\_{\text{Ck}} \end{bmatrix} \\ &= \begin{bmatrix} \underline{Y}\_{\text{Ak}} + \underline{Y}\_{\text{Lk}} + d\_{\text{C}} \underline{Y}\_{\text{Lk}} & -\underline{Y}\_{\text{Lk}} & -d\_{\text{C}} \underline{Y}\_{\text{Lk}} \\ -\underline{Y}\_{\text{Lk}} & \underline{Y}\_{\text{Sk}} + \underline{Y}\_{\text{Lk}} + \underline{Y}\_{\text{Lk}} & -\underline{Y}\_{\text{Lk}} \\ -d\_{\text{C}} \underline{Y}\_{\text{2k}} & -\underline{Y}\_{\text{Lk}} & \underline{Y}\_{\text{Sk}} + d\_{\text{C}} \underline{Y}\_{\text{Lk}} + \underline{Y}\_{\text{Lk}} \end{bmatrix}^{-1} \cdot \begin{bmatrix} \underline{I}\_{\text{Ak}} \\ \underline{I}\_{\text{Bk}} \\ \underline{I}\_{\text{R}}$$

where

176 Power Quality Harmonics Analysis and Real Measurements Data

design value [(1) or (3)]. Thus, the capacitor value considered in the harmonic study is *dC*·*C*, i.e. −*j*1/(*dC*·*C*·ω1·*k*) = −*j·*(*X*2/*k*)/*dC* = *Z*2*<sup>k</sup>*/*dC* where ω1 = 2π·*f*1 and *f*1 is the fundamental frequency of the supply voltage. This parameter allows examining the impact of the capacitor bank degradation caused by damage in the capacitors or in their fuses on the power system harmonic response. If *dC* = 1, the capacitor has the design value [(1) or (3)]

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

*Three-phase load*

*C*

*VCk*

*B*

*VBk*

*A*

*VAk*

*IBk*

*ICk*

**ZBus***<sup>k</sup>*

(*X*2/*k*)/*dC*

This Section examines the harmonic response of the system "observed" from the three-phase load. It implies analyzing the passive set formed by the utility supply system and the Steinmetz circuit (see Fig. 4). The system harmonic behavior is characterized by the equivalent harmonic impedance matrix, **ZBus***k*, which relates the *k*th harmonic three-phase voltages and currents at the three-phase load node, **V***k* = [*VAk VBk VCk*]T and **I***k* = [*IAk IBk ICk*]T. Thus, considering point *N* in Fig. 4 as the reference bus, this behavior can be characterized

*k·X*<sup>1</sup> *k·XL*

Fig. 3. Power system harmonic analysis in the presence of the Steinmetz circuit.

*<sup>N</sup> IAk*

*k·X*<sup>1</sup> *k·XL*

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

(*X*2/*k*)/*dC*

Fig. 4. Study of the parallel resonance in the presence of the Steinmetz circuit.

whereas if *dC* < 1, the capacitor value is lower than the design value.

*Utility supply system*

**3.1 Study of the parallel resonance** 

by the voltage node method,

*k·XS RS*


It can be observed that the diagonal and non-diagonal impedances of the harmonic impedance matrix **ZBus***k* (i.e., *ZAAk* to *ZCCk*) directly characterize the system harmonic behavior. Diagonal impedances are known as phase driving point impedances (Task force on Harmonic Modeling and Simulation, 1996) since they allow determining the contribution of the harmonic currents injected into any phase *F* (*IFk*) to the harmonic voltage of this phase (*VFk*). Non–diagonal impedances are the equivalent impedances between a phase and the rest of the phases since they allow determining the contribution of the harmonic currents injected into any phase *F* (*IFk*) to the harmonic voltage of any other phase *G* (*VGk*, with *G* ≠ *F*). Thus, the calculation of both sets of impedances is necessary because a resonance in either of them could cause a high level of distortion in the corresponding voltages and damage harmonic power quality.

As an example, a network as that in Fig. 4 was constructed in the laboratory and its harmonic response (i.e., **ZBus***k* matrix) was measured with the following per unit data (*UB* = 100 V and *SB* = 500 VA) and considering two cases (*dC* = 1 and 0.5):


Considering that the system fundamental frequency is 50 Hz, the measurements of the **ZBus***<sup>k</sup>* impedance magnitudes (i.e., |*ZAAk*| to |*ZCCk* |) with *X*1, apr and *X*2, apr are plotted in Fig. 5 for both cases (*dC* = 1 in solid lines and *dC* = 0.5 in broken lines). It can be noticed that

• The connection of the Steinmetz circuit causes a parallel resonance in the **Z**Bus*<sup>k</sup>* impedances that occurs in phases *A* and *C*, between which the capacitor is connected, and is located nearly at the same harmonic for all the impedances (labeled as *kp*, meas). This asymmetrical resonant behavior has an asymmetrical effect on the harmonic voltages (i.e., phases *A* and *C* have the highest harmonic impedance, and therefore the highest harmonic voltages.)

Characterization of Harmonic Resonances in

*IBk ICk*

**3.2 Study of the series resonance** 

**YBus***<sup>k</sup>*

*C*

*VBk*

*VCk*

*B*

*A*

**V**<sup>I</sup>

and

where

the Presence of the Steinmetz Circuit in Power Systems 179

*k·XS RS*

**I II**

Fig. 6. Study of the series resonance in the presence of the Steinmetz circuit.

harmonic three-phase currents and voltages at the node I, **I**<sup>I</sup>

*Ak Sk Sk*

*I Y Y*

this behavior can be characterized by the voltage node method,

0 00 00 0

−

3 2

*T k Sk Pk Lk C k*

*Y Y Y Y dY*

location and *YPk* = *ZPk*–1 = *g*LM# (|*ZP*1|,

0 00

I I-II I I

 = ⋅ 

**I YY V 0 YYV**

*k kk k*

II-I II II

*kk k*

*VAk Zpk*

This Section studies the harmonic response of the system "observed" from the utility supply system. It implies analyzing the passive set formed by the supply system impedances, the Steinmetz circuit and the three-phase load (see Fig. 6). The system harmonic behavior is characterized by the equivalent harmonic admittance matrix, **YBus***k*, which relates the *k*th

*<sup>k</sup>* = [*VAk VBk VCk*]T, respectively. Thus, considering point *N* in Fig. 6 as the reference bus,

*IAk N*

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

*<sup>k</sup>* = [*IAk IBk ICk*]T and

II II II

*Ak Bk Ck Ak Bk Ck*

*V V V V V V*

⋅

,

(6)

(*X*2/*k*)/*dC*

*k·X*<sup>1</sup> *k·XL*

11 2

2 3

1 2

=+++ (8)

*<sup>P</sup>*, *k*) is the three-phase load admittance. The function

00 0 0

0 00 0

*Y Y Y dY Y YY Y Y dY Y Y*

*Sk T k k C k Sk k Tk Lk Sk C k Lk T k*

00 0 0

*IY Y*

*Ck Sk Sk*

( ) <sup>1</sup> <sup>I</sup> I I-II II II-I I <sup>I</sup> , *Ak AAk ABk ACk Ak k Bk k k k k k k k BAk BBk BCk Bk Ck CAk CBk CCk Ck*

1 12 2 1

λ

*Y Y Y Y dY Y Y Y Y Y*

 = =− = = <sup>⋅</sup> **Bus I Y YY Y V Y V** (7)

> ; . *T k Sk Pk k C k T k Sk Pk k Lk*

=+++ =+++

Admittances *YSk*, *YLk*, *Y*1*<sup>k</sup>* and *dC*·*Y*2*<sup>k</sup>* were already introduced in the parallel resonance

*g*LM# (·) represents the admittance expressions of the load models 1 to 7 proposed in (Task

*I YYY V I YYY V I YYY V*

<sup>−</sup> <sup>−</sup> <sup>−</sup>

*Bk Sk Sk*

<sup>=</sup> <sup>−</sup> − − −− − −− −

*IY Y*

Fig. 5. Measured impedance - frequency matrix in the presence of the Steinmetz circuit with *X*1, apr = *X*2, apr = 2.323 pu (solid line: *dC* = 1; broken line: *dC* = 0.5).


The measurements of the **ZBus***k* impedance magnitudes (i.e. |*ZAAk*| to |*ZCCk* |) with *X*1 and *X*2 are not plotted for space reasons. In this case, the parallel resonance shifts to *kp*, meas ≈ 5.22 (*dC* = 1) and *kp*, meas ≈ 7.43 (*dC* = 0.5) but the general conclusions of the *X*1, apr and *X*2, apr case are true.

Fig. 6. Study of the series resonance in the presence of the Steinmetz circuit.

#### **3.2 Study of the series resonance**

This Section studies the harmonic response of the system "observed" from the utility supply system. It implies analyzing the passive set formed by the supply system impedances, the Steinmetz circuit and the three-phase load (see Fig. 6). The system harmonic behavior is characterized by the equivalent harmonic admittance matrix, **YBus***k*, which relates the *k*th harmonic three-phase currents and voltages at the node I, **I**<sup>I</sup> *<sup>k</sup>* = [*IAk IBk ICk*]T and **V**<sup>I</sup> *<sup>k</sup>* = [*VAk VBk VCk*]T, respectively. Thus, considering point *N* in Fig. 6 as the reference bus, this behavior can be characterized by the voltage node method,

I I-II I I II-I II II 11 2 1 2 2 3 00 0 0 0 00 0 00 0 0 0 00 00 0 0 00 *k kk k kk k Ak Sk Sk Bk Sk Sk Ck Sk Sk Sk T k k C k Sk k Tk Lk Sk C k Lk T k I Y Y IY Y IY Y Y Y Y dY Y YY Y Y dY Y Y* = ⋅ <sup>−</sup> <sup>−</sup> <sup>−</sup> <sup>=</sup> <sup>−</sup> − − −− − −− − **I YY V 0 YYV** II II II , *Ak Bk Ck Ak Bk Ck V V V V V V* ⋅ (6)

and

178 Power Quality Harmonics Analysis and Real Measurements Data

*kp*, meas ≈ 7.2

1.5

1.0

0.5

*kp*, meas ≈ 5.02


0

1.5

1.0

0.5


0

1.5

1.0

0.5


0

1

resonance.)

are true.

 *k*

3 5 7 9 11 1

*X*1, apr = *X*2, apr = 2.323 pu (solid line: *dC* = 1; broken line: *dC* = 0.5).

251 Hz is the frequency of the measured parallel resonance.)




Fig. 5. Measured impedance - frequency matrix in the presence of the Steinmetz circuit with

• In the case of *dC* = 1 (in solid lines), the connection of the Steinmetz circuit causes a parallel resonance measured close to the fifth harmonic (*kp*, meas ≈ 251/50 = 5.02, where

• If the Steinmetz circuit suffers capacitor bank degradation, the parallel resonance is shifted to higher frequencies. In the example, a 50% capacitor loss (i.e., *dC* = 0.5 in broken lines) shifts the parallel resonance close to the seventh harmonic (*ks*, meas ≈ 360/50 = 7.2, where 360 Hz is the frequency of the measured parallel

The measurements of the **ZBus***k* impedance magnitudes (i.e. |*ZAAk*| to |*ZCCk* |) with *X*1 and *X*2 are not plotted for space reasons. In this case, the parallel resonance shifts to *kp*, meas ≈ 5.22 (*dC* = 1) and *kp*, meas ≈ 7.43 (*dC* = 0.5) but the general conclusions of the *X*1, apr and *X*2, apr case

 *k* 3 5 7 9 11

*ZCB*| (pu)

*ZCA*| (pu)

*ZBA*| (pu)

$$\mathbf{I}\_{k}^{1} = \begin{bmatrix} \underline{I}\_{Ak} \\ \underline{I}\_{Ak} \\ \underline{I}\_{Ck} \end{bmatrix} = \left( \underline{\mathbf{Y}}\_{k}^{1} - \underline{\mathbf{Y}}\_{k}^{1\text{II}} \, \underline{\mathbf{Y}}\_{k}^{\text{II}} \, \, \underline{\mathbf{Y}}\_{k}^{\text{II}} \right) \underline{\mathbf{Y}}\_{k}^{1} = \underline{\mathbf{Y}}\_{\text{Bulk}} \underline{\mathbf{Y}}\_{k}^{1} = \begin{bmatrix} \underline{Y}\_{AAk} & \underline{Y}\_{Alk} & \underline{Y}\_{ACk} \\ \underline{Y}\_{BAk} & \underline{Y}\_{Blk} & \underline{Y}\_{Bck} \\ \underline{Y}\_{CAk} & \underline{Y}\_{Ck} & \underline{Y}\_{CCk} \end{bmatrix} \cdot \begin{bmatrix} \underline{V}\_{Ak} \\ \underline{V}\_{kb} \\ \underline{V}\_{cc} \end{bmatrix} \tag{7}$$

where

$$\begin{aligned} \underline{Y}\_{T1k} &= \underline{Y}\_{Sk} + \underline{Y}\_{pk} + \underline{Y}\_{1k} + d\_{\mathbb{C}} \underline{Y}\_{2k} &; \quad \underline{Y}\_{T2k} = \underline{Y}\_{Sk} + \underline{Y}\_{\mathbb{R}k} + \underline{Y}\_{1k} + \underline{Y}\_{\mathbb{L}k} \\\underline{Y}\_{T3k} &= \underline{Y}\_{Sk} + \underline{Y}\_{\mathbb{R}k} + \underline{Y}\_{\mathbb{L}k} + d\_{\mathbb{C}} \underline{Y}\_{\mathbb{2}k}. \end{aligned} \tag{8}$$

Admittances *YSk*, *YLk*, *Y*1*<sup>k</sup>* and *dC*·*Y*2*<sup>k</sup>* were already introduced in the parallel resonance location and *YPk* = *ZPk*–1 = *g*LM# (|*ZP*1|, λ*<sup>P</sup>*, *k*) is the three-phase load admittance. The function *g*LM# (·) represents the admittance expressions of the load models 1 to 7 proposed in (Task

Characterization of Harmonic Resonances in

*ks,* meas ≈ 7.3

*ks,* meas ≈ 5.1




2.5

2.0

1.0

1.5

0.5

0

2.5

2.0

1.0

1.5

0.5

0

2.5

2.0

1.0

1.5

0.5

1

0

 *k*

**4.1 Power system harmonic characterization** 

The most critical **Z**Bus*k* impedances are obtained from (5):

3579

the Presence of the Steinmetz Circuit in Power Systems 181




Fig. 7. Measured admittance - frequency matrix in the presence of the Steinmetz circuit with

In this Section, the magnitudes of the most critical **Z**Bus*<sup>k</sup>* impedances (i.e., |*ZAAk*|, |*ZCCk*|, |*ZACk*| and |*ZCAk*|) and **Y**Bus*<sup>k</sup>* admittances (i.e., |*YAAk*|, |*YCCk*|, |*YACk*| and |*YCAk*|) are

analytically studied in order to locate the parallel and series resonance, respectively.

*X*1, apr = 1.790 pu and *X*2, apr = 6.523 pu (solid line: *dC*= 1; broken line: *dC* = 0.5).

**4. Analytical study of power system harmonic response** 

1

 *k*

3579

*YCBk*| (pu)

*YCAk*| (pu)

*YBAk*| (pu)

force on Harmonic Modeling and Simulation, 2003), and |*ZP*1| and λ*<sup>P</sup>* are the magnitude and the displacement power factor of the load impedances at the fundamental frequency, respectively. For example, the expression *g*LM1 (|*ZP*1|, λ*P*, *k*) = 1/{|*ZP*1|·(λ*P* + *jk*(1 – λ*<sup>P</sup>*2 )1/2 )} corresponds to the series R-L impedance model, i.e. the model LM1 in (Task force on Harmonic Modeling and Simulation, 2003).

It can be observed that the diagonal and non-diagonal admittances of the harmonic admittance matrix **YBus***k* (i.e., *YAAk* to *YCCk*) directly characterize system harmonic behavior. Diagonal admittances allow determining the contribution of the harmonic voltages at any phase *F* (*VFk*) to the harmonic currents consumed at this phase (*IFk*). Non–diagonal admittances allow determining the contribution of the harmonic voltages at any phase *F* (*VFk*) to the harmonic currents consumed at any other phase *G* (*IGk*, with *G* ≠ *F*). Thus, the calculation of both sets of admittances is necessary because this resonance could lead to a high value of the admittance magnitude, magnify the harmonic currents consumed in the presence of background voltage distortion and damage harmonic power quality.

As an example, a network as that in Fig. 6 was constructed in the laboratory and its harmonic response (i.e., **YBus***k* matrix) was measured with the following per unit data (*UB* = 100 V and *SB* = 500 VA) and considering two cases (*dC* = 1 and 0.5):


Considering that the system fundamental frequency is 50 Hz, the measurements of the **YBus***<sup>k</sup>* admittance magnitudes (i.e. |*YAAk*| to |*YCCk* |) with *X*1, apr and *X*2, apr are plotted in Fig. 7 for both cases (*dC* = 1 in solid lines and *dC* = 0.5 in broken lines). It can be noted that


The measurements of the **YBus***k* admittance magnitudes (i.e., |*YAAk*| to |*YCCk* |) with *X*1 and *X*<sup>2</sup> are not plotted for space reasons. In this case, the series resonance shifts to *ks*, meas ≈ 6.31 (*dC* = 1) and *ks*, meas ≈ 8.83 (*dC* = 0.5) but the general conclusions of the *X*1, apr and *X*2, apr case are true.

and the displacement power factor of the load impedances at the fundamental frequency,

corresponds to the series R-L impedance model, i.e. the model LM1 in (Task force on

It can be observed that the diagonal and non-diagonal admittances of the harmonic admittance matrix **YBus***k* (i.e., *YAAk* to *YCCk*) directly characterize system harmonic behavior. Diagonal admittances allow determining the contribution of the harmonic voltages at any phase *F* (*VFk*) to the harmonic currents consumed at this phase (*IFk*). Non–diagonal admittances allow determining the contribution of the harmonic voltages at any phase *F* (*VFk*) to the harmonic currents consumed at any other phase *G* (*IGk*, with *G* ≠ *F*). Thus, the calculation of both sets of admittances is necessary because this resonance could lead to a high value of the admittance magnitude, magnify the harmonic currents consumed in the

As an example, a network as that in Fig. 6 was constructed in the laboratory and its harmonic response (i.e., **YBus***k* matrix) was measured with the following per unit data

• External balancing equipment: According to (1) and (3), two pairs of reactances were connected with the railroad substation, namely *X*1, apr = 1.790 pu and *X*2, apr = 6.523 pu and *X*1 = 1.698 pu and *X*2 = 10.03 pu. The former was calculated by neglecting the inductor resistance (1) and the latter was calculated by considering the actual value of

• 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

Considering that the system fundamental frequency is 50 Hz, the measurements of the **YBus***<sup>k</sup>* admittance magnitudes (i.e. |*YAAk*| to |*YCCk* |) with *X*1, apr and *X*2, apr are plotted in Fig. 7 for

• The connection of the Steinmetz circuit causes a series resonance in the **Y**Bus*<sup>k</sup>* admittances that occurs in phases *A* and *C*, between which the capacitor is connected, and is located nearly at the same harmonic for all the admittances (labeled as *ks*, meas). This asymmetrical resonant behavior has an asymmetrical effect on the harmonic consumed currents (i.e., phases *A* and *C* have the highest harmonic

• In the case of *dC* = 1 (in solid lines), the connection of the Steinmetz circuit causes a series resonance measured close to the fifth harmonic (*ks*, meas ≈ 255/50 = 5.1, where

• If the Steinmetz circuit suffers capacitor bank degradation, the series resonance is shifted to higher frequencies. In the example, a 50% capacitor loss (i.e., *dC* = 0.5 in broken lines) shifts the series resonance close to the seventh harmonic (*ks*, meas ≈ 365/50 = 7.3, where 365 Hz is the frequency of the measured series resonance.) The measurements of the **YBus***k* admittance magnitudes (i.e., |*YAAk*| to |*YCCk* |) with *X*1 and *X*<sup>2</sup> are not plotted for space reasons. In this case, the series resonance shifts to *ks*, meas ≈ 6.31 (*dC* = 1) and *ks*, meas ≈ 8.83 (*dC* = 0.5) but the general conclusions of the *X*1, apr and *X*2, apr case are true.

both cases (*dC* = 1 in solid lines and *dC* = 0.5 in broken lines). It can be noted that

admittance, and therefore the highest harmonic consumed currents.)

255 Hz is the frequency of the measured series resonance.)

τ

presence of background voltage distortion and damage harmonic power quality.

(*UB* = 100 V and *SB* = 500 VA) and considering two cases (*dC* = 1 and 0.5):

λ

λ

λ

*P*, *k*) = 1/{|*ZP*1|·(

<sup>1</sup> ≈ 0.1341/1.698 = 0.079).

*<sup>P</sup>* are the magnitude

λ*<sup>P</sup>*2 )1/2 )}

*P* + *jk*(1 –

force on Harmonic Modeling and Simulation, 2003), and |*ZP*1| and

respectively. For example, the expression *g*LM1 (|*ZP*1|,

Harmonic Modeling and Simulation, 2003).

• Supply system: *ZS*1 = 0.076 +*j*0.154 pu. • Railroad substation: *RL* = 1.464 pu, λ*L* = 0.95.

Modeling and Simulation, 2003).

this resistance (3) (*R*<sup>1</sup> ≈ 0.1342 pu, and therefore

Fig. 7. Measured admittance - frequency matrix in the presence of the Steinmetz circuit with *X*1, apr = 1.790 pu and *X*2, apr = 6.523 pu (solid line: *dC*= 1; broken line: *dC* = 0.5).
