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

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.

#### **4.1 Power system harmonic characterization**

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

Characterization of Harmonic Resonances in

admittances are written as

• the harmonic order *k*,

• the parameter

response studies.

rewritten from (14) as

and (11) as

(2),

the Presence of the Steinmetz Circuit in Power Systems 183

It must be noted that (9) and (13) depend on the supply system and the Steinmetz circuit admittances (i.e., *YSk*, *YLk*, *Y*1*<sup>k</sup>* and *dC*·*Y*2*<sup>k</sup>*) and (11) depends on the previous admittances and three-phase load admittances (i.e., *YSk*, *YLk*, *Y*1*<sup>k</sup>*, *dC*·*Y*2*<sup>k</sup>* and *YPk*). In the study, these

1

 τ

*L*, i.e. the single-phase load fundamental displacement power factor

λ*P*.

{ }

( ) <sup>1</sup> ,

*AAk S Sk Sk Stz k Lk Stz k*

*Z XY Y Y Y Y*

1 2

*P*

(15)

{ }

( ) ,

3 ( )

*S Sk Sk AAk Pk AAk*

*S Sk k Pk k*

*XYD Y D*

τ1,

It is worth pointing out that the resistance of the supply system is neglected (i.e., *ZSk* = *RS* + j*kXS* ≈ j*kXS*) and the resistance of the Steinmetz circuit is only considered in the inductor design [i.e., *Z*1*<sup>k</sup>*= *R*1 + j*kX*<sup>1</sup> ≈ j*kX*1 and *Z*2*<sup>k</sup>*= −*j·X*2/*k*, with *X*1 and *X*2 obtained from (3)]. This is because the real part of these impedances does not modify the series resonance frequency significantly (Sainz et al., 2007, 2009a) while the impact of *R*1 on Steinmetz circuit design modifies the resonance. This influence is not considered in the previous harmonic

In order to reduce the above number of variables, the **Z**Bus*<sup>k</sup>* impedances and **Y**Bus*<sup>k</sup>* admittances are normalized with respect to the supply system fundamental reactance *XS*. For example, the normalized magnitudes |*ZAAk*|N and |*YAAk*|N can be expressed from (9)

2 2

N N 3 ( )

where (15) has only terms *XS*·*Y*Sk, *XS*·*YLk*, *XS*·*Y*1*<sup>k</sup>*, *XS*·*dC*·*Y*2*<sup>k</sup>* and *XS*·*YPk*, which can be

*AAk S S AAk P*

<sup>+</sup> = = + ⋅

*XY Y N Y N Y XY X*

*S S S Sk k*

+ ++ = =

*X X XY D*

N 2

1 1 1 1 13 ; ; <sup>1</sup> <sup>3</sup>

{ } ( ) ( )

−− +

<sup>+</sup> ≈ − <sup>=</sup> ≈ =−

*C k Pk P P*

*<sup>d</sup> <sup>d</sup> d Y jk Yg Z k jX k <sup>R</sup>*

τ

13 3

*Yj Y Y j kX R jk jk X k <sup>R</sup>*

τ

 ττ

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

1 2 LM # 1

where (3) is used to obtain the Steinmetz circuit components *X*1 and *X*2, and *g*LM# (·) represents the three-phase load admittance models 1 to 7 proposed in (Task force on Harmonic Modeling and Simulation, 2003). Thus, it is observed that the **Z**Bus*<sup>k</sup>* impedances

*S LL L*

2

λ

and **Y**Bus*<sup>k</sup>* admittances are functions of eight variables, namely

• the supply system fundamental reactance *XS*,

• the R/X ratio of the Steinmetz circuit inductor

• the linear load fundamental displacement power factor

• the single-phase load resistance *RL*,

τ

*AAk*

*Z*

• the degradation parameter *dC*,

*Sk Lk k*

2 1

*C L <sup>L</sup> <sup>L</sup> <sup>C</sup>*

• the magnitude of the linear load fundamental impedance |*ZP*1| and

*L*

( ) ( )

+ ⋅ −

( ) ( )

; ,,, <sup>3</sup>

2 1

*L L*

 τ

λ

τ

(14)

λ*L*

1 1 1

λτ

 λ

$$\begin{aligned} \left| \underline{Z}\_{\rm{Akl}} \right| &= \left| \frac{\underline{Y}\_{\rm{Sk}}^{2} + \underline{Y}\_{\rm{Sk}} (\underline{Y}\_{\rm{Srt1k}} + \underline{Y}\_{\rm{Lk}}) + \underline{Y}\_{\rm{Stz2k}}}{\underline{Y}\_{\rm{sk}} \underline{D}\_{k}} \right| &; \quad \left| \underline{Z}\_{\rm{Ckt}} \right| = \left| \frac{\underline{Y}\_{\rm{Sk}}^{2} + \underline{Y}\_{\rm{Sk}} (\underline{Y}\_{\rm{Stz1k}} + \underline{Y}\_{\rm{Lk}}) + \underline{Y}\_{\rm{Stz2k}}}{\underline{Y}\_{\rm{sk}} \underline{D}\_{k}} \right| \\ \left| \underline{Z}\_{\rm{Atk}} \right| = \left| \underline{Z}\_{\rm{Ckt}} \right| &= \left| \frac{\underline{Y}\_{\rm{Stz2k}} + d\_{\rm{C}} \underline{Y}\_{\rm{s}k} \underline{Y}\_{\rm{Stk}}}{\underline{Y}\_{\rm{sk}} \underline{D}\_{k}} \right| \end{aligned} \tag{9}$$

where

$$\begin{aligned} \underline{Y}\_{\text{Stz1k}} &= \underline{Y}\_{1k} + d\_{\text{c}} \underline{Y}\_{\text{2k}} + \underline{Y}\_{\text{1k}} \qquad ; \qquad \underline{Y}\_{\text{Stz2k}} = \underline{Y}\_{1k} \underline{Y}\_{1k} + \underline{Y}\_{\text{1k}} d\_{\text{c}} \underline{Y}\_{\text{2k}} + \underline{Y}\_{\text{1k}} d\_{\text{c}} \underline{Y}\_{\text{2k}} \\\ \underline{D}\_{k} &= \underline{Y}\_{\text{Sk}}^{2} + 2 \underline{Y}\_{\text{Sk}} \underline{Y}\_{\text{8z1k}} + 3 \cdot \underline{Y}\_{\text{8z2k}}. \end{aligned} \tag{10}$$

The harmonic of the parallel resonance numerically obtained as the maximum value of the above impedance magnitudes is located nearly at the same harmonic for all the impedances (Sainz et al., 2007) and labeled as *kp*, n.

The most critical **Y**Bus*<sup>k</sup>* admittances are obtained from (7):

$$\begin{split} \left| \underline{Y}\_{Akl} \right| &= \left| \frac{\underline{Y}\_{\text{St}} \left( \underline{Y}\_{\text{St}} \underline{N}\_{Akl} + \underline{Y}\_{\text{R}} \underline{N}\_{Akl}^{(P)} \right)}{\underline{Y}\_{\text{St}} \underline{D}\_{k} + \underline{Y}\_{\text{R}} \cdot \underline{D}\_{k}^{(P)}} \right| \quad \quad \left| \underline{Y}\_{\text{Ck}} \right| = \left| \frac{\underline{Y}\_{\text{St}} \left( \underline{Y}\_{\text{St}} \underline{N}\_{\text{CCk}} + \underline{Y}\_{\text{R}} \underline{N}\_{\text{CCk}}^{(P)} \right)}{\underline{Y}\_{\text{St}} \underline{D}\_{k} + \underline{Y}\_{\text{R}} \cdot \underline{D}\_{k}^{(P)}} \right| \\ \left| \underline{Y}\_{\text{A}\text{Ck}} \right| = \left| \underline{Y}\_{\text{Ck}} \right| = \left| \frac{\underline{Y}\_{\text{S}k}^{2} \left( \underline{Y}\_{\text{S}k} d\_{\text{C}} \underline{Y}\_{\text{Z}k} + \underline{Y}\_{\text{S}\text{c2k}} + \underline{Y}\_{\text{R}} d\_{\text{C}} \underline{Y}\_{\text{2k}} \right)}{\underline{Y}\_{\text{S}k} \underline{D}\_{k} + \underline{Y}\_{\text{R}} \cdot \underline{D}\_{k}^{(P)}} \right| \end{split} \tag{11}$$

where

$$\begin{array}{llll}\underline{N}\_{AA} = \underline{Y}\_{\text{sl}}(\underline{Y}\_{\text{il}} + d\_{\text{c}}\underline{Y}\_{\text{2k}}) + 2\cdot\underline{Y}\_{\text{sl}z2k} & ; & \underline{N}\_{\text{CC1}} = \underline{Y}\_{\text{sl}}(\underline{Y}\_{\text{il}} + d\_{\text{c}}\underline{Y}\_{\text{2k}}) + 2\cdot\underline{Y}\_{\text{sl}z2k} \\\ \underline{N}^{(\underline{P})}\_{AA} = \underline{Y}\_{\text{R}}^2 + 2\cdot(\underline{Y}\_{\text{sl}} + \underline{Y}\_{\text{Sir}1k})\underline{Y}\_{\text{R}} + 3\cdot(\underline{Y}\_{\text{Sir}2k} + \underline{Y}\_{\text{sl}}(\underline{Y}\_{\text{1l}} + d\_{\text{c}}\underline{Y}\_{\text{2k}})) + \underline{Y}\_{\text{Sk}}(\underline{Y}\_{\text{Sl}} + 2\cdot\underline{Y}\_{\text{Lk}}) \\\ \underline{N}^{(\underline{P})}\_{cck} = \underline{Y}\_{\text{R}}^2 + 2\cdot(\underline{Y}\_{\text{Sk}} + \underline{Y}\_{\text{Sir}1k})\underline{Y}\_{\text{R}} + 3\cdot(\underline{Y}\_{\text{Sir}2k} + \underline{Y}\_{\text{Sk}}(\underline{Y}\_{\text{Lk}} + d\_{\text{c}}\underline{Y}\_{\text{2k}})) + \underline{Y}\_{\text{Sk}}(\underline{Y}\_{\text{Sk}} + 2\cdot\underline{Y}\_{\text{Lk}}) \\\ \underline{D}^{(\underline{P})}\_{\text{k}} = \underline{Y}\_{\text{R}}^2 + \left(3\underline{Y}\_{\text{S}k} + 2\cdot\underline{Y}\_{\text{Sir}1k}\right)\underline{Y}\_{\text{R}} + 3\cdot($$

The harmonic of the series resonance numerically obtained as the maximum value of the above admittance magnitudes is located nearly at the same harmonic for all the admittances (Sainz et al., in press) and labeled as *ks*, n.

Since the expressions of the **Y**Bus*<sup>k</sup>* admittances (11) are too complicated to be analytically analyzed, the admittance *YPk* is not considered in their determination (i.e., *YPk* = 0), and they are approximated to

$$\begin{aligned} \left| \underline{Y}\_{\text{AA}} \right| = \left| \underline{Y}\_{\text{AA, apx}} \right| = \left| \frac{\underline{Y}\_{\text{Si}} \underline{N}\_{\text{A} \text{A}}}{\underline{D}\_{k}} \right| \quad \left; \qquad \left| \underline{Y}\_{\text{CCk}} \right| = \left| \underline{Y}\_{\text{CCk}, apx} \right| = \left| \frac{\underline{Y}\_{\text{Si}} \underline{N}\_{\text{CCk}}}{\underline{D}\_{k}} \right| \\\ \left| \underline{Y}\_{\text{AC1}} \right| = \left| \underline{Y}\_{\text{CA}} \right| = \left| \underline{Y}\_{\text{AC1, apx}} \right| = \left| \underline{Y}\_{\text{AC1}, apx} \right| = \left| \frac{\underline{Y}\_{\text{SA}} (\underline{Y}\_{\text{AC}} d\_{\text{C}} \underline{Y}\_{\text{2k}} + \underline{Y}\_{\text{Sc2}} \underline{1})}{\underline{D}\_{k}} \right|. \end{aligned} \tag{13}$$

This approximation is based on the fact that, the three-phase load does not influence the series resonance significantly if large enough, (Sainz et al., 2009b). The harmonic of the series resonance numerically obtained as the maximum value of the above admittance magnitudes is located nearly at the same harmonic for all the admittances (Sainz et al., in press) and labeled as *ks*, napx.

It must be noted that (9) and (13) depend on the supply system and the Steinmetz circuit admittances (i.e., *YSk*, *YLk*, *Y*1*<sup>k</sup>* and *dC*·*Y*2*<sup>k</sup>*) and (11) depends on the previous admittances and three-phase load admittances (i.e., *YSk*, *YLk*, *Y*1*<sup>k</sup>*, *dC*·*Y*2*<sup>k</sup>* and *YPk*). In the study, these admittances are written as

$$\begin{aligned} \underline{Y}\_{ik} &= -j\frac{1}{kX\_{S}} \quad ; \quad \underline{Y}\_{\perp k} = \frac{1}{R\_{\perp}\left(1 + jk\tau\_{\perp}\right)} \quad ; \quad \underline{Y}\_{\perp k} = \frac{1}{jk \cdot X\_{\perp}} = -j\frac{1}{k} \Big(\frac{\lambda\_{\perp}\tau\_{\perp}}{\lambda\_{\perp}}\Big) \frac{1 + \sqrt{3}\tau\_{\perp}}{R\_{\perp}\left(\sqrt{3} - \tau\_{\perp}\right)}\\ \underline{d}\_{c}\underline{Y}\_{\perp k} &= \frac{d\_{c}}{-jX\_{\perp}/k} = jk\frac{d\_{c}\lambda\_{\perp}^{2}\left[\left(1 - \sqrt{3}\tau\_{\perp}\right) - \tau\_{\perp}\left(\sqrt{3} + \tau\_{\perp}\right)\right]}{R\_{\perp}\left(\sqrt{3} - \tau\_{\perp}\right)} \quad ; \quad \underline{Y}\_{\mathbb{R}} = \mathsf{g}\_{\perp \text{M4\pi}}\left(\left|\underline{Z}\_{\mathbb{R}^{3}}\right|, \lambda\_{\mathbb{R}}, k\right), \end{aligned} \tag{14}$$

where (3) is used to obtain the Steinmetz circuit components *X*1 and *X*2, and *g*LM# (·) represents the three-phase load admittance models 1 to 7 proposed in (Task force on Harmonic Modeling and Simulation, 2003). Thus, it is observed that the **Z**Bus*<sup>k</sup>* impedances and **Y**Bus*<sup>k</sup>* admittances are functions of eight variables, namely

• the harmonic order *k*,

182 Power Quality Harmonics Analysis and Real Measurements Data

*Y YY Y Y Y YY Y Y Z Z*

+ ++ + ++ = =

1 2 11 2

( ) ( ) ( ) ( )

12 2 1

*Sk AAk Sk CCk*

*k k*

*D D*

2 2

( ) .

*Sk Sk k Stz k C*

*D*

*k*

*P P*

(9)

(11)

(12)

(13)

( ) ( ) ;

*Sk Sk Stz k Lk Stz k Sk Sk Stz k k Stz k*

*Sk k Sk k*

*Y D Y D*

*D Y YY Y* (10)

( ) () ;

*Sk Sk AAk Pk AAk Sk Sk CCk Pk CCk*

*Sk k Pk k Sk k Pk k*

12 2 2 2

2 ( ) 3 ( ( )) ( 2 ) 2 ( ) 3 ( ( )) ( 2 )

1 2 12

*YD Y D YD Y D*

2 2

,

=+ + =+ +

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

;

11 2 2 1 212

*Stz k k C k Lk Stz k Lk k Lk k k k C C*

The harmonic of the parallel resonance numerically obtained as the maximum value of the above impedance magnitudes is located nearly at the same harmonic for all the impedances

+ + = = + ⋅ + ⋅

*Y YN YN Y YN YN Y Y*

*AAk P P CCk*

22 2 ( )

= + +⋅ = + +⋅ = +⋅ + +⋅ + + + +⋅ = +⋅ + +⋅ + + + +⋅

*N Y Y dY Y N Y Y dY Y N Y Y Y Y Y Y Y dY Y Y Y N Y Y Y Y Y Y Y dY Y Y Y*

*AAk Sk k C C k Stz k CCk Sk Lk k Stz k*

( )2 ; ( )2

*AAk Pk Sk Stz k Pk Stz k Sk k C k Sk Sk Lk*

*CCk Pk Sk Stz k Pk Stz k Sk Lk C k Sk Sk k*

The harmonic of the series resonance numerically obtained as the maximum value of the above admittance magnitudes is located nearly at the same harmonic for all the admittances

Since the expressions of the **Y**Bus*<sup>k</sup>* admittances (11) are too complicated to be analytically analyzed, the admittance *YPk* is not considered in their determination (i.e., *YPk* = 0), and they

≈ = ≈ =

*Y N Y N Y Y Y Y*

, apx , apx

This approximation is based on the fact that, the three-phase load does not influence the series resonance significantly if large enough, (Sainz et al., 2009b). The harmonic of the series resonance numerically obtained as the maximum value of the above admittance magnitudes is located nearly at the same harmonic for all the admittances (Sainz et al., in

;

<sup>+</sup> =≈ = =

, apx , apx

*Y Y dY Y YYY Y*

*AAk AAk CCk CCk*

( ) ,

*Sk Sk k Stz k Pk k C C*

*Sk k Pk k*

*YD Y D*

<sup>1</sup> 2 1 (3 2 ) 3 ( ) 4 . *<sup>P</sup> k Pk Sk Stz k Pk Y Y Y Y Y Y YY Sk Stz k Sk Stz k*

2 2

*Y D*

*Y dY Y Z Z*

<sup>+</sup> = =

= + +⋅

*k Sk Sk Stz k Stz k*

The most critical **Y**Bus*<sup>k</sup>* admittances are obtained from (7):

2

+ + = = + ⋅

*D* = + +⋅ +⋅ + + () 2 <sup>2</sup>

*ACk CAk ACk ACk*

*Y Y dY Y Y dY Y Y*

*ACk CAk P*

*ACk CAk*

2

(Sainz et al., 2007) and labeled as *kp*, n.

where

where

() 2

*P*

*P*

are approximated to

press) and labeled as *ks*, napx.

() 2

(Sainz et al., in press) and labeled as *ks*, n.

*Stz k C k Sk*

*Sk k*

1 2

2 3.

*AAk CCk*


It is worth pointing out that the resistance of the supply system is neglected (i.e., *ZSk* = *RS* + j*kXS* ≈ j*kXS*) and the resistance of the Steinmetz circuit is only considered in the inductor design [i.e., *Z*1*<sup>k</sup>*= *R*1 + j*kX*<sup>1</sup> ≈ j*kX*1 and *Z*2*<sup>k</sup>*= −*j·X*2/*k*, with *X*1 and *X*2 obtained from (3)]. This is because the real part of these impedances does not modify the series resonance frequency significantly (Sainz et al., 2007, 2009a) while the impact of *R*1 on Steinmetz circuit design modifies the resonance. This influence is not considered in the previous harmonic response studies.

In order to reduce the above number of variables, the **Z**Bus*<sup>k</sup>* impedances and **Y**Bus*<sup>k</sup>* admittances are normalized with respect to the supply system fundamental reactance *XS*. For example, the normalized magnitudes |*ZAAk*|N and |*YAAk*|N can be expressed from (9) and (11) as

$$\begin{aligned} \left| \underline{Z}\_{Akl} \right|\_{\rm N} &= \frac{\left| \underline{Z}\_{Akl} \right|}{X\_S} = \frac{1}{X\_S} \left| \frac{X\_S^2 \left[ \underline{Y}\_{Sk}^2 + \underline{Y}\_{Sk} (\underline{Y}\_{S1k} + \underline{Y}\_{Lk}) + \underline{Y}\_{S12k} \right]}{X\_S^2 \underline{Y}\_{Sk} \underline{D}\_k} \right| \\\ \left| \underline{Y}\_{Akl} \right|\_{\rm N} &= X\_S \left| \underline{Y}\_{Akl} \right|\_{\rm N} = X\_S \left| \frac{X\_S^3 \underline{Y}\_{Sk} \left( \underline{Y}\_{Sk} \underline{N}\_{Akl} + \underline{Y}\_{Pk} \underline{N}\_{Akl}^{(P)} \right)}{X\_S^3 \left[ \underline{Y}\_{Sk} \underline{D}\_k + \underline{Y}\_{Pk} \cdot \underline{D}\_k^{(P)} \right]} \right| \end{aligned} \tag{15}$$

where (15) has only terms *XS*·*Y*Sk, *XS*·*YLk*, *XS*·*Y*1*<sup>k</sup>*, *XS*·*dC*·*Y*2*<sup>k</sup>* and *XS*·*YPk*, which can be rewritten from (14) as

Characterization of Harmonic Resonances in

these denominators can be written as

τ

2

λ

λ τ

( ) <sup>2</sup>

∂ Δ

where

and

the Presence of the Steinmetz Circuit in Power Systems 185

The series resonance study in the next Section is only valid for z*P* > 20 because the approximation of not considering the three-phase load admittance (i.e., *YPk* = 0 ) is based on the fact that this load does not strongly influence the series resonance if z*P* is above 20. Nevertheless, the magnitude of the normalized admittances at the resonance point is low for

It is numerically verified that the parallel and series resonance, i.e. the maximum magnitude values of the normalized **Z**Bus*<sup>k</sup>* impedances and approximated **Y**Bus*<sup>k</sup>* admittances [obtained from (9) and (13)] with respect to the harmonic *k* respectively, coincide with the minimum value of their denominators for the whole range of system variables. Thus, from (9) and (13),

( ) ( ) ( )

*AAk CCk ACk*

= = ⋅

*Y Y kY*

2

*C*

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

− − = = <sup>+</sup> −− +

From (17), it is observed that the series resonances of |*YAAk*, apx|N and |*YCCk*, apx|N admittances match up because their denominators are the same. This is true for the series resonance of |*YACk*, apx|N admittance and the parallel resonance of |*ZAAk*|N, |*ZCCk*|N and |*ZACk*|N impedances. However, despite the discrepancy in the denominator degrees, it is numerically verified that the harmonic of the parallel and series resonance is roughly the same for all the impedances and admittances. Then, these resonances are located from the minimum value of the |*YACk*, apx|N, |*ZAAk*|N, |*ZCCk*|N and |*ZACk*|N denominator because it is the simplest. In the study, this denominator is labeled as |Δ*k*| for clarity and the harmonic of the parallel and series resonance numerically obtained as the minimum value of |Δ*k*| is labeled as *kr*, Δ for both resonances. This value is analytically located by equating to zero the

*L L*

*r r*

derivative of |Δ*k*|2 with respect to *k*, which can be arranged in the following form:

*d*

*L L L L L L*

( ) ( )

1 3 13 3

4 2 4 2 <sup>1</sup> 1 2 ,a 1 ,a 2 6 ( ) 0, *<sup>k</sup> Hkk Gk G k Gk G r r <sup>k</sup>*

∂ (20)

= ++ + +=

λ

*L L LL L*

3 3

= ++ = − ++ +

( ) ( ) ( ) , apx , apx , apx NN N

*AAk CCk ACk*

NN N

2

τ

; .

 ττ  τ

{ } ( ) ( )

 τ

> τ

*C*

*d*

(2 3) 3 ; ( ( 2) 2 3)

(17)

(18)

(19)

z*P* < 20 and the consumed currents do not increase significantly (Sainz et al., 2009a).

**4.2 Analytical location of the parallel and series resonance** 

2 2 12 34

1 1 12 1

(2 3) ; ( 2)

31 4 1

*<sup>x</sup> H r x H rx*

*x x* τ

> τ

*L L*

= + =− +

*kkHk H j Hk H*

= + +⋅ +

( ) ( ),

= ⋅ = ⋅ = ⋅

Den Den Den

Den Den Den

*<sup>x</sup> H r x x H xr r*

*kZ kZ kZ*

$$X\_{S}\underline{Y}\_{S} = -j\frac{1}{k} \quad , \quad X\_{S}\underline{Y}\_{Lk} = \frac{1}{r\_{\!\!\!L}(1+jk\tau\_{\!\!L})} \quad , \quad X\_{S}\underline{Y}\_{R} = \mathcal{g}\_{\text{1M}\#,N}\left(z\_{\mathbb{P}^{\mu}}\,\mathcal{A}\_{\text{p}}\,k\right)$$

$$X\_{S}\underline{Y}\_{\text{1k}} = \frac{X\_{S}}{jk\cdot X\_{1}} = -j\frac{1}{k\cdot x\_{1}} = -j\frac{1}{k}\left(\frac{\mathcal{A}\_{\text{p}}\tau\_{1}}{\mathcal{A}\_{1}}\right)^{2}\frac{1+\sqrt{3}\tau\_{\!\!L}}{r\_{\!\!\!L}\left(\sqrt{3}-\tau\_{1}\right)}\,\tag{16}$$

$$\text{y}\quad \mathcal{A} \qquad \qquad d\_{\text{c}}\lambda^{2}\left[\left(1-\sqrt{3}\tau\_{\!\!L}\right)-\tau\_{1}\left(\sqrt{3}+\tau\_{\!\!L}\right)\right]$$

$$X\_s d\_\subset \underline{Y}\_{2\lambda} = \frac{X\_s d\_\subset}{-jX\_2/k} = jk \frac{d\_\subset \lambda\_L^2}{x\_2} = jk \frac{d\_\subset \lambda\_L^2 \left\{ \left( 1 - \sqrt{3}\tau\_L \right) - \tau\_1 \left( \sqrt{3} + \tau\_L \right) \right\}}{r\_L \left( \sqrt{3} - \tau\_1 \right)},$$

where *g*LM#, N (·) represents the normalized expressions of the three-phase load models 1 to 7 proposed in (Task force on Harmonic Modeling and Simulation, 2003), *rL* = *RL* /*XS* and *zP* = |*ZP*1| /*XS*. The normalized expressions *g*LM#, N (·) are obtained and presented in (Sainz et al., 2009a) but are not included in the present text for space reasons. As an example, the normalized expression of model LM1 in (Task force on Harmonic Modeling and Simulation, 2003) is *g*LM1, N (z*p*, λ*P*, *k*) = 1/{z*p*·(λ*P* + *jk*(1−λ*<sup>P</sup>*2 )1/2 )} (Sainz et al., 2009a).

From (15), it is interesting to note that the normalization does not modify the parallel and series resonance (*kp*, n, *ks*, n and *ks*, napx), but the number of variables of the normalized **Z**Bus*<sup>k</sup>* impedances and **Y**Bus*<sup>k</sup>* admittances are reduced to seven (16), i.e.,


Moreover, the usual ranges of values of these variables can be obtained by relating them with known parameters to study resonances under power system operating conditions. Thus, the power system harmonic response is analyzed for the following variable ranges:


The ratios *rL* and z*P* are equal to the ratios λ*<sup>L</sup>*·*SS* /*SL* and *SS* /*SP* (Sainz et al., 2009a), where *SS* is the short-circuit power at the PCC bus, *SL* is the apparent power of the single-phase load and *SP* is the apparent power of the three-phase load. Thus, the range of these ratios is determined considering the usual values of the ratios *SS* /*SL* and *SS* /*SP* (Chen, 1994; Chen & Kuo, 1995) and the fundamental displacement power factors λ*L* and λ*P*.

In next Section, the normalized magnitudes of the most critical **Z**Bus*<sup>k</sup>* impedances (9) and approximated **Y**Bus*<sup>k</sup>* admittances (13) are analytically studied to obtain simple expressions for locating the parallel and series resonance. Thus, these expressions are functions of the following five variables only: *k*, *rL* = *RL* /*XS*, λ*L*, τ1 and *dC*.

The series resonance study in the next Section is only valid for z*P* > 20 because the approximation of not considering the three-phase load admittance (i.e., *YPk* = 0 ) is based on the fact that this load does not strongly influence the series resonance if z*P* is above 20. Nevertheless, the magnitude of the normalized admittances at the resonance point is low for z*P* < 20 and the consumed currents do not increase significantly (Sainz et al., 2009a).

#### **4.2 Analytical location of the parallel and series resonance**

It is numerically verified that the parallel and series resonance, i.e. the maximum magnitude values of the normalized **Z**Bus*<sup>k</sup>* impedances and approximated **Y**Bus*<sup>k</sup>* admittances [obtained from (9) and (13)] with respect to the harmonic *k* respectively, coincide with the minimum value of their denominators for the whole range of system variables. Thus, from (9) and (13), these denominators can be written as

$$\begin{split} \text{Den} \left( \left| \underline{Y}\_{\text{AA},\text{apx}} \right|\_{\text{N}} \right) &= \text{Den} \left( \left| \underline{Y}\_{\text{CC},\text{apx}} \right|\_{\text{N}} \right) = k \cdot \text{Den} \left( \left| \underline{Y}\_{\text{AC},\text{apx}} \right|\_{\text{N}} \right) \\ = k \cdot \text{Den} \left( \left| \underline{Z}\_{\text{AA}} \right|\_{\text{N}} \right) &= k \cdot \text{Den} \left( \left| \underline{Z}\_{\text{CC}} \right|\_{\text{N}} \right) = k \cdot \text{Den} \left( \left| \underline{Z}\_{\text{AC}} \right|\_{\text{N}} \right) \\ = k \left| k (H\_{1}k^{2} + H\_{2}) + j \cdot (H\_{3}k^{2} + H\_{4}) \right|\_{\text{N}} \end{split} \tag{17}$$

where

184 Power Quality Harmonics Analysis and Real Measurements Data

*S Sk S Lk S Pk P P L L*

*XY j XY XY g z k <sup>k</sup> r jk*

τ

λτ

λ

λ

where *g*LM#, N (·) represents the normalized expressions of the three-phase load models 1 to 7 proposed in (Task force on Harmonic Modeling and Simulation, 2003), *rL* = *RL* /*XS* and *zP* = |*ZP*1| /*XS*. The normalized expressions *g*LM#, N (·) are obtained and presented in (Sainz et al., 2009a) but are not included in the present text for space reasons. As an example, the normalized expression of model LM1 in (Task force on Harmonic Modeling and Simulation,

From (15), it is interesting to note that the normalization does not modify the parallel and series resonance (*kp*, n, *ks*, n and *ks*, napx), but the number of variables of the normalized **Z**Bus*<sup>k</sup>*

• the ratio of the single-phase load resistance to the supply system fundamental reactance

• the ratio of the linear load fundamental impedance magnitude to the supply system

Moreover, the usual ranges of values of these variables can be obtained by relating them with known parameters to study resonances under power system operating conditions. Thus, the power system harmonic response is analyzed for the following variable ranges:

λ

*<sup>P</sup>* = (0.9, ..., 1).

is the short-circuit power at the PCC bus, *SL* is the apparent power of the single-phase load and *SP* is the apparent power of the three-phase load. Thus, the range of these ratios is determined considering the usual values of the ratios *SS* /*SL* and *SS* /*SP* (Chen, 1994; Chen &

In next Section, the normalized magnitudes of the most critical **Z**Bus*<sup>k</sup>* impedances (9) and approximated **Y**Bus*<sup>k</sup>* admittances (13) are analytically studied to obtain simple expressions for locating the parallel and series resonance. Thus, these expressions are functions of the

1 and *dC*.

λ*L*, τ

λ

1 = (0, ..., 0.5).

λ

11 1 1 2

*S L L*

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

<sup>+</sup> = =− =− ⋅ ⋅ <sup>−</sup>

= == <sup>−</sup> <sup>−</sup>

λ

*<sup>X</sup> X Y j j jk X k x k <sup>r</sup>*

*<sup>d</sup> Xd d XdY jk jk jX k x <sup>r</sup>*

λ*P* + *jk*(1−

impedances and **Y**Bus*<sup>k</sup>* admittances are reduced to seven (16), i.e.,

2 2 1

1

*S k*

2

*S C k*

λ

τ

• the degradation parameter *dC*,

• Harmonic: *k* = (1, ..., 15).

• Steinmetz circuit inductor:

• Linear load: z*P* = (5, ..., 1000) and

*P*, *k*) = 1/{z*p*·(

• the R/X ratio of the Steinmetz circuit inductor

fundamental reactance z*P* = |*ZP*1| /*XS* and

• Single-phase load: *rL* = (5, ..., 1000) and

• Degradation parameter: *dC* = (0.25, ..., 1).

The ratios *rL* and z*P* are equal to the ratios

following five variables only: *k*, *rL* = *RL* /*XS*,

• the linear load fundamental displacement power factor

τ

Kuo, 1995) and the fundamental displacement power factors

2003) is *g*LM1, N (z*p*,

• the harmonic order *k*,

*rL* = *RL* /*XS*, • the parameter

(2),

2 1

*L*

*L*

1 1 13 , <sup>3</sup>

*C L <sup>L</sup> <sup>L</sup> SC C*

1 1 , , , , <sup>1</sup>

( ) ( )

LM #, N

λ

(16)

λ*L*

{ } ( ) ( ) ( )

−− +

τ

τ

1

, <sup>3</sup>

 τ

( )

 τ

13 3

*<sup>P</sup>*2 )1/2 )} (Sainz et al., 2009a).

*L*, i.e. the single-phase load fundamental displacement power factor

λ*P*.

> λ*L* and λ*P*.

*<sup>L</sup>*·*SS* /*SL* and *SS* /*SP* (Sainz et al., 2009a), where *SS*

τ1,

*<sup>L</sup>* = (0.9, ..., 1).

 ττ

$$\begin{aligned} H\_1 &= r\_L \tau\_L (2\mathbf{x}\_1 + 3) + 3\mathbf{x}\_1 \quad \; ; \quad H\_2 = -\left(\frac{\mathbf{x}\_2}{d\_\odot}\right) (\mathbf{x}\_1 (r\_L \tau\_L + 2) + 2r\_L \tau\_L + 3) \\\ H\_3 &= r\_L (2\mathbf{x}\_1 + 3) \quad \; ; \quad H\_4 = -\left(\frac{\mathbf{x}\_2}{d\_\odot}\right) r\_L (\mathbf{x}\_1 + 2) \end{aligned} \tag{18}$$

and

$$\mathbf{x}\_{1} = \left(\frac{\mathcal{\lambda}\_{\mathrm{L}}}{\mathcal{\lambda}\_{\mathrm{L}}\boldsymbol{\pi}\_{\mathrm{L}}}\right)^{2} \frac{r\_{\mathrm{L}}\left(\sqrt{3} - \boldsymbol{\pi}\_{\mathrm{L}}\right)}{1 + \sqrt{3}\boldsymbol{\pi}\_{\mathrm{L}}} \qquad ; \qquad \mathbf{x}\_{2} = \frac{r\_{\mathrm{L}}\left(\sqrt{3} - \boldsymbol{\pi}\_{\mathrm{L}}\right)}{\lambda\_{\mathrm{L}}^{2}\left\{\left(1 - \sqrt{3}\boldsymbol{\pi}\_{\mathrm{L}}\right) - \boldsymbol{\pi}\_{\mathrm{L}}\left(\sqrt{3} + \boldsymbol{\pi}\_{\mathrm{L}}\right)\right\}}. \tag{19}$$

From (17), it is observed that the series resonances of |*YAAk*, apx|N and |*YCCk*, apx|N admittances match up because their denominators are the same. This is true for the series resonance of |*YACk*, apx|N admittance and the parallel resonance of |*ZAAk*|N, |*ZCCk*|N and |*ZACk*|N impedances. However, despite the discrepancy in the denominator degrees, it is numerically verified that the harmonic of the parallel and series resonance is roughly the same for all the impedances and admittances. Then, these resonances are located from the minimum value of the |*YACk*, apx|N, |*ZAAk*|N, |*ZCCk*|N and |*ZACk*|N denominator because it is the simplest. In the study, this denominator is labeled as |Δ*k*| for clarity and the harmonic of the parallel and series resonance numerically obtained as the minimum value of |Δ*k*| is labeled as *kr*, Δ for both resonances. This value is analytically located by equating to zero the derivative of |Δ*k*|2 with respect to *k*, which can be arranged in the following form:

$$\frac{\partial \left( \left| \Delta\_k \right|^2 \right)}{\partial k} = 6H\_1 k (k^4 + G\_1 k^2 + G\_2) \qquad \Rightarrow \qquad k\_{r,a}^4 + G\_1 k\_{r,a}^2 + G\_2 = 0,\tag{20}$$

Characterization of Harmonic Resonances in

*kr*, a = 6.16 and 8.74 (

From these results, it is seen that

laboratory test of Section 3.

three-phase load models.

45

35

25

15

5

5

*kp*, n, *ks*, n, *kr*, a

• Series resonance: *kr*, a = 4.95 and 7.03 (

τ

correctly, i.e. *kr*, a ≈ *kp*, n and *kr*, a ≈ *ks*, napx.

the Presence of the Steinmetz Circuit in Power Systems 187

τ

1 = 7.9% and *dC* = 1.0 and 0.5, respectively).

• As the variable *zP* = 30.788/0.154 = 199.9 > 20, the numerical results obtained from

• The harmonic of the |*ZAAk*|N and |*YAAk*, apx|N (and therefore |*YAAk*|N) maximum values nearly coincides with the harmonic of the |Δ*k*| minimum value, *kr*, <sup>Δ</sup> ≈ *kp*, n and *kr*, <sup>Δ</sup> ≈ *ks*, napx, and that (22) provides the harmonic of the parallel and series resonance

• Although the resistances *RS* and *R*1 of the supply system and the Steinmetz circuit inductor are neglected in the analytical study [i.e., *ZSk* ≈ *j*·*k*·*XS* and *Z*1*<sup>k</sup>* ≈ *j*·*k*·*X*1 in (14)], the results are in good agreement with the experimental measurements in Sections 3.1 and 3.2, i.e. *kr*, a ≈ *kp*, meas and *kr*, a ≈ *ks*, meas. However, the magnitude values obtained numerically are greater than the experimental measurements (e.g.,

plot of Fig. 8 and |*ZAAk*| ≈ 1.1 pu for *kp*, meas = 7.2 in Fig. 5 or

• The influence of the resistance *R*1 on Steinmetz circuit design (3) shifts the parallel and series resonance to higher frequencies. This was also experimentally verified in the

Fig. 9 compares *kr*, a, with *kp*, n and *ks*, n. Considering the validity range of the involved variables, the values leading to the largest differences are used. It can be observed that *kr*, a provides the correct harmonic of the parallel and series resonance. The largest differences obtained are below 10% and correspond to *ks*, n when z*P* = 20, which is the lowest acceptable z*P* value to apply the *kr*, a analytical expression. Although only the linear load model LM1 is considered in the calculations, it is verified that the above conclusions are true for the other

10


*<sup>L</sup>* = 1.0, *dC* = 0.5,

*ks,* n (*zP* = 20,

*ks,* n (*zP* = 50,

*kp,* n, *ks,* n (*zP* = ∞)

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

τ<sup>1</sup> = 0.4

λ

λ

*<sup>P</sup>* = 0.95, LM1)

*<sup>P</sup>* = 0.95, LM1)

− *kr*, a|/*kres*(%)

0

The previous research unifies the study of the parallel and series resonance, providing an expression unique to their location. This expression is the same as in the series resonance case (Sainz et al., 2007), but substantially improves those obtained in the parallel resonance case (Sainz et al., in press). Moreover, the Steinmetz circuit inductor resistance is considered

2

4

6

8




plot of Fig. 8 and |*YAAk*| ≈ 2.1 pu for *kp*, meas = 7.2 in Fig. 7).

 *rL* 50 100 150 200 250 300

Fig. 9. Comparison between *kres* and *kr*, a (*kres* = *kp*, n or *ks*, n).

1 = 0 and *dC* = 1.0 and 0.5, respectively) and

τ

τ

1 = 0% and *dC* = 0.5

1 = 0% and *dC* = 0.5

where *kr*, a is the harmonic of the parallel and series resonance analytically obtained and

$$\mathbf{G}\_1 = \frac{\mathbf{2} \cdot \{\mathbf{2} H\_2 H\_1 + H\_3^2\}}{\mathbf{3} \cdot H\_1^2} \qquad ; \qquad \mathbf{G}\_2 = \frac{H\_2^2 + \mathbf{2} \cdot H\_4 H\_3}{\mathbf{3} \cdot H\_1^2}. \tag{21}$$

Thus, the root of equation (20) allows locating the parallel and series resonance:

$$k\_{r,a} = \sqrt{\frac{-G\_1 + \sqrt{G\_1^2 - 4 \cdot G\_2}}{2}}.\tag{22}$$

Fig. 8. Resonance location: a) Parallel resonance (power system data in Section 3.1: *XS* = 0.049 pu, *RL* = 1.341 pu and λ*<sup>L</sup>* = 1.0). b) Series resonance (power system data in Section 3.2: *XS* = 0.154 pu, *RL* = 1.464 pu, λ*L* = 0.95, |*ZP*1| = 30.788 pu, λ*P* = 0.95 and threephase load model LM1).

To illustrate the above study, Fig. 8 shows |*ZAAk*|N, |*YAAk*|N, |*YAAk*, apx|N and |Δ*k*| for the power systems presented in the laboratory tests of Sections 3.1 and 3.2, and the analytical results of the resonances (22) for these systems are

• Parallel resonance: *kr*, a = 4.94 and 7.05 (τ1 = 0 and *dC* = 1.0 and 0.5, respectively) and *kr*, a = 5.13 and 7.33 (τ1 = 6.0% and *dC* = 1.0 and 0.5, respectively).

• Series resonance: *kr*, a = 4.95 and 7.03 (τ1 = 0 and *dC* = 1.0 and 0.5, respectively) and *kr*, a = 6.16 and 8.74 (τ1 = 7.9% and *dC* = 1.0 and 0.5, respectively).

From these results, it is seen that

186 Power Quality Harmonics Analysis and Real Measurements Data

2 2 21 3 2 43

*H H*

2 11 2

> τ<sup>1</sup> = 6.0%

> τ<sup>1</sup> = 7.9%

<sup>4</sup> . <sup>2</sup> *<sup>r</sup>*

4

1 1

; . 3 3

⋅ + + ⋅ = = ⋅ ⋅ (21)

*GG G <sup>k</sup>* − + −⋅ <sup>=</sup> (22)

5

2.5



0

1

0

*ks,* n = 8.81

*kr,* Δ = 8.74 *ks,* napx = 8.75

*P* = 0.95 and three-

0.5



 *k* 567 9 8

λ

1 = 0 and *dC* = 1.0 and 0.5, respectively) and

*ks,* n = 6.22

*kr,* Δ = 6.16 *ks,* napx = 6.18

*<sup>L</sup>* = 1.0). b) Series resonance (power system data in

*L* = 0.95, |*ZP*1| = 30.788 pu,

*kp,* n = 5.13 *kp,* n = *kr,* Δ = 7.33

*kr,* Δ = 5.14

where *kr*, a is the harmonic of the parallel and series resonance analytically obtained and

1 2 2 2

Thus, the root of equation (20) allows locating the parallel and series resonance:

,a

τ<sup>1</sup> = 0%

*kr,* Δ = 7.05

τ<sup>1</sup> = 0%

*dC* = 0.5

(a)

35

17.5


(b)


N, |

*YAAk*, apx|N (pu)

0

0.8

0.4

0

4

*ks,* n = 5.00

*kr,* Δ = 4.95 *ks,* napx = 4.97

*XS* = 0.049 pu, *RL* = 1.341 pu and

phase load model LM1).

*kr*, a = 5.13 and 7.33 (

Section 3.2: *XS* = 0.154 pu, *RL* = 1.464 pu,

 *k* 567 9 8

results of the resonances (22) for these systems are • Parallel resonance: *kr*, a = 4.94 and 7.05 (

τ

*dC* = 1.0 *dC* = 0.5

*ks,* n = 7.09

*kr,* Δ = 7.03 *ks,* napx = 7.04

λ

Fig. 8. Resonance location: a) Parallel resonance (power system data in Section 3.1:

λ

To illustrate the above study, Fig. 8 shows |*ZAAk*|N, |*YAAk*|N, |*YAAk*, apx|N and |Δ*k*| for the power systems presented in the laboratory tests of Sections 3.1 and 3.2, and the analytical

τ

1 = 6.0% and *dC* = 1.0 and 0.5, respectively).

*kp,* n = *kr,* Δ = 4.94 *kp,* n = 7.04

*dC* = 1.0

2 (2 ) 2

*HH H H HH G G*


Fig. 9 compares *kr*, a, with *kp*, n and *ks*, n. Considering the validity range of the involved variables, the values leading to the largest differences are used. It can be observed that *kr*, a provides the correct harmonic of the parallel and series resonance. The largest differences obtained are below 10% and correspond to *ks*, n when z*P* = 20, which is the lowest acceptable z*P* value to apply the *kr*, a analytical expression. Although only the linear load model LM1 is considered in the calculations, it is verified that the above conclusions are true for the other three-phase load models.

Fig. 9. Comparison between *kres* and *kr*, a (*kres* = *kp*, n or *ks*, n).

The previous research unifies the study of the parallel and series resonance, providing an expression unique to their location. This expression is the same as in the series resonance case (Sainz et al., 2007), but substantially improves those obtained in the parallel resonance case (Sainz et al., in press). Moreover, the Steinmetz circuit inductor resistance is considered

Characterization of Harmonic Resonances in

**6. Examples** 

in the literature.

results.

range of relatively large harmonics.

the Presence of the Steinmetz Circuit in Power Systems 189

• The resonant behavior of the Steinmetz circuit with power system reactors occurs in a

• The resonances are located in the low-order harmonics only if the displacement power

impedance is small in comparison with the supply system reactances (i.e., small *rL*

2009a), the latter only occurs in weak power systems where the short-circuit power at the PCC bus, *SS*, is low compared to the apparent power of the single-phase load, *SL*.

from the unity value. It is also true if the Steinmetz circuit capacitor degrades, i.e. the

For the sake of illustration, two different implementations of the *kr*, a expression, (22), are developed. In the first, the analytical study in Section 4 is validated from laboratory measurements. Several experimental tests were made to check the usefulness of the *kr*, a expression in locating the parallel and series resonance. In the second, this expression is applied to locate the harmonic resonance of several power systems with a Steinmetz circuit

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

The harmonic response of the network in Fig. 4 was measured in the laboratory for two


λ

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

• Case 2: System data of Case 1 except the single-phase load fundamental displacement

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

circuit inductor is far from the zero value, i.e. its displacement power factor

Steinmetz circuit suffers capacitor loss and *dC* is also far from the unity value.

λ

λ

τ

*<sup>L</sup>* ≈ 1) and this

λ1 is far

*<sup>L</sup>*·*SS*/*SL* (Sainz et al.,

1 ratio of the Steinmetz

factor of the single-phase load impedance is close to the unity value (i.e.,

ratios). The former condition is common but, considering that *rL* =

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

**6.1.1 Experimental measurements of the parallel resonance** 

factor of the railroad substation, which becomes


• Case 1 (studied in Section 3.1):

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.

reactances also change, i.e. *X*1, apx = 1.640 pu and *X*2, apx = 5.975 pu (1).

• The resonances are shifted to high-order harmonics if the

in the analytical location of the resonances, making a contribution to previous studies. This resistance, as well as damping the impedance values, shifts the resonance frequencies because it influences Steinmetz circuit design (i.e., the determination of the Steinmetz circuit reactances).
