Effective electric power quantities and the sequence reference frame: A comparison study

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Effective electric power quantities and the sequence reference frame: A comparison study O. Boudebbouz ∗ , A. Boukadoum, A. Medoued Department of Electrical Engineering, Laboratoire d’électrotechnique de Skikda «LES», Université du 20 Août 1955, Skikda 21000, Algeria

a r t i c l e

i n f o

Article history: Received 3 November 2015 Received in revised form 13 May 2016 Accepted 25 May 2016 Available online xxx Keywords: Three-phase circuits Unbalance Harmonics Power quality IEEE Standard 1459 Sequence reference frame

a b s t r a c t More recently, unbalanced and distorted electric three phase quantities have been analyzed in the sequence reference frame, successively, by two elegant decomposition techniques referred to Girgis and Langella. Some comparisons regarding the matrices structures of the two methods have been drawn out in the Langella work. The latter also, makes it possible to evaluate the quantities defined in the IEEE Standard 1459-2010 by means of harmonic sequence components. Some previous author’s works have focused on the Girgis method in which they analyzed the harmonic resonance phenomena and evaluated the quantities defined in the IEEE Standard 1459-2010 in the sequence reference frame. In the present work, the authors follow up with the Langella method and they present more explicit sequence effective electric quantities based on the two methods. Some substantial comparisons between the aforementioned works are presented as well. © 2016 Elsevier B.V. All rights reserved.

1. Introduction For a long ninety years ago, the sequence reference frame initiated by Fortescue has, only, been adopted for linear unbalanced threephase power systems [1]. It sets down the theory of symmetrical components as well as the methods to be used in the steady state analysis of AC circuits. It is used daily by modern power engineers who deal with three-phase circuits. Analysis of three-phase circuits in an efficient and comprehensible way is the key importance of the Fortescue paper [2]. Some recent works have focused on the Fortescue method in nonlinear electric power systems [3–5]. In [3], the instantaneous approach has been applied to IEEE Standard 1459 power terms and quality indices. In this work, the sequence reference frame has been only considered for fundamental electric quantities. In [4], non-fundamental effective apparent power defined through an instantaneous power approach has been, only, proposed for balanced harmonics. In [5], forward generalization to harmonic domain of effective voltage expression contained in Section 3.2.2.8 of the IEEE Standard 1459-2010 [6], has been noted without showing any matrix used for obtaining the sequence harmonic components. Implicitly, we understand that the Fortescue transformation has been considered as an analysis tool in the harmonic domain. The Fortescue transformation has some limitations for nonlinear systems and more recently, two new methods have been proposed to the study the unbalance in the presence of harmonics or interharmonics [7,8]. The first one related to Girgis et al. proposed three matrices’ transformations [7]. Each matrix is applied to a sub-set of integer harmonic orders of which the corresponding balanced three-phase quantities have the same sequence. Unbalanced and distorted systems of related three-phase phasors at each harmonic order can be resolved into three symmetrical phasors called balanced, first unbalanced, and second unbalanced. They represent the sequence harmonic components of the original phasors. This new designation has some similarities with the classical one related to the Fortescue method but with more extent to the harmonic domain. The second one related to Langella et al. proposed a new unique transformation matrix that is capable of suitably extracting the balanced, first unbalanced, and second unbalanced components suitable for all of the harmonic and interharmonic orders [8]. This method uses a same designation as Girgis method but introduces sequence interharmonic components as well. A used transformation matrix has a generic property for all harmonic and interharmonic components and conserves some similarities with the Girgis method.

∗ Corresponding author. Tel.: +213 771 785796. E-mail address: [email protected] (O. Boudebbouz). http://dx.doi.org/10.1016/j.epsr.2016.05.027 0378-7796/© 2016 Elsevier B.V. All rights reserved.

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Some comparisons regarding the matrices structures of the two methods have been drawn out in the Langella work. The latter also, makes it possible to evaluate the quantities defined in the IEEE Standard 1459-2010 by means of harmonic sequence components. Some previous author’s works have focused on the Girgis method in which they analyzed the harmonic resonance phenomena and evaluated the quantities defined in the IEEE Standard 1459-2010 in the sequence reference frame [9,10]. In the present work, the authors follow up with the Langella method and present more explicit sequence electric quantities based on the two methods. Some substantial comparisons between the aforementioned works are presented as well. 2. Unbalance decomposition techniques More recently, new decomposition techniques related to Girgis et al. and Langella et all have been proposed to give harmonic sequence components corresponding to the overall range of harmonic orders [7,8]. They allow us to define, at each harmonic order, the balanced components (bn), the first unbalanced components (fu) and the second unbalanced components (su). Each method uses three matrices’ transformations which present some differences. However, they are able to separate a frequency domain which falls in same sub-sets of harmonic orders of which the corresponding balanced three-phase quantities have the same sequence. For perfectly balanced and distorted systems, the harmonics of order h = 1, 4, 7, 10, 13,. . ., are purely positive – sequence and are designed as subset S1 . The harmonics of order h = 2, 5, 8, 11,. . ., are purely negative – sequence and are designed as subset S2 . The harmonics of order h = 3, 6, 9,. . ., are purely zero – sequence and are designed as subset S0 . It is known that DC components are present in non-negligible proportions in AC circuits near traction and HVDC lines, in LV networks where inverters or compensators operate incorrectly, and as quasi-DC when currents are induced geomagnetically in transmission systems. In [7], the DC components have been considered as zero sequence components whereas in [8] they have not been mentioned. In the present paper, the DC components are ignored. 2.1. Girgis method (GM) For any range of harmonic orders of interest, the HSCs proposed by GM are found by the application of the three matrices, T¯ G1 , T¯ G2 and T¯ G0 as follow:

⎡

T¯ G1 =

1 3

1

⎢ ⎣ 1 ˛2 1

⎡ T¯ G2 =

1 3

1 3

˛

1

1

1 ˛2

˛

⎤ ⎥ ⎦

˛

˛2 ⎦

1

1

1

1

⎢ ⎢1 ⎢ ⎣ 1

(1)

⎤

⎢ ⎣1 ⎡

T¯ G0 =

˛2

˛

⎥

1

√ −1 − 3 2 √ −1 + 3 2

(2)

1

√ −1 + 3 2 √ −1 − 3 2

⎤ ⎥ ⎥ ⎥ ⎦

(3)

˛ = ej(2/3) the rotational operator. The application of the matrices (1)–(3) show evidence of the presence of the imbalance at each harmonic order, including the fundamental one. For each harmonic order, the phasors of the voltages (currents) in the sequence reference frame can be found using:

⎡ ¯h ⎤ ⎡ ¯h⎤ Va Vbn ⎢ ¯h ⎥ ¯h⎢ ¯h⎥ ⎣ Vfu ⎦ = G ⎣ Vb ⎦

(4)

V¯ ch

h V¯ su

¯ h represents one of the three matrices, T¯ G1 , T¯ G2 and T¯ G0 according to the harmonic order. T¯ G1 is applied to harmonics of subsets S1 in G which the harmonics orders are h = 3m − 2. T¯ G2 is applied to harmonics of subsets S2 in which the harmonics orders are h = 3m − 1. T¯ G0 is applied to harmonics of subsets S0 in which the harmonics orders are h = 3m. m is a nonzero integer number. 2.2. Langella method (LM) The LM represents a generic process with a unique transformation matrix according to the harmonic or interharmonic orders. In the compact form, we have:

⎡ T¯ Lr =

1 3

1

˛k

˛2k

⎤

⎢ ⎥ ⎣ 1 ˛k+1 ˛2k+2 ⎦ 1 ˛k+2

(5)

˛2k+1

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With ˛(.) = ej(2(.)/3) , k = rωf /ω1 , where r = 1, 2, 3,. . ., N; k assumes integer values for the harmonics of fundamental angular frequency ω1 and noninteger values for the remaining interharmonics. The operator ˛k has been presented as the generalization of the classical one noted ˛. If we are, only, interested with integer harmonic orders, generated matrices can be derived from (5) as follows:

⎡

T¯ L1 =

1 3

1

⎢ ⎣ 1 ˛2

1 3

1 3

˛ 1

1 ˛2

˛

⎣1

⎤ ⎥ ⎦

(6)

⎤ ⎦

1

1

1

˛

˛2

1

1

1

⎣1

˛

˛2 ⎦

1

˛2

˛

⎡ T¯ L0 =

˛2

1

1

⎡ T¯ L2 =

˛

(7)

⎤ (8)

Similar to the GM, we can write:

⎡ ¯h ⎤ ⎡ ¯h⎤ Va Vbn ⎢ ¯ h ⎥ ¯h ⎢ ¯ h ⎥ ⎣ Vfu ⎦ = L ⎣ Vb ⎦

(9)

V¯ ch

h V¯ su

L¯ h represents one of the three matrices, T¯ L1 , T¯ L2 and T¯ L0 according to the harmonic order and they are applied to subsets S1 , S2 and S0 respectively. 3. Updated electric power quantities contained in the IEEE Standard 1459-2010 3.1. Author’s proposals According to the matrices (1)–(3), the authors proposed in [10], more coherent expressions for effective electric quantities as follow:

  H

1 2 1 )2 + (V 1 )2 + (Vsu ) Ve = (V + bn fu 2 h≥2

 h ) (Vbn

2

h )2 + (Vfu

(V h ) + su 2

2

 +

H

 h )2 (Vfu

h )2 + (Vsu

+

h ) (Vbn

2

h=3,6...

2

 (10)

/ 3, 6. . . h=

  H 

1 )2 + (I 1 )2 + 4(I 1 )2 + (I Ie = su bn fu h≥2



2

2

h ) + (I h ) + 4(I h ) (Ibn su fu

2



+

H 

2

2

h ) + (I h ) + 4(I h ) (Ifu su bn

2

 (11)

h=3,6...

h= / 3, 6. . . From the above expressions, the effective apparent power can be found as: Se = 3Ve Ie . Eqs. (10) and (11) have been rewritten as:



Ve =

 Ie =

(Ve1 )2 + (VeH12 )2 + (VeH0 )2

(Ie1 )2 + (IeH12 )2 + (IeH0 )2

(12) (13)

H12 represents subsets S1 and S2 except the fundamental component. H0 represents the subset S0 . Based on (12) and (13), the effective apparent power would be given as follow: 2 2 Se = 9Ve2 Ie2 = 9((Ve1 )2 + (VeH12 )2 + (VeH0 )2 )((Ie1 )2 + (IeH12 )2 + (IeH0 )2 ) = Se1 + SeN

(14)

2 2 2 Se1 = 9(Ve1 )(Ie1 )

(15)

while

and 2 2 2 2 2 2 2 2 2 2 2 2 = 9{(V )2 (I SeN eH12 ) + (Ve1 ) (IeH0 ) + (VeH12 ) (Ie1 ) + (VeH0 ) (Ie1 ) + (VeH12 ) (IeH12 ) + (VeH0 ) (IeH12 ) e1

+ (VeH12 )2 (IeH0 )2 + (VeH0 )2 (IeH0 )2 }

(16)

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3.2. Langella’s proposals In [8], Langella et al. proposed expressions of effective voltages and currents (17) and (18) but with implicit sequence components of the neutral current and line to line voltages.

  1 2

Ve =

2 Vbn

2 + Vfu

2 + Vsu

 2 + I2 + I2 + Ibn su fu

Ie =

+

2 + V2 + V2 Vbnll full sull

 (17)

3

2 + I2 Ibn nu

(18)

3

where Vbnll , Vfull and Vsull are the balanced, the first and the second unbalanced line-to-line voltages for overall range of harmonic orders of interest. In (17), the line-to-line voltages are given without noting that second unbalanced, first unbalanced and balanced components of subsets S1 , S2 and S0 , respectively, would be, necessarily, nil. On the other hand, the neutral current as shown in (18) is simply represented by balanced component Inb and unbalanced one Inu without decomposing it into the corresponding sequence harmonic components. We believe that this proposal would be better detailed since the neutral current only contains components of the zero sequence. The latter is considered as balanced component for triplen harmonics and an unbalanced one as zero sequence for harmonics of subsets S1 and S2 . Then, for a single harmonic order, the neutral current contains the corresponding zero sequence component and the remainder sequence components are, necessarily, nil. It is, also, noted that the equivalent current in Girgis work has been taken from [11] that ignores the neutral current. 3.3. Langella’s proposals update All of the matrices proposed by Girgis and Langella satisfy the well known orthogonal property: ∗

T

T

∗

¯ h ] [G ¯ h ] = [G ¯ h ] [G ¯ h] = [G ∗

T

T

∗

[L¯ h ] [L¯ h ] = [L¯ h ] [L¯ h ] =

1 I 3

(19)

1 I 3

(20)

where * represents the conjugate, and I is the identity matrix. The use of the Langella matrices (6)–(8) and the same analysis as in [10] allow us to define the expressions of the generalized effective voltage and current in the sequence reference frame, as follow:

     H H H 2 2 2 2 (V h ) (V h ) (V 1 ) (V h ) h 2 h 2 h 2 h 2 h 2 h 2 1 2 1 2 ) + (Vfu ) + su + (Vbn ) + (Vfu ) + su + (Vbn ) + (Vsu ) + fu + (Vfu ) + (Vsu ) + bn Ve = (Vbn 2

2

2

h=4,7...

h=2,5...

2

 H H H  2   h 2 h 2   h 2 h 2   h 2 h 2  h 2 h 2 h 2 1 1 2 1 2 ) + (Ifu ) + 4(Isu ) + (Ibn ) + (Ifu ) + 4(Isu ) + (Ibn ) + (Isu ) + 4(Ifu ) + (Ifu ) + (Isu ) + 4(Ibn ) Ie = (Ibn h=4.7...

h=2,5...

(21)

h=3,6...

(22)

h=3,6...

The generalized effective current is derived considering the neutral current which is given from: in = ia + ib + ic

(23)

In [10], the phasor of the neutral current has been derived as: h , I¯nh = 3I¯su

h = 3m − 1 or

h , I¯nh = 3I¯bn

h = 3m

h = 3m − 2

(24)

The use of the Langella matrices and the same analysis as in [10], the phasor of neutral current can be derived as: h , I¯nh = 3I¯su

h = 3m − 2

h , 3I¯fu

h = 3m − 1

I¯nh

=

h , I¯nh = 3I¯bn

(25)

h = 3m

Eqs. (21) and (22) can be written as:



(Ve1 )2 + (VeH1 )2 + (VeH2 )2 + (VeH0 )2

Ve =

 Ie =

(Ie1 )2 + (IeH1 )2 + (IeH2 )2 + (IeH0 )2

(26) (27)

H1 and H2 represent, respectively, subsets S1 and S2 except the fundamental component. H0 represents the subset S0 . Based on (26) and (27), the effective apparent power would be given as follow: 2 2 Se2 = 9Ve2 Ie2 = ((Ve1 )2 + (VeH1 )2 + (VeH2 )2 + (VeH0 )2 )((Ie1 )2 + (IeH1 )2 + (IeH2 )2 + (IeH0 )2 ) = Se1 + SeN

(28)

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while 2 Se1 = 9(Ve1 )2 (Ie1 )2

and

 2 SeN =9

(29)

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

(VeH1 ) (Ie1 ) + (VeH2 ) (Ie1 ) + (VeH0 ) (Ie1 ) + (Ve1 ) (IeH1 ) + (VeH1 ) (IeH1 ) + (VeH2 ) (IeH1 ) + (VeH0 ) (IeH1 ) + (Ve1 ) (IeH2 ) 2

2

2

2

2

2

2

2

2

2

2

2

2

+ (VeH1 ) (IeH2 ) + (VeH2 ) (IeH2 ) + (VeH0 ) (IeH2 ) + (Ve1 ) (IeH0 ) + (VeH1 ) (IeH0 ) + (VeH2 ) (IeH0 ) + (VeH0 ) (IeH0 )

2

2

(30)

Scattered terms of effective apparent power (15)–(16) and (29)–(30) are able to characterize the harmonic pollution in more convenient way. In deregulation power systems, this finding can be used in determining the main contributors to the poor power quality. 4. Further comparisons The relevant comparisons can be pointed out on the line-to-line voltages derived from GM and LM. 4.1. Equations based GM The relationships between line-to-line voltages in the phase reference frame with those in the sequence reference frame can be derived for harmonics of subsets S1 , S2 and S0 as follow: For h = 3m − 2 h h h h h h h h h V¯ ab = V¯ ah − V¯ bh = (V¯ bn + V¯ fu + V¯ su ) − (˛2 V¯ bn + ˛V¯ fu + V¯ su ) = (1 − ˛2 )V¯ bn + (1 − ˛)V¯ fu h h h h h h h h h V¯ bc = V¯ bh − V¯ ch = (˛2 V¯ bn + ˛V¯ fu + V¯ su ) − (˛V¯ bn + ˛2 V¯ fu + V¯ su ) = ˛2 (1 − ˛2 )V¯ bn + ˛(1 − ˛)V¯ fu h h h h h h h h h = V¯ ch − V¯ ah = (˛V¯ bn + ˛2 V¯ fu + V¯ su ) − (V¯ bn + V¯ fu + V¯ su ) = ˛(1 − ˛2 )V¯ bn + ˛2 (1 − ˛)V¯ fu V¯ ca

For h = 3m − 1 h h h h h h h h h = V¯ ah − V¯ bh = (V¯ bn + V¯ fu + V¯ su ) − (˛V¯ bn + ˛2 V¯ fu + V¯ su ) = (1 − ˛)V¯ bn + (1 − ˛2 )V¯ fu V¯ ab h h h h h h h h h V¯ ca = V¯ ch − V¯ ah = (˛2 V¯ bn + ˛V¯ fu + V¯ su ) − (V¯ bn + V¯ fu + V¯ su ) = ˛2 (1 − ˛)V¯ bn + ˛(1 − ˛)V¯ fu h h h h h h h h h V¯ ca = V¯ ch − V¯ ah = (˛2 V¯ bn + ˛V¯ fu + V¯ su ) − (V¯ bn + V¯ fu + V¯ su ) = ˛2 (1 − ˛)V¯ bn + ˛(1 − ˛)V¯ fu

For h = 3m



h h h h V¯ ab = V¯ ah − V¯ bh = (V¯ bn + V¯ fu + V¯ su )−

 h = V¯ ch − V¯ ah = V¯ ca

h V¯ bn +

−1 + 2

h V¯ bn +

−1 + 2

 h V¯ ca = V¯ ch − V¯ ah =

√ 3

√ 3

−1 − 2

h V¯ fu +

−1 − 2

⎡ ⎡ ¯h ⎤ ⎡ ⎤ √ j Vab 1 1 h 3e 6 V¯ bn ⎢ ¯ h ⎥ ⎣ ˛2 ˛ ⎦ ⎢ ⎢ = V ⎣ bc ⎦ ⎣  h V¯ ca

˛

√

3e

−j

−1 − 2

h V¯ fu +

In matrix form, we can write:

˛2

h + V¯ bn

√ 3

√ 3

√ 3

h + V¯ fu

−1 + 2

h V¯ ca

√ ⎡ −1 − 3 ⎡ ¯h ⎤ Vab 2 ⎢ ⎢ ¯h ⎥ ⎢ ⎣ Vbc ⎦ = ⎢ 1 1 ⎣ √ h V¯ ca

−1 + 2

3

−1 − 2

=

h h h − (V¯ bn + V¯ fu + V¯ su )=

−3 + 2

h h h − (V¯ bn + V¯ fu + V¯ su )=

−3 + 2

 h V¯ su

√ √ 3+ 3 h 3− 3 h V¯ fu + V¯ su 2 2 √ 3

√ 3

h V¯ fu +

−3 − 2

h V¯ fu +

−3 − 2

√ 3

√ 3

h V¯ su

h V¯ su

⎤ ⎥ ⎥ For h = 3m − 2 ⎦

(31)

6 V¯ h

fu

√ j h 3e 6 V¯ fu √ ⎤ −1 + 3  2 ⎥ √3ej V¯ h

˛

 h V¯ su

 h V¯ su

⎤ ⎡ ⎡ ¯h ⎤ ⎡ ⎤ √ −j  Vab 1 1 h 3e 6 V¯ bn ⎥ ⎢ ¯ h ⎥ ⎣ ˛ ˛2 ⎦ ⎢ ⎥ For h = 3m − 1 ⎢ ⎣ Vbc ⎦ = ⎦ ⎣  ˛2

√ 3

⎥ ⎥ √ ⎦ 3

fu

√ h 3V¯ su

(32)

 For h = 3m

(33)

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Table 1 Percent harmonic voltage and current phasors base values: Va1 = 271.03 [V]; Ia1 = 99.98 [A]. h

Vah (%) ◦

ˇah

Vbh (%) ◦

ˇbh

1

3

100

10.28

5

−0.74

6.76

104.49

10.53

−121.2 103.73

8.69

ˇch

121.3

9.7

◦

Iah (%)

100

ah

−22

◦

Ibh (%)

−120.4

Ich (%) h◦ ˇa,b,c :

5.79 4.3

6.58 136.5

34.90 −175

79.77

42.30

99.49

65.09

0

8.58 125.2

157.7

68.84

93.49

◦

bh

146.7

167.4

100

0

9 7.44

142.3

6.28

Vch (%)

7 4.92

27.85 −65 45.81 −167.9

0

0

8.64 −47.4 11.05 −49.19 8.22 −47.37 5.93 48 40.59 41.89 0

The voltage phase angles in degree.

4.2. Equations based LM The same relationships as ones derived by means of GM are shown in (34) and (35). We only define those at 3m – 1 and 3m orders knowing that ones at 3m − 2 orders are the same for both methods (i.e., GM and LM).

⎤ ⎡ ⎡ ¯h ⎤ ⎡ ⎤ √ −j  Vab 1 1 h 3e 6 V¯ bn ⎥ ⎢ ¯ h ⎥ ⎣ ˛ ˛2 ⎦ ⎢ ⎥ For h = 3m − 1 ⎢ ⎣ Vbc ⎦ = ⎦ ⎣  h V¯ ca

˛2

˛

√ j h 3e 6 V¯ su

⎡ ⎡ ¯h ⎤ ⎡ ⎤ √ j Vab 1 1 h 3e 6 V¯ fu ⎢ ¯ h ⎥ ⎣ ˛2 ˛ ⎦ ⎢ ⎢ ⎣ Vbc ⎦ = ⎣  h V¯ ca

˛

˛2

√ −j h 3e 6 V¯ su

(34)

⎤ ⎥ ⎥ For h = 3m ⎦

(35)

From (31) to (35), substantial comparison between the two methods can be carried out. It is noted that triplen harmonics present relevant differences. 5. Example This example is taken from Ref. [12]. The corresponding voltage and current phasors are shown in Table 1. Resulted sequences voltages of triplen harmonics are shown in Table 2. Fig. 1 shows phasors of three phase voltages at third harmonic order. The corresponding first and second unbalanced voltages according to Girgis and Langella methods are shown in Figs. 2 and 3, respectively. The LM contains the same information as those obtained by means of the Fortescue decomposition but extended for all harmonic components. Positive and negative sequence triplen harmonics are classified, respectively, as second and first unbalanced components by LM and they haven’t been univocally classified by GM. From (33) and Fig. 2a and b, first and second unbalanced components based GM appear in a new symmetry respective of a rotational angle equal ‘␲’. 6. Updated indices characterization The updated indices proposed in [10] can be obtained by means of the Langella method. However, they give different values compared with the Girgis method, especially, for harmonics of subset S0 . This difference can inform us about the ability of both methods to interpret harmonic unbalance. It, also, helps us to establish a substantial comparison between them. Table 3 shows cumulative first and second harmonic unbalance factors according to GM and LM where their corresponding expressions are:

Table 2 Sequences RMS voltages of triplen harmonics in volts. Harmonic orders

h Vbn h Vfu h Vsu h Vfull h Vsull

GM

LM

h=3

h=9

h=3

h=9

26.6426

25.2118

26.6426

25.2118

0.8760

3.1358

1.6472

2.4428

2.1388

1.3192

1.6213

2.3678

1.5172

5.4314

2.8535

4.2311

3.7046

2.2850

2.8081

4.1011

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Fig. 1. Phasors of three-phase voltages at third harmonic. Phase (a) in blue, phase (b) in red, phase (c) in black. (For interpretation of reference to color in this figure legend, the reader is referred to this web version of this article.)

Fig. 2. Phasors of sequence voltages at 3rd harmonic order according to GM. (a) first unbalanced voltages, (b) second unbalanced voltages.

Fig. 3. Phasors of sequence voltages at 3rd harmonic order according to LM. (a) first unbalanced voltages, (b) second unbalanced voltages.

Please cite this article in press as: O. Boudebbouz, et al., Effective electric power quantities and the sequence reference frame: A comparison study, Electr. Power Syst. Res. (2016), http://dx.doi.org/10.1016/j.epsr.2016.05.027

G Model

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Table 3 Cumulative first and second harmonic unbalance factors. Cumulative first harmonic unbalance factor based GM

Cumulative first harmonic unbalance factor based LM

Cumulative second harmonic unbalance factor based GM

Cumulative second harmonic unbalance factor based LM

G0 MUFfu 1.1694%

L0 MUFfu 1.0583%

G0 MUFsu 0.9026%

L0 MUFsu 1.0307%

Cumulative first harmonic unbalance factor:



G0 ,L0 MUFfu

N m=1

=

3m ) (Vfu

2

(36)

1 Vbn

Cumulative second harmonic unbalance factor



G0 ,L0 MUFsu

=

N m=1

3m ) (Vsu

2

1 Vbn

(37)

L0 and MUF L0 give precious information about the penetration of positive and negative The main remarks about the results are that MUFfu su sequence components extracted from unbalanced three-phase triplen harmonics. Moreover, they can be seen, differently, in diagnosis analysis for induction motors for example. The cited factors interpret an opposite contribution on the motor operation. The torques L0 is the contribution of the negative sequence and MUF L0 is the one of the interaction will result in almost no effect because the MUFfu su positive sequence. Their values are almost the same. G0 and MUF G0 only interpret zero sequence behavior. For this case, further works are needed for clearly show how about However, MUFfu su the torques interaction in induction motors would be affected by such unbalanced three phase triplen harmonics.

7. Conclusion The Langella method seems to be in agreement with the Fortescue transformation. The corresponding effective electric quantities based sequence components developed in the present paper are in harmony with those proposed in the IEEE Standard 1459 for fundamental frequency. The GM is in agreement with Fortescue transformation only for harmonics at orders 3m − 1 and 3m − 2. Those at orders 3m are derived in a new way. The corresponding line to line voltages are derived in a new symmetry respective of a rotational angle equal ‘␲’. The corresponding updated indices derived from the two methods can be very useful in power systems analysis and rotating machines operation, especially, for power quality monitoring and diagnosis purposes. However, further practical and experimental works are recommended, particularly, for triplen harmonics to make disparity between both methods more clear. The sequence reference frame remains a key tool for modeling linear electrical systems and their qualitative studies as well. Its extension to the harmonic domain promises more accurate analysis knowing that the actual electric systems are potentially nonlinear and unbalanced. Acknowledgements This work was supported by Laboratoire d’électrotechnique de Skikda (LES) of Université du 20 Août 1955, Skikda, Algeria. References [1] C.L. Fortescue, Method of symmetrical coordinates applied to the solution of polyphase networks, Atlantic City NJ, June 28, 1918. [2] S.S. Venkata, G.T. Heydt, N. Balijepalli, Education/industry relations and power engineering’s impact on everyday life ‘high impact papers in power engineering (1900–1999)’, IEEE Power Eng. Rev. (April 2001). [3] N. Munoz-Galeano, J.C. Alfonso-Gil, S. Orts-Grau, S. Seguí-Chilet, F.J. Gimeno-Sales, Instantaneous approach to IEEE Std. 1459 power terms and quality indices, Electr Power Syst. Res. 125 (2015) 228–234, http://dx.doi.org/10.1016/j.epsr.2015.04.012. [4] N. Munoz-Galeano, J.C. Alfonso-Gil, S. Orts-Grau, S. Segui-Chilet, F.J. Gimeno-Sales, Non-fundamental effective apparent power defined through an instantaneous power approach, Elect. Power Energy Syst. 33 (2011) 1711–1720, http://dx.doi.org/10.1016/j.ijepes.2011.08.013. [5] A.E. Emanuel, Power Definitions and the Physical Mechanism of Power Flow, John Wiley & Sons Ltd., UK, 2010. [6] IEEE Standard: Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, IEEE Std 1459-2010 (Revision of IEEE Std 1459-2000). [7] T. Zheng, E.B. Makram, A.A. Girgis, Evaluating power system unbalance in the presence of harmonics, IEEE Trans. Power Deliv. 18 (2) (2003) 393–397. [8] R. Langella, A. Testa, A.E. Emanuel, Unbalance definition for electrical power systems in the presence of harmonics and interharmonics, IEEE Trans. Instrum. Meas. 61 (10) (2012) 2622–2631. [9] O. Boudebbouz, A. Boukadoum, S. Leulmi, Harmonic sequence impedances: a tool for the three-phase frequency scan technique, Arab J. Sci. Eng. 36 (2011) 1277–1286, http://dx.doi.org/10.1007/s13369-011-0127-8. [10] O. Boudebbouz, A. Boukadoum, S. Leulmi, ‘Effective apparent power definition based on sequence components for non sinusoidal electric power quantities’, Electr. Power Syst. Res. 117 (2014) 210–218, http://dx.doi.org/10.1016/j.epsr.2014.08.017. [11] Practical definitions for powers in systems with nonsinusoidal forms and unbalanced: a discussion, IEEE Trans. Power Del. 11 (1) (1996) 79–101. [12] IEEE Trial-Use Standard: Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Non-sinusoidal, Balanced or Unbalanced Conditions, IEEE Std 1459-2000, Approved September 2002.

Please cite this article in press as: O. Boudebbouz, et al., Effective electric power quantities and the sequence reference frame: A comparison study, Electr. Power Syst. Res. (2016), http://dx.doi.org/10.1016/j.epsr.2016.05.027