ECE 5984: Power Distribution System AnalysisECE 5984: Power Distribution System Analysis Lecture 5: Shunt Admittance of Distribution Lines Reference: Textbook, Chapter 5, and Glover-Sarma-Overbye•

ECE 5984: Power Distribution System Analysis

Lecture 5: Shunt Admittance of Distribution Lines Reference: Textbook, Chapter 5, and Glover-Sarma-Overbye

Instructor: V. Kekatos

1

•  Charge in conductors induces electric field [Gauss’ law]

•  Induced voltage difference modeled as capacitance

•  Capacitance calculated as charge-to-voltage ratios

Line capacitance

2

V12 =q

2⇡✏ln

D2

D1

✏ = ✏r✏0

✏0 = 8.854 · 10�12 [F/m]

permittivity

•  Generalization to multiple conductors through superposition

[Glover-Sarma-Overybye]

Vij =1

2⇡✏

NX

k=1

qk lnDkj

Dki[V]

•  Voltage difference due to charge q [Cb/m]

Transmission line

3

•  Equidistant conductors (D) of radius r

•  Interested in phase voltages

•  Shunt capacitance for phase conductor a

Vab =1

2⇡✏(qa � qb) ln

D

r

•  Assuming (a1) symmetry (or transposition); and (a2) balanced charges

Van =Vab + Vac

3=

1

2⇡✏qa ln

D

r<latexit sha1_base64="peMEcIKtuSLuoNpg5h6Cwgumj1s=">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</latexit><latexit sha1_base64="peMEcIKtuSLuoNpg5h6Cwgumj1s=">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</latexit><latexit sha1_base64="peMEcIKtuSLuoNpg5h6Cwgumj1s=">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</latexit><latexit sha1_base64="peMEcIKtuSLuoNpg5h6Cwgumj1s=">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</latexit>

Can =qaVan

=2⇡✏

ln Dr

[F/m]<latexit sha1_base64="omNbwXfeFbMN6ofDEoXkHSxou2Q=">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</latexit><latexit sha1_base64="omNbwXfeFbMN6ofDEoXkHSxou2Q=">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</latexit><latexit sha1_base64="omNbwXfeFbMN6ofDEoXkHSxou2Q=">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</latexit><latexit sha1_base64="omNbwXfeFbMN6ofDEoXkHSxou2Q=">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</latexit>

[Kersting]

[Glover-Sarma-Overybye]

Method of images

4

•  Earth is modeled by mirror conductor carrying opposite charges

•  Voltage difference across conductor and mirror

•  Due to symmetry

Vii0 =1

2⇡✏

"NX

k=1

qk lnSki

Dki�

NX

k=1

qk lnDki

Ski

#

Vin =Vii0

2=

1

2⇡✏

NX

n=1

qk lnSki

Dki

Primitive potential coefficient matrix

5

•  Phase voltages are linear combinations of charges at all non-dirt conductors

•  Trick: If di is a complex number denoting the location of conductor i

Vin =NX

k=1

Pikqk, where Pik := 11.177 · ln Ski

Dki[mile/µF]

•  Primitive potential coefficient matrix

v�

vn

�=

P�� P�n

P>�n Pnn

� q�

qn

�P [mile/µF]

not phasors

with Pik as matrix entries

Dki = |dk � di| and Ski = |dk � d⇤i |<latexit sha1_base64="A4/6SzEnipcNk9oPI9xiV9Kq8AI=">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</latexit><latexit sha1_base64="A4/6SzEnipcNk9oPI9xiV9Kq8AI=">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</latexit><latexit sha1_base64="A4/6SzEnipcNk9oPI9xiV9Kq8AI=">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</latexit><latexit sha1_base64="A4/6SzEnipcNk9oPI9xiV9Kq8AI=">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</latexit>

Phase potential and capacitance matrices

6

•  Kron reduction to eliminate qn since

•  Primitive potential coefficient matrix

vn = 0

v�

vn

�=

P�� P�n

P>�n Pnn

� q�

qn

�

•  Phase shunt admittance matrix

•  Phase capacitance matrix

positive diagonal and negative off-diagonal entries

Ysh = j!C� µS/mile<latexit sha1_base64="HRNKIXAoDgIp2B0FL039Lgo2ymk=">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</latexit><latexit sha1_base64="HRNKIXAoDgIp2B0FL039Lgo2ymk=">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</latexit><latexit sha1_base64="HRNKIXAoDgIp2B0FL039Lgo2ymk=">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</latexit><latexit sha1_base64="HRNKIXAoDgIp2B0FL039Lgo2ymk=">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</latexit>

v� = P�q� where P� = P�� �P�nP�1nnP

>�n [mile/µF]

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•  Phase potential coefficient matrix

q� = C�v� where C� = P�1� µF/mile

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Parallel lines

7

Procedure generalizes to parallel lines

n

1-a

2-c

Line 1

Line 2

1-b 1-c

2-b2-a

2.5 ft 4.5 ft

2.0 ft 3.0 ft

6.0 ft

29 ft

Step 1: Find distances between conductors

Step 2: Find distances between conductors and mirror conductors

D =

S =

Parallel lines (cont’d)

8

Step 3: Form primitive potential coefficient matrix Pij := 11.177 · ln Sji

Dji[mile/µF]

Step 4: Kron reduction to get the phase potential coefficient matrix

v�1

v�2

�=

P�1�1 P�1�2

P>�1�2

P�1�2

� q�1

q�2

�

Step 5: Invert to get the (coupled) phase capacitance matrix

C =

P�1�1 P�1�2

P>�1�2

P�1�2

��1

Parallel lines (cont’d)

9

Step 6: Convert capacitance to (coupled) phase shunt admittance matrix

Ysh,12 = j!C = µS/mile

•  Compare shunt admittance values to series impedance from Lecture 5…

Concentric neutral underground cables

10

•  Due to shielding, electric fields are confined within cables

•  No coupling between phase cables, and between cables and earth

•  All neutral strands are at the same potential (ground)

•  Voltage between phase conductor and ground (e.g., neutral strand #1)

2Rs

1

2

3

k

k � 1

Vpg =1

2⇡✏

"qp ln

R

Rp+

kX

n=1

qn lnD1n

R

#

k : # concentric neutrals

R : radius of concentric arrangement

Rp : radius of phase conductor

Rs : radius of neutral strand

D1n : distance between strand 1 and strand n

2Rp

Concentric neutral underground cables (cont’d)

11

•  Equal charge on neutral strands

•  Using formula for bundled conductors uniformly spaced on the perimeter

2Rs

1

2

3

k

k � 1•  Distances between strands

D1n = |R�Rej2⇡(n�1)

k |, n = 2, . . . , k

Vpg =1

2⇡✏

"qp ln

R

Rp+

kX

n=1

qn lnD1n

R

#

2Rp

qn = �qpk, 8n = 1, . . . , k

Cpg =qpVpg

=2⇡✏

ln RRp

� 1k ln kRs

R

Tape-shielded cables

12

•  Limiting case of concentric neutrals for

2Rp

k ! 1

Cpg =qpVpg

=2⇡✏

ln RRp

•  For either cables, no capacitive coupling across phases or circuits in parallel lines

•  Phase admittance matrix is diagonal Ysh =

•  For both overhead and underground lines, shunt admittances are typically ignored

Sequence admittance

13

•  Similarly to series impedances

•  Sequence shunt admittance matrix

•  Diagonal for underground or transposed overhead lines

ip = Ysh,�v� () is = Ysh,svs

Ysh,s := A�1s Ysh,�As =

2

4y00 y01 y02y01 y11 y12y02 y12 y22

3

5

•  In fact, for underground lines with three identical cables Ysh,s = Ysh,�<latexit sha1_base64="pWnWxqkycgbHrTI+5YVFEy7d6yc=">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</latexit><latexit sha1_base64="pWnWxqkycgbHrTI+5YVFEy7d6yc=">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</latexit><latexit sha1_base64="pWnWxqkycgbHrTI+5YVFEy7d6yc=">AAACXnicbZHNSsNAEMe38avWr7ZeBC/BIngoJRFBL0LBi8cKtipNKJvtpF3cbMLupFhCXsSn8apv4M1HcdP04EcHlv3xn5ndnf8GieAaHeezYq2tb2xuVbdrO7t7+wf1RnOg41Qx6LNYxOoxoBoEl9BHjgIeEwU0CgQ8BM83Rf5hBkrzWN7jPAE/ohPJQ84oGmlUv/AiitMgzJ7yUeYhvGCmp3lb59erE14y5fmo3nI6ziLs/+AuoUWW0Rs1Kk1vHLM0AolMUK2HrpOgn1GFnAnIa16qIaHsmU5gaFDSCLSfLcbL7VOjjO0wVmZJtBfqyo72eMYTXaKeTUpAarzws5fysJ99GY20nkdBewKy2M1Nxcz6b00hrsoNUwyv/IzLJEWQrHxomAobY7vw2h5zBQzF3ABliptZbTalijI0P1IzLrp/PfsPg/OO63Tcu4tW93zpZ5UckxNyRlxySbrklvRInzDySt7IO/mofFmb1p51UJZalWXPIfkV1tE3HzG6GQ==</latexit><latexit sha1_base64="pWnWxqkycgbHrTI+5YVFEy7d6yc=">AAACXnicbZHNSsNAEMe38avWr7ZeBC/BIngoJRFBL0LBi8cKtipNKJvtpF3cbMLupFhCXsSn8apv4M1HcdP04EcHlv3xn5ndnf8GieAaHeezYq2tb2xuVbdrO7t7+wf1RnOg41Qx6LNYxOoxoBoEl9BHjgIeEwU0CgQ8BM83Rf5hBkrzWN7jPAE/ohPJQ84oGmlUv/AiitMgzJ7yUeYhvGCmp3lb59erE14y5fmo3nI6ziLs/+AuoUWW0Rs1Kk1vHLM0AolMUK2HrpOgn1GFnAnIa16qIaHsmU5gaFDSCLSfLcbL7VOjjO0wVmZJtBfqyo72eMYTXaKeTUpAarzws5fysJ99GY20nkdBewKy2M1Nxcz6b00hrsoNUwyv/IzLJEWQrHxomAobY7vw2h5zBQzF3ABliptZbTalijI0P1IzLrp/PfsPg/OO63Tcu4tW93zpZ5UckxNyRlxySbrklvRInzDySt7IO/mofFmb1p51UJZalWXPIfkV1tE3HzG6GQ==</latexit>

•  Find primitive potential coefficient matrix

Summary

14

•  Kron reduction to get the phase potential coefficient matrix

•  For underground cables, the capacitance matrix is a scaled identity

•  Inversion to get the phase capacitance matrix

•  Shunt admittance is typically ignored

•  Find distances between (mirror) conductors