Power System Series Resonance Studies by Modified Admittance Scan
Felix O. Kalunta, MNSE, Frank N. Okafor, FNSE, Member, IEEE, Osita U. Omeje
Abstract – This paper presents a modified method of
formulating the loop admittance matrix which is
deployed to identify the series resonant frequencies in
large electrical networks involving numerous shunt
capacitances. Three matrices were assembled from the
network R, L and C elements and later synthesized to
obtain the network loop admittance matrix. Its
application to a sample network has shown the
practicability and effectiveness of this method. This paper
also seeks to demonstrate the impact of certain factors
like cable capacitance and skin effect on the value of
resonant frequencies.
Index Terms – Harmonic Resonance, Loop Admittance
Matrix, Power Quality and Series Resonance.
I. INTRODUCTION
The application of capacitor banks in the power industry has
yielded some utility benefits such as power factor correction,
voltage support and release of system capacity. However,
their interaction with system inductive supply circuit [1, 2]
causes power quality problems by way of amplifying high
order harmonics. These could lead to overheating, failure of
the capacitor banks themselves or blowing of power
transformer units resulting in constant interruption in
production schedules. Another concern is that the broadband
spectrum emitted in the process could result in emission
limits being exceeded for non-characteristic harmonics and
for inter-harmonics.
The challenge is how to theoretically predict the resonant
frequencies based on which appropriate mitigation measures
can be included at the planning stage or the magnitude of
amplified currents for the power equipment to be de-rated in
order to withstand this exigency. There is also the need to
indicate which of the system component is under the threat of
series resonance [3]. The papers [4 – 5] describes the use of
frequency scan technique to detect the possible resonant
frequencies in the electrical network while [5, 6] presented a
harmonic resonance mode assessment based on the analysis
of eigenvalues and eigenvectors of the system bus impedance
matrix. Further details of modal analysis in the study of
resonance are treated in the technical literature [9-11].
_______________________________________________
Felix O. Kalunta is currently pursuing his Ph.D degree with the
Department of Electrical/Electronic Engineering, University of
Lagos, Nigeria. He is on study leave from Federal Institute of
industrial Research Lagos, Nigeria (e-mail: felka3@yahoo.co.uk).
Osita U. Omeje is currently pursuing his Ph.D degree with the
Department of Electrical/Electronic Engineering, University of
Lagos, Nigeria (e-mail: ositaomeje@gmail.com)
Frank N. Okafor is a professor with the Electrical/Electronic
Engineering Department, University of Lagos, Nigeria. (e-mail:
cfrankok@yahoo.com).
These methods are primarily suitable for parallel resonance
problem with little adaptation to the analysis of series
resonance. The reason lies in the close relationship between
loop impedance and the occurrence of series resonance.
An attempt to apply the modal analysis to series resonance
problem was also made [8] but yielded an incomplete
solution. A dummy branch method was later incorporated
into the modal analysis but the approach seems more like a
short circuit study which does not reflect the actual series
resonance scenario. The loop admittance scan technique is an
adaptation of frequency scan for locating the resonance peaks
in a series resonance problem. Calculations are performed to
determine the loop admittance matrix of the concerned
network, while the driving point admittances are the
determinants of the resonant frequencies.
The complexities involved in the calculation of the loop
admittance matrix for large power networks especially when
cable capacitance is involved have necessitated the need for a
modified admittance scan. In such cases, matrix of network
elements R, L and C are easily assembled by computer
programming and later synthesized to form a loop impedance
matrix. This is the approach adopted in this paper. The
formation of dummy loops in the calculation of the network
matrices is applied in order to account for the shunt
capacitances without altering the network topology. Other
factors that affect resonance characteristics like skin effect of
cables are also investigated in this paper. This study will
account for skin effect by calculating the resistance R of the
cable at various discrete frequencies using an equation that
varies according to cable type [11].
𝑅 = 𝑅1 ( 0.187 + 0.532√ℎ ), where h≥2.35 ----------- (1)
where 𝑅1 is the resistance of the cable at the fundamental
frequency and h is the harmonic order.
II. MODAL ANALYSIS APPLIED TO SERIES
RESONANCE
The determination of series resonant frequencies is based on
mesh analysis at each harmonic frequency h in per unit.
Resonance mode analysis in this case is based on the fact that
the loop impedance matrix of a power network becomes
singular at resonant frequencies. This requires the calculation
of the eigenvalues of the system as well as their sensitivities
to changes in system parameters. Imagine that a system
experiences resonance at frequency h according to the
frequency scan analysis. It implies that some elements of the
loop current vector have large values at h. This in turn
implies that the inverse of the [𝑍ℎ ] matrix has large elements.
This phenomenon is primarily caused by the fact that one of
the eigen-values of the Z – matrix is close to zero. In fact, if
the system had no damping, the Z – matrix would become
singular due to one of its eigenvalues becoming zero. The
above reasoning leads us to believe that the characteristics of
the smallest eigenvalue of the [𝑍ℎ ] matrix could contain
useful information about the cause of the harmonic
resonance. The above analysis can be formally stated as
follows:
[𝑉]ℎ = [𝑍𝑙𝑜𝑜𝑝 ]ℎ [𝐼𝑙𝑜𝑜𝑝 ]ℎ
The current vector for each harmonic order is as follows
[𝐼𝑙𝑜𝑜𝑝 ]ℎ = [𝑍𝑙𝑜𝑜𝑝 ]−1
ℎ [𝑉]ℎ , h = 1, 2, 3… n ---------- (2)
Where[𝑍𝑙𝑜𝑜𝑝 ]ℎ , is the loop impedance matrix
[𝑉]ℎ is the harmonic loop voltage vector.
As usual, the loop impedance matrix can be decomposed
into the following form:
[𝑍] = [L][D][T] is the eigen-decomposition of the
[𝑍] matrix at frequency h.
[L] and [T] are the left and right eigenvector matrices of [Z]
[D] - diagonal eigenvalue matrix of Z.
𝑇= 𝐿−1 is due to the fact that 𝑌 is symmetric.
Equation (2) now becomes
[𝐼]=[𝐿][𝐷]−1 [𝑇] [𝑉] ------------------------ (3)
If [𝑇][𝑉] and [𝑇][𝐼] are defined as the modal voltage and
current vectors respectively. It can be seen that admittance
scan equation has been transformed into the following form:
λ1−1 0
𝐽1
0 0 𝑉𝑚1
−1
𝐽2
0 λ2
0 0 [𝑉𝑚2 ] ------------- (4)
[ ]=
⋮
⋮
0
0 ⋱ 0
−1 𝑉
𝐽𝑛
[ 0
0 0 λ𝑛 ] 𝑚𝑛
This can be abbreviated as,
∴ [𝐽𝑘 ]ℎ = [λ−1
𝑘 ]ℎ [𝑉𝑚𝑘 ]ℎ -------------------- (5)
The inverse of the eigen value, λ𝑘−1 , has the unit of
admittance and is named modal admittance. One can easily
see that if 𝜆1 = 0 or is very small, a small applied mode 1
voltage will lead to a large mode 1 current. On the other
hand, the other modal currents will not be affected since they
have no 'coupling' with mode 1 voltage. In other words, one
can easily identify the 'locations' of resonance frequencies in
the modal domain. After identifying the critical mode of
resonance, it is possible to find the 'participation' of each loop
in the resonance. This can be done using the well-known
participation factor theory described in [10].
III. FORMULATION OF LOOP ADMITTANCE SCAN
The admittance scan is a series resonance counterpart of the
bus impedance scan. The scan is performed in one network
mesh at a time. A sinusoidal voltage of unit amplitude
∆𝑉 = 1, and of certain harmonic frequency is inserted into
this mesh and the corresponding loop current is calculated.
The process is repeated for other harmonic frequencies in per
unit. For the purpose of illustration, consider a radial network
in fig. 1 containing capacitor banks with a non-linear load
connected to bus J through a transformer and a long cable.
The network is partitioned along the point of common
coupling between the consumer distribution network and
public utility supply as in fig. 2. The consumer side of the
network is modeled at each harmonic frequency h, and the
system supply side is reduced to its Thevenin equivalent also
at each harmonic frequency. There are two buses where
voltage amplification could be excited by parallel resonance
and also three meshes or loops where current amplification
can occur by series resonance.
Fig. 1: A simple consumer premises supplied from the public
utility network
Fig. 2: The equivalent circuit model of the 2 – bus consumer
network
The application of loop analysis to fig. 2 produces the
following equation,
[𝑉]ℎ = [𝑍𝑙𝑜𝑜𝑝 ]ℎ [𝐼𝑙𝑜𝑜𝑝 ]ℎ
The current vector at each harmonic frequency, h is as
follows
[𝐼𝑙𝑜𝑜𝑝 ]ℎ = [𝑍𝑙𝑜𝑜𝑝 ]−1
ℎ [𝑉]ℎ , h = 1, 2, 3 … n ------------- (6)
The matrix [𝑍𝑙𝑜𝑜𝑝 ]ℎ is known as the loop impedance matrix.
The inverse of the loop impedance matrix is known as loop
admittance matrix, 𝑌𝑙𝑜𝑜𝑝
𝑌𝑙𝑜𝑜𝑝 = [𝑍𝑙𝑜𝑜𝑝 ]−1 ---------------------------- (7)
This matrix is the counterpart of bus impedance matrix and is
therefore useful in the determination of the frequencies at
which harmonic series resonance occurs.
Equation (6) can now be written as
𝐼1 𝑙𝑜𝑜𝑝
𝑌11
[𝐼2 ]
= [𝑌21
𝐼3
𝑌31
𝑌13 𝑙𝑜𝑜𝑝 𝑉1
𝑌23 ]
[𝑉2 ] ------------------ (8)
𝑉3
𝑌33
𝑌12
𝑌22
𝑌32
For the purpose of resonance analysis, only the driving point
𝑙
admittances 𝑌𝑘𝑘
are required.
𝐼
𝑙
Therefore 𝑌𝑘𝑘
= [𝑉𝑘 ]
𝑘
𝑉𝑗 =0
, ∀𝑘 ≠𝑗
[𝐼𝑙𝑜𝑜𝑝 ]ℎ𝑘 = [𝑌𝑙𝑜𝑜𝑝 ]ℎ𝑘𝑘 [𝑉𝑙𝑜𝑜𝑝 ]ℎ𝑘 , 𝑉𝑗 = 0 ∀ 𝑘 ≠ 𝑗 ---- (9)
Suppose 𝑉𝑘 = 1𝑝. 𝑢 for all h =1, 2, 3 … n
[𝐼𝑙𝑜𝑜𝑝 ]ℎ𝑘 = [𝑌𝑙𝑜𝑜𝑝 ]ℎ𝑘𝑘 ------------------------ (10)
A graph of [𝐼𝑙𝑜𝑜𝑝 ]𝑘 against h is equivalent to a graph of
[𝑌𝑙𝑜𝑜𝑝 ]𝑘𝑘 against h. Therefore, driving point admittance
versus frequency plot is obtained for each mesh in the
network. Therefore, driving point admittance versus
frequency plot is obtained for each mesh in the network.
This technique when applied for the determination of series
resonant frequencies in power networks can be referred to as
admittance scan. The off – diagonal or transfer admittances
could be considered in the admittance scan when dealing
with many harmonic sources applied simultaneously at
different network meshes.
The current change in the k-th mesh due to the insertion of
unit harmonic voltages in meshes k, i and j is stated as
𝐶𝑝−1 = the inverse shunt capacitance at p-th dummy loop.
𝐼𝑘 = 𝑌𝑘𝑘 + 𝑌𝑘𝑖 + 𝑌𝑘𝑗 ------------------ (11)
IV. MODIFICATION OF NETWORK MATRICES TO
ACCOUNT FOR THE SHUNT CAPACITANCES
Procedure:
1. The three (𝑛 × 𝑛) network matrices are assembled as usual
in the absence of the shunt capacitances. These are
designated as 𝑅𝑜𝑙𝑑 , 𝐿𝑜𝑙𝑑 and 𝐺𝑜𝑙𝑑 respectively.
Fig 3: Equivalent circuit diagram showing two dummy loops
for shunt capacitors C1 and C2
2. All parallel connection of capacitances can be combined
together by summation.
3. Dummy loops m in number are created at each node where
the shunt capacitances exist such that the total number of
loops becomes(𝑛 + 𝑚), see fig 3. The dummy loops are
assigned loop numbers 𝑛 + 1, 𝑛 + 2 , … 𝑛 + 𝑚
4. Extra rows and columns corresponding to the number of
the created dummy loops are added to each of the three
network matrices to form a partitioned matrix as shown,
𝑅𝑛𝑒𝑤 = [
𝑅𝑜𝑙𝑑
𝐴′
𝐿𝑜𝑙𝑑
𝐵′
𝐵
] ---------------------- (13)
𝐻
𝐺𝑜𝑙𝑑
𝑈′
𝑈
] ----------------------- (14)
𝑉
𝐿𝑛𝑒𝑤 = [
𝐺𝑛𝑒𝑤 = [
𝐴
] ------------------- (12)
𝑄
5. The dummy loops are eliminated using the kron –
reduction formular,
𝑅𝑛𝑒𝑤 = 𝑅𝑜𝑙𝑑 − 𝐴𝑄𝐴𝑇 , 𝐿𝑛𝑒𝑤 = 𝐿𝑜𝑙𝑑 − 𝐵𝐻𝐵𝑇
and 𝐺𝑛𝑒𝑤 = 𝐺𝑜𝑙𝑑 − 𝑈𝑉𝑈 𝑇
Fig. 4: The sequential process of carrying out an admittance
scan based on the creation of dummy loops
6. The matrix of network elements R, L and G are finally
synthesized according to equation (15) to obtain the loop
impedance matrix. This is repeated for each harmonic
frequency, h.
The updated matrices obtained from application of the above
procedure to fig. 3 are,
[𝑍𝑙𝑜𝑜𝑝 ] = [𝑅𝑛𝑒𝑤 ] + 𝑗𝜔1 ℎ[𝐿𝑛𝑒𝑤 ] − 𝑗[𝐺𝑛𝑒𝑤 ]/𝜔1 ℎ - (15)
Where L – 𝑛 × 𝑛 loop inductance matrix
R – 𝑛 × 𝑛 loop resistance matrix
G – 𝑛 × 𝑛 loop inverse capacitance matrix
𝜔1 – fundamental frequency in rad/s
For a dummy loop p created between two actual loops 𝑖 and 𝑗
𝐴𝑖𝑝 = 𝐴𝑝𝑖 = −𝑅𝑠𝑝 , 𝐴𝑗𝑝 = 𝐴𝑝𝑗 = 𝑅𝑠𝑝 , 𝑄𝑝𝑝 = 𝑅𝑠𝑝 ,
elsewhere the entries are zero.
Similarly, 𝐵𝑖𝑝 = 𝐵𝑝𝑖 = −𝐿𝑠𝑝 , 𝐵𝑗𝑝 = 𝐵𝑝𝑗 = 𝐿𝑠𝑝 , 𝐻𝑝𝑝 = 𝐿𝑠𝑝
and 𝑉𝑝𝑝 = 𝐶𝑝−1 , elsewhere the entries are zero.
Where R sp = the total resistance in the branch common to
loops i, j and p
Lsp = the total inductance in the branch common to loops i, j
and p
𝑅𝐿𝑜𝑜𝑝
𝑅11
𝑅21
= 𝑅31
−𝑅𝑠1
[ 0
𝑅12
𝑅22
𝑅32
𝑅𝑠1
0
𝐿𝐿𝑜𝑜𝑝
𝐿11
𝐿21
= 𝐿31
−𝐿𝑠1
[ 0
𝐿12
𝐿22
𝐿32
𝐿𝑠1
−𝐿𝑠1
𝐺𝐿𝑜𝑜𝑝
𝐺11
𝐺21
= 𝐺31
0
[ 0
𝐺12
𝐺22
𝐺32
0
0
𝑅11
𝑅23
𝑅33
0
0
𝐿11
𝐿23
𝐿33
0
𝐿𝑠1
𝐺11
𝐺23
𝐺33
0
0
−𝑅𝑠1
𝑅𝑠1
0
𝑅𝑠1
0
0
0
0 --------------- (16)
0
0]
−𝐿𝑠1
𝐿𝑠1
0
𝐿𝑠1
0
0
−𝐿𝑠2
𝐿𝑠2 --------- (17)
0
𝐿𝑠2 ]
0
0
0
𝐶1−1
0
0
0
0 ------------ (18)
0
𝐶2−1 ]
Equation (15) is then applied after the process of eliminating
the dummy loops,
12
𝑌𝑙𝑜𝑜𝑝 = [𝑍𝑙𝑜𝑜𝑝 ]𝑛𝑒𝑤 −1 ------------------ (19)
The entire process of carrying out an admittance scan based
on the creation of dummy loops is described in the flow chart
in fig. 4. The modified admittance scan and modal loop
analysis has been applied to the equivalent circuit shown in
fig.2, and the results are displayed in Fig. 7 and 8
respectively. With the circuit, series resonance result could
be compared to that of parallel resonance solved by
frequency scan and Resonance Mode Analysis (see fig. 5 and
6).
Modal Admittance (pu)
10
[𝑍𝑙𝑜𝑜𝑝 ] = [𝑅𝑛𝑒𝑤 ] + 𝑗𝜔1 ℎ[𝐿𝑛𝑒𝑤 ] − 𝑗[𝐺𝑛𝑒𝑤 ]/𝜔1 ℎ
8
6
4
2
0
0
10
20
30
40
50
60
Frequency (p.u)
Fig. 7: Results of modal Loop analysis on test system
4.5
45
bus1
bus2
bus3
40
Loop1
Loop2
Loop3
4
35
3.5
30
Admittance (pu)
Impedance (pu)
Loop1
Loop2
Loop3
25
20
15
10
3
2.5
2
1.5
5
1
0
0.5
0
10
20
30
40
50
60
Frequency (p.u)
0
0
10
20
30
40
50
Frequency (p.u)
Fig 5: Bus Impedance Scan Results of the test system
Fig. 8: Loop Admittance Scan results of the test system.
70
bus1
bus2
bus3
Modal Impedance (pu)
60
V. APPLICATION TO A DISTRIBUTION NETWORK
50
The network diagram in fig.9 represents an 11kV
underground radial distribution network which feeds nine
load centers containing two major harmonic sources and four
capacitor banks. It is an expansion of the distribution network
used as case study in Reference [7]. The load centers are
connected by 35mm2 x 3, 11kV underground cables whose
parameters are as follows,
40
30
20
10
0
0
10
20
30
40
50
60
Frequency (p.u)
Fig. 6: Results of bus Resonance mode analysis (RMA)
The results indicate that the modified loop admittance scan
compares favourably to its counterparts: modal-loop analysis
and bus impedance scan. The key resonance modes
according to the results displayed in Fig. 5 – 8 occur at
frequencies (pu) = 4, 17 and 27. The modal admittance scan
(fig.7) produces only one resonance peak for each mesh
whereas about three peaks are captured in each mesh in the
results of modified loop admittance scan. This achievement
is because the network topological structure was not altered
in the process of applying the proposed technique in
localizing the series resonant frequencies.
Resistance/ph/km = 0.243Ω,
Inductance/ph/km = 5.023 x 10 - 4 H
Capacitance/ph/km = 2.456 x 10-8 F
The frequency response characteristics resulting from
application of modified loop admittance scan across the
entire distribution network are shown in fig. 10 – 13.
60
Fig. 12: Frequency response considering the effect of cable
capacitance
Fig. 9: Diagram showing the 11kV radial distribution
network in a consumer premises
Fig.13: Frequency response considering cable capacitance
and skin effect
The results indicate the occurrence of series resonance peaks
at the various meshes M1 – M9 as indicated.
Fig 10: Frequency response neglecting the effect of cable
capacitance and skin effect
In figure 10 where the effects of cable capacitance and skin
effect are neglected, the resonant frequencies in per unit are
h = 6, 7, 14 and 15.
In figure 11 where the effect of aggregate harmonic sources
is considered, the resonant frequencies are the same as in
fig. 10.
In figure 12 where only the effect of cable capacitance is
considered, the resonance frequencies in per unit are h = 6, 7
and 14.
In figure 13 where the effects of cable capacitance and skin
effect are considered, the resonance frequencies in per unit
are h = 6, 7 and 14.
Fig 11: Frequency response for aggregate sources neglecting
the effect of cable capacitance and skin effect
Comparison between figures 10 - 13 indicates that cable
capacitance contributes immensely to the shifting of the
resonant frequencies in M3 – M6 while the skin effect only
reduces the magnitude of loop admittance and widens the
frequency curve. Since resistance instead of inductance or
capacitance (that is the main contributors to system
resonance) varies according to the skin effect, consequently,
considering the skin effect in the calculations does not shift
the resonant frequencies. However, the increased resistance
could dampen the admittance peaks at resonance points
thereby decreasing the branch currents. Skin effect also
increases the bandwidth of resonance. This implies the
greater chance of current amplification occurring within the
neighbourhood of the resonant peaks. Comparison between
fig. 10 and 11 indicates that the application of many
harmonic frequency sources only impacts on the magnitude
of the driving point admittances and therefore is of no
consequence to this study which is focused on the resonant
frequencies.
VI. CONCLUSION
The modified loop admittance scan proves to be an
acceptable method for capturing all the dominant frequencies
involved in series resonance without any restriction to the
utilized frequency step. The application of a single unit
voltage is sufficient in the definition of admittance scan.
Specific achievements are recorded in this study: the
determination of series resonant frequencies by the
application of loop admittance matrix, the use of matrix
partitioning to treat independently the connection of shunt
capacitors and the contribution of skin effect as well as cable
capacitance on these results. It is noted that skin effect does
not contribute to the shifting of resonant frequencies but only
cable capacitance. Hence, in the calculation of the resonance
peaks, the contribution of skin effect can be neglected.
However, skin effect makes a significant contribution to the
value of branch currents which necessitates its consideration.
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