### Presentation at Clemson. Agee

```EXTENSIONS TO THE INHERENT
STRUCTURAL THEORY OF POWER
NETWORKS, AND APPLICATIONS
Prof. John T. Agee
Head of the Control and Process Control Cluster
Department of Electrical Engineering
Tshwane University of Technology. Pretoria. South
Africa
Others
• Mr. Humble Tajudeen Sikiru
• Prof. A. A. Jimoh
• Prof. Alex Haman
• Prof. Roger Ceschi
Well Known
• Generally, generator would be located near
sources of primary energy
• There are main electricity consumption points or
load centres supports industrial and commercial
activities as manufacturing, mining, etc
• In a simplistic manner, a power system network
consists of the network of transmission lines, the
A Re-statement of the Nature of
Power Systems
• Can we view power system networks as
interconnections of sources, sinks and circuit
elements?
Alternatively
intensive power system analysis techniques
based on non-linear load flow equations, ......
• Can power systems be analysed using simple
circuit analysis laws?
Summary of Presentation
• Thought-provoking comments on the classical
• The inherent structural theory of power
systems networks(ISTN): in history
• Our recent extension of the ISTN with the
introduction of new indices
• Illustrate the use of some of our ISTN indices
in power system analysis.
A Q-V Sensitivity Presentation
• Consider
I  YbusV
• Where I: injected currents, Ybus : bus
...Q-V
N
I i   YikVk
j 1
Si  Pi  jQi  Vi I i*
N
Pi  Vi  YikVk cos( i   k   ik ), k  1,2, N
k 1
N
Qi  Vi  YikVk sin( i   k   ik ), k  1,2, N
k 1
The Lesson from the Q-V Methods
• The load flow methods introduce nonlinearities
that are not inherent in the original problem
• May thus add several orders of complexity in
arriving at a solution of the problem
• The sub-optimality of solutions of some power
flow problems, arise from the method of
solution: and may not be inherent in the
problem itself
......
• If the complexity of a power system network is
increased by the volatility of microgrids/
distributed generation/intermittent renewable
sources, shall classical load flow methods
improve or complicate the ease of solution of
network problems?
The Theory of the Inherent Structural
Characteristics of Transmission Networks (ISTN)
• The earliest thoughts in this regard, were
formulated by Laughton (1964)
• This approach argues that, the interactions of
voltages V, and currents I (and hence power
flows)in a power system networks are
governed by ohms law of the form V=ZI or
I=YZ
Classical ISTN
• That variations of V or I creates variations of the
other.
• That Z (Y) remains constant in a given network
• That the behaviour of the network is preserved in
the structure of its Y matrix: the Y matrix thus
contains all the information on the inherent
(electrical) structure or behaviour of the
network.
Success of the Classical ISTN
• Several successful applications of classical
ISTN have been reported:
– location of capacitors & harmonic filters
– Power quality studies
– Generator allocation
– Identification of weak nodes in power systems
Challenges of Classical ISTN
• Was not very successful in the analysis of
highly interconnected networks
• Extensions of this theory, providing the socalled T-index is also found to be highly
complex in practical applications
Recent Extensions to the ISTN Theory
• Realised that buses in a power system do not
have the same play: generator impedances YG,
load impedances YL and transmission line or
contributions to the I-V behaviour of the
network.
New ISTN Terminology
• Parallels were drawn with nuclear forces:
proto-proton attraction (affinity), electronelectron attractions, and proton-electron
attractions
• A related partitioning of the Y matrix of the
network:
YGG
Y 
YLG
YGL 

YLL 
Hence
 I G  YGG

 I  Y
 L   LG
YGL  VG 



YLL  VL 
INSTN Indices
• Re-write
VL   Z LL FLG   I L 

 I   K A  V 
 G   GL GG   G 
Z LL  YLL1 , FLG  YLL1YLG
K GL  YGL YLL1
AGG  YGG  YGL YLL1YLG
1. The Ideal Generator Contribution
•
KGL  YGLYLL1
absolute values give the ideal
generator contribution, of each generator, at
V Z I F V
L
LL
L
LG
G
• The summation of each row is approximately
equal to unity (Thukaram & Vyjayanthi, 2009)
2. Generator-Generator Attraction Region
•
AGG  YGG  YGLYLL1YLG
• The eigenvalues of AGG define the ‘structural
impact of the generator-electrical attraction
region’
• The generator associated with the least
eigenvalue has the highest impact on
generator voltages
.... Impact of Generator-Generator
Attraction Region
• Now,
1
GG
VG  A [ IG  KGL I L ]
• Decompose
with

AGG  WW

r
w 
j 1
j
j
w*j
as appropriate eigenvalues
r
• Yielding
*
VG  
j 1
*
j
wj w
j
[ I G  K GL I L ]
3. Generator Affinity
• Re-write
VG   Z GG H GL   I G 

 I  W C  V 
 L   LG LL   L 
Z GG  YGG1 ,WLG  YLG YGG1
H GL  YGG YGL1
CLL  YLL  YLG YGG1 YGL
... Generator Affinity
•
represents the influence of
generators over load buses and is termed the
“generator affinity”
HGL  YGGYGL1
• The absolute value of the summation of each
row of this matrix is approximately equal to
unity
•
CLL  YLL  YLGYGG1YGL
• The eigenvalues of CLL determine the
“structural impact of the electrical load
attraction region” or how load buses affect
• The load bus with the lowest eigenvalue
participation in CLL affects load voltages most.
.... Structural Impact of Load Electrical
Attraction Region
• Now,
1
LL
VL  C [ I L  WLG I G ]
• Decompose
with

C LL  MM

r
m  m
i 1
i
i
*
i
as appropriate eigenvalues
n
• Yielding
*
VL  
i 1
*
i
mi m
i
[ I L  WLG I G ]
Summary of ISTN Indices
• Ideal generator contribution, derived from FLG, gives how a
• Structural impact of the load electrical attraction region:
system networks & and can be used to counter the
limitation of the ideal generator contribution
• Structural impact of generator electrical attraction:
valuable in identifying generators that are located at
structurally weal nodes
• Generator affinity: clarifies which load buses will be
supplied with larger power levels, based on their low
APPLICATIONS
Southwest England 40 bus network
Device Location
Number of
Compensators
Proposed method
bus number bus
number
Classical Q-V sensitivity
bus number bus number
1
33
29
2
33, 36
29, 37
3
33, 36, 38
4
33, 36, 38, 25
29, 37, 33
29, 37, 33, 35
Standard Deviation of Voltages
Number of SVCs
1
2
Proposed method
Classical
0.0193
0.0122
Q-V sensitivity
0.0235
0.0198
3
0.0123
0.0121
4
0.0120
0.0118
Example
• Topologically strong versus topologically weak
networks
IEEE 30 bus network
Eigenvalue
• The Y matrix has a zero
eigenvalue ( actually,
less than a given
precision value)
S?N
Bus Number
Ranking
1
0
1st
2
0.7370
2nd
3
1.5385
3rd
4
2.3619
4th
5
3.1080
5th
• Eigen vectors
Not Easily Improved
Southwest England 40 bus network
....
• The smallest eigenvalue (in absolute value) is greater that the precision
defined for this test network and it is 0.0045. Eigen vector
```