extensions to the inherent structural theory of power networks, and applications

EXTENSIONS TO THE INHERENT STRUCTURAL THEORY OF POWER NETWORKS, AND APPLICATIONS

Prof. John T. AgeeHead of the Control and Process Control Cluster

Department of Electrical EngineeringTshwane 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 generators and load centres.

A Re-statement of the Nature of Power Systems

• Can we view power system networks as interconnections of sources, sinks and circuit elements?

Alternatively

• Instead of the traditional, computationally 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 load flow approach

• 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

• Where I: injected currents, Ybus : bus admittance matrix and, V:nodal voltages

VYI bus

...Q-V

NkVYVQ

NkVYVP

IVjQPS

VYI

ikki

N

kkikii

ikki

N

kkikii

iiiii

N

jkiki

,2,1),sin(

,2,1),cos(

1

1

*

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 so-called 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 generator-load impedances YGL had different contributions to the I-V behaviour of the network.

New ISTN Terminology

• Parallels were drawn with nuclear forces: proto-proton attraction (affinity), electron-electron attractions, and proton-electron attractions

• A related partitioning of the Y matrix of the network:

LLLG

GLGG

YYYY

Y

Hence

L

G

LLLG

GLGG

L

G

VV

YYYY

II

INSTN Indices

• Re-write

G

L

GGGL

LGLL

G

L

VI

AKFZ

IV

LGLLGLGGGG

LLGLGL

LGLLLGLLLL

YYYYA

YYK

YYFYZ

1

1

11,

1. The Ideal Generator Contribution

• absolute values give the ideal

generator contribution, of each generator, at load buses

• The summation of each row is approximately equal to unity (Thukaram & Vyjayanthi, 2009)

1 LLGLGL YYK

GLGLLLL VFIZV

2. Generator-Generator Attraction Region

•

• 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

LGLLGLGGGG YYYYA 1

.... Impact of Generator-Generator Attraction Region

• Now,

• Decompose

with as appropriate eigenvalues

• Yielding

][1 LGLGGGG IKIAV

*

1

*jj

r

jjGG wwWWA

][1

*

LGLG

r

j j

jjG IKI

wwV

3. Generator Affinity

• Re-write

L

G

LLLG

GLGG

L

G

VI

CWHZ

IV

GLGGLGLLLL

GLGGGL

GGLGLGGGGG

YYYYC

YYH

YYWYZ

1

1

11 ,

... Generator Affinity

• represents the influence of generators over load buses and is termed the “generator affinity”

• The absolute value of the summation of each row of this matrix is approximately equal to unity

1 GLGGGL YYH

4. Load-load Electrical Attraction

• represents the equivalent load buses admittance or load-load attraction

• The eigenvalues of CLL determine the “structural impact of the electrical load attraction region” or how load buses affect load voltage

• The load bus with the lowest eigenvalue participation in CLL affects load voltages most.

GLGGLGLLLL YYYYC 1

.... Structural Impact of Load Electrical Attraction Region

• Now,

• Decompose

with as appropriate eigenvalues

• Yielding

][1 GLGLLLL IWICV

*

1

*ii

r

iiLL mmMMC

][1

*

GLGL

n

i i

iiL IWImmV

Summary of ISTN Indices• Ideal generator contribution, derived from FLG, gives how a

generator contributes to load voltage at a given load bus

• Structural impact of the load electrical attraction region: captures the critical behaviour of load-load buses to power 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 impedance links

APPLICATIONS

Southwest England 40 bus network

Device Location

Number of Compensators

Proposed methodbus number bus number

Classical Q-V sensitivitybus number bus number

1 33 29

2 33, 36 29, 37

3 33, 36, 38 29, 37, 33

4 33, 36, 38, 25 29, 37, 33, 35

Standard Deviation of Voltages

Number of SVCs Proposed method Classical

Q-V sensitivity

1 0.0193 0.0235

2 0.0122 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)

• Eigen vectors

S?N Bus Number Ranking1 0 1st2 0.7370 2nd3 1.5385 3rd4 2.3619 4th5 3.1080 5th

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