EE531 Power Network Modeling and Simulation Development and efficient solution of largescale computa

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EE531 Power Network Modeling and
SimulationDevelopment and efficient solution of
large-scale computational problems relevant to
power systems.

• Satish J. Ranade
• Fall 2008

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Lecture 1

• Course Overview
• Mechanics/Administrative
• Review of Network Analysis
• http//de.nmsu.edu/lectures/EE531/EE531main.htm

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Course Objective

• First graduate course in power systems sequence
• Advanced power system modeling, computation and
  analysis
• How to formulate and solve
• Methods suitable for large-scale systems
• Apply to power flow, short circuit, stability and
  network transients
• Use in all future application classes

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Graduate Program

• EE332 Power I
• EE431/542 Power II
• EE493/543 Power III
• EE494/544 Distribution(S08)
• EE432/537 Power Electronics(F08)
• EE 531 Power Network Modeling and Analysis
• EE532 Power System Stability/Transients(F09)
• EE533 Power System Operations(F08)
• EE534 Protection ( S08)
• EE535 Reliability (F08)
• EE 544 Distribution(S08)

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EE531 Topics Basic Theory
Will this work in theory?
Contributed by Kelly Garcia
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EE531 Topics

• Network analysis Ybus and Zbus
• Coupled elements
• Phase domain modeling
• Component models
• Lines Carsons Equations
• Transformers
• Synchronous and Induction Machines

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EE531 Topics

• Solution of Linear Equations
• LDU factorization
• Indirect Methods
• Partial Inverses
• Application to fault studies
• Nonlinear Equations
• Application to power flow study
• Continuation power flow
• Sparse matrix methods

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EE531 Topics

• Solution of Ordinary Differential Equations
• Explicit methods
• Implicit Methods
• Application to stability studies
• Application to network transients
• Optimization

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EE531 Grading, etc.

• GRADING POLICY
•  
• Homework 20
• Assigned each week, due following Monday
• Tests 3 60
• Projects 20
•  
• The grading scale is absolute 90 - 100 A, 80
  - 89 B, 70 - 79 C, 60 - 69 D , lt
  60 F

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EE531 Grading, etc.

• Course Text
•   Mariesa Crow Computational Methods for
  Electric Power Systems, CRC Press, Boca Raton,
  FL, 2002 ISBN 0-8493-1352-x 9000
•  
• Software
• Matlab, Mathcad, or Visual Basic
• Lectures Posted in MOV format at
• http//de.nmsu.edu/lectures/EE531/EE531main.htm
• Notes, Examples, Blue paper at
• www.ece.nmsu.ed/sranade

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Modeling and Analysis

• Network Equations
• Fundamental modeling issue
• Relationship between voltages and currents
• KVL Component model or primitive
• KCL Component model or primitive
• Bus Admittance/Impedance matrix Most Common
• Review for simple networks
• Building block for more complex modeling
• Network Equations get embedded in larger problem
  formulations
• Many problems have same form(topology) as network
  equations

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Network Equations

• Context
• Will look at single phase network
• ac steady state linear
• May have mutually coupled elements
• Could be the one of the sequence networks
• Building block for phase domain model

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Network Matrices
Single phase model ac steady state Relation
between E1, E2 and V1 V2 V3 Can calculate all
other things from V1 V2 V3
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Network Matrices
Can do KVL or KCL KCL(Admittance formulation)
most common
1
2
Voltage to Current Source Conversion
3
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Network Matrices
1
2
z12
Admittance Matrix I Y V I vector of
injected currents I1E1/Zg1 I2E3/Zg3
I30 V vector of line-ground voltage I1 Y11
Y12 Y13 V1 I2 Y21 Y22 Y23 V2 I3
Y31 Y32 Y33 V3
3
I2
I1
V1
V3
zg2
z13
z23
zg1
V2
I1(V1-V2)/z12 (V1-V3)/z13V1/Zg1 I1V1(1/z121/
z131/Zg1) V2(-1/z12)V3(-1/z13)
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Network Matrices
Admittance Matrix I Y V I vector of injected
currents V vector of line-ground
voltage Absent mutual coupling between
elements Diagonal Yii ? Admittances of
elements connected to node i Off Diagonal Yij
-? admittances of elements directly between
node i and node j
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Network Matrices
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Network Matrices
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Network Solution
I Y V
Not trivial for Large Matrices- will spend time
on this
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Admittance Matrix General Development

• Graphs and Incidence Matrices
• Primitive Impedance and Component Equations
• Y Ct yprim C
• Extend to cases where there is mutual coupling

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Admittance Matrix General Development

• Graphs and Incidence Matrices

A directed graph is a collection of nodes and
branches, such that each branch joins two
distinct nodes the branches are assumed to be
oriented. The graph describes The topology of our
network with assumed Directions for current and
voltage drop
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Admittance Matrix General Development

• Graphs and Incidence Matrices

Incidence Matrix C Let N number of nodes B
number of branches C is a BxN matrix
Branch 1
Branch 3
Branch 2
Branch 5
Branch 4
Nodes 1 2 3 branches
1 -1 0 1
0 1 -1 2 C -1 0 1 3 1
0 0 4 0 1 0 5
Cij 1 if Branch i is incident to node j
and directed away -1 if Branch i is
incident to node j directed towards
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Admittance Matrix General Development

• Primitive matrix

V1 -
The primitive model describes the behavior of a
component The simplest component is a series
impedance between Two nodes branch from node 1
to node 2 The primitive relation between branch
current and voltage is I1 V1/z12 V1y12 I1
is the branch current and V1 is the voltage drop
across the branch Let I denote the vector of
branch currents and let V denote the vector of
branch voltage drops. The primitive equations can
be written as V Z I or I Y V Z Is the
primitive impedance matrix and Y the primitive
admittance matrix
1
2
z12
3
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Admittance Matrix General Development

• Primitive matrix

In our system there are 5 branches ( 3 lines, 2
generator impedances). Absent mutual coupling Z
and Y are diagonal z12 0 0 0 0 0 z23 0 0 0 Z
0 0 z31 0 0 0 0 0 zg1 0 0 0 0 0 zg2
1/z12 0 0 0 0 0 1/z23 0 0 0 Y 0 0
1/z31 0 0 0 0 0 1/zg1 0 0 0 0 0 1/zg2
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Admittance Matrix General Development

• We want the network equations, i.e.,
• Relation between nodal injections
• and node currents
• branch voltages V related to node
• voltages V as
• V C V e.g., V1 V1-V2
• Nodal current injections are related
• to branch currents by KCL
• I CT I e.g., I1 I1-I3Ig1
• Since I Y V primitive
• I CT Y V CT Y C V
• So the admittance matrix Y CT Y C

V1 -
1
2
z12
I1
I3
3
z13
Ig1
z23
I1
V1
I2
V2
zg2
V3
zg1
Nodes 1 2 3 branches
1 -1 0 1
0 1 -1 2 C -1 0 1 3 1
0 0 4 0 1 0 5
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Admittance Matrix General Development

• Do the product CT Y C

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Admittance Matrix General Development
1 0 -1 1 0 y12 0 0 0 0 1
-1 0 -1 1 0 0 1 0 y23 0 0 0
0 1 -1 0 -1 1 0 0 0 0 y31 0
0 -1 0 1 0 0 0 yg1 0 1
0 0 0 0 0 0 yg2 0 1 0
First row of CTY First Row of CTYC y12 0
-y31 yg1 0 1 -1 0 y12y31yg1
-y12 -y31 0 1 -1 -1 0
1 1 0 0 0 1 0 Compare
direct construction rule!
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Network Matrices Mutually Coupled Elements
Our simple rules for building the Admittance
matrix do not work when branches are mutually
coupled
Z1
Mutually coupled impedances
M
Z2

• Mutual coupling arises in many ways
• When lines share a common right of way
  current/voltage in one line induces voltage in
  the other line
• In phase domain analysis phase of a line are of
  course mutually coupled
• Transformers

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Network Matrices Mutually Coupled Elements

• Mutual coupling arises in many ways

A

• Magnetic
• Capacitive
• Conductive

B
C
N
H
E
A
IA
B
Vag
Vag
C

N
Ig
Vg -
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Network Matrices Mutually Coupled Elements
Simplest case- sequence model of two overhead
lines in proximity
-------------- Vi---------------------- -
Ii
Ii
Zii
i,j branch m,n node numbers


Zij
Ij
Ii
Vm
Vm


Zjj
Vn
Vn
The primitive branch voltage drop is given by
the impedance matrix equation
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Network Matrices Mutually Coupled Elements
zii self impedance of line i zjj self impedance
of line j zij mutual impedance ( note polarity
with currents entering dots)
The corresponding primitive admittance matrix
form is
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Network Matrices Mutually Coupled Elements
With coupling between line1 from 1-2 and line3
from 1-3 Suppose the primitive matrix for these
lines is given as y11 y31 Indices go with
y31 y33 line number here
y11 0 y13 0 0 0 1/z23 0 0 0 Y
y31 0 y33 0 0 0 0 0 1/zg1 0
0 0 0 0 1/zg2
the primitive admittance matrix for the system
is
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Network Matrices Mutually Coupled Elements
1
2
Branch 1
Branch 3
Y CT y C
Branch 2
3
Cij 1 if Branch I is incident to
node j and directed away -1 if
Branch I is incident to node j
directed towards
Nodes--1 2 3 branches 1 -1 0 1
0 1 -1 2 C 1 0 -1 3 1
0 0 4 0 1 0 5
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Network Matrices Mutually Coupled Elements
So the admittance matrix Y CT y C
Without coupling notice significant changes
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Network Matrices Mutually Coupled Elements
We can extend the rules for building Y to
handle coupled elements see mathcad file for an
example If line 1 from bus i to bus j is
coupled to the line 2 from bus k to bus l
1 2
i j k l
Add -y11 to Yij term Add y11 to Yji term Add
-y22 to Ykl term Add -y22 to Ylk term
Add y11 to Yii term Add y11 to Yjj term Add y22
to Ykk term Add y22 to Yll term
Add y12 to Yik term Add y12 to Yki term Add y12
to Yjl term Add y12 to Ylj term
Add -y12 to Yil term Add y12 to Yli term Add
y12 to Yjk term Add y12 to Ykj term
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Network Matrices Mutually Coupled
Elements Coupling-free Synthesis
1 2
i j k l
i j k l
y11
1 2
y12
y12
-y12
-y12
y22
Add -y11 to Yij term Add y11 to Yji term Add
-y22 to Ykl term Add -y22 to Ylk term
Add y11 to Yii term Add y11 to Yjj term Add y22
to Ykk term Add y22 to Yll term
Equivalent circuit Without mutual
coupling Standard rules for building Y apply
Add y12 to Yik term Add y12 to Yki term Add y12
to Yjl term Add y12 to Ylj term
Add -y12 to Yil term Add y12 to Yli term Add
y12 to Yjk term Add y12 to Ykj term
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Y with mutual coupling
Line 1 i1 j2 Line 2
k1 l3
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Lecture 1 Summary

• Admittance(Y) and Impedance(ZY-1) matrices
  useful in network analysis
• Reviewed rule for building Y in networks without
  mutual coupling
• Reviewed graph-theoretic method for building Y
• Generalizing to include coupling
• Next up
• Properties/Interpretation of Y and Z
• Applications
• Network Reduction
• Component models and primitives