Basics of modelling

1. MODELLING OF SYSTEMS
Prepared by
Ashfaq Qureshi
Assistant Professor
Electrical Engineering Department
Shankersinh Vaghela Bapu Institute of Technology, Gandhinagar

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What is modelling???
• Systematic representation of system
• Visual representation of system
• Simplification of system
• Approximation of system
Without altering essential
properties of the system

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Type of models
• Graphic model
• Physical model
• Analytical model
• Number model
• Computational model
• Mathematical Model
I asked Fermi whether he was not impressed by the agreement
between our calculated numbers and his measured numbers. He
replied, How many arbitrary parameters did you use for your
calculations?" I thought for a moment about our cut-of
procedures and said, Four." He said, I remember my friend
Johnny von Neumann used to say, with four parameters I can fit
an elephant, and with five I can make him wiggle his trunk.“
Freeman Dyson on describing the predictions of his model for
meson-proton scattering to Enrico Fermo in 1953. [01]

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Mathematical Modelling
• Mathematical model is a
description of a system using
mathematical concepts and
language.
• The process of developing a
mathematical model is termed
mathematical modeling.
• It is the engineering practice to
use mathematical modelling

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Principles of Mathematical Modelling
• Why?
• What?
• Given?
• Assume?
• How?
• Predict?
• Valid?
• Verified?
• Improve?
• Use?

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Which kind of mathematics I have to deal
with??
• Differential equations (First order, second order and higher order
)
• Partial differential equations
• Linear models
• Algebra and geometry
• Polynomial equations
• Bifurcations
• Nonlinearities
• Waves
• Probability and statistics

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The commonly utilized approach
State Space Approach
𝑋= [A] X + [B]
u
Y= [C] X + [D]
u
1. Input Variable
(u)
2. State Variable
(X)
3. Output (Y)
Input variable (u) and
output variable
depends on system and
user
State variables can be
decided from presence
and location of energy
storing elements and
information storage

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Process for formulating mathematical equations in
electrical circuit
• Step 1:- Identify energy (memory) storing components
• Step 2:- Identify state variables
• Step 3:- Consider dynamic equation of energy (memory) storing
components
• Step 4:- Arrange the equations in required format

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Samples modelling of components
Ta t( ) Ts t( ) 0
Ta t( ) Ts t( )
 t( ) s t( ) a t( )
Ta t( ) = through - variable
angular rate difference = across-variable

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Samples modelling of components
v t( )
R
C
t
v t( )
d
d

1
L 0
t
tv t( )



d r t( )
y t( ) K 1 e
 1 t
 sin  1 t  1 

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Let’s “MODEL”
Step 1:- Identify energy (memory) storing
components
“C”

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Let’s “MODEL”
Step 2:- Identify state variables
“VC”

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Let’s “MODEL”
Step 3:- Consider dynamic equation of energy (memory)
storing components
C*dVC/dt= IC
C*dVC/dt= (Vi-VC)/ R

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Let’s “MODEL”
C*dVC/dt= (Vi-VC)/ R
Rearranging,
VC= - [1/R*C] VC + [1/R*C] Vi
VC = [1] VC + [0] Vi
state matrix [A] = - [1/R*C] input matrix [B] = - [1/R*C]
output matrix [C] = [1]
direct transition (feedthrough) matrix
[D] = [0]

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Try Yourself

16. Try “Laplace
Transformation”
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What to do for modelling of power system
• Depends on what is the purpose of modelling
• The structure is modelled by graph theory
• The load can be modelled by static or dynamic
• Consideration of scalar and vector quantities
• Load flow study uses static modelling
• Transient and fault study use dynamic modelling
• Static and dynamic state estimation

18. Reference
1. http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf
2. http://www.comap.com/modelingHB/Modeling_HB_Sample.pdf
3. http://mtm.ufsc.br/~daniel/matap/IntMatMod.pdf
4. http://www3.nd.edu/~msen/Teaching/EngAn/EngAnNotes.pdf
5. http://www.eolss.net/sample-chapters/c02/e6-03b-03-02.pdf
6. http://www.eeh.ee.ethz.ch/uploads/tx_ethstudies/modelling_hs08_script_02.pdf
7. http://nptel.ac.in
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Thank
Y u