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MODELLING OF SYSTEMS
Prepared by
Ashfaq Qureshi
Assistant Professor
Electrical Engineering Department
Shankersinh Vaghela Bapu Institute of Technology, Gandhinagar
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 2
What is modelling???• Systematic representation of system
• Visual representation of system
• Simplification of system
• Approximation of system
Without altering essential properties of the system
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 3
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]
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 4
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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 5
Principles of Mathematical Modelling
• Why?• What?• Given?• Assume?• How?
• Predict?• Valid?• Verified? • Improve?• Use?
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 6
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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 7
The commonly utilized approachState Space Approach
= [A] X + [B] u
Y= [C] X + [D] u1. 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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 8
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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 9
Samples modelling of componentsTa t( ) Ts t( ) 0
Ta t( ) Ts t( )
t( ) s t( ) a t( )
Ta t( ) = through - variable
angular rate difference = across-variable
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 10
Samples modelling of components
v t( )R
Ctv t( )d
d
1L 0
ttv t( )
d r t( )
y t( ) K 1 e 1 t
sin 1 t 1
05/03/2023 11Shankersinh Vaghela Bapu Institute of Technology
Let’s “MODEL”
Step 1:- Identify energy (memory) storing components
“C”
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 12
Let’s “MODEL”
Step 2:- Identify state variables
“VC”
05/03/2023 13Shankersinh Vaghela Bapu Institute of Technology
Let’s “MODEL”
Step 3:- Consider dynamic equation of energy (memory) storing components
C*dVC/dt= IC
C*dVC/dt= (Vi-VC)/ R
05/03/2023 14Shankersinh Vaghela Bapu Institute of Technology
Let’s “MODEL”C*dVC/dt= (Vi-VC)/ R
Rearranging, = - [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]
05/03/2023 15Shankersinh Vaghela Bapu Institute of Technology
Try Yourself
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 16
Try “Laplace Transformation”
05/03/2023 17Shankersinh Vaghela Bapu Institute of Technology
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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 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
05/03/2023 Shankersinh Vaghela Bapu Institute of Technology 19
Thank Y u
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