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 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

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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 componentsTa 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

Ctv t( )d

d

1L 0

ttv 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, = - [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

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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

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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