Power_system_transients.pdf

Fundamental Notions

• Transients� electrical manifestation of a sudden change in circuit

conditions• switch opens

• switch closes

• network fault

� transient period is very short relative to the time spent in steady state operation, but is extremely important

• greatest stresses on circuit components

• excessive currents and voltages

• circuit damage

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Understanding Transients

• Most power engineers have only the haziest concepts of what is happening in the circuit during transients

• Some view the subject as bordering the occult

• Transient canbe understood

• Course objective: Learn to� calculate and assess behaviors, and

� prevent or control damaging transient energies



Basics of Transients

• circuit parameters: R, L, and C� all components and devices have all three parameters

� these attribute differ to a greater or lesser degree

� under steady-state conditions, one will dominate

� in transient conditions the conditions are very different

• lumped vs. distributed parameters� in reality, parameters are distributed in nature

� in circuit analysis, we frequently lump the parameters for ease of analysis

� under many conditions, good analysis requires using distributed parameters



Lumped Parameter Modeling

• When are the resistance, inductance, and capacitance effects important or necessary to model?

a capacitor? an inductor?



Energy Concepts

• electric field energy storage:

• magnetic field energy storage:

• energy dissipation:

221 IL

221 VC

2IR



Energy Balance

• Under steady-state conditions� dc circuit – the energy stored is constant

� ac circuit – the energy is being transferred cyclically between the L’s and C’s of the circuit

• Sudden changes in a circuit causes a redistribution of energy� energy redistribution cannot take place instantaneously

� change in the magnetic field requires a change in current, but is opposed by the induced emf:

� change in the electric field requires a change in the charge or the voltage, but is opposed by the flow of current:

dt

dILe =

dt

dQ

dt

dVCI ==



Energy Conservation Law

• Redistribution of energy following a circuit change takes a finite time

• All circuits at all times are governed by the conservation of energy

dt

dE

dt

dE

dt

dE

EEE

ndissipatiostoragesupply

ndissipatiostoragesupply

+=

+=



Circuit Modeling

• set down the differential or algebraic equations� use Kirchhoff’s voltage and current laws

• example:switch closing

V

Sw R

C V1

I

∫∫

∫

=−

=−

==

+=

RC

dt

VV

dVdt

dVRCVV

dt

dVC

dt

dQI

dtIC

IRV

1

1

11

1

1



Circuit Modeling

( )

( )( )

( )( ) RCt

RCt

eVVVV

VVA

Vgiven

eAVV

constRC

tVV

RC

dt

VV

dV

−

−

−−=∴

−=

−=

+−=−

=− ∫∫

0

0

0:

ln

11

1

1

1

1

1

1

Time

V

V1

V1(0)



Lump Circuit Characteristics

• Three basic circuit types:

CC

R R

L

L

RCt

e−

LtR

e−

LCtj

LCtj

ee−

±



Lump Circuit Characteristics

• the only kind of response that is evoked when a linear electric circuit comprising lumped elements is disturbed takes the form of exponential functions or combinations of exponential functions with real or imaginary components (sine or cosine functions)

• for circuits with more elements, the responses may be more complicated but they are no more complex



Superposition

• very powerful tool for solving problems� in a linear system, if a stimulus S1 produces a response R1

and a stimulus S2 produces a response R2, then S1 and S2

applied simultaneously will evoke a response R1+R2

• switching operations� an event in which a new path for current is created or an

existing path is eliminated• including faults, circuit breaker operations, and lightning strikes

� transients are a response to switching operations



Superposition and Switches

• Exampleopening a switch

E

Z1

Z2Contacts part

I1 I2( )( ) ( )

φφφω

φω

>←=+−⋅−−=

−=

tII

tutAI

tAI

0

sin

sin

21

2

1

Time

Time



Superposition

• Example� initially, C1 is charged to 100 kV and C2 is uncharged

� switch S is closed and at 40 s later the gap G sparks over

� determine the current in R2 and the voltage on C1

immediately after the sparkover

C15 F

C20.5 F

R1100

R21.0 k

GS



Power_system_transients.pdf

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