Advanced Power SystemsAdvanced Power Systems

Dr. Kar

U of Windsor

Dr. Kar271 Essex HallEmail: nkar@uwindsor.caOffice Hour: Thursday, 12:00-2:00 pm

http://www.uwindsor.ca/users/n/nkar/88-514.nsf

GA: TBAB20 Essex Hall

Email: TBA & TBA Office Hour: -----

Course Text Book:

Electric Machinery Fundamentals by Stephen J. Chapman, 4th Edition, McGraw-Hill, 2005

Electric Motor Drives – Modeling, Analysis and Control by R. Krishnan Pren. Hall Inc., NJ, 2001

Power Electronics – Converters, Applications and Design by N. Mohan, J. Wiley & Son Inc., NJ, 2003

Power System Stability and Control by P. Kundur, McGraw Hill Inc., 1993 Research papers

Grading Policy:

Attendance (5%)Project (20%)Midterm Exam (30%)Final Exam (45%)

Course Content

Working principles, construction, mathematical modeling,

operating characteristics and control techniques for synchronous

machines

Working principles, construction, mathematical modeling,

operating characteristics and control techniques for induction

motors

Introduction to power switching devices

Rectifiers and inverters

Variable frequency PWM-VSI drives for induction motors

Control of High Voltage Direct Current (HVDC) systems

Exam Dates

Midterm Exam:

Final Exam:

Term Projects

Group 1:Student 1 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca)Project Title:

Group 2:Student 1 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca)

Project Title:

Group 3:Student 1 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca)

Synchronous Machines

Construction Working principles Mathematical modeling Operating characteristics

CONSTRUCTION

1. Most hydraulic turbines have to turn at low speeds (between 50 and 300 r/min)

2. A large number of poles are required on the rotor

Hydrogenerator

Turbine Hydro (water)

D 10 m

Non-uniform air-gap

N

S S

N

d-axis

q-axis

Salient-Pole Synchronous Generator

Salient-Pole Synchronous Generator

Stator

Salient-pole rotor

Cylindrical-Rotor Synchronous Generator

Stator

Cylindrical rotor

Damper Windings

Operation Principle

The rotor of the generator is driven by a prime-mover

A dc current is flowing in the rotor winding which produces a rotating magnetic field within the machine

The rotating magnetic field induces a three-phase voltage in the stator winding of the generator

Electrical Frequency

Electrical frequency produced is locked or synchronized to the mechanical speed of rotation of a synchronous generator:

 

 

where fe = electrical frequency in Hz

P = number of poles

nm= mechanical speed of the rotor, in r/min

120

Pnf

me

Direct & Quadrature Axes

Stator

Uniform air-gap

Stator winding

Rotor

Rotor winding

N

S

Turbogenerator

d-axis

q-axis

PU System

QuantityBase

QuantityActualValuePU

base

ohmPU

base

basebase

base

base

base

base

base

basebasebasebase

basebasebasebasebasebase

base

basebase

Z

ZZ

V

IY

VA

V

S

V

I

VZXR

IVVASQP

V

VAI

22

Per unit system, a system of dimensionless parameters, is used for computational convenience and for readily comparing the performance of a set of transformers or a set of electrical machines.

Where ‘actual quantity’ is a value in volts, amperes, ohms, etc. [VA]base and [V]base are chosen first.

Classical Model of Synchronous Generator

The leakage reactance of the armature coils, Xl

Armature reaction or synchronous reactance, Xs

The resistance of the armature coils, Ra

If saliency is neglected, Xd = Xq = Xs

Ia

E Vt 0o

jXsjXl Ra

+

+

Equivalent circuit of a cylindrical-rotor synchronous machine

Phasor Diagram

E

Ia

IaXl

IaXs

q-axis

Vt

IaRa

d-axis

The following are the parameters in per unit on machine rating of a 555 MVA, 24 kV, 0.9 p.f., 60 Hz, 3600 RPM generator

Lad=1.66 Laq=1.61 Ll=0.15 Ra=0.003

(a)When the generator is delivering rated MVA at 0.9 p. f. (lag) and rated terminal voltage, compute the following:

(i) Internal angle δi in electrical degrees(ii) Per unit values of ed, eq, id, iq, ifd

(iii) Air-gap torque Te in per unit and in Newton-meters

(b) Compute the internal angle δi and field current ifd using the following equivalent circuit

Direct and Quadrature Axes

The direct (d) axis is centered magnetically in the center of the north pole

The quadrature axis (q) axis is 90o ahead of the d-axis : angle between the d-axis and the axis of phase a Machine parameters in abc can then be converted into d/q frame using Mathematical equations for synchronous machines can be obtained

from the d- and q-axis equivalent circuits Advantage: machine parameters vary with rotor position w.r.t. stator, ,

thus making analysis harder in the abc axis frame. Whereas, in the d/q reference frame, parameters are constant with time or .

Disadvantage: only balanced systems can be analyzed using d/q-axis system

d- and q-Axis Equivalent Circuitsd- and q-Axis Equivalent Circuits

Ifd

Xfd

Rfd

Xl

pd

Ikd1 Imd

Vtd

Ra Id

Xkd1

Xmd

Rkd1

q

d-axis

vfd

+

-

pfd

+

+ -

-

pkd1

Xld

pq

Ikq1 Imq

Vtq

Ra Iq

Xkq1

Xmq

Rkq1

q-axis

+

-pkq1

Imd=-Id+Ifd+Ikd1

Imq=-Iq+Ikq1

Small disturbances in a power system

o Gradual changes in loadso Manual or automatic changes of excitationo Irregularities in prime-mover input, etc.

Importance of steady-state stability

o Knowledge of steady-state stability provides valuable information about the dynamic characteristics of different power system components and assists in their design

- Power system planning

- Power system operation

- Post-disturbance analysis

Related Terms

o Generator Modeling using the d- and q-axis equivalent circuitso Transmission System Modeling with a RL circuito A Small Disturbance is a disturbance for which the set of equations

describing the power system may be linearized for the purpose of analysiso Steady-State Stability is the ability to maintain synchronism when the

system is subjected to small disturbanceso Loss of synchronism is the usual symptom of loss of stabilityo Infinite Bus is a system with constant voltage and constant frequency,

which is the rest of the power systemo Eigen values and eigen vectors are used to identify system steady-state

stability condition

The Flux Equations

fdmdkdmddlmdd iXiXiXX 1

fdmdkdkdmddmdkd iXiXXiX 111

fdfdmdkdmddmdfd iXXiXiX 1

1kqmqqlmqq iXiXX

111 kqkqmqqmqkq iXXiX

Rearranged Flux Linkage equations

1

1

1

1

1

1

kq

q

fd

kd

d

kqmqmq

mqlmq

fdmdmdmd

mdkdmdmd

mdmdlmd

kq

q

fd

kd

d

i

i

i

i

i

XXX

XXX

XXXX

XXXX

XXXX

The Voltage Equations

qdatdd iRvp

0

1

1110

1kdkdkd iRp

fdfdfdfd iRvp 0

1

dqatqq iRvp

0

1

1110

1kqkqkq iRp

……………..(1)

The Mechanical Equations

dqqde

em

IIT

TTHdt

ddt

d

20

0

where

……………..(2)

Linearized Form of the Machine Model

qddqdqqde

em

kqkqkq

ddqatqq

fdfdfdfd

kdkdkd

qqdatdd

IIIIT

TTH

iR

iRv

iRv

iR

iRv

0000

0

1110

0

0

0

0

1110

0

0

0

2

1

1

1

1

1

……………..(3)

Terminal Voltage

The d- and q-axis components of the machine terminal voltage

can be described by the following equations:

where, Vt is the machine terminal voltage in per unit.

The linearized form of Vtd and Vtq are:

cos

sin

ttq

ttd

Vv

Vv

………………………….(4)

0

0

sin

cos

ttq

ttd

Vv

Vv ……………………….…(5)

Substituting ∆Vtd and ∆Vtq in the flux equations:

qddqdqqde

em

kqkqkq

ddqatq

fdfdfdfd

kdkdkd

qqdatd

IIIIT

TTH

iR

iRV

iRv

iR

iRV

0000

0

1110

0

00

0

0

1110

0

00

0

2

1

sin1

1

1

cos1

……..(6)

Rearranging the flux equations in a matrix form:

UBIRXSX

m

fd

T

vU

1

1

kq

q

fd

kd

d

I

I

I

I

I

I

1

1

kq

q

fd

kd

d

X

1

1

kq

q

fd

kd

d

X

where,

………………...…..(7)

and…

0002

02

0

1000000

0000000

sin0000

0000000

cos0000

0000000

0000

0000

0000

H

I

H

I

V

V

S

dq

dt

qt

02

02

0

00000

0000

0000

0000

0000

0000

0000

10

0

10

0

0

HH

R

R

R

R

R

R

dq

kq

a

kd

a

fd

H

B

20

00

00

00

0

0

0

Flux Linkage Equations (from the d- and q-axis equivalent circuits)

1

1

1

1

1

1

000

000

00

00

00

kq

q

fd

kd

d

kqmqmq

mqlmq

fdmdmdmd

mdkdmdmd

mdmdlmd

kq

q

fd

kd

d

i

i

i

i

i

XXX

XXX

XXXX

XXXX

XXXX

1

1

1

1

1

1

000

000

00

00

00

kq

q

fd

kd

d

kqmqmq

mqlmq

fdmdmdmd

mdkdmdmd

mdmdlmd

kq

q

fd

kd

d

i

i

i

i

i

XXX

XXX

XXXX

XXXX

XXXX

Linearized flux linkage equations:

and thus,

1

1

1

1

1

1

1

1

1

1

1

1

1

1

00000

00000

0000

0000

0000

000

000

00

00

00

kq

q

fd

kd

d

reac

kq

q

fd

kd

d

kqmqmq

mqlmq

fdmdmdmd

mdkdmdmd

mdmdlmd

kq

q

fd

kd

d

kqmqmq

mqlmq

fdmdmdmd

mdkdmdmd

mdmdlmd

kq

q

fd

kd

d

X

XXX

XXX

XXXX

XXXX

XXXX

XXX

XXX

XXXX

XXXX

XXXX

i

i

i

i

i

………………………………………...(8)

UBXA

UBXXRS

UBXXRXS

UBIRXSX

reac

reac

1

1

XXX

i

i

i

i

i

I reac

kq

q

fd

kd

d

reac

kq

q

fd

kd

d

1

1

1

1

1

1

1 reacXRSA

where,

: from (8)

: inserting (8) into (7)

: system state matrix………..(9)

System to be Studied

Infinite Bus

Generator

Vt

It

System State Matrix and Eigen Values

1 reacXRSA

1

2

j

System State Matrix:

Eigen Values: j21,

Eigen Values

o Eigen values are the roots of the characteristic equation

o Number of eigen values is equal to the order of the characteristic equation or number of state variables

o Eigen values describe the system response ( ) to any disturbance

UBXAX

te 1

Analyzing the Eigen Values of the System State Matrix

o Compute the eigen values of the system state matrix, Ao The eigen values will give necessary information about the steady-state

stability of the systemo Stable System: If the real parts of ALL the eigen values are negative

Example:

o A system with the above eigen values is on the verge of instability

0005.0

0.215.0,

3

21

j

Machine Parameters

Machine parameters Per unit values

d-axis magnetizing reactance, Xmd 1.189

q-axis magnetizing reactance, Xmq 0.7164

Armature leakage reactance, Xl 0.100

Field circuit leakage reactance, Xfd 0.276

First d-axis damper circuit leakage reactance, Xkd1 0.181

First q-axis damper circuit leakage reactance, Xkq1 0.193

Armature winding resistance, Ra 0.0186

Field winding resistance, Rfd 0.0058

First d-axis damper winding resistance, Rkd1 0.062

First q-axis damper winding resistance, Rkq1 0.052

Salient-pole synchronous generator

3kVA, 220V, 4-pole, 60 Hz and 1800 r/min