Post on 31-Dec-2015
description
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