Download d- and q-Axis Equivalent Circuits

Advanced Power Systems
Dr. Kar
U of Windsor
Dr. Kar
271 Essex Hall
Email: [email protected]
Office Hour: Thursday, 12:00-2:00 pm
http://www.uwindsor.ca/users/n/nkar/88-514.nsf
GA: TBA
B20 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
Project
Midterm Exam
Final Exam
(5%)
(20%)
(30%)
(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 ([email protected])
Student 2 ([email protected])
Student 3 ([email protected])
Project Title:
Group 2:
Student 1 ([email protected])
Student 2 ([email protected])
Student 3 ([email protected])
Project Title:
Group 3:
Student 1 ([email protected])
Student 2 ([email protected])
Student 3 ([email protected])
Synchronous Machines




Construction
Working principles
Mathematical modeling
Operating characteristics
CONSTRUCTION
Salient-Pole Synchronous Generator
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
d-axis
Nonuniform airgap
N
D  10 m
q-axis
Turbin
e
Hydro (water)
Hydrogenerator
S
S
N
Salient-Pole Synchronous Generator
Stator
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:
fe 
nm P
120
where fe = electrical frequency in Hz
P = number of poles
nm= mechanical speed of the rotor, in r/min
Direct & Quadrature Axes
d-axis
Stator winding
N
Uniform air-gap
Stator
q-axis
Rotor winding
Rotor
S
Turbogenerator
PU System
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.
PU Value 
Actual Quantity
Base Quantity
Where ‘actual quantity’ is a value in volts, amperes, ohms, etc.
[VA]base and [V]base are chosen first.
I base 
VAbase
V base
Pbase  Qbase  S base  VAbase  V base I base
Rbase  X base  Z base
Ybase 
Z
PU
I base
V base

Z
ohm
Z base
2
2
V base V base
V base



I base S base VAbase
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
jXs
jXl
Ra
+
+
E d
Ia
Vt
0o
Equivalent circuit of a cylindrical-rotor synchronous machine
Phasor Diagram
q-axis
E
IaXs
d
f
IaRa
Ia
d-axis
Vt
IaXl
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
q: angle between the d-axis and the axis of phase a
Machine parameters in abc can then be converted into d/q frame using q
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, q,
thus making analysis harder in the abc axis frame. Whereas, in the d/q
reference frame, parameters are constant with time or q.
Disadvantage: only balanced systems can be analyzed using d/q-axis
system
d- and q-Axis Equivalent Circuits
+
+
Rfd
pykd1
-
pyfd
+
vfd
yq
Xl
Xfd
Ifd
Ikd1
Ra
Id
Imd
Rkd1
Xmd
Vtd
pyd
Xkd1
-
-
d-axis
Imd=-Id+Ifd+Ikd1
- yd
Xl
+
pykq1
-
Ikq1
Rkq1
Ra
Imq=-Iq+Ikq1
Iq
Imq
Xmq
pyq
Xkq1
q-axis
Vtq
Small disturbances in a power system
o
o
o
Gradual changes in loads
Manual or automatic changes of excitation
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 circuits
o Transmission System Modeling with a RL circuit
o A Small Disturbance is a disturbance for which the set of equations
describing the power system may be linearized for the purpose of analysis
o Steady-State Stability is the ability to maintain synchronism when the
system is subjected to small disturbances
o Loss of synchronism is the usual symptom of loss of stability
o Infinite Bus is a system with constant voltage and constant frequency,
which is the rest of the power system
o Eigen values and eigen vectors are used to identify system steady-state
stability condition
The Flux Equations


y d  - X md  X l id  X md ikd 1  X md i fd




y kd 1  - X md id  X md  X kd 1 ikd 1  X md i fd


y fd  - X md id  X md ikd 1  X md  X fd i fd


y q  - X mq  X l iq  X mq ikq1


y kq1  - X mq iq  X mq  X kq1 ikq1
Rearranged Flux Linkage equations
 y d  -  X md  X l 
y   - X
md
 kd1  
y fd    - X md

 
y
q

 
y kq1  

 
X md
 X md  X kd1 
X md

X md
X md
X md  X fd


- X mq  X l
- X mq

X mq
- X mq  X kq1

  id 
 i 
  kd1 
  i fd 
 
  iq 
 ikq1 
 

The Voltage Equations
1
0
1
0
1
0
1
0
1
0
py d   vtd  Ra id y q
py kd1   - Rkd1 ikd1
 
p y fd  v fd - R fd i fd
 
p y q  vtq  Ra iq -y d


p y kq1  - Rkq1 ikq1
……………..(1)
The Mechanical Equations
dd
  - 0
dt
d  0
Tm - Te 

dt 2 H
where
Te  y d I q -y q I d
……………..(2)
Linearized Form of the Machine Model
y q0
y d  vtd  Ra id  y q 

0
0
1

1

0
y kd1  - Rkd1 ikd1

1
0
y
fd
 v fd - R fd i fd

1
0
y q  vtq  Ra iq - y d -
y d0

0

1
0
y kq1  - Rkq1 ikq1

 d  
 
0
2H
Tm - Te 
Te  y d 0 I q  I q 0 y d -y q 0 I d - I d 0 y q
……………..(3)
Terminal Voltage
The d- and q-axis components of the machine terminal voltage
can be described by the following equations:
vtd  Vt sin d
vtq  Vt cosd
………………………….(4)
where, Vt is the machine terminal voltage in per unit.
The linearized form of Vtd and Vtq are:
vtd  Vt cosd 0  d
vtq  -Vt sin d 0  d
……………………….…(5)
Substituting ∆Vtd and ∆Vtq in the flux equations:
y q0
y d  Vt cos d 0  d  Ra id  y q 

0
0
1

1

0
y kd1  - Rkd1 ikd1

1
0
y
fd
 v fd - R fd i fd

1
0
y q  -Vt sin d 0  d  Ra iq - y d -
y d0

0

1
0
y kq1  - Rkq1 ikq1

 d  
 
0
2H
Tm - Te 
Te  y d 0 I q  I q 0 y d -y q 0 I d - I d 0 y q
……..(6)
Rearranging the flux equations in a matrix form:
 
………………...…..(7)
 X   S X   R I   B U 


where,
  
 y d 
  
kd1 
y
 y 
fd 

    
 X    y q 

   
 y kq1 
  
 d 
  
   
 y d 
y 
 kd1 

  y fd 
 X    y 
q 

 

  y kq1 



d


  


 Id 


   I kd1 
  

 I    I fd 
  

 

I
q




  I kq1 
v fd 
U    
 Tm 
and…
0
0
0
0

0
0

S   0 - 0
0
0

0
0
0 - 0 I q 0

2H
- 0 R fd
 0

 0

R   0
 0

 0
 0

0
0
0
0
0
0
0
0Vt cos d 0
0
0
0
0
0
0
0 - 0Vt sin d 0
0
0
0
0
0 I d 0
0
2H
0
0
0
0
0
0
0
0
0
0 Ra
0
0
0
0
- 0 Rkd1
0
0
0 Ra
0
0
0
0
0
0
0y q 0
2H
0
- 0y d 0
2H
0 
y q0 

0 

-y d 0 
0 

1 
0 


0 

0 

0 
- 0 Rkq1 

0 
0 

0
0
0

B   0

0
0

0 
0 

0 

0 
0 
2H 
Flux Linkage Equations (from the d- and q-axis equivalent circuits)
 y d  -  X md  X l 
y   - X
md
 kd1  
y fd    - X md

 
y
0
q

 
y kq1  
0
X md
 X md  X kd1 
X md
0
0
X md
0
0
  id 
 i 
X md
0
0
  kd1 
X md  X fd 
0
0
  i fd 
 
0
- X mq  X l 
X mq
  iq 
0
- X mq
- X mq  X kq1  ikq1 
Linearized flux linkage equations:
 y d  -  X md  X l 
y   - X
md
 kd1  
 y fd    - X md

 

y
0
q

 
 y kq1  
0
X md
 X md  X kd1 
X md
0
0
X md
0
0
  id 
 i 
X md
0
0
  kd1 
X md  X fd 
0
0
  i fd 


0
- X mq  X l 
X mq

i
q


0
- X mq
- X mq  X kq1   ikq1 
and thus,
 id  -  X md  X l 
i   - X
md
 kd1  
 i fd    - X md

 
0
 iq  
 ikq1  
0
-  X md  X l 
 -X
md

  - X md

0


0
X md
 X md  X kd1 
X md
0
0
X md
 X md  X kd1 
X md
0
0
 y d 
y 
 kd1 
 y fd 
  X reac -1  y q 
 y kq1 


 d 
  
X md
0
0


X md
0
0

X md  X fd 
0
0


0
- X mq  X l 
X mq

0
- X mq
- X mq  X kq1 
X md
0
0
X md
0
0
X md  X fd 
0
0
0
- X mq  X l 
X mq
0
- X mq
- X mq  X kq1 
-1
 y d 
y 
 kd1 
 y fd 


 y q 
 y kq1 
0
0
0
0
0
………………………………………...(8)
 y d 
0 y kd1 


0  y fd 

0  y q 


0  y kq1 


0  d 
  
 y d 
y 
 id 
 kd1 
i 
 y fd 
 kd1 
I    i fd    X reac -1  y q    X reac -1X 



i
 y kq1 
q




 ikq1 

d


  
: from (8)
 
 X   S X   R I   B U 


 S X   R  X reac  X   B U 
-1


 S   R  X reac -1 X   B U 
: inserting (8) into (7)
  AX   B U 
where,
A  S   RX reac -1 
………..(9)
: system state matrix
System to be Studied
Vt
It
Generator
Infinite Bus
System State Matrix and Eigen Values
System State Matrix: A  S   R X reac -1 
Eigen Values: 1, 2  -  j
j
1
q
2

Eigen Values
o
Eigen values are the roots of the characteristic equation
 
 X    AX   B U 


o
o
Number of eigen values is equal to the order of the characteristic
equation or number of state variables
t
Eigen values describe the system response (e 1 ) to any disturbance
Analyzing the Eigen Values of the System State Matrix
o
o
o
Compute the eigen values of the system state matrix, A
The eigen values will give necessary information about the steady-state
stability of the system
Stable System: If the real parts of ALL the eigen values are negative
Example:
o
1 , 2  -0.15  j 2.0
3  -0.0005
A system with the above eigen values is on the verge of instability
Machine Parameters
Salient-pole synchronous generator
3kVA, 220V, 4-pole, 60 Hz and 1800 r/min
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