Transcript Power System Dynamics -- Postgraduate Course of Tsinghua Univ

Power System Dynamics
-- Postgraduate Course of
Tsinghua Univ. Graduate School at Shenzhen
NI Yixin
Associate Professor
Dept. of EEE, HKU
<[email protected]>
Introduction
0.1 Requirements of modern power systems (P. S. )
0.2 Recent trends of P. S.
0.3 Complexity of modern P. S.
0.4 Definitions of different types of P. S. stability
0.5 Computer-aid P. S. stability analysis
0.6 Contents of our course
Introduction (1)
0.1 Requirements of modern power systems (P. S. )

Satisfying load demands (as a power source)

Good quality: voltage magnitude, symmetric three phase
voltages, low harmonics, standard frequency etc. (as a 3phase ac voltage source)

Economic operation

Secure and reliable operation with flexible controllability

Loss of any one element will not cause any operation
limit violations (voltage, current, power, frequency,
etc. ) and all demands are still satisfied.

For a set of specific large disturbances, the system will
keep stable after disturbances.

Good energy management systems (EMS)
Introduction (2)
0.2 Recent trends of P. S.




Systems interconnection: to obtain more benefits.
It may lead to new stability issues ( e.g. low-frequency
power oscillation on the tie lines; SSR caused by seriescompensated lines etc. ).
Systems are often heavily loaded and very stressed.
System stability under disturbances is of great concern.
New technology applications in power systems. (e.g.
computer/ modern control theory/ optimization theory/
IT/ AI tech. etc. )
Power electronics applications: provides flexible
controller in power systems. ( e. g. HVDC transmission
systems, STATCOM, UPFC, TCSC, etc.)
Introduction (3)
0.3 Complexity of modern P. S.
Large scale,

Hierarchical and distributed structure,

Non-storable electric energy,

Fluctuate and random loads,

Highly nonlinear dynamic behavior,

Unforeseen emergencies,

Fast transients which may lead to system collapse in
seconds or minutes,

Complicated control and their coordination requests.
-- Modern P. S. is much more complicated than ever and in
the meantime it plays a significant role in modern society.

Introduction (4)
Some viewpoints of Dr. Kundur (author of the ref. book ):
--- The complexity of power systems is continually increasing
because of the growth in interconnections and use of new
technologies. At the same time, financial and regulatory
constrains have forced utilities to operate the systems nearly
at stability limits.
--- Of all the complex phenomena on power systems, power
system stability is the most intricate to understand and
challenging to analyze. Electric power systems of the 21
century will present an even more formidable challenge as
they are forced to operate closer to their stability limit.
Introduction (5)
0.4 Definitions of different types of P. S. stability

P. S. stability: the property of a P. S. that enable it to
remain in a state of operating equilibrium under normal
operating conditions and to return to an acceptable state
of equilibrium after being disturbed.

Classification of stability

Based on size of disturbance:



large disturbance stability ( transient stability, IEEE):
nonlinear system models
small disturbance/signal stability ( steady-state stability,
IEEE): linearized system models
The time span considered:



transient stability: 0 to 10 seconds
mid-term stability: 10 seconds to a few minutes
long-term stability(dynamics): a few minutes to 1 hour
Introduction (6)
0.4 Definitions of different types of P. S. stability (cont.)

Classification of stability (cont.)

Based on physical nature of stability:
 Synchronous operation (or angle) stability:
 insufficient synchronizing torque -- nonoscillatory instability

insufficient damping torque -- oscillatory
instability
 Voltage stability:

insufficient reactive power and voltage
controllability
 Subsynchronous oscillation (SSO) stability

insufficient damping torque in SSO
Introduction (7)
0.5 Computer-aid P. S. stability analysis
Introduction (8)
0.6 Contents of the course
Introduction
Part I: Power system element models
1. Synchronous machine models
2. Excitation system models
3. Prime mover and speed governor models
4. Load models
5. Transmission line and transformer models
Part II: Power system dynamics: theory and analysis
6. Transient stability and time simulation
7. Steady-state stability and eigenvalue analysis
8. Low-frequency oscillation and control
9. *Voltage stability
10. *Subsynchronous oscillation
11. Improvement of system stability
Summary
Part I
Power system element models
Chapter 1
Synchronous machine models
(a)
Chapter 1 Synchronous
machine (S. M.) models
1.1 Ideal S. M. and its model in abc coordinates
1.1.1 Ideal S. M. definition

Note:
* S. M. is a rotating magnetic element with complex dynamic
behavior. It is the heart of P. S. It
* It provides active and reactive power to loads and has
strong power, frequency and voltage regulation/control
capability .
* To study S. M., mathematic models are developed for S. M.
* Special assumptions are made to simplify the modeling.
Chapter 1 Synchronous
machine (S. M.) models
1.1.1 Ideal S. M. definition (cont.):

Assumptions for ideal S. M.

Machine magnetic permeability (m) is a constant with
magnetic saturation neglected. Eddy current, hysteresis,
and skin effects are neglected, so the machine is linear.

Symmetric rotor structure in direct (d) and quadratic (q)
axes.

Symmetric stator winding structure: the three stator
windings are 120 (electric) degrees apart in space with
same structure.

The stator and rotor have smooth surface with tooth and
slot effects neglected. All windings generate sinusoidal
distributed magnetic field.
Chapter 1 Synchronous
machine (S. M.) models
1.1.2 Voltage equations in abc coordinates
 Positive direction setting:
 dq and abc axes, speed
direction
 Angle definition:
 a   : (d leading ahead a)
 b   a  120, a  240
 c   a  240, a  120



Y directions for abcfDQ
windings
i directions for abcfDQ
u directions for abcfDQ
(uD=uQ=0)
Chapter 1 Synchronous
machine (S. M.) models
ua  pΨ a  ra ia
1.1.2 Voltage equations in abc coordinates (cont.)  u  pΨ  r i
 b
b
b b
 u  pΨ  r i

Voltage equations for abc windings:
c
c c
 c
where p= d / dt, t in sec.
rabc: stator winding resistance, in W.
iabc : stator winding current, in A.
uabc: stator winding phase voltage, in V.
yabc: stator winding flux linkage, in Wb.
Note: * pyabc: generate emf in abc windings
* uabc~iabc: in generator conventional direction.
* iabc~ yabc: positive iabc generates negative yabc
respectively
Chapter 1 Synchronous
machine (S. M.) models
1.1.2 Voltage equations in abc coordinates (cont.)
 Voltage equations for fDQ windings:
u f  pΨ f  rf i f
rfDQ: rotor winding resistance, in W.
f: field winding,
D: damping winding in d-axis,
Q: damping winding in q-axis.
ifDG, ufDG, yfDG: rotor winding currents,
voltages and flux linkages in A, V, Wb.
Note: * uD=uQ=0
* ufDQ~ifDQ: in load convention
* ifDG ~yfDG: positive ifDG generates
positive yfDG respectively
* q-axis leads d-axis by 90 (electr.) deg.

 u D  pΨ D  rD iD  0
 u  pΨ  r i  0
Q
Q Q
 Q
Chapter 1 Synchronous
machine (S. M.) models
1.1.2 Voltage equations in abc coordinates (cont.)
 Voltage equations in matrix format:
u  pΨ  ri
u  (ua , ub , uc , u f , u D , uQ ) T
Ψ  (Ψ a ,Ψ b ,Ψ c ,Ψ f ,Ψ D ,Ψ Q ) T
r  diag(ra , ra , ra , rf , rD , rQ ) T
i  (ia , ib , ic , i f , iD , iQ ) T
where ‘–’ before iabc is caused by generator convention
of stator windings.
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates
Ψ a   Laa
Ψ   L
 b   ba
Ψ c   Lca
  
 
Ψ f   L fa
  
Ψ D   LDa
Ψ   L
 Q   Qa
Lab
Lbb
Lac
Lbc
Laf
Lbf
LaD
LbD
Lcb
Lcc
Lcf
LcD
L fb
LDb
LQb
L fc
LDc
LQc
L ff
LDf
LQf
L fD
LDD
LQD
 y abc   L1133
or 
=

y fDQ   L2133
LaQ   ia 
LbQ   ib 
LcQ   ic 
 
 
L fQ   i f 
 
LDQ   iD 
LQQ   iQ 
L1233   iabc 
; y (61)  L(66) i(61)


L22 33   i fDQ 
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (cont.)
 In Flux linkage eqn.:
Lij ( i, j = a, b, c, f, D, Q ): self and mutual
inductances,
L11 : stator winding self and mutual inductance,
L22 : rotor winding self and mutual inductances,
L12 , L21 : mutual inductances among
stator and rotor windings ,
y, i : same definition as voltage eqn..
Note: * Positive iabc generates negative yabc
respectively.
* The negative signs of iabc make Laa, Lbb, Lcc> 0.
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (cont.)
 Stator winding self/mutual inductance (L11)

Stator winding self inductance (Laa, Lbb, Lcc)
Ψa
Laa 
 0 (ib , ic , i f , iD , iQ  0)
ia
Laa: reach max @ d-a aligning (when a=0, 180°)
reach min @ d-a perpendicular (when a=90, 270°)
Laa~ a: ‘sin’-curve, with period of 180°
Laa  LS  Lt cos 2 a  LS  Lt cos 2
Lbb  LS  Lt cos 2 b  LS  Lt cos 2(  120)
Lcc  LS  Lt cos 2 c  LS  Lt cos 2(  120)
(Ls>Lt>0, for round rotor: Lt=0)
(See appendix 1 of the text book for derivation)
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (cont.)
 Stator winding self/mutual inductance (L11)

Stator winding mutual inductance
Lab 
ya
ib
 0 (ia ,c , fDQ  0); Lba 
yb
ia
Lab: reach max |.| when a= -30, 150°
reach min |.| when a= 60, 240°
Laa~ a: ‘sin’-curve, with period of 180°
Lab = Lba   M s  Lt cos 2( a  30)
 ( M s  Lt cos 2(  30))
Lbc = Lcb   (M s  Lt cos 2(  90))
Lca = Lac  ( M s  Lt cos 2(  150))
(Ms>Lt>0, for round rotor: Lt=0)
(See appendix 1 of the text book for derivation)
 Lab  0 (ib ,c , fDQ  0)
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (cont.)

Rotor winding self/mutual inductance (L22)

Rotor winding self inductance (constant: why?)
Lff = Lf = const. >0
LDD = LD = const. >0
LQQ = LQ = const. >0

Rotor winding mutual inductance
LfQ = LfQ = 0, LDQ = LQD = 0
LfD = LDf = MR = const. > 0
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (cont.)
 Stator and rotor winding mutual inductance (L12; L21 )
 abc~f: (Mf=const.>0, period: 360°, max. when d-abc align)
Laf  L fa  M f cos a  M f cos
Lbf  L fb  M f cos(  120)
Lcf  L fc  M f cos(  120)



abc~D: similar to abc~f, MfMD>0
abc~Q:(MQ=const.>0, period: 360°,
max. when q-abc align)
LaQ  LQa  M Q cos( a  90)   M Q sin 
LbQ  LQb   M Q sin(  120)
LcQ  LQc   M Q sin(  120)
Chapter 1 Synchronous
machine (S. M.) models
1.1.3 Flux linkage equations in abc coordinates (summary)
 Time varying L-matrix : related to rotor position
 L11 (abc~abc): 180° period; L12, L21(abc~fDG): 360° period.
 Non-sparse L-matrix: most mutual inductances  0
 L-matrix: non-user friendly, lead to abc  dq0 coordinates!
Ψ a   Laa
Ψ   L
 b   ba
 Ψ c   Lca
  
 
Ψ f   L fa
  
Ψ D   LDa
Ψ   L
 Q   Qa
Lab
Lbb
Lac
Lbc
Laf
Lbf
LaD
LbD
Lcb
Lcc
Lcf
LcD
L fb
LDb
LQb
L fc
LDc
LQc
Lf
MR
0
MR
LD
0
LaQ   ia 
LbQ   ib 
LcQ   ic 
 
 
0   if 
 
0   iD 
LQ   iQ 
Chapter 1 Synchronous
machine (S. M.) models
1.1.4 Generator power, torque and motion eqns.
 Instantaneous output power eqn. (Pe in W)
Pe  u a ia  ub ib  u c ic

Electromagnetic torque eqn. (Te in N-m,  in rad.)
1 T dL
1
Te   p P i
i  pP
[y a (ib  ic )  y b (ic  ia )  y c (ia  ib )]
2 d
3
 0 1 1
1 T 
 p P : number of pole pairs,

 pP
y abc  1 0 1  iabc 
T
T
T
3

i
:
(i
,
i
)
 1 1 0 
 (61) abc fDQ




Chapter 1 Synchronous
machine (S. M.) models
1.1.4 Generator power, torque and motion eqns. (cont.)

Rotor motion eqns.
 According to Newton’s law, we have:
 dw m
J
 Tm  Te

 dt

 d m  w
m

dt

where Tm: input mechanical torque of generator (in N-m)
Te: output electromagnetic torque (in N-m)
wm/m: rotor mechanical speed/angle (in rad/s, rad.)
we/e: rotor electrical speed/angle (in rad/s, rad.),
w m  w e / pP  w / pP ;  m   e / pP   / pP
J: rotor moment of inertia (also called rotational inertia)
2

m
r
J= i i i Kg-m2
In the manufacturer’s handbook, J is given by [GD2], in ton-m2.
[GD2] (ton-m2)  103/4 J (Kg-m2).
Chapter 1 Synchronous
machine (S. M.) models
1.1.4 Generator power, torque and motion eqns. (cont.)
 Rotor motion eqns. (cont.)
 1 dw
 dw m
J
 Tm  Te
J

T

T
m
e


 dt
 pP dt



d

d
m


 wm
w


dt

dt

(w m  w e / pP  w / pP ;  m   e / pP   / pP )
Te  pP
1
[y a (ib  ic )  y b (ic  ia )  y c (ia  ib )]
3
Chapter 1 Synchronous
machine (S. M.) models
1.1.5 Summary of S. M. model in abc coordinates and SI units:
 6 volt. DEs. (abcfDQ): u  py  ri
 6 flux linkage AEs. (abcfDQ):
y  Li
 2 rotor motion eqns. (w, :
1 dw
d
J
 Tm  Te ;
w
pP dt
dt
Te  pP

Totally 14 eqns. with 8 DEs and 6 AEs.
8th order nonlinear model.
 8 state variables are: y (61) and w,  (related to 8 DEs)
Totally 19 variables: u: 4 (vD=vQ=0), i: 6, y: 6, plus (Tm, w, .
If 5 variables are known, remaining 14 variables can be solved.
Usually uf and Tm are known (as input signals), 3 network interface
eqns. (3 vabc-iabc relations from network) are known.




1
[y a (ib  ic )  y b (ic  ia )  y c (ia  ib )]
3
Chapter 1 Synchronous
machine (S. M.) models
1.1.5 Summary of S. M. model in abc coordinates (cont.)

Request of transformation of S. M. model:

abc to dq0 coordinates: Park’s transformation, Park’s
eqns.

per unit system and S. M. pu model

Reduced-order practical models:
-- Neglect stator abc winding transients (8th order 5th
order). It can interface with network Y-matrix in Aes.
-- Introduce practical variables (E’dq, E”dq, Ef etc.)