4.3 Direct Methods for Transient Stability Analysis
Spring 2016
Instructor: Kai Sun
Guest lecturer: Bin Wang
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Content
•One‐Machine‐Infinite‐Bus (OMIB) Equivalent method (EEAC/SIME)
•Transient energy function for a multi‐machine system
•TEF based direct methods (CUEP, PEBS, BCU, etc.)
•Linear decoupling based direct method
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OMIB Equivalent Based Method (EEAC or SIME) [2]‐[4]
Main Idea:
• According to rotor angle curves over a time window (e.g. obtained from simulation), partition machines into 2 groups
– Critical machines (CMs)
– Non‐critical machines (NMs)
• Only n‐1 ways of partitioning need to be studied.
• For each way of partitioning, construct a 2‐machine equivalent and consequently an OMIB equivalent, such that conclusions of the EAC can be applied.
[2]
Y. Xue, et al, "A Simple Direct Method for Fast Transient
Stability Assessment of Large Power Systems". IEEE Trans.
PWRS, PWRS3: 400–412, 1988.
[3] Y. Xue, et al, "Extended Equal-Area Criterion Revisited".
IEEE Trans. PWRS, PWRS7: 10101022,1992.
[4] M. Pavella, et al, “Transient Stability of Power Systems: a
Unified Approach to Assessment and Control”, Kluwer, 2000.
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Main Steps [4]
max
26
Applications to real systems [4]
27
Application in Commercialized Software
From TSAT v.14 User Manual by Powertech Labs 28
TEF Method for a Multi‐machine Power System • Simplifications on the model are needed:
– All generators in the classical model and all loads as constant impedances
– Neglect system damping
n
2 H i 
2
 i  Pmi  Ei Gii   Ei E j  Bij sin  ij  Gij cos  ij 
0
j 1
j i
 Pmi  E Gii    Cij sin  ij  Dij cos  ij 
n
2
i
j 1
j i
 Pmi  Pei
Define the center of inertia (COI) and motions relative to the COI:
n

COI 
H 
i 1
n
i i
H
i 1
1

HT
n
H 
i 1
i i
i
n
2 H T  COI  PCOI   ( Pmi  Pei )
2 H i i  Pmi  Pei 
i 1
COT  COI 0
i   i 0
Hi
PCOI
HT
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Assume a linear
integration path [1]
Defining the post‐disturbance TEF
d ( i   j )
n
n

i
Hi
1
2


V   J i i     ( Pmi  Pei 
PCOI )d  i
2 i 1
HT
i 1  S
def
( icl   jcl )  ( iS   jS )
 ijcl   ijS
d ij
 k  d ij
i
V
KE ,i
(i )
i
=VKE
V
PE ,i
i
(i )
V
Magnetic ,ij
i, j
(ij )
V
Dissipated ,ij
(trajectory of i   j )
i, j
VPE
• A general procedure of the TEF method:
1. Run time-domain simulation up to the instant of fault clearing (tcl) to obtain
angles and speeds of all generators, which are used to calculate V(xcl)
2. Calculate the critical energy Vcr for the post-disturbance system (this is the
most difficult step for a large-scale systems; Vcr may be defined as the
maximum VPE at the closest or controlling UEP)
[1] J.N. Qiang, Clarifications on the Integration Path of Transient Energy Function,
3. Check Vcr-V(xcl)
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IEEE Trans Power Systems, May 2005
Recap of Direct Methods
• Goal: Determine the system stability using initial states x0 of post‐fault trajectory
• Methodology: – Define a transient energy function (TEF) V(x)
– Calculate the initial energy of the system V (x0)
– Find the critical point xcr to estimate the critical energy Vcr V(xcr)
– If V (x0) < Vcr, then the system is stable; otherwise, the system is unstable. – Stability margin index V/pu
Normalized SMI 
V (xcr )  V (x 0 )
100%
Vk (x0 )
• Key problems:
– How to define V(x)?
– How to find x0?
– How to find xcr?
Pre‐fault
Post‐fault
Fault‐on
Time/s
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Terminology
• Equilibrium point (EP)
– Xspre, Xs, X1, X2, Xcl and Xco
• Stable equilibrium point (SEP)
– Xspre, Xs
• Unstable equilibrium point (UEP)
– X1, X2, Xcl and Xco
• Stable/unstable manifold of an EP x0
X spre
• Exit point Xe
• Critical clearing time (CCT)
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Stability Region, Stability Boundary and CUEP • Stability region: The region where any trajectory initialized from any point in the region will converge to the SEP
• Stability boundary: The union of the stable manifold of the UEPs whose unstable manifolds contain trajectories approaching the SEP
• Controlling UEP (CUEP): The UEP whose stable manifold contains the exit point, which is contingency‐dependent. The CUEP xco is the xcr.
[1] Hsiao-dong Chiang and Luis F.C. Alberto, Stability Regions of Nonlinear Dynamical Systems. New Jersey: Cambridge
University Press, 2015
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Transient Energy Function‐Network Reduction Model
• Simplifications on the model
– Constant impedance load model
– Classical generator model
• A commonly used TEF of a multi‐machine system
Kinetic
Magnetic
Potential
Dissipated
V ( ,  )  0  Hi   Pmi  i   si    Cij  cos  ij  cos  sij    
2
i
i
i
i
j
• The above TEF:
– Generally is NOT a Lyapunov function
– Is dependent on system trajectory
i
j
i  j
 si  sj
Dij cos  ij d  i   j 
Potential
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A Commonly Used TEF
• The dissipated energy term can be:
– Ignored, then the commonly used TEF becomes a Lyapunov function
– Approximated by a certain simple function. For example, under linear integration path assumption
i  j

si  sj
Dij cos ij d  i   j   Dij
i   j   si   sj 
ij   sij
 sin 
ij
  sij 
Test the approximation on potential energy [2] (based on a network‐
reduction model)
First‐swing stability
Multi‐swing stability
[2] Athay, T.; Podmore, R.; Virmani, S., "A Practical Method for the Direct Analysis of Transient Stability," in Power Apparatus
35
and Systems, IEEE Transactions on , vol.PAS-98, no.2, pp.573-584, March 1979
Transient Energy Function‐Network Reduction Model
• Local Lyapunov function exists for lossy power systems
– Can determine the local stability of an EP
– Cannot help determine the stability region
• Lyapunov function does NOT exist for a general lossy power system [1]
– For systems with certain losses, there could be an energy function which can help determine the stability
– Any effort on analytical TEF needs to check the existence of such TEF
• Generalized TEF:
– Allows positive derivatives along system trajectories in some bounded sets
– Could be applicable to more complicated power system stability models than classical system model
[1] Hsiao-Dong Chiang, "Study of the existence of energy functions for power systems with losses," in Circuits and Systems,
IEEE Transactions on , vol.36, no.11, pp.1423-1429, Nov 1989
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Transient Energy Function‐Network Preserving Model
• Advantages:
– Allow for more realistic dynamic models in power systems, e.g. load
and generator
– Transfer conductance is significantly smaller than that of the network‐
reduction model. Then, the commonly used TEF is “close” to a Lyapunov function
• Disadvantages:
– Need to handle DAE systems rather than ODE in network‐reduction model
– Jump behaviors and singular surfaces inherent in the DAE are difficult to handle both numerically and analytically
• Practical handling:
– Singular perturbation approach can provide an ODE corresponding the given DAE.
– Analysis on the ODE and transform the results back to the DAE system
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A Commonly Used TEF
V  0  H i   Pmi  i   si    Cij  cos  ij  cos  sij    Dij
2
i
i
i
i
j
i
 i   j   si   sj 
j
 ij   sij
 sin 
ij
  sij 
x0
X spre
Conservative
0
Overestimated
V(x)
V(xcr) Vcr
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Thinking on Contingency Dependency
• Small signal stability
V.S.
Transient stability
Stability boundary
Liebig's law of the minimum
(Bucket theory)
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The Conceptual Controlling UEP Method
• Assumptions:
– Xspre locates inside the stability region
of the Xs
– The TEF is a Lyapunov function V
• Key step:
– Find the CUEP xco for given fault-on
trajectory, then xcr xco
• First-swing stability
– First‐swing stable
stable
– First‐swing unstable
Conservative
x0
X spre
unstable
Overestimated
Fault clearing time
0
Actual CCT
CCT base on first-swing stability
CCT base on multi-swing stability
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Some Existing CUEP Methods
• Controlling (corresponding or relevant) UEP method
• Development of the CUEP concept:
– The closest UEP who has minimum TEF
– The UEP closest to the fault-on trajectory
– The UEP “in the direction” of fault-on trajectory
– The UEP related to the machine(s) that first go out of
synchronism when the fault is sustained
• Problems:
– The computed UEP is not the CUEP
– The resulting direct methods can be either
overestimated or very conservative
– The involved computations are heuristic and have no
theoretical foundations
Conservative
0
(
V(xcl)
x0
X spre
Overestimated
V(x)
)
Vcr
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The BCU Method
• Boundary of stability region based CUEP method [3]
• Key steps:
– Detect the exit point Xe as the local
maximum of potential energy
– Integrate the reduced system from exit
point to the minimum gradient point
(MGP), i.e. first local minimum of  fi ( )
i
in  in

Original: 
1
1

P
P






 in H mi ei H PCOI
i
T

1
1
Reduced: in 
 Pmi  Pei   PCOI  fi ( )
Hi
HT
x0
X spre
– Use MGP as initial point of a certain
iterative algorithm, e.g. NewtonRaphson, to solve the CUEP
[3] Hsiao-Dong Chiang; Wu, F.F.; Varaiya, P.P., "A BCU method for direct analysis of power system transient stability,"
in Power Systems, IEEE Transactions on , vol.9, no.3, pp.1194-1208, Aug 1994
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The BCU Method
• Illustration of exit point and MGP
Softwares:
1. DIRECT 4.0 developed by EPRI in 1995 and included in PSAPAC by HKU
2. TEPCO-BCU Package by TEPCO and Bigwood since 1997
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The PEBS Method
• Potential Energy Boundary Surface method [4]
• Key steps:
– Detect the exit point Xe as the local
maximum of potential energy
– Use the constant energy surface as a local
approximation of the stability boundary of
the post-fault system, i.e. xcr xe:
x0
( ,  ) : V ( ,  )  V ( exit )
• Problems:
– The resulting direct methods can be either
overestimated or conservative
– V(xexit) may take values from ( )
Conservative
0
(
V(xco)
X spre
Overestimated
)
V(x)
Vcr
[4] Kakimoto, Naoto; Ohnogi, Yukio; Matsuda, Hisao; Shibuya, Hiroshi, "Transient Stability Analysis of Large-Scale Power System by
Lyapunov's Direct Method," in Power Apparatus and Systems, IEEE Transactions on , vol.PAS-103, no.1, pp.160-167, Jan. 1984 44
The PEBS Method
• When the potential energy surface is flatter between the exit point and the CUEP,
V(xexit) will be close to V(xco). Then, the PEBS method will more likely be on the
conservative side.
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Examples
• Closest UEP v.s. CUEP on 39-bus system [2]:
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Examples
• BCU v.s. PEBS on the IEEE 9-bus system:
Case #
Fault‐bus
Tripped line
CCT by simulation
/s
CCT by BCU
/s
CCT by PEBS
/s
1
7
7‐5
0.179
0.174
0.187
2
7
8‐7
0.195
0.171
0.207
3
5
7‐5
0.353
0.346
0.343
4
4
4‐6
0.329
0.323
0.324
5
9
6‐9
0.231
CUEP not found
0.252
6
9
9‐8
0.249
0.226
0.249
7
8
9‐8
0.324
CUEP not found
0.340
8
8
8‐7
0.297
0.212
0.330
9
6
4‐6
0.493
0.477
0.487
10
6
6‐9
0.430
0.419
0.421
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Linear Decoupling Based Direct Method [5]
• Decoupability assumption: A multi‐machine power system can be decoupled into a set of SMIB systems.
G1
GN
SMIB‐1
SMIB‐2
GN‐1
Original coupled system
…
G3
…
…
…
G2
SMIB‐(N‐2))
SMIB‐(N‐1)
Decoupled systems
[5] Bin Wang; Kai Sun; Xiaowen Su, "A decoupling based direct method for power system transient stability analysis," in Power &
Energy Society General Meeting, 2015 IEEE , vol., no., pp.1-5, 26-30 July 2015
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Linear Decoupling Based Direct Method
• Determination of the decoupled SMIB systems
Original coupled system
Decoupled systems
m


0 
2





P
E
G
(
C
sin
D
cos
)
1 
 M ,1 1 1  1 j
1j
1j
1j 
2H1 
j 1, j 1



m

  0  P  E2G 
(C2 j sin 2 j  D2 j cos2 j ) 

2
2 2
M ,2
2H2 

j 1, j 2




m

 
m  0  PM ,m  Em2Gm   (Cmj sin mj  Dmj cosmj ) 
2Hm 

j 1, j m

q1  1 sin(q1  q10 )  sin q10   0
q2   2 sin(q2  q20 )  sin q20   0
qm1   m1 sin(qm1  qm1,0 )  sin qm1,0   0
Linearization at the equilibrium
• Stability analysis
– TEF can be applied to each SMIB, which is equivalent to EAC.
– The smallest margin among all SMIBs can be used to estimate the stability of the original multi‐machine system.
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Linear Decoupling Based Direct Method
• Example on IEEE 9‐bus system
• Test one
– Three‐phase fault on line 4‐5
– Two modes: 0.8Hz and 1.7Hz
– CCT = 0.197s
• Test two
– Rank all line‐tripping contingencies.
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Future Directions of Direct Methods
•
•
•
•
Reliable estimation of xcr using computationally efficient algorithms
Better TEF such that V(xcr) Vcr with less conservativeness
Pre-analysis on contingencies
Hybrid methods based on time-domain simulation
x0
X spre
[6] J.N. Qiang, Clarifications on the Integration Path of Transient Energy Function, IEEE Trans Power Systems, May 2005
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Survey by PSERC on Transient Stability Assessment Tools
Vendors
Source: V. Vittal, et al, “On-Line Transient Stability Assessment Scoping Study,” PSERC Report, Feb 2005
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