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Spring 2016Instructor: Kai Sun

Guest lecturer: Bin Wang

4.3 Direct Methods for Transient Stability Analysis

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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]

[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.

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.

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Main Steps [4]

max

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Applications to real systems [4]

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Application in Commercialized SoftwareFrom TSAT v.14 User Manual by Powertech Labs 

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Define the center of inertia (COI) and motions relative to the COI:

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

2

1

2

10

2 sin cos

sin cos

ni

i mi i ii i j ij ij ij ij

mi i

n

ij ij ij ij eii mi

j

jj i

i

i

j

P E G P

H P E G E E B

C P

G

D

1

1

1

1

n

i i ni

COI i iniT

ii

HH

HH

1

0

2 ( )

n

T COI COI mi eii

COT COI

H P P P

0

2

ii i mi ei COI

T

i i

HH P P PH

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Defining the post‐disturbance TEF

, ( )KE i ii

V , ( )PE i i

i

V ,,

( )Magnetic ij iji j

V ,,

(trajectory of )Dissipated ij i ji j

V =VKE 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)

3. Check Vcr-V(xcl)

def2

1 1

1 ( )2

i

Si

n ni

i i mi ei COI ii i T

HV J P P P dH

Assume a linear integration path [1]

( )

( ) ( )i j

cl cl S Si j i j

ijcl Sij ij

ij

d

d

k d

[1] J.N. Qiang, Clarifications on the Integration Path of Transient Energy Function, IEEE Trans Power Systems, May 2005

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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

• Key problems:– How to define V(x)? – How to find x0?– How to find xcr?

Post‐faultPre‐fault

Fault‐on

Time/s

V/pu0

0

( ) ( )Normalized 100%( )

cr

k

V VSMIV

x xx

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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 

• Exit point Xe• Critical clearing time (CCT)

x0

presX

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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

• The above TEF:– Generally is NOT a Lyapunov function– Is dependent on system trajectory

20( , ) cos cos cos di j

si sji i mi i si ij ij sij ij ij i j

i i i j i jV H P C D

MagneticPotentialKinetic Dissipated

Potential

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First‐swing stability Multi‐swing stability

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

Test the approximation on potential energy [2] (based on a network‐reduction model)

cos d sini j

si sj

i j si sjij ij i j ij ij sij

ij sij

D D

[2] Athay, T.; Podmore, R.; Virmani, S., "A Practical Method for the Direct Analysis of Transient Stability," in Power Apparatus and Systems, IEEE Transactions on , vol.PAS-98, no.2, pp.573-584, March 1979

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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. loadand 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

20 cos cos sini j si sj

i i mi i si ij ij sij ij ij siji i i j i j ij sij

V H P C D

x0

presX

V(x)0 VcrV(xcr)

Conservative Overestimated

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Thinking on Contingency Dependency

• Small signal stability V.S. Transient stability

Liebig's law of the minimum

(Bucket theory)

Stability boundary

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The Conceptual Controlling UEP Method

• Assumptions:– Xs

pre 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 unstable

x0

presX

0Actual CCT

Conservative Overestimated

CCT base on first-swing stability

CCT base on multi-swing stability

Fault clearing time

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x0

presX

Some Existing CUEP Methods

• 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

• Controlling (corresponding or relevant) UEP method

V(x)0 VcrV(xcl)

Conservative Overestimated( )

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The BCU Method• Boundary of stability region based CUEP method [3]

x0

presX

• 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

– Use MGP as initial point of a certain iterative algorithm, e.g. Newton-Raphson, to solve the CUEP

( )ii

f

Original: 1 1

in in

in mi ei COIi T

P P PH H

1 1Reduced: ( )in mi ei COI ii T

P P P fH H

[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:

• Problems:– The resulting direct methods can be either

overestimated or conservative– V(xexit) may take values from ( )

( , ) : ( , ) ( )exitV V

x0

presX

V(x)0 VcrV(xco)

( )Conservative Overestimated

[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

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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

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

• BCU v.s. PEBS on the IEEE 9-bus system:

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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.

Original coupled system Decoupled systems

SMIB‐1

SMIB‐2

SMIB‐(N‐2))

SMIB‐(N‐1)

…… …

G1

G2

G3GN‐1

GN

…

[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

• 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.

201 ,1 1 1 1 1 1 1

1, 11

202 ,2 2 2 2 2 2 2

1, 22

20,

1,

( sin cos )2

( sin cos )2

( sin cos )2

m

M j j j jj j

m

M j j j jj j

m

m M m m m mj mj mj mjj j mm

P E G C DH

P E G C DH

P E G C DH

0sin)sin(

0sin)sin(0sin)sin(

0,10,1111

2020222

1010111

mmmmm qqqq

qqqqqqqq

Original coupled system Decoupled systems

Linearization at the equilibrium

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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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x0

presX

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

[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

Source: V. Vittal, et al, “On-Line Transient Stability Assessment Scoping Study,” PSERC Report, Feb 2005

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