Network Structure in Swing
Mode Bifurcations

Motivation & Background
– Practical Goal: examine loss of stability
mechanisms associated w/ heavily loaded
transmission corridors.
– Expect presence of low frequency, interarea
swing modes across transmission corridor.
– Can bifurcation tools developed for voltage
analysis be adapted to this scenario (are voltage
& angle instabilities really that different)?
Key Ideas

Voltage methods typically assume one
degree of freedom path in “parameter
space” (e.g. load), or seek “closest” point in
parameter space at which bifurcations occur

Alternative: leave larger # of degrees of
freedom in parameter space, but constrain
structure of eigenvector at bifurcation.
Key Questions

Is there a priori knowledge of form of
eigenvector of interest for “mode” of
instability we’re after?

Precisely what formulation for matrix
who’s eigenvector/eigenvalue is constrained
(e.g., what generator model, what load
model, how is DAE structure treated, etc.)
Caveats (at present...)

Development to date uses only very simple,
classical model for generators.

Previous work in voltage stability shows
examples in which “earlier” loss of stability
missed by such a simple model (e.g.
Rajagopalan et al, Trans. on P.S. ‘92).
Review - Relation of PF Jacobian
and Linearized Dynamic Model

This issue well treated in existing literature,
but still useful to develop notation suited to
generalized eigenvalue problem.

Structure in linearization easiest to see if we
keep all phase angles as variables; neglect
damping/governor; assume lossless
transmission & symmetric PF Jac. Relax
many of these assumptions in computations.
Review - Relation of PF Jacobian
and Linearized Dynamic Model

Form of nonlinear DAE model
•
M = L 1(P - P (, V))
•
L 1 = 
I
N
0 = L 2(P - P (, V))
I
N
0 = (Q (V) - Q (, V))
I
N
Review - Relation of PF Jacobian
and Linearized Dynamic Model

Requisite variable/function definitions:
V  IR n, V := vector of bus voltage magnitudes;
 IR n,  := vector of bus voltage phase angles
relative to an arbitrary synchronous
reference frame of frequency  0 (no
reference angle is deleted);
  IR m,  := vector of generator frequency
deviations, relative to synchronous
frequency  0;
Review - Relation of PF Jacobian
and Linearized Dynamic Model

variable/function definitions:
M IR mxm , M := diagonal matrix of normalized
generator inertias;
PI IR n, PI:= vector of net active power injection at
each bus; assumed constant;
QI: IRn-m IRn-m, QI(V) := vector valued function of
net reactive power injection at load buses,
normalized by voltage magnitude;
Review - Relation of PF Jacobian
and Linearized Dynamic Model

variable/function definitions:
L 1 := rows 1 through m of an nxn identity matrix;
L 2 := rows m+1 through n of an nxn identity matrix;
PN(, V) := vector-valued function of active power
absorbed by network at each bus
QN(, V) := vector-valued function of reactive
power absorbed by network at load buses,
normalized by voltage magnitude
Linearized DAE/”Singular
System” Form

Write linearization as:
~ • ~
Ex = Rx
Component Definitions

where:


~
E= 

~
R=



M mxm
0
0
I mxm
0
0
0
-I mxm
Imxm
0
0
0

0

0 2(n-m)x2(n-m)
0

0
S
I2(n-m)x2(n-m)
0
Component Definitions

and:
Imxm
S=  0
0

J12 
J22 
0
0
J11
J21
J: IR nx IR n-m IR (2n-m)x(2n-m), J =
 P

Q
 
N
N


{Q - Q }
VL 
N
P
VL
N
I
Relation to Reduced Dimension
Symmetric Problem

Consider reduced dimension, symmetric
generalized eigenvalue problem defined by
pair (E, J), where:
 M
E= 
 0


0 2(n-m)x2(n-m)
0
Relation to Reduced Dimension
Symmetric Problem

FACT: Finite generalized eigenvalues of
(E, J) completely determine finite
~ ~
generalized eigenvalues of (E R)
Relation to Reduced Dimension
Symmetric Problem

In particular,
(E, J ) has a finite generalized eigenvalue

~ ~
(E, R), has finite generalized eigenvalue
with = j .


Key Observation

In seeking bifurcation in full linearized
dynamics, we may work with reduced
dimension, symmetric generalized
eigenvalue problem whose structure is
determined by PF Jacobian & inertias.

When computation (sparsity) not a concern,
-1
-1
equivalent to e.v.’s of M [J11 - J12J22J21]
Role of Network Structure
Question: what is a mechanism by which
[J11 - J12J-122J21] might drop rank?


First, observe that under lossless network
approximation, the reduced Jacobian has
admittance matrix structure; i.e. diagonal
elements equal to – {sum of off-diagonal
elements}.
Role of Network Structure

Given this admittance matrix structure,
-1
reduced PF Jacobian[J11 - J12J22J21]
has associated network graph.

A mechanism for loss of rank can then be
identified: branches forming a cutset all
have weights of zero.
Role of Network Structure

Eigenvector associated with new zero
eigenvalue is identifiable by inspection:

J
 11
-
  1 
-1
J12 J22 J21   -1 


=0
where  is a positive real constant, and
partition of eigenvector is across the cutset.
Role of Network Structure

Returning to associated generalized
eigenvalue problem, to preserve sparsity,
one would have:
 J
 J11
 21
J12   1 



–1
=0
J22 
 w
Role of Network Structure

Finally, in original generalized eigenvalue
problem for full dynamics, the new


eigenvector has structure [ 1 , –  1 ] in
components associated with generator
phase angles.

Strongly suggests an inter area swing mode,
with gens on one side of cutset 180º out of
phase with those on other side.
Summary so far...

Exploiting on a number of simplifying
assumptions (lossless network, symmetric
PF Jacobian, classical gen model...),
identify candidate structure for eigenvector
associated with a “new” eigenvalue at zero.

Look for limiting operating conditions that
yield J realizing this bifurcation & e-vector.
Computational Formulation

Very analogous to early “direct” methods of
finding loading levels associated with
Jacobian singularity in voltage collapse
literature (e.g., Alvarado/Jung, 88).

But instead of leaving eigenvector
components associated with zero
eigenvalue as free variables, we constrain
components associated with gen angles.
Computational Formulation

Must compensate with “extra” degrees of
freedom.

For example to follow, generation dispatch
selected as new variables. Clearly, many
other possible choices...
Computational Formulation

Final observation: while it is convenient to
keep all angles as variables in original
analysis, in computation we select a
reference angle and eliminate that variable.

Resulting structure of gen angle e-vector
 
components becomes [ 0 , 1 ]
Computational Formulation

Simultaneous equations to be solved:
~~
0 = f (, V)
~ ~
~T T
T
0 = J((, V)[0, 1 , w ]

Note that f tilde terms are power balance
equations, deleting gen buses. Once angles
& voltages solved, gen dispatch is output.
Computational Formulation

Solution method is full Newton Raphson.

Aside: the Jacobian of these constraint
equations involves 2nd order derivative of
PF equations. Solutions routines developed
offer very compact & efficient vector
evaluations of higher order PF derivative.
Case Study

Based on modified form of IEEE 14 bus
test system.
14 Bus
Bus Test
Test System
System
14
13
13
19
19
12
12
G
G
#
#
14
14
13
13
11
11
17
17
11
11
12
12
Transmission Line
Line #'s
#'s
-- Transmission
#'s
-- Bus
Bus #'s
#
#
20
20
10
10
1
1
18
18
16
16
2
2
G
G
5
5
7
7
7
7
8
8
4
4
Cutset
Here
5
5
2
2
6
6
4
4
G
G
3
3
8
8
G
G
9
9
10
10
1
1
15
15
9
9
6
6
14
14
3
3
G
G
Case Study
N-R Initialization: initial operating point
selected heuristically at present. Simply
begin from op. pnt. that loads up a
transmission corridor, with gens each side.
 Here choice has gens 1, 2, 3 on one side,
gens 6, 8 on other side.
 Model has rotational damping added as
rough approximation to governor action.

Table 1: Original & Critical Operating Pnts.
Bus #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Original Op. Point
Critical Op. Point
Voltage/Phase(degrees)
1.0600 0
1.0450 -4.7545
1.0800 -2.8056
0.9196 -25.1819
0.9476 -23.6900
1.0200 -70.0082
0.9514 -62.1503
1.0400 -58.2712
0.9407 -76.0226
0.9426 -83.9820
0.9852 -87.8051
0.9478 -90.6433
1.0056 -82.1261
0.9960 -89.9647
Voltage/Phase(degrees)
1.0600 0
1.0450 -3.3284
1.0800 -6.4522
0.8351 -28.9282
0.8733 -26.8531
1.0200 -84.2610
0.8663 -79.5870
1.0400 -79.4164
0.8562 -94.5890
0.8702 -103.0148
0.9412 -105.1052
0.9385 -105.5251
0.9884 -97.3412
0.9410 -108.4568
Selected Generalized Eigenvectors of the Full Dynamic Model
State deviation described by
component
Freq@Bus 1
Freq@Bus 2
Freq@Bus 3
Freq@Bus 6
Freq@Bus 8
Angle@Bus 1
Angle@Bus 2
Angle@Bus 3
Angle@Bus 6
Angle@Bus 8
Angle@Bus 4
Angle@Bus 5
Angle@Bus 7
Angle@Bus 9
Angle@Bus 10
Angle@Bus 11
Angle@Bus 12
Angle@Bus 13
Angle@Bus 14
Original Op. Pnt;
Evector for lambda =
-0.0189 + 3.6342i
-0.4033-0.0048i
-0.3917-0.0047i
-0.4287-0.0051i
0.3068+0.0026i
0.4964+0.0054i
-0.0007+0.1110i
-0.0007+0.1078i
-0.0008+0.1180i
0.0003-0.0844i
0.0008-0.1366i
-0.0005+0.0625i
-0.0005+0.0639i
0.0004-0.0868i
0.0005-0.1030i
0.0005-0.1130i
0.0004-0.1060i
0.0003-0.0895i
0.0003-0.0912i
0.0005-0.1136i
Critical Op. Pnt;
Evector for lambda =
0.0000 (original)
0.0000-0.0000i
0.0000+0.0000i
0.0000-0.0000i
0.0000-0.0000i
0.0000+0.0000i
0.2675-0.0000i
0.2675+0.0000i
0.2675+0.0000i
0.2672-0.0000i
0.2672+0.0000i
0.2674+0.0000i
0.2674+0.0000i
0.2672-0.0000i
0.2671-0.0000i
0.2671-0.0000i
0.2671-0.0000i
0.2672+0.0000i
0.2672+0.0000i
0.2671-0.0000i
Critical Op. Pnt;
Evector for lambda =
0.0000 (newly created)
-0.0000-0.0000i
-0.0000-0.0000i
-0.0000+0.0000i
0.0000-0.0000i
0.0000+0.0000i
-0.2019+0.0000i
-0.2019-0.0000i
-0.2019-0.0000i
0.2359+0.0000i
0.2359-0.0000i
-0.1326-0.0000i
-0.1295-0.0000i
0.2364+0.0000i
0.3156+0.0000i
0.3448+0.0000i
0.3103+0.0000i
0.2517-0.0000i
0.2576-0.0000i
0.3388+0.0000i
State deviation
Original Op. Pnt;
described by
Evector for lambda =
component
-0.0189 + 3.6342i
0.0003-0.0569i
Voltage(pu)@Bus 4
Voltage(pu)@Bus 5 0.0003-0.0502i
Voltage(pu)@Bus 7 0.0003-0.0548i
Voltage(pu)@Bus 9 0.0003-0.0564i
Voltage(pu)@Bus 10 0.0003-0.0483i
Voltage(pu)@Bus 11 0.0002-0.0272i
Voltage(pu)@Bus 12 0.0000-0.0052i
Voltage(pu)@Bus 13 0.0001-0.0101i
Voltage(pu)@Bus 14 0.0002-0.0364i
Critical Op. Pnt;
Evector for lambda =
0.0000 (original)
-0.0001-0.0000i
-0.0001-0.0000i
-0.0001-0.0000i
-0.0001-0.0000i
-0.0001-0.0000i
-0.0001-0.0000i
-0.0000-0.0000i
-0.0000-0.0000i
-0.0001-0.0000i
Critical Op. Pnt;
Evector for lambda =
0.0000 (newly created)
0.1399+0.0000i
0.1249+0.0000i
0.1490+0.0000i
0.1546+0.0000i
0.1339+0.0000i
0.0791+0.0000i
0.0159-0.0000i
0.0302-0.0000i
0.1020+0.0000i
More Eigenvectors of the Full Dynamic Model
State deviation
described by
eigenvector
component
Freq@Bus 1
Freq@Bus 2
Freq@Bus 3
Freq@Bus 6
Freq@Bus 8
Angle@Bus 1
Angle@Bus 2
Angle@Bus 3
Angle@Bus 6
Angle@Bus 8
Angle@Bus 4
Angle@Bus 5
Angle@Bus 7
Angle@Bus 9
Angle@Bus 10
Angle@Bus 11
Angle@Bus 12
Angle@Bus 13
Angle@Bus 14
Critical Op. Pnt;
Evector for lambda =
-0.3957
-0.0181+0.0000i
-0.0181+0.0000i
-0.0181-0.0000i
0.0000-0.0000i
0.0000+0.0000i
0.4384-0.0000i
0.4384+0.0000i
0.4384+0.0000i
-0.0000+0.0000i
-0.0000+0.0000i
0.3689+0.0000i
0.3659+0.0000i
-0.0005-0.0000i
-0.0799-0.0000i
-0.1091-0.0000i
-0.0745-0.0000i
-0.0159+0.0000i
-0.0218+0.0000i
-0.1031-0.0000i
Critical Op. Pnt;
Evector for lambda =
-0.2294
-0.0000+0.0000i
-0.0000+0.0000i
-0.0000-0.0000i
-0.0103-0.0000i
-0.0103+0.0000i
0.0000+0.0000i
0.0000-0.0000i
0.0000-0.0000i
0.2924-0.0000i
0.2924+0.0000i
0.0463+0.0000i
0.0484+0.0000i
0.2928-0.0000i
0.3457+0.0000i
0.3652-0.0000i
0.3421+0.0000i
0.3030+0.0000i
0.3069-0.0000i
0.3612-0.0000i
Future Work
Key question 1: must systems inevitably
encounter loss of stability via flux
decay/voltage control mode (as identified in
Rajagopalan et al) before this type of
bifurcation?
 Hypothesis: perhaps not if good reactive
support throughout system as transmission
corridor is loaded up.

Future Work

Key question 2: possibility of same
weakness as direct point of collapse
calculations in voltage literature - many
generators hitting reactive power limits
along the loading path.

Answer will be closely related to that of
question 1!
Conclusions

Simple exercise to shift focus back from
bifurcations primarily associated w/
voltage, to bifurcations primarily associated
with swing mode.

Key idea: hypothesize a form for
eigenvector, restrict search for bifurcation
point to display that eigenvector.
Conclusions

While further is clearly development
needed, method here could provide simple
computation to identify a stability
constraint on ATC across a transmission
corridor.