Network Structure in Swing Mode Bifurcations u Motivation & Background –Practical Goal: examine...

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.


Form of nonlinear DAE model

M• = L1(P

I - PN(, V))L1

• =

0 = L2(PI - PN(, V))

0 = (QI(V) - QN(, V))


Requisite variable/function definitions:V n, V := vector of bus voltage magnitudes; n, := vector of bus voltage phase angles

relative to an arbitrary synchronousreference frame of frequency 0 (noreference angle is deleted);

m, := vector of generator frequencydeviations, relative to synchronousfrequency 0;

RIRI

RI


variable/function definitions:

M mxm, M := diagonal matrix of normalizedgenerator inertias;

PI n, PI:= vector of net active power injection ateach bus; assumed constant;

QI: n-m n-m, QI(V) := vector valued function ofnet reactive power injection at load buses,normalized by voltage magnitude;

RI

RI

RI RI


variable/function definitions:

L1 := rows 1 through m of an nxn identity matrix;L2 := rows m+1 through n of an nxn identity matrix;

PN(, V) := vector-valued function of active powerabsorbed by network at each bus

QN(, V) := vector-valued function of reactivepower absorbed by network at load buses,normalized by voltage magnitude

Linearized DAE/”Singular System” Form

Write linearization as:

~E •

x = ~Rx

Component Definitions where:

Mmxm 0 0

0 I mxm 0

0 0 02(n-m)x2(n-m)

E~

=

0 -Imxm 0

Imxm 0 0

0 0 I2(n-m)x2(n-m)

SR~

=

Component Definitions

and:

J: nx n-m (2n-m)x(2n-m), J =

Imxm 0 0

0 J11 J12

0 J21 J22

S =

RI RI RI

P

N

V P

N

Q

N

V {Q

N - Q

I}

L

L

Relation to Reduced Dimension Symmetric Problem

Consider reduced dimension, symmetric generalized eigenvalue problem defined by pair (E, J), where:

M 0

0 02(n-m)x2(n-m)

E =


FACT: Finite generalized eigenvalues of (E, J) completely determine finite generalized eigenvalues of (

~E ~R)


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, equivalent to e.v.’s of M-1[J

11 - J

12J

22-1

J21

]

Role of Network Structure

Question: what is a mechanism by which 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}.

[J11

- J12

J22-1

J21

]


Given this admittance matrix structure, reduced PF Jacobianhas associated network graph.

A mechanism for loss of rank can then be identified: branches forming a cutset all have weights of zero.

[J11

- J12

J22-1

J21

]


Eigenvector associated with new zero eigenvalue is identifiable by inspection:

where is a positive real constant, and partition of eigenvector is across the cutset.

J11

- J12

J22-1

J21

1-1

= 0


Returning to associated generalized eigenvalue problem, to preserve sparsity, one would have:

J11

J12

J21

J22

1–1 w

= 0


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...


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 ]


Simultaneous equations to be solved:

Note that f tilde terms are power balance equations, deleting gen buses. Once angles & voltages solved, gen dispatch is output.

0 = f~

(~

, V)

0 = J~

((~

, V)[0, 1T, w~T]T


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.

10

11

12

13

14

1

2

3

45

6

7

8

9

G

G

GG

G

1

#

2

3

4

5

6

7 8

910

1112

13

1415

16

17

18

19 20 - Transmission Line #'s# - Bus #'s

14 Bus Test System

10

11

12

13

14

1

2

3

45

6

7

8

9

G

G

GG

G

1

#

2

3

4

5

6

7 8

910

1112

13

1415

16

17

18

19 20 - Transmission Line #'s# - Bus #'s

14 Bus Test System

Cutset Here

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.Original Op. Point Critical Op. Point

Bus # Voltage/Phase(degrees) Voltage/Phase(degrees)1 1.0600 0 1.0600 02 1.0450 -4.7545 1.0450 -3.32843 1.0800 -2.8056 1.0800 -6.45224 0.9196 -25.1819 0.8351 -28.92825 0.9476 -23.6900 0.8733 -26.85316 1.0200 -70.0082 1.0200 -84.26107 0.9514 -62.1503 0.8663 -79.58708 1.0400 -58.2712 1.0400 -79.41649 0.9407 -76.0226 0.8562 -94.589010 0.9426 -83.9820 0.8702 -103.014811 0.9852 -87.8051 0.9412 -105.105212 0.9478 -90.6433 0.9385 -105.525113 1.0056 -82.1261 0.9884 -97.341214 0.9960 -89.9647 0.9410 -108.4568

Selected Generalized Eigenvectors of the Full Dynamic ModelState deviation described bycomponent

Original Op. Pnt;Evector for lambda =-0.0189 + 3.6342i

Critical Op. Pnt;Evector for lambda =

0.0000 (original)


0.0000 (newly created)Freq@Bus 1 -0.4033-0.0048i 0.0000-0.0000i -0.0000-0.0000i

Freq@Bus 2 -0.3917-0.0047i 0.0000+0.0000i -0.0000-0.0000i

Freq@Bus 3 -0.4287-0.0051i 0.0000-0.0000i -0.0000+0.0000i

Freq@Bus 6 0.3068+0.0026i 0.0000-0.0000i 0.0000-0.0000i

Freq@Bus 8 0.4964+0.0054i 0.0000+0.0000i 0.0000+0.0000i

Angle@Bus 1 -0.0007+0.1110i 0.2675-0.0000i -0.2019+0.0000i

Angle@Bus 2 -0.0007+0.1078i 0.2675+0.0000i -0.2019-0.0000i

Angle@Bus 3 -0.0008+0.1180i 0.2675+0.0000i -0.2019-0.0000i

Angle@Bus 6 0.0003-0.0844i 0.2672-0.0000i 0.2359+0.0000i

Angle@Bus 8 0.0008-0.1366i 0.2672+0.0000i 0.2359-0.0000i

Angle@Bus 4 -0.0005+0.0625i 0.2674+0.0000i -0.1326-0.0000i

Angle@Bus 5 -0.0005+0.0639i 0.2674+0.0000i -0.1295-0.0000i

Angle@Bus 7 0.0004-0.0868i 0.2672-0.0000i 0.2364+0.0000i

Angle@Bus 9 0.0005-0.1030i 0.2671-0.0000i 0.3156+0.0000i

Angle@Bus 10 0.0005-0.1130i 0.2671-0.0000i 0.3448+0.0000i

Angle@Bus 11 0.0004-0.1060i 0.2671-0.0000i 0.3103+0.0000i

Angle@Bus 12 0.0003-0.0895i 0.2672+0.0000i 0.2517-0.0000i

Angle@Bus 13 0.0003-0.0912i 0.2672+0.0000i 0.2576-0.0000i

Angle@Bus 14 0.0005-0.1136i 0.2671-0.0000i 0.3388+0.0000i

State deviationdescribed bycomponent

Original Op. Pnt;Evector for lambda =-0.0189 + 3.6342i


0.0000 (original)


0.0000 (newly created)Voltage(pu)@Bus 40.0003-0.0569i -0.0001-0.0000i 0.1399+0.0000i

Voltage(pu)@Bus 50.0003-0.0502i -0.0001-0.0000i 0.1249+0.0000i





Voltage(pu)@Bus 120.0000-0.0052i -0.0000-0.0000i 0.0159-0.0000i

Voltage(pu)@Bus 130.0001-0.0101i -0.0000-0.0000i 0.0302-0.0000i


More Eigenvectors of the Full Dynamic ModelState deviationdescribed byeigenvectorcomponent


-0.3957


-0.2294

Freq@Bus 1 -0.0181+0.0000i -0.0000+0.0000iFreq@Bus 2 -0.0181+0.0000i -0.0000+0.0000iFreq@Bus 3 -0.0181-0.0000i -0.0000-0.0000iFreq@Bus 6 0.0000-0.0000i -0.0103-0.0000iFreq@Bus 8 0.0000+0.0000i -0.0103+0.0000iAngle@Bus 1 0.4384-0.0000i 0.0000+0.0000iAngle@Bus 2 0.4384+0.0000i 0.0000-0.0000iAngle@Bus 3 0.4384+0.0000i 0.0000-0.0000iAngle@Bus 6 -0.0000+0.0000i 0.2924-0.0000iAngle@Bus 8 -0.0000+0.0000i 0.2924+0.0000iAngle@Bus 4 0.3689+0.0000i 0.0463+0.0000iAngle@Bus 5 0.3659+0.0000i 0.0484+0.0000iAngle@Bus 7 -0.0005-0.0000i 0.2928-0.0000iAngle@Bus 9 -0.0799-0.0000i 0.3457+0.0000iAngle@Bus 10 -0.1091-0.0000i 0.3652-0.0000iAngle@Bus 11 -0.0745-0.0000i 0.3421+0.0000iAngle@Bus 12 -0.0159+0.0000i 0.3030+0.0000iAngle@Bus 13 -0.0218+0.0000i 0.3069-0.0000iAngle@Bus 14 -0.1031-0.0000i 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.

Network Structure in Swing Mode Bifurcations u Motivation & Background –Practical Goal: examine...

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