1. Network Structure in Swing Mode Bifurcations u 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)?

2. Key Ideas u 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 u Alternative: leave larger # of degrees of freedom in parameter space, but constrain structure of eigenvector at bifurcation.

3. Key Questions u Is there a priori knowledge of form of eigenvector of interest for “mode” of instability we’re after? u 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.)

4. Caveats (at present...) u Development to date uses only very simple, classical model for generators. u 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).

5. Review - Relation of PF Jacobian and Linearized Dynamic Model u This issue well treated in existing literature, but still useful to develop notation suited to generalized eigenvalue problem. u 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.

6. Review - Relation of PF Jacobian and Linearized Dynamic Model u Form of nonlinear DAE model

7. Review - Relation of PF Jacobian and Linearized Dynamic Model u Requisite variable/function definitions:

8. Review - Relation of PF Jacobian and Linearized Dynamic Model u variable/function definitions:

9. Review - Relation of PF Jacobian and Linearized Dynamic Model u variable/function definitions:

10. Linearized DAE/”Singular System” Form u Write linearization as:

11. Component Definitions u where:

12. Component Definitions u and: J : n x n-m  (2n-m)x(2n-m), J =    I mxm 00 0 J 11 J 12 0 J 21 J 22    S = RIRIRI      P N  V  P N   Q N  V  {Q N - Q I }    L L

13. Relation to Reduced Dimension Symmetric Problem u Consider reduced dimension, symmetric generalized eigenvalue problem defined by pair (E, J), where:

14. Relation to Reduced Dimension Symmetric Problem u FACT: Finite generalized eigenvalues of (E, J) completely determine finite generalized eigenvalues of

15. Relation to Reduced Dimension Symmetric Problem u In particular,

16. Key Observation u 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. u When computation (sparsity) not a concern, equivalent to e.v.’s of

17. Role of Network Structure u Question: what is a mechanism by which might drop rank? u 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}.

18. Role of Network Structure u Given this admittance matrix structure, reduced PF Jacobian has associated network graph. u A mechanism for loss of rank can then be identified: branches forming a cutset all have weights of zero.

19. Role of Network Structure u Eigenvector associated with new zero eigenvalue is identifiable by inspection: where is a positive real constant, and partition of eigenvector is across the cutset.

20. Role of Network Structure u Returning to associated generalized eigenvalue problem, to preserve sparsity, one would have:

21. Role of Network Structure u Finally, in original generalized eigenvalue problem for full dynamics, the new eigenvector has structure [ 1, – 1 ] in components associated with generator phase angles. u Strongly suggests an inter area swing mode, with gens on one side of cutset 180º out of phase with those on other side.

22. Summary so far... u 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. u Look for limiting operating conditions that yield J realizing this bifurcation & e-vector.

23. Computational Formulation u Very analogous to early “direct” methods of finding loading levels associated with Jacobian singularity in voltage collapse literature (e.g., Alvarado/Jung, 88). u But instead of leaving eigenvector components associated with zero eigenvalue as free variables, we constrain components associated with gen angles.

24. Computational Formulation u Must compensate with “extra” degrees of freedom. u For example to follow, generation dispatch selected as new variables. Clearly, many other possible choices...

25. Computational Formulation u 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. u Resulting structure of gen angle e-vector components becomes [ 0, 1 ]

26. Computational Formulation u Simultaneous equations to be solved: u Note that f tilde terms are power balance equations, deleting gen buses. Once angles & voltages solved, gen dispatch is output.

27. Computational Formulation u Solution method is full Newton Raphson. u 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.

28. Case Study u Based on modified form of IEEE 14 bus test system.

29. 10 11 12 13 14 1 2 3 4 5 6 7 8 9 G G G G G 1 # 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 - Transmission Line #'s # - Bus #'s 14 Bus Test System Cutset Here

30. Case Study u 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. u Here choice has gens 1, 2, 3 on one side, gens 6, 8 on other side. u Model has rotational damping added as rough approximation to governor action.

35. Future Work u 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? u Hypothesis: perhaps not if good reactive support throughout system as transmission corridor is loaded up.

36. Future Work u 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. u Answer will be closely related to that of question 1!

37. Conclusions u Simple exercise to shift focus back from bifurcations primarily associated w/ voltage, to bifurcations primarily associated with swing mode. u Key idea: hypothesize a form for eigenvector, restrict search for bifurcation point to display that eigenvector.

38. Conclusions u While further is clearly development needed, method here could provide simple computation to identify a stability constraint on ATC across a transmission corridor.