Small-signal analysis#

With small_signal_analysis=True the operating point is linearized before the time stepping: the algebraic variables are eliminated, the reduced state matrix is built, and its eigenvalues, participation factors and a modal report are attached to the returned model. The analysis lives entirely at t = 0 s, so a short horizon keeps the run cheap.

The system is the standard IEEE 39-bus benchmark with ten subtransient machines. The run prints the modal report, sorted by damping ratio, so the critical electromechanical modes top the list:

[1]:

import hermess

dae = hermess.simulate(
    "ieee39", T_end=0.5, ts=5e-3, line_dyn=False,
    small_signal_analysis=True, small_signal_figures=False,
)


Small-signal modal analysis: 80 modes (0 unstable) over 110 states, sorted by damping ratio (most critical first).
| #   | Eigenvalue (σ±jω)   | f [Hz]   | ζ [%]   | Flag   | Dominant states (top 4, participation %)                                       |
|-----|---------------------|----------|---------|--------|--------------------------------------------------------------------------------|
| 1   | -0.000              | —        | +0.0    | zero   | SM@31:delta (10%), SM@34:delta (10%), SM@32:delta (10%), SM@30:delta (10%)     |
| 2   | -0.271 ± 7.026j     | 1.118    | +3.9    | low ζ  | SM@32:omega (25%), SM@32:delta (25%), SM@31:omega (21%), SM@31:delta (21%)     |
| 3   | -0.345 ± 7.892j     | 1.256    | +4.4    | low ζ  | SM@33:omega (33%), SM@33:delta (33%), SM@35:omega (5%), SM@35:delta (5%)       |
| 4   | -0.274 ± 5.978j     | 0.951    | +4.6    | low ζ  | SM@38:omega (15%), SM@38:delta (15%), SM@31:omega (12%), SM@31:delta (12%)     |
| 5   | -0.182 ± 3.896j     | 0.620    | +4.7    | low ζ  | SM@39:omega (15%), SM@39:delta (15%), SM@34:omega (7%), SM@34:delta (6%)       |
| 6   | -0.377 ± 8.005j     | 1.274    | +4.7    | low ζ  | SM@36:omega (28%), SM@36:delta (28%), SM@35:omega (15%), SM@35:delta (15%)     |
| 7   | -0.301 ± 6.351j     | 1.011    | +4.7    | low ζ  | SM@35:omega (16%), SM@35:delta (16%), SM@36:omega (9%), SM@36:delta (9%)       |
| 8   | -0.411 ± 7.500j     | 1.194    | +5.5    |        | SM@37:omega (28%), SM@37:delta (28%), SM@30:omega (6%), SM@30:delta (6%)       |
| 9   | -0.329 ± 5.629j     | 0.896    | +5.8    |        | SM@34:omega (21%), SM@34:delta (21%), SM@38:omega (19%), SM@38:delta (19%)     |
| 10  | -0.750 ± 9.906j     | 1.577    | +7.6    |        | SM@30:omega (35%), SM@30:delta (35%), SM@37:omega (10%), SM@37:delta (10%)     |
| 11  | -0.459 ± 1.683j     | 0.268    | +26.3   |        | SM@38:e_qprim (25%), SM@38:Rf (13%), SM@38:Efd (7%), SM@36:e_qprim (5%)        |
| 12  | -0.493 ± 1.089j     | 0.173    | +41.2   |        | SM@39:e_qprim (31%), SM@39:Rf (23%), SM@36:e_qprim (8%), SM@36:Rf (5%)         |
| 13  | -0.552 ± 1.211j     | 0.193    | +41.5   |        | SM@38:e_qprim (20%), SM@38:Rf (12%), SM@36:e_qprim (8%), SM@39:e_qprim (6%)    |
| 14  | -5.311 ± 9.109j     | 1.450    | +50.4   |        | SM@37:Vr (44%), SM@37:Efd (43%), SM@37:Rf (12%), SM@37:e_qprim (1%)            |
| 15  | -5.280 ± 8.815j     | 1.403    | +51.4   |        | SM@32:Vr (34%), SM@32:Efd (33%), SM@33:Vr (10%), SM@32:Rf (9%)                 |
| 16  | -0.448 ± 0.745j     | 0.119    | +51.5   |        | SM@31:e_qprim (17%), SM@30:e_qprim (14%), SM@31:Rf (13%), SM@32:e_qprim (11%)  |
| 17  | -5.292 ± 8.760j     | 1.394    | +51.7   |        | SM@33:Vr (32%), SM@33:Efd (31%), SM@33:Rf (9%), SM@32:Vr (9%)                  |
| 18  | -0.489 ± 0.803j     | 0.128    | +52.0   |        | SM@30:e_qprim (24%), SM@30:Rf (17%), SM@34:e_qprim (7%), SM@30:Efd (5%)        |
| 19  | -5.293 ± 8.563j     | 1.363    | +52.6   |        | SM@35:Vr (41%), SM@35:Efd (40%), SM@35:Rf (11%), SM@33:Vr (2%)                 |
| 20  | -0.332 ± 0.496j     | 0.079    | +55.6   |        | SM@37:e_qprim (43%), SM@37:Rf (36%), SM@37:Efd (5%), SM@30:e_qprim (3%)        |
| 21  | -0.568 ± 0.837j     | 0.133    | +56.1   |        | SM@36:e_qprim (21%), SM@34:e_qprim (17%), SM@36:Rf (13%), SM@34:Rf (11%)       |
| 22  | -5.262 ± 7.668j     | 1.220    | +56.6   |        | SM@39:Vr (40%), SM@39:Efd (39%), SM@39:Rf (13%), SM@39:e_qprim (3%)            |
| 23  | -5.227 ± 7.487j     | 1.192    | +57.2   |        | SM@31:Vr (41%), SM@31:Efd (40%), SM@31:Rf (13%), SM@31:e_qprim (1%)            |
| 24  | -0.380 ± 0.506j     | 0.080    | +60.1   |        | SM@35:e_qprim (36%), SM@35:Rf (30%), SM@33:e_qprim (5%), SM@36:e_qprim (5%)    |
| 25  | -0.413 ± 0.524j     | 0.083    | +61.9   |        | SM@32:e_qprim (25%), SM@32:Rf (22%), SM@31:e_qprim (18%), SM@31:Rf (14%)       |
| 26  | -5.245 ± 6.284j     | 1.000    | +64.1   |        | SM@30:Vr (41%), SM@30:Efd (39%), SM@30:Rf (14%), SM@30:e_qsec (2%)             |
| 27  | -0.456 ± 0.514j     | 0.082    | +66.3   |        | SM@33:e_qprim (27%), SM@33:Rf (24%), SM@34:e_qprim (15%), SM@34:Rf (8%)        |
| 28  | -5.068 ± 5.514j     | 0.878    | +67.7   |        | SM@34:Vr (37%), SM@34:Efd (35%), SM@34:Rf (14%), SM@36:Vr (4%)                 |
| 29  | -5.087 ± 5.345j     | 0.851    | +68.9   |        | SM@36:Vr (30%), SM@36:Efd (28%), SM@36:Rf (12%), SM@38:Vr (7%)                 |
| 30  | -5.037 ± 5.134j     | 0.817    | +70.0   |        | SM@38:Vr (29%), SM@38:Efd (28%), SM@38:Rf (13%), SM@38:e_qprim (6%)            |
| 31  | -5.126 ± 0.197j     | 0.031    | +99.9   |        | SM@35:e_dprim (33%), SM@35:psv (24%), SM@37:e_dprim (12%), SM@37:psv (10%)     |
| 32  | -241.142            | —        | +100.0  |        | SM@34:e_dsec (98%), SM@34:e_dprim (2%), SM@35:e_dsec (0%), SM@38:e_dsec (0%)   |
| 33  | -257.630            | —        | +100.0  |        | SM@37:e_dsec (83%), SM@38:e_dsec (14%), SM@35:e_dsec (2%), SM@37:e_dprim (1%)  |
| 34  | -264.580            | —        | +100.0  |        | SM@35:e_dsec (92%), SM@38:e_dsec (7%), SM@35:e_dprim (1%), SM@37:e_dsec (0%)   |
| 35  | -261.869            | —        | +100.0  |        | SM@38:e_dsec (78%), SM@37:e_dsec (17%), SM@35:e_dsec (5%), SM@37:e_dprim (0%)  |
| 36  | -144.644            | —        | +100.0  |        | SM@39:e_dsec (100%), SM@39:e_dprim (0%), SM@30:e_dsec (0%), SM@30:e_dprim (0%) |
| 37  | -100.429            | —        | +100.0  |        | SM@30:e_dsec (99%), SM@30:e_dprim (1%), SM@39:e_dsec (0%), SM@30:omega (0%)    |
| 38  | -70.527             | —        | +100.0  |        | SM@32:e_dsec (80%), SM@31:e_dsec (12%), SM@36:e_dsec (5%), SM@33:e_dsec (3%)   |
| 39  | -71.833             | —        | +100.0  |        | SM@36:e_dsec (70%), SM@33:e_dsec (18%), SM@32:e_dsec (10%), SM@31:e_dsec (1%)  |
| 40  | -72.386             | —        | +100.0  |        | SM@31:e_dsec (70%), SM@36:e_dsec (12%), SM@32:e_dsec (9%), SM@33:e_dsec (9%)   |
| 41  | -72.480             | —        | +100.0  |        | SM@33:e_dsec (69%), SM@31:e_dsec (17%), SM@36:e_dsec (13%), SM@32:e_dsec (1%)  |
| 42  | -19.257             | —        | +100.0  |        | SM@34:e_qsec (92%), SM@33:e_qsec (3%), SM@34:e_qprim (3%), SM@34:Vr (0%)       |
| 43  | -17.689             | —        | +100.0  |        | SM@33:e_qsec (54%), SM@36:e_qsec (27%), SM@32:e_qsec (10%), SM@34:e_qsec (3%)  |
| 44  | -18.102             | —        | +100.0  |        | SM@32:e_qsec (84%), SM@33:e_qsec (7%), SM@36:e_qsec (4%), SM@32:e_qprim (2%)   |
| 45  | -18.001             | —        | +100.0  |        | SM@36:e_qsec (64%), SM@33:e_qsec (31%), SM@36:e_qprim (2%), SM@33:e_qprim (1%) |
| 46  | -15.649             | —        | +100.0  |        | SM@31:e_qsec (91%), SM@37:e_qsec (2%), SM@31:e_qprim (2%), SM@38:e_qsec (1%)   |
| 47  | -15.302             | —        | +100.0  |        | SM@37:e_qsec (81%), SM@38:e_qsec (13%), SM@37:e_qprim (2%), SM@31:e_qsec (1%)  |
| 48  | -14.917             | —        | +100.0  |        | SM@38:e_qsec (78%), SM@37:e_qsec (11%), SM@38:e_qprim (2%), SM@35:e_qsec (2%)  |
| 49  | -13.870             | —        | +100.0  |        | SM@35:e_qsec (80%), SM@39:e_qsec (11%), SM@38:e_qsec (2%), SM@35:e_qprim (2%)  |
| 50  | -14.143             | —        | +100.0  |        | SM@39:e_qsec (82%), SM@35:e_qsec (12%), SM@39:e_qprim (2%), SM@39:Vr (1%)      |
| 51  | -9.892              | —        | +100.0  |        | SM@30:e_qsec (92%), SM@30:e_qprim (3%), SM@30:Vr (2%), SM@30:Efd (1%)          |
| 52  | -6.837              | —        | +100.0  |        | SM@34:e_dprim (87%), SM@34:omega (2%), SM@34:delta (2%), SM@35:e_dprim (2%)    |
| 53  | -5.413              | —        | +100.0  |        | SM@37:e_dprim (42%), SM@35:e_dprim (21%), SM@37:psv (21%), SM@35:psv (6%)      |
| 54  | -5.271              | —        | +100.0  |        | SM@37:psv (42%), SM@37:e_dprim (20%), SM@35:e_dprim (14%), SM@35:psv (11%)     |
| 55  | -5.218              | —        | +100.0  |        | SM@34:psv (35%), SM@35:psv (14%), SM@33:psv (8%), SM@36:psv (7%)               |
| 56  | -5.169              | —        | +100.0  |        | SM@34:psv (46%), SM@35:psv (10%), SM@38:psv (9%), SM@36:psv (8%)               |
| 57  | -5.015              | —        | +100.0  |        | SM@39:psv (96%), SM@30:psv (3%), SM@39:pm (1%), SM@39:omega (0%)               |
| 58  | -5.045              | —        | +100.0  |        | SM@30:psv (84%), SM@38:psv (4%), SM@32:psv (2%), SM@39:psv (2%)                |
| 59  | -5.120              | —        | +100.0  |        | SM@31:psv (45%), SM@38:psv (42%), SM@32:psv (5%), SM@31:pm (2%)                |
| 60  | -5.115              | —        | +100.0  |        | SM@36:psv (49%), SM@38:psv (22%), SM@31:psv (16%), SM@33:psv (5%)              |
| 61  | -5.095              | —        | +100.0  |        | SM@33:psv (60%), SM@36:psv (19%), SM@32:psv (7%), SM@31:psv (5%)               |
| 62  | -5.090              | —        | +100.0  |        | SM@32:psv (68%), SM@33:psv (15%), SM@31:psv (11%), SM@32:pm (2%)               |
| 63  | -0.139              | —        | +100.0  |        | SM@39:omega (29%), SM@39:delta (25%), SM@37:delta (3%), SM@34:delta (3%)       |
| 64  | -1.419              | —        | +100.0  |        | SM@39:e_dprim (93%), SM@30:omega (0%), SM@32:omega (0%), SM@30:pm (0%)         |
| 65  | -2.183              | —        | +100.0  |        | SM@34:pm (11%), SM@36:pm (8%), SM@35:pm (8%), SM@33:pm (8%)                    |
| 66  | -1.312              | —        | +100.0  |        | SM@32:e_dprim (87%), SM@32:Rf (7%), SM@32:e_qprim (1%), SM@31:e_dprim (1%)     |
| 67  | -1.184              | —        | +100.0  |        | SM@30:e_dprim (91%), SM@30:Rf (3%), SM@30:omega (1%), SM@30:e_qprim (1%)       |
| 68  | -0.866              | —        | +100.0  |        | SM@36:e_dprim (92%), SM@36:Rf (5%), SM@36:Efd (1%), SM@36:e_qprim (1%)         |
| 69  | -1.006              | —        | +100.0  |        | SM@38:e_dprim (93%), SM@38:Rf (3%), SM@38:e_qprim (1%), SM@31:e_dprim (1%)     |
| 70  | -2.499              | —        | +100.0  |        | SM@30:pm (78%), SM@39:pm (15%), SM@30:e_dprim (2%), SM@37:pm (1%)              |
| 71  | -2.481              | —        | +100.0  |        | SM@39:pm (78%), SM@30:pm (10%), SM@32:pm (4%), SM@37:pm (2%)                   |
| 72  | -0.925              | —        | +100.0  |        | SM@33:e_dprim (89%), SM@33:Rf (5%), SM@33:e_qprim (1%), SM@31:e_dprim (1%)     |
| 73  | -0.967              | —        | +100.0  |        | SM@31:e_dprim (91%), SM@31:Rf (3%), SM@33:e_dprim (1%), SM@31:e_qprim (1%)     |
| 74  | -2.397              | —        | +100.0  |        | SM@34:pm (67%), SM@37:pm (6%), SM@31:pm (6%), SM@38:pm (5%)                    |
| 75  | -2.451              | —        | +100.0  |        | SM@32:pm (46%), SM@33:pm (38%), SM@31:pm (8%), SM@34:pm (2%)                   |
| 76  | -2.422              | —        | +100.0  |        | SM@35:pm (34%), SM@36:pm (24%), SM@37:pm (18%), SM@38:pm (7%)                  |
| 77  | -2.425              | —        | +100.0  |        | SM@31:pm (54%), SM@37:pm (20%), SM@38:pm (9%), SM@32:pm (8%)                   |
| 78  | -2.433              | —        | +100.0  |        | SM@38:pm (58%), SM@37:pm (35%), SM@38:psv (2%), SM@38:delta (1%)               |
| 79  | -2.447              | —        | +100.0  |        | SM@35:pm (46%), SM@36:pm (40%), SM@33:pm (5%), SM@32:pm (2%)                   |
| 80  | -2.449              | —        | +100.0  |        | SM@33:pm (38%), SM@32:pm (23%), SM@36:pm (17%), SM@31:pm (10%)                 |

Simulation: 100%|██████████| 100.0/100% [00:00<00:00]

The s-plane shows the same information at a glance. Modes to the right of the dashed damping cone are the ones to watch, the swing modes between 0.6 and 1.6 Hz:

[2]:

dae.plot_eigenvalues(damping_ref=0.05)

../_images/examples_small_signal_3_0.png

The participation factors say which states carry a mode, i.e. where a damping problem lives and which machine a controller should act on. Mode 2 of the report is the least damped oscillatory mode; its participation is shared across machines on both sides of the network, the signature of an inter-area mode rather than a local one:

[3]:

dae.participation_table(mode=2)

[3]:

	state	participation
0	SM@32:omega	0.2546
1	SM@32:delta	0.2531
2	SM@31:omega	0.2134
3	SM@31:delta	0.2119
4	SM@32:e_qprim	0.0113
5	SM@31:e_qprim	0.0090
6	SM@32:e_dprim	0.0071
7	SM@31:e_qsec	0.0069
8	SM@32:e_qsec	0.0065
9	SM@35:omega	0.0032

Everything shown here is also available as plain data (dae.A, dae.eigenvalues, dae.state_names, and res.small_signal after extract_results), so a parameter study, a damping-versus-gain sweep for example, is a loop over simulate calls away.