In the previous section, it was shown how the net-damping criterion can provide direct infor- mation on whether a SISO system is stable or unstable. However there has been no information relating the criterion to poorly-damped or near-instability conditions.
5.5.1
Damping in a multi-pole system
As mentioned in Section (3.1), the damping of a system can be strictly defined only for 2ndorder systems as the one described in (3.1). When it comes to multi-pole systems, it is not possible to provide a similarly strict definition of the system’s damping. A step-wise excitation of the system excites all of its eigenmodes (given the fact the unit step contains the full frequency spectrum) and the total system response consists of their superposition.
However, the behavior of a multi-pole system is normally dominated by its dominant poles (if these exist), which dictate the main properties of its response to a perturbation. Furthermore, poles with very low damping have, by definition, a very small absolute real part, becoming potentially dominant as they find themselves very close to the imaginary axis. In such cases, the final response of the system will be mostly dictated by those poorly-damped poles and it is here suggested, in a non-strict manner, that their damping factor can be regarded as the damping of the complete system.
5.5.2
Net-damping in poorly-damped configurations
Typically, the encirclement by the Nyquist plot of -1 occurs at low frequencies and in the neigh- borhood of resonances [87]. These resonances are usually identified with poorly-damped poles that move towards the RHP of the complex plane due to a change of a critical parameter (e.g. transferred power). When the system is on the verge of instability, the Nyquist curve intersects with the point -1. This occurs at a frequency ωcrit with the corresponding closed-loop system
having either a real pole at the origin of the s−plane or a pair of marginally-stable complex- conjugate poles with a damped natural frequencyωd = ωcrit. If these poles have not yet become
unstable but are close enough to the imaginary axis, the Nyquist curve will cross the real axis on the right of -1 but in close proximity to it. This occurs at a frequencyωNthat is closely related
5.5. Correlation between net-damping and damping factor If the system is marginally stable, its net-damping at the frequencyωN = ωcrit = ωdis equal to
zero
D(ωN) = D(ωcrit) = DF(ωcrit) + DG(ωcrit) = 0 (5.47)
Based on the previous analysis, it is here suggested that it is possible to correlate the level of net damping of a system measured atωN, with the existence of poorly-damped poles that are close
to instability. The closer these poles approach the imaginary axis, the more the net damping
D(ωN) should approach zero until the poles become marginally stable and D(ωN) = 0. The
value that quantifies the level of damping for these poles is their damping factor. The closer the latter is to zero, the less damped the poles and the system is closer to instability.
The objective of this analysis, is to provide a way though solely a frequency analysis of the sys- tem to determine whether there are poorly-damped poles critically close to the imaginary axis, without actually finding the poles of the system and the frequency characteristics of the poorly- damped poles. For this reason, four different scenarios are examined where the two-terminal VSC-HVDC system appears to have poorly-damped poles whose damping decreases with the change of a system parameter or operational condition, until they almost become marginally stable. In all cases, the damping of these poles is plotted in conjunction with the measurement of the net damping at the frequencyωN where the Nyquist curve crosses the real axis closest
to -1. As for the previous sections, the DVC is at all times considered to feature the power- feedforward term.
The four different cases use the basic values as defined in Table 3.1 with the custom differences being identified in the following way
- Case 1: The system features overhead dc-transmission lines with their properties defined in Table 2.1 and their length is varied from 50-230 km.
- Case 2: The system features overhead dc-transmission lines and the controller bandwidths
adandaf are equal and varied from 200-630 rad/s.
- Case 3: The system features overhead dc-transmission lines of 230 km in length and the transferred power at the inverter Station 2 is varied from 0-1000 MW.
- Case 4: The system features cable dc-transmission lines with their properties defined in Table 2.1 and their length is varied from 26-43 km.
Each of the graphs in Fig. 5.11 shows the pole movement of the system for an increasing trend of the chosen variable, with the concerned poles being encircled. In the first three cases, the dampingDF(ωN) of the VSC input admittance is positive at ωN and therefore for the system
to be stable, the net-damping should be positive. This is confirmed in Figures 5.11(a)-5.11(c) where the systems are already known to be stable and the measured net damping is indeed
Chapter 5. Stability in two-terminal VSC-HVDC systems: frequency-domain analysis
−1.5 −1 −0.5 0
−2 0 2
Real part (pu)
Imaginary part (pu)
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 D ( ω ) at ω N [pu]
Damping of concerned poles concerned poles
(a) Results for Case 1 scenario.
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 −4 −2 0 2 4
Real part (pu)
Imaginary part (pu)
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 D ( ω ) at ω N [pu]
Damping of concerned poles concerned poles
(b) Results for Case 2 scenario.
−1.5 −1 −0.5 0 −1 −0.5 0 0.5 1
Real part (pu)
Imaginary part (pu)
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 D ( ω ) at ω N [pu]
Damping of concerned poles concerned poles
(c) Results for Case 3 scenario.
−0.4 −0.3 −0.2 −0.1 0
−10 0 10
Real part (pu)
Imaginary part (pu)
0 0.005 0.01 0.015 −0.04 −0.02 0 D ( ω ) at ω N [pu]
Damping of concerned poles concerned poles
(d) Results for Case 4 scenario.
Fig. 5.11 Frequency analysis of poorly-damped systems. Four scenarios are examined with a different variable of the system changing in each of them. In the pole movement, ”∗” corresponds to the starting value and ”” to the final value of the variable. The fifth pole associated with the current-controller bandwidthacc is far to the left and is not shown here.
5.6. Stability improvement