#### Mitigating destabilizing effects of line dynamics in droop controlled inverters

Ph.D. Student: Gurupraanesh Raman | Advisor: Jimmy C.-H. Peng | Project Duration: 2018-19

Summary

This work demonstrates that small-signal instability under droop control occurs due to two separate phenomena: P-V/Q-f cross coupling and line dynamics. While the former has been exhaustively studied in the past, the latter has not. We first show that instability still can occur in systems where all cross-coupling has been eliminated, and that this is attributable to the distribution system dynamics. We then analyze such a decoupled system to identify the remaining instability factors, all of which arise from the line dynamics. The distribution system lag factor is found to be the sole factor that needs to be addressed to guarantee stability in decoupled systems. Finally, we propose a general formula-based lead compensator to compensate the distribution system lag. The proposed design requires no hardware modifications, and is demonstrated to guaranteed stability in decoupled systems, and significantly expand the stability region in cross-coupled systems.

A novel approach to study the effect of line dynamics

Prior works have reported that the the use of static power flow models for the interconnection network leads to over-optimistic stability assessment. However, this effect was not analyzed in a formal manner. Here, we developed the so-called fifth-order model for a multi-inverter system that does incorporate line dynamics, while at the same time eliminating P-f/Q-V cross-coupling by adopting the "generalized droop" law. Here, the power measurement is carried out in a rotated measuring frame, whose rotation angle depends on the R/X ratio of the system. We found that such systems can indeed experience instability at higher droop values, despite the removal of the cross-coupling effect. Our analysis of the small-signal model of this system found that there remain some instability factors, those that disappear if the third-order model is adopted, which neglects the line dynamics.

Figure 1. Bode plot of 1/DSLF for various R/X ratios indicating significant phase lag at power frequency.

Figure 2. Bode plots of the original filter F0 and the proposed filter for various R/X ratios. The proposed filter provides phase lead at power frequency.

Our novel approach of examining the line dynamics' effect under generalized droop control meant that we could clearly see what are the instability factors, and how the resulting destabilization can be mitigated without empirically tuned compensators.

We observed three phenomena arising from the line dynamics: distribution system lag factor (DSLF), EM-induced cross-coupling and EM-induced damping. Separating these and analyzing, we observed that the latter two effects have a minor effect on the location of the low-frequency poles, but do not really move the poles towards the right-hand side. In contrast, we found that the DSLF does destabilize the poles, and that if it were unity, the system remains stable for any droop gain.

The destabilization due to the DSLF originates from the phase lag this term contributes to the droop control loop. As seen from Fig. 1, this is significant only at lower R/X ratios, where the time-constant L/R of the distribution lines are higher. This is an interesting result in that it indicates that the line dynamics has an opposite dependence on the R/X ratio compared to P-V/Q-f cross-coupling.

Mitigating the destabilization

To guarantee small-signal stability under generalized droop, we simply multiplied the existing first-order power filter in the droop loop with DSLF. This yields us a modified filter which provides exactly the required amount of phase lead at the power frequency, essentially eliminating the effect of the DSLF. This is demonstrated in Fig. 2. Such a formula-based filter/compensator design is far preferrable to empirical design rules, and makes the process independent of the system topology, inverter ratings and droop coefficients. Given any R/X, the filter can be directly obtained given no further information.

On applying the proposed filter, we observe its effectiveness by plotting its poles. Fig. 3 shows that the low-frequency poles under the original filter were angled right-wards, leading to the possibility of instability as the droop gains are increased. However, for the proposed filter, the poles are oriented left-wards. As the droop gains are increased, the stability actually improves. This reversal in trend is mainly caused by the EM-induced damping, in the absence of which the poles under the proposed filter will lie on the straight line at -1/2T_c, where T_c is the time-constant of the original filter.

Figure 3. Juxtaposition of the system poles under generalized droop control for R/X=1 with the conventional filter (blue x) and the proposed filter (red o). The zoomed version of (a) is shown in (b), focusing on the critical poles

Figure 4. Critical poles of the system with the proposed filter indicating guaranteed stability for (a) variation in kf and (b) variation in R/X ratio.

The impact of canceling the DSLF is evident from the stability regions shown in Figure 5. A stability region is the subset of the droop gain hyperspace where the system is positively damped. In the case of generalized droop, the only instability factor is the distribution system lag. When this is eliminated, the stability region becomes infinite.

Figure 5. Stability region under generalized droop control for various R/X ratios with (a) original filter and (b) the proposed filter (the entire plane is theoretically stable for (b)).

Figure 6. Bode plots of the original filter F0 and the proposed filter for various R/X ratios. The proposed filter provides phase lead at power frequency.

To verify that the line dynamics and cross-coupling are separate and independent instability phenomena, we now take up the conventional droop control, where cross-coupling does exist. For this case, we would expect that the stability will improve significantly for lower R/X ratios, where the line dynamics is the dominant effect. For higher R/X ratios, we expect it to not change very significantly, because the line dynamics effect is very weak, and its mitigation does not at all affect the P-V/Q-f cross-coupling. Figure 6 corroborates these expectations.

The results from Figure 6 offers further insights into which effect- line dynamics or cross-coupling, is dominant for practical systems. Practical distribution systems typically have R/X ratios around 1. While the extensive research focus in the past has been on cross-coupling, the actual dominant effect to address is line dynamics.