Multimode damping controller

The derivation of the multimode damping controller from the locally measured signals involves estimating the critical oscillation modes accurately in spite of uncertainties in the operating points and parameters of the power system using the modified RLS algorithm described in Section 4.3. Based on the power system configuration, the corresponding phase shift is added to each estimated mode to get the required damping signal component. By using the damping signals from each mode separately, the required active and reactive current references for POD are generated using a proportional controller similar to the discussion in Section 6.3.

is shown in Fig. 6.12, where the frequency estimate of the PLL (ωt1) and the active power flow

(Pt1) at the compensator location are used to set up the POD controller. For the investigated

system, a phase shift of0◦_and90◦_{have been used for the oscillatory mode estimates inω} t1and

Pt1, respectively. The control scheme for the second compensator is set up similarly using the

locally measured signals,ωt2 andPt2.

Fig. 6.12 Block diagram of the multimode POD controller for compensator 1.

By using the estimate of each oscillation mode in the multimode damping controller, the in- jected active and reactive power from the compensators will consist of only the frequency of the oscillation mode to be damped. This decouples the performance of the damping controller on the various modes. As each mode is a global variable in the system dynamics, the use of multiple compensators that are designed to damp a particular oscillation mode results in a net additive damping thus avoiding any negative interaction between the compensators.

6.5.2

Stability analysis

The dynamic model of the system in Fig. 6.11 is implemented to investigate the effectiveness of the independent multimode damping controller. While the full controller in Fig. 6.12 is non- linear, a modal analysis of the whole system can be made by using the RLS algorithm in its steady-state form, similar to the one in (4.33). Thus, the whole system dynamic can be expressed as

∆ ˙x = AG∆x + BG∆T + BC∆u (6.33)

where the state (∆x), mechanical torque (∆T) and control input (∆u) vectors are given by

∆x = ∆ωg1 ∆δg1 ∆ωg2 ∆δg2 ∆ωg3 ∆δg3 T ∆T = ∆Tm1 ∆Tm2 ∆Tm3 T ∆u = id f1 i q f1 idf2 i q f2 T

The matrices AG, BG and BC are calculated from the steady-state operating point and the

injected currents are controlled as in Fig. 6.12. Using (6.33), the performance of the control method for independent damping of the oscillation modes for various cases will be investigated in the following.

As an example, the transient impedance of all generators is chosen to be 0.3 pu and a steady- state operating point with Pg1 = 0.4 pu, Pg2 = −0.5 pu, Pg3 = 0.1 pu; the terminal voltage

for all the generators is set to 1 pu. The compensators are connected at two specific locations as shown in Fig. 6.11. The steady-state reactance of the transmission lines and leakage reactance of the transformers in pu are as shown in the figure. Choosing the inertia constant of the generators asHg1 = Hg2 = 6.5 s and Hg3 = 2.2 s, a small-signal analysis around the selected operating

point results in two oscillation modes with frequencies 0.98 Hz (mode 1) and 1.69 Hz (mode 2) when both compensators are in idle mode. The first mode involves mainly oscillation of Generator 1 against Generator 2, whereas the second mode involves oscillation of Generator 1 and 3 against Generator 2. The second oscillation mode is mainly caused by the mechanical behavior of Generator 3 and therefore the power output of the third generator will comprise mainly of this frequency.

The first test is made to investigate how the damping controller works when each compensator is used separately. The movement of the poles are plotted in Fig. 6.13 when the gains of the first compensator change from zero to [KP1 = KQ1 = KP2 = KQ2 = −0.30] whereas Fig. 6.14

shows the movement of poles when the gains of the second compensator change from zero to [KP1 = KQ1= −0.24 and KP2= KQ2 = −0.10]. The results show that controlling only mode

1 results in positive damping to mode 1 without affecting the damping of mode 2. Similarly, controlling only mode 2 results in positive damping to mode 2 without affecting the damping of mode 1. This confirms the validity of the controller to provide damping at the critical oscillation mode of interest without affecting the system damping at the other mode.

Fig. 6.13 Movement of oscillatory mode poles using compensator 1 when controlling mode 1 oscillation only (plot (a)), mode 2 oscillation only (plot (b)) and both modes (plot (c)); Black curves represent mode 1 poles and gray curves represent mode 2 poles; Poles start at ’_{◦’ and move} toward ’⊲’.

Fig. 6.14 Movement of oscillatory mode poles using compensator 2 when controlling mode 1 oscillation only (plot (a)), mode 2 oscillation only (plot (b)) and both modes (plot (c)); Black curves represent mode 1 poles and gray curves represent mode 2 poles; Poles start at ’_{◦’ and move} toward ’⊲’.

Even if the same control method is used for the two compensators and their gains are adjusted to obtain similar damping for the previous results, it should be observed that the impact of each compensator to damp a particular mode differs, depending on the observability and con- trollability of the oscillation mode at the compensator location. Another set of tests is made to investigate the damping performance of the controller when the two compensators are active at the same time with the compensator gains chosen similar to the results in Figs. 6.13 and 6.14. It is possible to see from the results in Fig. 6.15 that independent damping of each mode is achieved in this case as well. It is also possible to see that the action of the two compensators adds up in increasing the damping of the specific oscillation mode, i.e. the two compensators perform well when operating separately or together without any risk of negative interaction. With the proposed method and by using each compensator to control the mode where the ob- servability and controllability of that mode is higher at the connection point, the use of multiple compensators can maximize the damping that can be provided to the system. In the case of active power use for POD, this could mean that the amount of total energy needed to damp an oscillation mode can be reduced by using distributed compensators.

Fig. 6.15 Movement of oscillatory mode poles using compensator 1 and compensator 2 when controlling mode 1 oscillation only (plot (a)), mode 2 oscillation only (plot (b)) and both modes (plot (c)); Black curves represent mode 1 poles and gray curves represent mode 2 poles; Poles start at ’_◦’ and move toward ’⊲’.

Finally, the damping analysis is made assuming an error of 0.2 Hz in the mode frequencies considered in the control algorithm to represent a lack of full knowledge of the system para- meters. For this, a mode frequency of 1.18 Hz and 1.89 Hz is assumed instead of the actual values of 0.98 Hz and 1.69 Hz for oscillation mode 1 and oscillation mode 2, respectively. The performance of the control method is summarized in Figs. 6.16 - 6.17 when using Compen- sator 2. In this case, the movement of the poles is plotted when the gains of Compensator 2 are changed from zero to [KP1 = KQ1 = KP2 = KQ2 = −0.30]. It can be observed by com-

paring the two figures that the damping performance of each mode is reduced and interactions between the modes arise that could lead to a negative damping of the uncontrolled mode as in Fig. 6.17 (plot (a) gray curves). To avoid this, a mode frequency adaptation mechanism as de- scribed in Fig. 4.12 in Section 4.3.3 is necessary and this will be verified in Section 7.2.2 using time-domain simulations.

Fig. 6.16 Movement of oscillatory mode poles using compensator 2 and with accurate knowledge of mode frequencies when controlling mode 1 oscillation only (plot (a)), mode 2 oscillation only (plot (b)) and both modes (plot (c)); Black curves represent mode 1 poles and gray curves represent mode 2 poles; Poles start at ’_{◦’ and move toward ’⊲’.}

Fig. 6.17 Movement of oscillatory mode poles using compensator 2 and with inaccurate knowledge of mode frequencies when controlling mode 1 oscillation only (plot (a)), mode 2 oscillation only (plot (b)) and both modes (plot (c)); Black curves represent mode 1 poles and gray curves represent mode 2 poles; Poles start at ’◦’ and move toward ’⊲’.