PSSs in Multi-Machine Applications
9.4 Determination of the PSS parameters based on the P-Vr approach with speed perturbations as the stabilizing signal
9.4.2 Transfer function of the PSS of generator i in a multi-machine system The basic concepts for the determination of the parameters of a PSS in a single machine sys-
configurations. Consequently, in multi-machine systems, individual PSS designs that are based on the synthesized P-Vr transfer function using the methodology adopted in Section 5.10 are also robust over a wide range of operating conditions. Typically, this applies for generator real pow- er outputs exceeding 0.5 pu. An examination of Figures 5.21, 5.22 and Tables 5.5 and 5.6 reveals that the mode shifts are essentially real, an observation which supports the above statement.
The robustness and application of the P-Vr characteristic has been demonstrated and veri- fied for generators on very large systems [20].
For the multi-machine system the P-Vr characteristics of each generator, , are calculated in a similar fashion to that for the single- machine infinite-bus system, except that the characteristics are calculated for the entire net- work with the shaft dynamics of all machines disabled. The calculation is similar to that de- scribed in Section 5.10.3 in which rows and columns of the A, B and C matrices associated with the speed states in the states equations (3.9) are eliminated; the D matrix is usually a null matrix. The relationship between perturbations in electric power (or torque) as the output quantity and voltage reference as the input quantity can then be formed, and the frequency response evaluated for the set of encompassing operating conditions and over the range of modal frequencies.
The derivation of the synthesized transfer function,
, (9.10) which is selected from the family of P-Vr frequency response characteristics as the most suitable basis for the tuning of the PSS, has been covered in Section 5.10.6.
9.4.2 Transfer function of the PSS of generator i in a multi-machine system
Sec. 9.4 PSS parameters based on the P-Vr approach 467 where is the damping gain; is the PSS compensation block; and
are the transfer functions of the washout and low-pass filters, respectively.
Figure 9.10 Model of generator i fitted with a PSS in a multi-machine power system As has been outlined earlier in Section 5.14:
1. The aim of the tuning procedure is to introduce on the generator shaft a damping torque (a torque proportional to machine speed); this causes the modes of rotor oscil- lation to be shifted directly to the left1 in the complex s-plane.
2. The compensation transfer function is tuned to achieve the desired left-shift in the complex s-plane of the relevant modes of rotor oscillation.
3. The damping gain (on machine MVA rating) of the PSS determines the extent of the left-shift.
Based on item 1, the ideal transfer function between speed and the electrical damping torque perturbations due to the action of the PSS i over the range of modal frequen- cies should ideally be:
, (9.12)
where is a damping torque coefficient and is a real number (p.u. on generator MVA rat- ing). The transfer function compensates in magnitude as well as phase for the synthe-
1. By ‘direct left-shift’ is implied that the mode shift is , . As explained in Chapter 13, deviations from the ‘direct left-shift’ of modes are mainly due to interactions between multi-machine PSSs and non-real generator participation factors.
ki Gci s GWi s GLPi s
OP WE R YS TS ME
/(sMi)
o/s
AVRi Di
i
Pmi
i Pei
Vri
Vti
Pdi
PSSi
Machine i
Vsi
–j0 0 Gci s
ki
i
Pei
Pei s PSSi = Deii s Dei
Gci s
sized P-Vr transfer function of machine i, , defined in (9.10). With rotor speed being used as the input signal to the PSS, whose output is , the expression (9.12) for
can also be expressed in terms of the P-Vr and PSS transfer functions as:
; (9.13)
hence, rearranging (9.13), we find
. (9.14)
It follows from an examination of (9.14) that
and , (9.15)
Note from (9.15) that is a damping torque coefficient. Assuming that the synthesized transfer function is of the general form
, (9.16)
then, from (9.15), the compensation block transfer function is
, (9.17)
where are parameters determined from the synthesized P-Vr characteristic in the tuning procedure. Note that in the form of (9.16) (i) real and complex zeros can be accommodated in the synthesized P-Vr transfer function; (ii) the coefficient of
is unity.
Substitution of (9.17) in (9.11), and incorporating the washout and low pass filters, yields the PSS transfer function:
. (9.18) For generator i, is the time constant of the washout filter; are the time con- stants of the low-pass filter which may be added (i) to ensure is proper, (ii) to mit- igate against excitation of the torsional modes of the rotating turbine/generator/exciter shaft system.
HPVrS
i s
Vsi s Dei
Dei Pei s
Vsi s
--- Vsi s
i s ---
HPVrS
i s kiGci s
= =
kiGci s
Dei HPVrS
i s
=
ki = Dei Gci s 1 HPVrS
i s
= ki
HPVrS
i s HPVrS
i s kci 1 sTb1
+ i
1+c1is c+ 2is2
1 sTa1
+ i
---
=
Gci s 1 kci
--- 1+c1is c+ 2is2 1 sTa1 + i
1 sTb1
+ i
---
=
kci c1i c2i Ta1
i · Tb1
i
s0
HPSSi s ki sTWi 1+sTWi --- 1
kci
--- 1+c1is c+ 2is2 1 sTa1 + i
1 sTb1
+ i
---
1
1+sT1i
1+sT2i
---
=
TWi T1iT2i
HPSSi s
Sec. 9.5 Synchronising and damping torque coefficients 469 Earlier the gain has been referred to as the ‘damping gain’ of the PSS. The gain is the DC gain of P-Vr characteristic of generator i. If the washout filter is ignored the DC gain of the PSS transfer function is ; conventionally this is referred to as the ‘PSS Gain’. (How- ever, the PSS gain has been attributed little meaning because the significance of the gain has not been recognized.)
Note, assuming that the synthesized P-Vr characteristic for generator i closely matches that for the selected operating condition, , it follows that
, (9.19)
(i.e. equal to the damping gain) over the modal frequency range1. In the next chapter, by examining the damping torque coefficient, this result will be used to confirm that the de- signed damping gain ki of PSS i is, in fact, achieved (see Section 10.6).
In the multi-machine PSS tuning methodology the PSS is designed not only to swamp any negative (destabilizing) inherent damping torque coefficients on that machine over the range of frequencies of the rotor modes, but also to provide sufficient damping so that the asso- ciated damping criteria of the multi-machine system are satisfied [9], [13]. These issues, to- gether with the contribution to damping by stabilizers installed on FACTS devices, are considered in a later chapters.
Examples of the application of the P-Vr approach to the tuning of PSSs in multi-machine power systems have been presented in several publications [9], [13], [16], [17], [18] and [19].
In Chapter 10 the tuning the PSSs of generators in an inherently unstable 14-generator, mul- ti-machine power system is described. Based on this system, the features of the PSS tuning technique discussed above are illustrated through an example for which the complete system data is provided.