RESONANT CHARACTERISTICS OF THE ELECTRICAL COMPENSATION PROVIDED BY A TWO-LEVEL SSSC
6.2 Subsynchronous Resonance (SSR)
6.2.1 Introduction
When series capacitance is inserted into a transmission line to increase the power transfer capability, it forms a resonant circuit with the line resistance and inductance which resonates at its natural electrical frequency after a change in system variables. Under such conditions, the interaction between a turbine-generator and a series compensated transmission line may lead to a form of dynamic instability called subsynchronous resonance (SSR) [18]. SSR has been formally defined by the IEEE Subsynchronous Resonance Working Group [81] as
"... encompassing the oscillatory attributes of electrical and mechanical variables associated with turbine-generators when coupled to a series capacitor compensated transmission system... ".
In the presence of SSR the turbine shafts are subjected to large amplitude torque oscillations.
These torque oscillations not only have the potential to cause shaft fatigue and shorten the life of the turbine, they can also cause shaft failure as illustrated by two famous incidents at Mohave Power Station in Nevada, U.S.A. [82]. It is therefore important to understand the effect that an SSSC will have on the transmission system before actual implementation of such a device, even though no physical capacitors are placed in series with the transmission line.
6.2.2 Mechanisms Of SSR
As previously described, when a conventional capacitor is used to compensate an RL transmission line to reduce its series reactance, the transmission line becomes a resonant RLC network. The line therefore exhibits a resonant minimum in its impedance at a natural frequency
!er
given as1
(6.1)
where
fo
XL
=
2nfoLX = 1
c 2rtfoC
is the synchronous frequency of the system;
is the inductive reactance of the line at the system frequency; and
is the capacitive reactance inserted in series with the line at the system frequency.
R
~
o.JU-
j XL'----II- . i X~ -1 z 1---0
1
wC f
XL
= jwL
I Z I
=Ir-R-2-+-(-W-L-_~-1=-)-2
V wC
fo fer
Xc
!"-?"''----~'---:---R IZI
=
R+
j(XL-XC)Fig.6.1: Conventional RLC line impedance elements as afunction offrequency [18].
Fig. 6.1 shows the total magnitude of the RLC line impedance as a function of frequency.
Because the inductive reactance XL increases and capacitive reactance Xc decreases with increasing frequency, there is always one impedance minimum and this minimum value occurs at the natural frequency of oscillation fer (as shown in Equation (6.1)), where the inductive and capacitive reactances are equal and cancel each other out(XTOTAL
=
XL - Xc=
0) leaving a small resistance in the transmission line to limit the resonant frequency current flow. In practice, the compensation ratio _C_X is always less than unity, therefore the resonant frequency, which isXL
dependent on the level of compensation of the line inductance, is lower than the system frequency fo, i.e·feris subsynchronous [18].
There are two mechanisms through which SSR can manifest itself, namely, the induction generator effect and torsional interaction. The induction generator effect is predominantly an electrical resonance, which occurs when the compensation ratio is high and the effective system resistance is too low at the natural frequency fen resulting in the flow of subsynchronous current becoming self-sustaining. The second mechanism, torsional interaction, is a form of self-excitation that occurs due to the interaction between the electrical system and mechanical dynamics of a turbine-generator. This particular mechanism can only occur in a generator with a multi-inertia shaft. In such a turbine-generator system, the turbine stages are connected by torsionally flexible shafts so that there are a number of natural torsional oscillatory modes. Each of these torsional modes has a distinct mechanical natural frequency of oscillationfn. When the series compensated transmission line possesses an electrical resonance.!en it results In aIr-gap fluxes in the generator at complementary frequencies offo ±fer. The electrical and mechanical systems will interact when the subsynchronous complementary frequencyfo -
!er
is close to one of the mechanical resonant modes. This interaction can result in oscillations that become self-excited and the subsynchronous currents are large. In the case where the damping of torsional modes in a turbine-generator is exceptionally low, a phenomenon called shaft torque amplification may occur. This further effect of SSR is capable of producing shaft torques of much larger amplitudes following a system disturbance than those developed in uncompensated transmission lines.In both mechanisms, the existence of physical capacitors is instrumental in introducing an electrical resonance in the transmission line and hence causing SSR. The question remains whether an inverter-based series compensator like the SSSC is able to cause a similar electrical resonance in transmission line. The following subsection reviews the literature concerning this matter.
6.2.3 Literature Review
This section now presents a review of the technical literature and examines the issue of whether or not the inverter-based series compensator is also a candidate for causing electrical resonance in the transmission line and therefore SSR.
Since the solid-state, inverter-based series capacitive compensator was first introduced by Gyugyi [13] to replace conventional capacitors by injecting a controllable synchronous ac voltage in series with the transmission line, various papers [16,26] have stated that these series compensators are immune to network resonances because no physical capacitors are placed on to the
transmission line. It was claimed that since the inverter-based series compensator injects the required quadrature voltage into the line only at the system frequency, it therefore exhibits a theoretically zero impedance at all other frequencies. Hence, with proper implementation, it cannot cause subsynchronous resonance because of the absence of the transmission line impedance minimum. Subsequently, Gyugyi mentioned in [17] that the "series compensation by a synchronous voltage source that can be restricted to the fundamental frequency is superior to that obtained with series capacitive compensation in that it is, with proper implementation, unable to produce undesired electrical resonances with the transmission network, and for this reason it cannot cause subsynchronous resonance." Later on, Gyugyi et al. [21] reiterated that the SSSC's claimed immunity to classical network resonances is one of its advantages. However, no impedance characteristics of the inverter-based series compensator compensated transmission line or any relevant results were presented to validate the above statement.
References [35,83] by Ooi et al. proposed a method of controlling a single inverter to act as an advanced series compensator but the magnitude of compensating reactance injected into the line could not be controlled. Ooi et al. [35,83] considered the potential of transmission line resonance and proposed the inclusion of a notch filter feedback circuit to the series compensator so that it, as an active filter, removes all low order frequencies in the transmission line. However, again, no frequency domain analysis on the transmission line impedance was shown in both papers and therefore such a solution to the problem of SSR has still not been fully evaluated.
While Sen did not include any resonance related topic III his two papers [27,34] on the development of the two-level SSSC and UPFC respectively, Ghosh [55] on the other hand, stated that with these FACTS series compensators "the system operates satisfactorily without any SSR for balanced and unbalanced faults." Despite the unanimous agreement in such claims, little detailed analysis of the resonant characteristics of the inverter-based series compensator has been presented in each case.
Nevertheless, it has been demonstrated [18,20,33] by analysis of transmission line impedance in the frequency domain that an inverter-based series compensator causes a similar resonant impedance effect to that caused by conventional series compensating capacitors, and therefore
"has the potential to cause SSR". However, the subsynchronous frequency at which the resonant minimum occurs in the impedance of the SSSC-compensation is different from that compensated by the conventional capacitors. Reference [33] also stated that even though the inverter-based series compensator can potentially excite SSR, it is "inherently more SSR stable than conventional series compensating capacitors."
Reference [77] by the author described a frequency-domain impedance characteristic study on the two-level SSSC when used to compensate a transmission system based on that in [34]. The conclusion drawn in [77] in regards to SSR is in agreement with [18,33]: the SSSC was shown to cause a resonant minimum in the transmission line impedance similar to that caused by conventional series compensating capacitors, and is therefore "prone to subsynchronous
"
resonance.
The above review shows not a disagreement in the field of SSR, but rather a development in the closer understanding of one aspect of the behaviour of the inverter-based series compensators.
The following section furthers the analysis on a two-level SSSC in the frequency domain to investigate the resonant characteristics of such a device when placed in series with the transmission line of a benchmark SSR study system.