A capacitor in series with the inductance of the transmission iine forms a series-resonant circuit with a natural frequency given by
where is the reactance of the capacitor in each phase and X, is the to- tal reactance of the line at the power frequency f. Since the degree of compensation, usually is in the range is usually less than the power frequency, and we say that the system has a subharmonic resonance or "mode." must include the equivalent series reactance of the generators and loads connected at the ends of the line. In practice these have complex frequency-response characteristics, as does the line it- self, and for accurate estimation of the resonance phenomena a more de- tailed circuit model of the power system must often be used.
The first effect of a subharmonic resonance is that during any distur- bance, transient currents are excited at the subharmonic resonant fre- quency These currents are superimposed on the power-frequency component, and are usually damped out within a few cycles by the resis- tances of the line and of the loads and generators connected to it. A n ex- ample is shown Figure 6. In this case the damping is said to be posi- tive, and the subharmonic mode is stable. It should be noted that the subharmonic mode is only one of the natural modes of a power system. Other natural modes have resonant frequencies above the power frequen- cy, and these can be troublesome if they occur at or near integer-order harmonic frequencies when sources of harmonic currents large rectifiers) are connected Chapter In general any disturbance, including any switching operation, will excite all the natural modes of the system, in different degrees. Usually all the resulting transient currents are positively damped, again in varying degrees.
Under certain conditions the subharmonic mode associated with series capacitors can experience a destabilizing influence from polyphase ac rotating machines. In extreme cases it can even become unstable in the absence of corrective measures. The destabilizing influence shows itself as a negative resistance in the equivalent circuit of synchronous and induction machines. If saliency is neglected, the per-phase equivalent cir- cuit of a synchronous machine is of the form shown in Figure 13. The generated (power-frequency) emf has been omitted from this circuit
Stator Total Leakage Resistance
Series Capacitors
FIGURE 13. Simplified of synchronous for subhar-
monic effects.
while we consider the subharmonic frequency alone. Also the field wind- ing has been omitted for simplicity. Suppose that subharmonic currents have been transiently excited by a disturbance in the external system. In general they will be unbalanced between the three phases, but if we resolve them into symmetrical components the positive-sequence com- ponents flowing in the machine stator will set up a magnetic flux which rotates in the same direction as the rotor, but at the angular velocity electrical radians per second. The rotor is rotating at f elec
faster than the subharmonic field. It is said to be slipping relative to this field, the slip being given by
Since
<
the slip is negative, and the rotor behaves much like that of an induction motor running above synchronous speed. The resistance of the (or solid rotor in the case of high-speed synchronous generators) referred to the stator is as in the induction motor equivalent circuit, and this is a negative quantity which contributes nega- tive damping or when added in series with the stator and external system resistances, The machine is therefore capable ofing mechanical energy into electrical energy associated with the subhar- monic mode.
If is very small, as can happen when the degree of series compens is large and approaches then can become large enough overcome the positive resistances in the system. The subharmonic is then unstable, and will grow to dangerous levels of current and vo as a result of the smallest disturbance. This situation rarely arises in tice, but when it does. corrective measures are necessary. These can
7.7. Resonance Effects with Series Capacitors 271 any or all of those which are applied to prevent subsynchronous resonance (discussed below).
The electrical subharmonic natural mode is rarely troublesome except where resonance can occur. As it rotates in the backward direction relative to the rotor and the main field, the subhar- field produces an alternating torque on the rotor at the frequency If this difference-frequency coincides with one of the natural tor- sional resonances of the machine's shaft system, torsional oscillations can be excited. This condition is known as subsynchronous resonance. SSR is a combined natural mode or resonance. Like the purely electrical subharmonic mode. it can be stable or unstable. depend- ing on the degree of damping. Although the negative resistance effect in synchronous machines can have a destabilizing influence. instability of the subsynchronous mode is more likely to be a result of phase shifts in the circuit external to the generator whose shaft is oscillating. The oscil- lation produces a frequency modulation of the power frequency subharmonic and harmonic sidebands, and the subharmonic sidebands may be made unstable by these phase shifts.
The machines which are most susceptible to SSR are large
stage steam turbines, which typically have four or five torsional modes in the frequency range 0-60 Hz. The lowest torsional frequency is the "swing" frequency in which the entire system of turbine cylinders the generator about synchronous speed as one inertia. Torsional resonances at higher frequencies involve the twisting of the shaft in different mode shapes, and resonant frequencies can extend up to hun- dreds of Hz. The damping of these modes is generally extremely
The consequences of an SSR condition can be dangerous in the short term, if the oscillations are unstable and build up sufficiently to break the shaft. But even if the oscillations are relatively well damped, disturbances
(like switching, fault clearing, can use up the fatigue life of shaft. This slow deterioration is called "low-cycle fatigue," and in recent years considerable effort has been made to understand it
The corrective measures for SSR are:
Tripping sections of line, or bypassing series capacitors, using pro- tective relays sensitive to very small incipient levels of subhar- rnonic current.
The installation of special subharmonic filter circuits. These can take the form of blocking filters (parallel-resonance type) in series with the power line; or damping circuits in parallel with the series capacitors.
The use of excitation control (modulation of field current) turbine-generators phased so as to provide increased positive damping at the subharmonic frequency.
272 7.8. Summary
4. The use of a static compensator whose reference voltage is modu- lated with such a phase as to provide increased positive damping at the subharmonic frequency.
Chapter 8
Synchronous Condensers 8.2. Condenser Design Features
0 I I
1950 1960 1970 1980 YEAR
FIGURE 1. Growth in hydrogen-cooled condenser ratings
8.2. CONDENSER DESIGN FEATURES
Functionally, a synchronous condenser is simply a synchronous machine that is brought up to speed and synchronized to the power system. After the unit is synchronized the field is controlled to either generate or absorb reactive power as needed by the power system. The synchronous con- denser falls into the class of active shunt compensators discussed in Chapter 2.
The majority of synchronous condenser installations are outdoor design and operate unattended with automatic controls for startup. shutdown. and on-line monitoring. Historically both air-cooled and hydrogen-cooled condensers have been used extensively; however, nearly of the large sizes in the United States are hydrogen-cooled. There is in addition a large (345 MVA) water-cooled unit in service.'"'
Figure 2 shows a hydrogen-cooled condenser together with major auxiliary components. In addition to the automatic control and protection equipment, the control building houses the excitation control equipment and motor control equipment.
The condenser is contained entirely within a gas-tight enclosure, with no running seals. which makes it inherently suitable for outdoor installa- tion. All leads required outside the shell are through individual bushings.
FIGURE 2. hydrogen-cooled synchronous condenser. Control building and cooling tower background.
The slip rings for excitation power and the starting motor, if one is used,
.
are contained in a separate compartment within the main shell, with an inflatable seal for use when the unit is at rest. This permits changing without the necessity of purging the entire condenser.The condenser pit area directly below the condenser, houses a number of auxiliaries, including the lube oil system, high-pressure bearing lift pump for reducing friction during starting, hydrogen accessory equip- ment, carbon-dioxide purging equipment and the neutral grounding equipment.
Figure 3 shows the saiient-pole type of rotor construction generally used. In addition to providing damping of rotor oscillations, the amor- tisseur bars carry the rotor circulating currents during startup where re- duced voltage starting is used. For that type of starting "complete amor- tisseurs" are required, with interpole connections of the arnortisseur bar groups as shown in Figure 3.
A number of fundamental design advances, including higher operating speeds and higher hydrogen pressures, have accompanied the extension of ratings to the larger sizes. Large condensers of the type shown Fig- ure 2 are typically rated 900 rpm and operate with 30-psig hydrogen pres- sure.
Figure 4 shows the essential elements of the condenser installation. These include connections to the system and auxiliary systems, discussed later. A key element is the excitation control equipment which to a large
FIGURE 3. Synchronous condenser rotor.
AUXILIARY POWER
LUBE OIL
COOLING WATER
HYDROGEN CONTROL
FIGURE 4. Condenser major auxiliary systems
8.3. Basic Electrical Characteristics 277