Q8Economic Operation of Power System

(A) Effect of Excitation Change and Load Sharing During Parallel Operation

Effect of excitation change: when two or more generators are operating in parallel (whether on an isolated bus with other generators, or connected to a strong/infinite bus), increasing one generator's field excitation current primarily increases its own reactive power (lagging kVAR) output, without significantly changing the total active power it supplies (provided its prime-mover/governor setting is unchanged) - this is because, for a generator connected to a bus of essentially fixed voltage and frequency, increasing excitation primarily increases the generator's internal EMF magnitude (affecting the reactive-power-determining voltage-magnitude relationship) without directly affecting the power angle (which is what determines active power transfer, and which is instead controlled by the prime-mover input). If excitation is increased on one unit among several operating in parallel without any redistribution among the others, that unit will absorb a disproportionately larger share of the total reactive load (operating at a lower, more lagging power factor), while other parallel units correspondingly reduce their own reactive contribution (potentially even being driven toward a leading power factor if the excitation change is large enough), all while the total active power shared among the units remains essentially unchanged, unless the units' governor settings are also separately adjusted.

Load sharing and load current sharing: the sharing of total system active power load among parallel-connected generators is governed by each unit's governor droop characteristic and speed-changer setting (exactly as discussed in the corresponding companion economic-operation subject's droop-sharing problems), with each unit's active power output determined by where its own droop line intersects the common operating frequency. The sharing of reactive power load (and hence the sharing of total load CURRENT, since load current has both active and reactive components) among parallel units is similarly governed by each unit's excitation/AVR droop characteristic (a similar voltage-versus-reactive-power droop relationship, analogous to the frequency-versus-active-power droop of the governor) - proper reactive power/load-current sharing among parallel generators requires each unit's AVR to be configured with an appropriate reactive-droop (or cross-current-compensation) characteristic, since without such a deliberate droop characteristic, small differences in generator terminal voltage setpoints among parallel units can otherwise cause large, poorly-controlled circulating reactive currents to flow between the units even when their total combined reactive output correctly matches the system's total reactive demand.

(B) Operating Characteristics of Cylindrical Alternator Rotor

A cylindrical (round, non-salient-pole) rotor is a smooth, cylindrical rotor construction used predominantly in high-speed synchronous generators (turbo-alternators driven by steam or gas turbines, typically operating at 3000 or 3600 rpm for 2-pole or, less commonly, 4-pole machines), with the field winding distributed in slots machined into the otherwise smooth cylindrical rotor surface, in contrast to the distinctly projecting, salient poles used in lower-speed hydro-generator rotor construction.

Uniform air gap and its consequences: since the cylindrical rotor presents an essentially uniform, constant air-gap length to the stator at every angular position around its periphery (unlike a salient-pole rotor, whose air gap varies substantially between the pole-face regions and the inter-polar gaps), the cylindrical-rotor machine's magnetic reluctance is essentially uniform in all directions, meaning it has only a single synchronous reactance value (Xs) applicable regardless of the specific relative angular position between the rotor's field axis and the stator's armature-reaction field axis - this is in direct contrast to a salient-pole machine, which exhibits two distinct reactance values (direct-axis reactance Xd and quadrature-axis reactance Xq, differing because of the non-uniform air gap), requiring a more complex two-reactance (two-axis) theory for accurate performance analysis.

Power-angle characteristic: this single-reactance property gives the cylindrical-rotor machine a simpler power-angle relationship, P=(EV/Xs)sin(delta), a pure sinusoidal function of the power angle delta with a single maximum at delta=90 degrees, whereas a salient-pole machine's power-angle characteristic includes an additional reluctance-power term (proportional to sin(2*delta)) arising from its differing d-axis and q-axis reactances, giving it a somewhat different (though qualitatively similar) power-angle curve shape with its maximum occurring at some angle somewhat less than 90 degrees.

Voltage waveform quality: the cylindrical rotor's uniform air gap and distributed field winding (rather than concentrated field coils on projecting poles) together produce a smoother, more nearly sinusoidal air-gap flux distribution, and hence a lower-harmonic-content (cleaner) output voltage waveform, compared to a salient-pole machine, an additional advantage of the cylindrical-rotor construction particularly valued in large turbo-alternators supplying power directly to extensive transmission networks, where waveform quality (low harmonic distortion) is an important power-quality consideration. The cylindrical rotor's smooth, robust, mechanically-symmetric construction is also specifically well suited to withstanding the very high centrifugal stresses encountered at the high rotational speeds (3000/3600 rpm) typical of steam/gas-turbine-driven generators, which a projecting-pole (salient-pole) rotor construction could not reliably withstand at such speeds, explaining why cylindrical-rotor construction is essentially universal for high-speed turbo-alternators, while salient-pole construction remains standard for lower-speed hydro-generators, whose lower rotational speed does not impose the same severe mechanical constraint.

It is useful to note the qualitative distinction between excitation-driven and governor-driven load sharing more explicitly: increasing a generator's field excitation while it remains connected to a strong (infinite) bus does not increase its active power output (since active power delivered is set by the mechanical input from the prime mover), but it does increase the machine's internal EMF magnitude, causing it to supply more reactive power (operate at a more lagging power factor, over-excited) to the system; conversely, under-exciting the machine causes it to absorb reactive power from the system (operate leading, under-excited), a behaviour exploited operationally when a synchronous generator or synchronous condenser is used specifically for reactive power / voltage support purposes on the system rather than primarily for active power (real energy) supply.

The cylindrical-rotor (round-rotor, non-salient-pole) alternator construction, typically used for high-speed steam and gas turbine-driven generators (as opposed to the salient-pole construction preferred for lower-speed hydro generators), possesses a uniform air-gap and hence essentially equal direct-axis and quadrature-axis synchronous reactances (Xd is approximately equal to Xq), which considerably simplifies its steady-state power-angle characteristic compared to the salient-pole machine: the well-known non-salient-pole power equation P = (EV/Xs)sin(delta) contains only this single fundamental sine term (dependent solely on the excitation EMF E, terminal voltage V, synchronous reactance Xs, and power angle delta), without the additional double-angle reluctance-power term (proportional to sin(2*delta)) that appears in the salient-pole machine's power-angle equation due to its saliency-induced reactance asymmetry (Xd different from Xq).

This simpler power-angle relationship for the cylindrical-rotor machine also yields a correspondingly simple maximum steady-state power transfer limit, occurring at a power angle of exactly delta=90 degrees (where sin(delta)=1, giving Pmax = E*V/Xs), beyond which the machine loses synchronism with the system (pulls out of step) - this maximum power limit, together with the machine's excitation limits (both an upper limit set by rotor winding thermal capability and a lower limit set by steady-state stability and under-excitation protection considerations), together define the practical steady-state operating envelope (commonly displayed as a capability curve) within which a cylindrical-rotor alternator can be safely and stably operated in parallel with the rest of the power system.