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### 5725ch4

1. 1. 4 Large and Medium Power Synchronous Generators: Topologies and Steady State 4.1 Introduction ........................................................................4-2 4.2 Construction Elements .......................................................4-2 The Stator Windings 4.3 Excitation Magnetic Field...................................................4-8 4.4 The Two-Reaction Principle of Synchronous Generators..........................................................................4-12 4.5 The Armature Reaction Field and Synchronous Reactances ..........................................................................4-14 4.6 Equations for Steady State with Balanced Load .............4-18 4.7 The Phasor Diagram.........................................................4-21 4.8 Inclusion of Core Losses in the Steady-State Model .................................................................................4-21 4.9 Autonomous Operation of Synchronous Generators..........................................................................4-26 The No-Load Saturation Curve: E1(If ); n = ct., I1 = 0 • The Short-Circuit Saturation Curve I1 = f(If ); V1 = 0, n1 = nr = ct. • Zero-Power Factor Saturation Curve V1(IF); I1 = ct., cosϕ1 = 0, n1 = nr • V1 – I1 Characteristic, IF = ct., cosϕ1 = ct., n1 = nr = ct. 4.10 Synchronous Generator Operation at Power Grid (in Parallel) ........................................................................4-37 The Power/Angle Characteristic: Pe (δV) • The V-Shaped Curves: I1(IF), P1 = ct., V1 = ct., n = ct. • The Reactive Power Capability Curves • Deﬁning Static and Dynamic Stability of Synchronous Generators 4.11 Unbalanced-Load Steady-State Operation ......................4-44 4.12 Measuring Xd, Xq, Z–, Z0 ...................................................4-46 4.13 The Phase-to-Phase Short-Circuit ...................................4-48 4.14 The Synchronous Condenser ...........................................4-53 4.15 Summary............................................................................4-54 References .....................................................................................4-56 4-1© 2006 by Taylor & Francis Group, LLC
2. 2. 4-2 Synchronous Generators4.1 IntroductionBy large powers, we mean here powers above 1 MW per unit, where in general, the rotor magnetic ﬁeldis produced with electromagnetic excitation. There are a few megawatt (MW) power permanent magnet(PM)-rotor synchronous generators (SGs). Almost all electric energy generation is performed through SGs with power per unit up to 1500 MVAin thermal power plants and up to 700 MW per unit in hydropower plants. SGs in the MW and tenthof MW range are used in diesel engine power groups for cogeneration and on locomotives and on ships.We will begin with a description of basic conﬁgurations, their main components, and principles ofoperation, and then describe the steady-state operation in detail.4.2 Construction ElementsThe basic parts of an SG are the stator, the rotor, the framing (with cooling system), and the excitationsystem. The stator is provided with a magnetic core made of silicon steel sheets (generally 0.55 mm thick) inwhich uniform slots are stamped. Single, standard, magnetic sheet steel is produced up to 1 m in diameterin the form of a complete circle (Figure 4.1). Large turbogenerators and most hydrogenerators have statorouter diameters well in excess of 1 m (up to 18 m); thus, the cores are made of 6 to 42 segments percircle (Figure 4.2).FIGURE 4.1 Single piece stator core. mp b a a Stator a segmentFIGURE 4.2 Divided stator core made of segments.© 2006 by Taylor & Francis Group, LLC
3. 3. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-3 The stator may also be split radially into two or more sections to allow handling and permit transportwith windings in slots. The windings in slots are inserted section by section, and their connection isperformed at the power plant site. When the stator with Ns slots is divided, and the number of slot pitches per segment is mp, the numberof segments ms is such that N s = ms ⋅ m p (4.1) Each segment is attached to the frame through two key-bars or dove-tail wedges that are uniformlydistributed along the periphery (Figure 4.2). In two successive layers (laminations), the segments areoffset by half a segment. The distance between wedges b is as follows: b = m p / 2 = 2a (4.2)This distance between wedges allows for offsetting the segments in subsequent layers by half a segment.Also, only one tool for stamping is required, because all segments are identical. To avoid winding damagedue to vibration, each segment should start and end in the middle of a tooth and span over an evennumber of slot pitches. For the stator divided into S sectors, two types of segments are usually used. One type has mp slotpitches, and the other has np slot pitches, such that Ns = Km p + n p ; n p < m p ; m p = 6 − 13 (4.3) SWith np = 0, the ﬁrst case is obtained, and, in fact, the number of segments per stator sector is an integer.This is not always possible, and thus, two types of segments are required. The offset of segments in subsequent layers is mp/2 if mp is even, (mp ± 1)/2 if mp is odd, and mp/3if mp is divisible by three. In the particular case that np = mp/2, we may cut the main segment in twoto obtain the second one, which again would require only one stamping tool. For more details, seeReference [1]. The slots of large and medium power SGs are rectangular and open (Figure 4.3a). The double-layer winding, usually made of magnetic wires with rectangular cross-section, is “kept”inside the open slot by a wedge made of insulator material or from a magnetic material with a lowequivalent tangential permeability that is μr times larger than that of air. The magnetic wedge may bemade of magnetic powders or of laminations, with a rectangular prolonged hole (Figure 4.3b), “gluedtogether” with a thermally and mechanically resilient resin.4.2.1 The Stator WindingsThe stator slots are provided with coils connected to form a three-phase winding. The winding of eachphase produces an airgap ﬁxed magnetic ﬁeld with 2p1 half-periods per revolution. With Dis as theinternal stator diameter, the pole pitch τ, that is the half-period of winding magnetomotive force (mmf),is as follows: τ = πDis / 2 p1 (4.4)The phase windings are phase shifted by (2/3)τ along the stator periphery and are symmetric. The averagenumber of slots per pole per phase q is© 2006 by Taylor & Francis Group, LLC
4. 4. 4-4 Synchronous Generators Slot linear Single (tooth insulation) turn coil Upper 2 turn coil Inter layer layer coil insulation Lower Stator layer coil open slot Flux barrier Elastic Magnetic wedge strip Elastic strip Magnetic wedge Wos (b) (a)FIGURE 4.3 (a) Stator slotting and (b) magnetic wedge. τ N S N S A XFIGURE 4.4 Lap winding (four poles) with q = 2, phase A only. Ns q= (4.5) 2 p1 ⋅ 3The number q may be an integer, with a low number of poles (2p1 < 8–10), or it may be a fractionarynumber: q = a+b/c (4.6)Fractionary q windings are used mainly in SGs with a large number of poles, where a necessarily lowinteger q (q ≤ 3) would produce too high a harmonics content in the generator electromagnetic ﬁeld (emf). Large and medium power SGs make use of typical lap (multiturn coil) windings (Figure 4.4) or ofbar-wave (single-turn coil) windings (Figure 4.5). The coils of phase A in Figure 4.4 and Figure 4.5 are all in series. A single current path is thus available(a = 1). It is feasible to have a current paths in parallel, especially in large power machines (line voltageis generally below 24 kV). With Wph turns in series (per current path), we have the following relationship: W ph ⋅ a Ns = 3 (4.7) ncwith nc equal to the turns per coil.© 2006 by Taylor & Francis Group, LLC
5. 5. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-5 τ N S N S A XFIGURE 4.5 Basic wave-bar winding with q = 2, phase A only. The coils may be multiturn lap coils or uniturn (bar) type, in wave coils. A general comparison between the two types of windings (both with integer or fractionary q) revealsthe following: • The multiturn coils (nc > 1) allow for greater ﬂexibility when choosing the number of slots Ns for a given number of current paths a. • Multiturn coils are, however, manufacturing-wise, limited to 0.3 m long lamination stacks and pole pitches τ < 0.8–1 m. • Multiturn coils need bending ﬂexibility, as they are placed with one side in the bottom layer and with the other one in the top layer; bending needs to be done without damaging the electric insulation, which, in turn, has to be ﬂexible enough for the purpose. • Bar coils are used for heavy currents (above 1500 A). Wave-bar coils imply a smaller number of connectors (Figure 4.5) and, thus, are less costly. The lap-bar coils allow for short pitching to reduce emf harmonics, while wave-bar coils imply 100% average pitch coils. • To avoid excessive eddy current (skin) effects in deep coil sides, transposition of individual strands is required. In multiturn coils (nc ≥ 2), one semi-Roebel transposition is enough, while in single- bar coils, full Roebel transposition is required. • Switching or lightning strokes along the transmission lines to the SG produce steep-fronted voltage impulses between neighboring turns in the multiturn coil; thus, additional insulation is required. This is not so for the bar (single-turn) coils, for which only interlayer and slot insulation are provided. • Accidental short-circuit in multiturn coil windings with a ≥ 2 current path in parallel produce a circulating current between current paths. This unbalance in path currents may be sufﬁcient to trip the pertinent circuit balance relay. This is not so for the bar coils, where the unbalance is less pronounced. • Though slightly more expensive, the technical advantages of bar (single-turn) coils should make them the favorite solution in most cases. Alternating current (AC) windings for SGs may be built not only in two layers, but also in one layer.In this latter case, it will be necessary to use 100% pitch coils that have longer end connections, unlessbar coils are used. Stator end windings have to be mechanically supported so as to avoid mechanical deformation duringsevere transients, due to electrodynamic large forces between them, and between them as a whole andthe rotor excitation end windings. As such forces are generally radial, the support for end windingstypically looks as shown in Figure 4.6. Note that more on AC winding speciﬁcs are included in Chapter7, which is dedicated to SG design. Here, we derive only the fundamental mmf wave of three-phasestator windings. The mmf of a single-phase four-pole winding with 100% pitch coils may be approximated with a step-like periodic function if the slot openings are neglected (Figure 4.7). For the case in Figure 4.7 with q =2 and 100% pitch coils, the mmf distribution is rectangular with only one step per half-period. Withchorded coils or q > 2, more steps would be visible in the mmf. That is, the distribution then better© 2006 by Taylor & Francis Group, LLC
6. 6. 4-6 Synchronous Generators Stator frame plate Pressure ﬁnger on stator stack teeth Resin bracket Resin rings in segments End windings Stator core Shaft directionFIGURE 4.6 Typical support system for stator end windings. nc nc nc nc nc nc nc nc nc nc nc nc nc nc nc nc AA AA AA AA AA AA AA AA 1 2 7 8 13 14 19 20 FA(x) 2ncIA τ x/τFIGURE 4.7 Stator phase mmf distribution (2p = 4, q = 2).approximates a sinusoid waveform. In general, the phase mmf fundamental distribution for steady statemay be written as follows: π F1A ( x , t ) = F1m ⋅ cos x ⋅ cos ω1t (4.8) τ W1 K W 1 I F1m = 2 2 (4.9) πp1where W1 = the number of turns per phase in series I = the phase current (RMS) p1 = the number of pole pairs KW1 = the winding factor: sin π / 6 ⎛ y π⎞ K W1 = ⋅ sin ⎜ (4.10) q ⋅ sin π / 6q ⎝ τ 2⎟ ⎠with y/τ = coil pitch/pole pitch (y/τ > 2/3).© 2006 by Taylor & Francis Group, LLC
7. 7. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-7 Equation 4.8 is strictly valid for integer q. An equation similar to Equation 4.8 may be written for the νth space harmonic: π FυA ( x , t ) = Fυm cos υ x cos ( ω1t ) τ 2 2 W1 K W ν I Fνm = (4.11) πp1 ν sin νπ / 6 y νπ KWν = ⋅ sin (4.12) q ⋅ sin ( νπ / 6q ) τ 2 Phase B and phase C mmf expressions are similar to Equation 4.8 but with 2π/3 space and time lags.Finally, the total mmf (with space harmonics) produced by a three-phase winding is as follows [2]: 3W1I 2 K W υ ⎡ ⎛ υπ 2π ⎞ ⎛ υπ 2π ⎞ ⎤ Fυ ( x , t ) = ⎢ K BI cos ⎜ − ω1t − ( υ − 1) ⎟ − K BII cos ⎜ + ω1t − ( υ + 1) ⎟ ⎥ (4.13) πp1υ ⎣ ⎝ τ 3 ⎠ ⎝ τ 3 ⎠⎦with sin ( υ − 1) π K BI = 3 ⋅ sin ( υ − 1) π / 3 (4.14) sin ( υ + 1) π K BII = 3 ⋅ sin ( υ + 1) π / 3 Equation 4.13 is valid for integer q. For ν = 1, the fundamental is obtained. Due to full symmetry, with q integer, only odd harmonics exist. For ν = 1, KBI = 1, KBII = 0, so themmf fundamental represents a forward-traveling wave with the following peripheral speed: dx τω1 = = 2τf 1 (4.15) dt π The harmonic orders are ν = 3K ± 1. For ν = 7, 13, 19, …, dx/dt = 2τf1/ν and for ν = 5, 11, 17, …,dx/dt = –2τf1/ν. That is, the ﬁrst ones are direct-traveling waves, while the second ones are backward-traveling waves. Coil chording (y/τ < 1) and increased q may reduce harmonics amplitude (reduced Kwν),but the price is a reduction in the mmf fundamental (KW1 decreases). The rotors of large SGs may be built with salient poles (for 2p1 > 4) or with nonsalient poles (2p1 =2, 4). The solid iron core of the nonsalient pole rotor (Figure 4.8a) is made of 12 to 20 cm thick (axially)rolled steel discs spigoted to each other to form a solid ring by using axial through-bolts. Shaft ends areadded (Figure 4.9). Salient poles (Figure 4.8b) may be made of lamination packs tightened axially bythrough-bolts and end plates and ﬁxed to the rotor pole wheel by hammer-tail key bars. In general, peripheral speeds around 110 m/sec are feasible only with solid rotors made by forgedsteel. The ﬁeld coils in slots (Figure 4.8a) are protected from centrifugal forces by slot wedges that aremade either of strong resins or of conducting material (copper), and the end-windings need bandages.© 2006 by Taylor & Francis Group, LLC
8. 8. 4-8 Synchronous Generators d d N Pole body Damper cageDamper cage q N Field coil q Field coil S S Pole wheel Solid rotor core (spider) Shaft d N q N d Shaft S S N 2p1 = 2 q q S d 2p1 = 8 (a) (b)FIGURE 4.8 Rotor conﬁgurations: (a) with nonsalient poles 2p1 = 2 and (b) with salient poles 2p1 = 8. Through Rolled bolts steel disc Stub shaft SpigotFIGURE 4.9 Solid rotor. The interpole area in salient pole rotors (Figure 4.8b) is used to mechanically ﬁx the ﬁeld coil sidesso that they do not move or vibrate while the rotor rotates at its maximum allowable speed. Nonsalient pole (high-speed) rotors show small magnetic anisotropy. That is, the magnetic reluctanceof airgap along pole (longitudinal) axis d, and along interpole (transverse) axis q, is about the same,except for the case of severe magnetic saturation conditions. In contrast, salient pole rotors experience a rather large (1.5 to 1 and more) magnetic saliency ratiobetween axis d and axis q. The damper cage bars placed in special rotor pole slots may be connectedtogether through end rings (Figure 4.10). Such a complete damper cage may be decomposed in twoﬁctitious cages, one with the magnetic axis along the d axis and the other along the q axis (Figure 4.10),both with partial end rings (Figure 4.10).4.3 Excitation Magnetic FieldThe airgap magnetic ﬁeld produced by the direct current (DC) ﬁeld (excitation) coils has a circumferentialdistribution that depends on the type of the rotor, with salient or nonsalient poles, and on the airgapvariation along the rotor pole span. For the time being, let us consider that the airgap is constant underthe rotor pole and the presence of stator slot openings is considered through the Carter coefﬁcient KC1,which increases the airgap [2]:© 2006 by Taylor & Francis Group, LLC
9. 9. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-9 d d d qr q 2p = 4 +FIGURE 4.10 The damper cage and its d axis and q axis ﬁctitious components. τs K C1 ≈ > 1, τ s − stator _ slot _ pitch (4.16) τs − γ1 g ⎡ ⎛ Wos ⎞ ⎤ ⎢ ⎜ ⎟ 2 ⎥ ⎝ g ⎠ ⎛W ⎞ γ1 = ⎢ − ln 1 + ⎜ os ⎟ ⎥ 4 (4.17) π ⎢ ⎛W ⎞ ⎝ g ⎠ ⎥ ⎢ tan ⎜ os ⎟ ⎥ ⎢ ⎣ ⎝ g ⎠ ⎥ ⎦with Wos equal to the stator slot opening and g equal to the airgap. The ﬂux lines produced by the ﬁeld coils (Figure 4.11) resemble the ﬁeld coil mmfs FF(x), as the airgapunder the pole is considered constant (Figure 4.12). The approximate distribution of no-load or ﬁeld-winding-produced airgap ﬂux density in Figure 4.12 was obtained through Ampere’s law. For salient poles: μ0W f I f τp BgFm = , for : x < (4.18) K c g (1 + K S 0 ) 2and BgFm = 0 otherwise (Figure 4.12a). WF turns/coil/pole WCF turns/coil/(slot) g τp 2p1 = 4 2p1 = 2FIGURE 4.11 Basic ﬁeld-winding ﬂux lines through airgap and stator.© 2006 by Taylor & Francis Group, LLC
10. 10. 4-10 Synchronous Generators FFm = WFIF Airgap ﬂux density Field BgFm winding mmf/pole τP τ (a) BgFm FFm = (ncp WCFIF)/2 τp τ (b)FIGURE 4.12 Field-winding mmf and airgap ﬂux density: (a) salient pole rotor and (b) nonsalient pole rotor. In practice, BgFm = 0.6 – 0.8 T. Fourier decomposition of this rectangular distribution yields the following: π BgF υ ( x ) = K F υ ⋅ BgFm cos υ x ; υ = 1, 3, 5,... (4.19) τ 4 τp π K Fυ = sin υ (4.20) π τ 2 Only the fundamental is useful. Both the fundamental distribution (ν = 1) and the space harmonicsdepend on the ratio τp/τ (pole span/pole pitch). In general, τp/τ ≈ 0.6–0.72. Also, to reduce the harmonicscontent, the airgap may be modiﬁed (increased), from the pole middle toward the pole ends, as an inversefunction of cos πx/τ: −τ p τp g( x) = g , for : <x< (4.21) π 2 2 cos x τ In practice, Equation 4.21 is not easy to generate, but approximations of it, easy to manufacture, areadopted.© 2006 by Taylor & Francis Group, LLC
11. 11. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-11 Reducing the no-load airgap ﬂux-density harmonics causes a reduction of time harmonics in the statoremf (or no-load stator phase voltage). For the nonsalient pole rotor (Figure 4.12b): np μ0 WCF I f τp BgFm = 2 , for : x < (4.22) K C g (1 + K S 0 ) 2and stepwise varying otherwise (Figure 4.12b). KS0 is the magnetic saturation factor that accounts forstator and rotor iron magnetic reluctance of the ﬁeld paths; np – slots per rotor pole. υπ BgFυ ( x ) = K Fυ ⋅ BgFM ⋅cos x (4.23) τ τp π cos υ K Fυ 8 ≈ 2 2 τ 2 (4.24) υπ ⎛ τp ⎞ ⎜1 − υ τ ⎟ ⎝ ⎠ It is obvious that, in this case, the ﬂux density harmonics are lower; thus, constant airgap (cylindricalrotor) is feasible in all practical cases. Let us consider only the fundamentals of the no-load ﬂux density in the airgap: π BgF1 ( x r ) = BgFm1 cos xr (4.25) τ For constant rotor speed, the rotor coordinate xr is related to the stator coordinate xs as follows: π π x r = x s − ω r t − θ0 (4.26) τ τ The rotor rotates at angular speed ωr (in electrical terms: ωr = p1Ωr – Ωr mechanical angular velocity).θ0 is an arbitrary initial angle; let θ0 = 0. With Equation 4.26, Equation 4.25 becomes ⎛π ⎞ BgF 1 ( x s , t ) = BgFm1 cos ⎜ x s − ω r t ⎟ (4.27) ⎝τ ⎠ So, the excitation airgap ﬂux density represents a forward-traveling wave at rotor speed. This travelingwave moves in front of the stator coils at the tangential velocity us: dx s τω r us = = (4.28) dt π It is now evident that, with the rotor driven by a prime mover at speed ωr , and the stator phases open,the excitation airgap magnetic ﬁeld induces an emf in the stator windings:© 2006 by Taylor & Francis Group, LLC
12. 12. 4-12 Synchronous Generators τ + 2 E A1 ( t ) = − W1K W 1 lstack BgF 1 ( x s , t )dx s ∫ d (4.29) dt τ − 2and, ﬁnally, E A1 ( t ) = E1 2 cosω r t (4.30) ⎛ω ⎞ τ2 E1m = π 2 ⎜ r ⎟ BgFm1lstackW1K W 1 (4.31) ⎝ 2π ⎠ πwith lstack equal to the stator stack length. As the three phases are fully symmetric, the emfs in them are as follows: ⎡ 2π ⎤ E A , B,C ,1 ( t ) = E1m 2 cos ⎢ω r t − ( i − 1) ⎥ ⎣ 3 ⎦ (4.32) i = 1, 2, 3So, we notice that the excitation coil currents in the rotor are producing at no load (open stator phases)three symmetric emfs with frequency ωr that is given by the rotor speed Ωr = ωr /p1.4.4 The Two-Reaction Principle of Synchronous GeneratorsLet us now suppose that an excited SG is driven on no load at speed ωr . When a balanced three-phaseload is connected to the stator (Figure 4.13a), the presence of emfs at frequency ωr will naturally producecurrents of the same frequency. The phase shift between the emfs and the phase current ψ is dependenton load nature (power factor) and on machine parameters, not yet mentioned (Figure 4.13b). Thesinusoidal emfs and currents are represented as simple phasors in Figure 4.13b. Because of the magneticanisotropy of the rotor along axes d and q, it helps to decompose each phase current into two components:one in phase with the emf and the other one at 90° with respect to the former: IAq, IBq, ICq, and, respectively,IAd, IBd, ICd. EA1 Slip rings Prime IAd mover Brushes ωr IAq IA IF IC ICd IBq ICq IB ZL EC1 EB1 IBd (a) (b)FIGURE 4.13 Illustration of synchronous generator principle: (a) the synchronous generator on load and (b) theemf and current phasors.© 2006 by Taylor & Francis Group, LLC
13. 13. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-13 As already proven in the paragraph on windings, three-phase symmetric windings ﬂowed by balancedcurrents of frequency ωr will produce traveling mmfs (Equation 4.13): ⎛π ⎞ Fd ( x , t ) = − Fdm cos ⎜ x s − ω r t ⎟ (4.33) ⎝τ ⎠ 3 2 I d W1 K W 1 Fdm = ; I d = I Ad = I Bd = I Cd (4.34) πp1 ⎛π π⎞ Fq (x , t ) = Fqm cos ⎜ x s − ω r t − ⎟ (4.35) ⎝τ 2⎠ 3 2 I q W1 K W 1 Fqm = ; I q = I Aq = I Bq = I Cq (4.36) πp1 In essence, the d-axis stator currents produce an mmf aligned to the excitation airgap ﬂux densitywave (Equation 4.26) but opposite in sign (for the situation in Figure 4.13b). This means that the d-axismmf component produces a magnetic ﬁeld “ﬁxed” to the rotor and ﬂowing along axis d as the excitationﬁeld does. In contrast, the q-axis stator current components produce an mmf with a magnetic ﬁeld that is again“ﬁxed” to the rotor but ﬂowing along axis q. The emfs produced by motion in the stator windings might be viewed as produced by a ﬁctitiousthree-phase AC winding ﬂowed by symmetric currents IFA, IFB, IFC of frequency ωr: E A , B,C = − jω r M FA I FA , B,C (4.37) From what we already discussed in this paragraph, ωr 2 μ 0WF I f K F 1 E A (t ) = 2π K W 1 lstack τ ⋅ cos ω1t 2π π K c g (1 + K S 0 ) (4.38) np WF = WCF ; for nonsalient pole rotor , see (4.22) 2The ﬁctitious currents IFA, IFB, IFC are considered to have the root mean squared (RMS) value of If in thereal ﬁeld winding. From Equation 4.37 and Equation 4.38: 2 W1WF K W 1τ ⋅ lstack M FA = μ 0 K F1 (4.39) π K C g (1 + K S 0 )MFA is called the mutual rotational inductance between the ﬁeld and armature (stator) phase windings. The positioning of the ﬁctitious IF (per phase) in the phasor diagram (according to Equation 4.37)and that of the stator phase current phasor I (in the ﬁrst or second quadrant for generator operationand in the third or fourth quadrant for motor operation) are shown in Figure 4.14. The generator–motor divide is determined solely by the electromagnetic (active) power:© 2006 by Taylor & Francis Group, LLC
14. 14. 4-14 Synchronous Generators jq E Iq I I G G IF d Id M M I IFIGURE 4.14 Generator and motor operation modes. ( ) > 0 generator, < 0 motor Pelm = 3Re E ⋅ I * (4.40)The reactive power Qelm is ( ) <> 0( generator / motor ) Qelm = 3 Im ag E ⋅ I * (4.41)The reactive power may be either positive (delivered) or negative (drawn) for both motor and generatoroperation. For reactive power “production,” Id should be opposite from IF , that is, the longitudinal armaturereaction airgap ﬁeld will oppose the excitation airgap ﬁeld. It is said that only with demagnetizinglongitudinal armature reaction — machine overexcitation — can the generator (motor) “produce”reactive power. So, for constant active power load, the reactive power “produced” by the synchronousmachine may be increased by increasing the ﬁeld current IF . On the contrary, with underexcitation, thereactive power becomes negative; it is “absorbed.” This extraordinary feature of the synchronous machinemakes it suitable for voltage control, in power systems, through reactive power control via IF control.On the other hand, the frequency ωr , tied to speed, Ωr = ωr/p1, is controlled through the prime movergovernor, as discussed in Chapter 3. For constant frequency power output, speed has to be constant.This is so because the two traveling ﬁelds — that of excitation and, respectively, that of armaturewindings — interact to produce constant (nonzero-average) electromagnetic torque only at standstillwith each other. This is expressed in Equation 4.40 by the condition that the frequency of E1 – ωr – be equal to thefrequency of stator current I1 – ω1 = ωr – to produce nonzero active power. In fact, Equation 4.40 is validonly when ωr = ω1, but in essence, the average instantaneous electromagnetic power is nonzero only insuch conditions.4.5 The Armature Reaction Field and Synchronous ReactancesAs during steady state magnetic ﬁeld waves in the airgap that are produced by the rotor (excitation) andstator (armature) are relatively at standstill, it follows that the stator currents do not induce voltages(currents) in the ﬁeld coils on the rotor. The armature reaction (stator) ﬁeld wave travels at rotor speed;the longitudinal IaA, IaB, IaC and transverse IqA, IqB, IqC armature current (reaction) ﬁelds are ﬁxed to the© 2006 by Taylor & Francis Group, LLC
15. 15. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-15rotor: one along axis d and the other along axis q. So, for these currents, the machine reacts with themagnetization reluctances of the airgap and of stator and rotor iron with no rotor-induced currents. The trajectories of armature reaction d and q ﬁelds and their distributions are shown in Figure 4.15a,Figure 4.15b, Figure 4.16a, and Figure 4.16b, respectively. The armature reaction mmfs Fd1 and Fq1 havea sinusoidal space distribution (only the fundamental reaction is considered), but their airgap ﬂuxdensities do not have a sinusoidal space distribution. For constant airgap zones, such as it is under theconstant airgap salient pole rotors, the airgap ﬂux density is sinusoidal. In the interpole zone of a salientpole machine, the equivalent airgap is large, and the ﬂux density decreases quickly (Figure 4.15 andFigure 4.16). ωr d q ωr (a) Excitation ﬂux density d Excitation mmf Longitudinal armature mmf (T) Longitudinal 0.8 armature ﬂux density Fundamental Badl Longitudinal armature Bad ﬂux density τ τ d (b)FIGURE 4.15 Longitudinal (d axis) armature reaction: (a) armature reaction ﬂux paths and (b) airgap ﬂux densityand mmfs.© 2006 by Taylor & Francis Group, LLC
16. 16. 4-16 Synchronous Generators ωr d q ωr (a) Transverse armature mmf Transverse armature q airgap ﬂux density Baq (T) 0.5 Transverse armature airgap ﬂux density fundamental Baql q τ τ (b)FIGURE 4.16 Transverse (q axis) armature reaction: (a) armature reaction ﬂux paths and (b) airgap ﬂux densityand mmf. Only with the ﬁnite element method (FEM) can the correct ﬂux density distribution of armature (orexcitation, or combined) mmfs be computed. For the time being, let us consider that for the d axis mmf,the interpolar airgap is inﬁnite, and for the q axis mmf, it is gq = 6g. In axis q, the transverse armaturemmf is at maximum, and it is not practical to consider that the airgap in that zone is inﬁnite, as thatwould lead to large errors. This is not so for d axis mmf, which is small toward axis q, and the inﬁniteairgap approximate is tolerable. We should notice that the q-axis armature reaction ﬁeld is far from a sinusoid. This is so only forsalient pole rotor SGs. Under steady state, however, we operate only with fundamentals, and with respectto them, we deﬁne the reactances and other variables. So, we now extract the fundamentals of Bad andBaq to ﬁnd the Bad1 and Baq1:© 2006 by Taylor & Francis Group, LLC
17. 17. Large and Medium Power Synchronous Generators: Topologies and Steady State 4-17 τ ⎛π ⎞ Bad ( x r ) sin ⎜ x r ⎟ dx r ∫ 1 Bad1 = (4.42) τ ⎝τ ⎠ 0with τ − τp τ + τp Bad = 0, for : 0 < x r < and < x <τ 2 2 π (4.43) μ 0 Fdm sin xr τ − τp τ + τp Bad = τ for < xr < K c g (1 + K sd ) 2 2and ﬁnally, μ 0 Fdm K d1 τp 1 τp Bad1 = ; K d1 ≈ + sin π (4.44) K c g (1 + K sd ) τ π τIn a similar way, π μ 0 Fqm sin xr Baq = τ , for : 0 ≤ x ≤ τ p and τ + τ p < x < τ ( K c g 1 + K sq ) r 2 2 (4.45) π μ 0 Fqm sin x r τp τ + τp Baq = τ for < xr < ( K c g q 1 + K sq )2 2 μ 0 Fqm K q1 Baq1 = ; Kc g (4.46) 1τp τp 2 ⎛ τp π ⎞ K q1 = − sin π + cos ⎜ ⎟ τ π τ 3π ⎝ τ 2⎠Notice that the integration variable was xr, referring to rotor coordinates. Equation 4.44 and Equation 4.46 warrant the following remarks: • The fundamental armature reaction ﬂux density in axes d and q are proportional to the respective stator mmfs and inversely proportional to airgap and magnetic saturation equivalent factors Ksd and Ksq (typically, Ksd ≠ Ksq). • Bad1 and Baq1 are also proportional to equivalent armature reaction coefﬁcients Kd1 and Kq1. Both smaller than unity (Kd1 < 1, Kq1 < 1), they account for airgap nonuniformity (slotting is considered only by the Carter coefﬁcient). Other than that, Bad1 and Baq1 formulae are similar to the airgap ﬂux density fundamental Ba1 in an uniform airgap machine with same stator, Ba1: μ 0 F1 3 2 W1 K W 1 I1 Ba1 = ; F1 = (4.47) K c g (1 + K S ) πp1© 2006 by Taylor & Francis Group, LLC