synchronous generators chuong (5)

© 2006 by Taylor & Francis Group, LLC 5-1 5 Synchronous Generators: Modeling for (and) Transients 5.1 Introduction 5-2 5.2 The Phase-Variable Model 5-3 5.3 The d–q Model 5-8 5.4 The per Unit (P.U.) d–q Model 5-15 5.5 The Steady State via the d–q Model 5-17 5.6 The General Equivalent Circuits 5-21 5.7 Magnetic Saturation Inclusion in the d–q Model 5-23 The Single d–q Magnetization Curves Model • The Multiple d–q Magnetization Curves Model 5.8 The Operational Parameters 5-28 5.9 Electromagnetic Transients 5-30 5.10 The Sudden Three-Phase Short-Circuit from No Load 5-32 5.11 Standstill Time Domain Response Provoked Transients 5-36 5.12 Standstill Frequency Response 5-39 5.13 Asynchronous Running 5-40 5.14 Simplified Models for Power System Studies 5-46 Neglecting the Stator Flux Transients • Neglecting the Stator Transients and the Rotor Damper Winding Effects • Neglecting All Electrical Transients 5.15 Mechanical Transients 5-48 Response to Step Shaft Torque Input • Forced Oscillations 5.16 Small Disturbance Electromechanical Transients 5-52 5.17 Large Disturbance Transients Modeling 5-56 Line-to-Line Fault • Line-to-Neutral Fault 5.18 Finite Element SG Modeling 5-60 5.19 SG Transient Modeling for Control Design 5-61 5.20 Summary 5-65 References 5-68 © 2006 by Taylor & Francis Group, LLC 5-2 Synchronous Generators 5.1 Introduction The previous chapter dealt with the principles of synchronous generators (SGs) and steady state based on the two-reaction theory. In essence, the concept of traveling field (rotor) and stator magnetomotive forces (mmfs) and airgap fields at standstill with each other has been used. By decomposing each stator phase current under steady state into two components, one in phase with the electromagnetic field (emf) and the other phase shifted by 90 °, two stator mmfs, both traveling at rotor speed, were identified. One produces an airgap field with its maximum aligned to the rotor poles ( d axis), while the other is aligned to the q axis (between poles). The d and q axes magnetization inductances X dm and X qm are thus defined. The voltage equations with balanced three-phase stator currents under steady state are then obtained. Further on, this equation will be exploited to derive all performance aspects for steady state when no currents are induced into the rotor damper winding, and the field-winding current is direct . Though unbalanced load steady state was also investigated, the negative sequence impedance Z – could not be explained theoretically; thus, a basic experiment to measure it was described in the previous chapter. Further on, during transients, when the stator current amplitude and frequency, rotor damper and field currents, and speed vary, a more general (advanced) model is required to handle the machine behavior properly. Advanced models for transients include the following: • Phase-variable model • Orthogonal-axis ( d–q) model • Finite-element (FE)/circuit model The first two are essentially lumped circuit models, while the third is a coupled, field (distributed parameter) and circuit, model. Also, the first two are analytical models, while the third is a numerical model. The presence of a solid iron rotor core, damper windings, and distributed field coils on the rotor of nonsalient rotor pole SGs (turbogenerators, 2 p 1 = 2,4), further complicates the FE/circuit model to account for the eddy currents in the solid iron rotor, so influenced by the local magnetic saturation level. In view of such a complex problem, in this chapter, we are going to start with the phase coordinate model with inductances (some of them) that are dependent on rotor position, that is, on time. To get rid of rotor position dependence on self and mutual (stator/rotor) inductances, the d–q model is used. Its derivation is straightforward through the Park matrix transform. The d–q model is then exploited to describe the steady state. Further on, the operational parameters are presented and used to portray electromagnetic (constant speed) transients, such as the three-phase sudden short- circuit. An extended discussion on magnetic saturation inclusion into the d–q model is then housed and illustrated for steady state and transients. The electromechanical transients (speed varies also) are presented for both small perturbations (through linearization) and for large perturbations, respectively. For the latter case, numerical solutions of state-space equations are required and illustrated. Mechanical (or slow) transients such as SG free or forced “oscillations” are presented for electromag- netic steady state. Simplified d–q models, adequate for power system stability studies, are introduced and justified in some detail. Illustrative examples are worked out. The asynchronous running is also presented, as it is the regime that evidentiates the asynchronous (damping) torque that is so critical to SG stability and control. Though the operational parameters with s = ωj lead to various SG parameters and time constants, their analytical expressions are given in the design chapter (Chapter 7), and their measurement is presented as part of Chapter 8, on testing. This chapter ends with some FE/coupled circuit models related to SG steady state and transients. © 2006 by Taylor & Francis Group, LLC Synchronous Generators: Modeling for (and) Transients 5-3 5.2 The Phase-Variable Model The phase-variable model is a circuit model. Consequently, the SG is described by a set of three stator circuits coupled through motion with two (or a multiple of two) orthogonally placed ( d and q) damper windings and a field winding (along axis d: of largest magnetic permeance; see Figure 5.1). The stator and rotor circuits are magnetically coupled with each other. It should be noticed that the convention of voltage–current signs (directions) is based on the respective circuit nature: source on the stator and sink on the rotor. This is in agreement with Poynting vector direction, toward the circuit for the sink and outward for the source (Figure 5.1). The phase-voltage equations, in stator coordinates for the stator, and rotor coordinates for the rotor, are simply missing any “apparent” motion-induced voltages: (5.1) The rotor quantities are not yet reduced to the stator. The essential parts missing in Equation 5.1 are the flux linkage and current relationships, that is, self- and mutual inductances between the six coupled circuits in Figure. 5.1. For example, FIGURE 5.1 Phase-variable circuit model with single damper cage. b d V b V fd V a I a a I fd I D I Q I a V c q c ω r ω r I fd V fd 2 P = Sink (motor) Source (generator) H E E × H I a V a 2 P = H X E E × H iR v d dt iR v d dt iR v d dt i As a A BS b B CS c c +=− +=− +=− Ψ Ψ Ψ DDD D QQ Q ff f f R d dt iR d dt IR V d dt =− =− −=− Ψ Ψ Ψ © 2006 by Taylor & Francis Group, LLC 5-4 Synchronous Generators (5.2) Let us now define the stator phase self- and mutual inductances L AA , L BB , L CC , L AB , L BC , and L CA for a salient-pole rotor SG. For the time being, consider the stator and rotor magnetic cores to have infinite magnetic permeability. As already demonstrated in Chapter 4, the magnetic permeance of airgap along axes d and q differ (Figure 5.2). The phase A mmf has a sinusoidal space distribution, because all space harmonics are neglected. The magnetic permeance of the airgap is maximum in axis d, P d , and minimum in axis q and may be approximated to the following: (5.3) So, the airgap self-inductance of phase A depends on that of a uniform airgap machine (single-phase fed) and on the ratio of the permeance P(θ er )/(P 0 + P 2 ) (see Chapter 4): (5.4) (5.5) Also, (5.6) To complete the definition of the self-inductance of phase A, the phase leakage inductance L sl has to be added (the same for all three phases if they are fully symmetric): (5.7) Ideally, for a nonsalient pole rotor SG, L 2 = 0 but, in reality, a small saliency still exists due to a more accentuated magnetic saturation level along axis q, where the distributed field coil slots are located. FIGURE 5.2 The airgap permeance per pole versus rotor position. Ψ A AA a AB b AC c Af f AD D AQ Q LI LI LI LI LI LI=+++++ PPP PP PP er er dq dq () cos cosθθ=+ = + + − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 02 2 22 2θθ er LWKPP AAg W er = () + () 4 2 2 11 2 02 π θcos PP l g PP l g gg stack ed stack eq ed02 0 02 0 += −= < μτ μτ ;; eeq LLL AAg er =+ 02 2cos θ LLLL AA sl er =++ 02 2cos θ g e (θ er ) θ er = p 1 θ r θ er = 0 θ er = 90°−90°θ er = 180° τ - Pole pitch l stack - Stack length g e (θ er ) - Variable equivalent airgap (l stack ) P g (θ er ) = μ 0 τl stack g e (θ er ) θ er © 2006 by Taylor & Francis Group, LLC Synchronous Generators: Modeling for (and) Transients 5-5 In a similar way, (5.8) (5.9) The mutual inductance between phases is considered to be only in relation to airgap permeances. It is evident that, with ideally (sinusoidally) distributed windings, L AB (θ er ) varies with θ er as L CC and again has two components (to a first approximation): (5.10) Now, as phases A and B are 120 ° phase shifted, it follows that (5.11) The variable part of L AB is similar to that of Equation 5.9 and thus, (5.12) Relationships 5.11 and 5.12 are valid for ideal conditions. In reality, there are some small differences, even for symmetric windings. Further, (5.13) (5.14) FE analysis of field distribution with only one phase supplied with direct current (DC) could provide ground for more exact approximations of self- and mutual stator inductance dependence on θ er . Based on this, additional terms in cos(4 θ er ) , even 6θ er , may be added. For fractionary q windings, more intricate θ er dependences may be developed. The mutual inductances between stator phases and rotor circuits are straightforward, as they vary with cos( θ er ) and sin(θ er ). (5.15) LLLL BB sl er =++ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 02 2 2 3 cos θ π LLLL CC sl er =++ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 02 2 2 3 cos θ π LLL L AB BA AB AB er == + − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 02 2 2 3 cos θ π LL L AB00 0 2 32 ≈=−cos π LL AB22 = LL L L AC CA er ==−+ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 0 2 2 2 2 3 cos θ π LL L L BC CB er ==−+ 0 2 2 2cos θ LM LM LM Af f er Bf f er Cf f = =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = cos cos cos θ θ π2 3 θθ π er + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 © 2006 by Taylor & Francis Group, LLC 5-6 Synchronous Generators (5.15 cont.) Notice that (5.16) L dm and L qm were defined in Chapter 4 with all stator phases on, and M f is the maximum of field/armature inductance also derived in Chapter 4. We may now define the SG phase-variable 6 × 6 matrix : (5.17) A mutual coupling leakage inductance L fDl also occurs between the field winding f and the d-axis cage winding D in salient-pole rotors. The zeroes in Equation 5.17 reflect the zero coupling between orthogonal windings in the absence of magnetic saturation. are typical main (airgap permeance) self- inductances of rotor circuits. are the leakage inductances of rotor circuits in axes d and q. The resistance matrix is of diagonal type: θ θ π AD D er BD D er LM LM = =− ⎛ 2 3 cos cos ⎝⎝ ⎜ ⎞ ⎠ ⎟ =+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− LM LM L CD D er AQ Q er B cos sin θ π θ 2 3 QQQer CQ Q er M LM =− − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− + ⎛ ⎝ ⎜ ⎞ sin sin θ π θ π 2 3 2 3 ⎠⎠ ⎟ L LL L LL dm qm dm qm 0 2 2 2 = + () = − () L ABCfDQ er θ () LL L fm r Dm r Qm r ,, LL L fl r Dl r Ql r ,, © 2006 by Taylor & Francis Group, LLC Synchronous Generators: Modeling for (and) Transients 5-7 (5.18) Provided core losses, space harmonics, magnetic saturation, and frequency (skin) effects in the rotor core and damper cage are all neglected, the voltage/current matrix equation fully represents the SG at constant speed: (5.19) with (5.20) (5.21) The minus sign for V f arises from the motor association of signs convention for rotor. The first term on the right side of Equation 5.19 represents the transformer-induced voltages, and the second term refers to the motion-induced voltages. Multiplying Equation 5.19 by [I ABCfDQ ] T yields the following: (5.22) The instantaneous power balance equation (Equation 5.22) serves to identify the electromagnetic power that is related to the motion-induced voltages: (5.23) P elm should be positive for the generator regime. The electromagnetic torque T e opposes motion when positive (generator model) and is as follows: (5.24) The equation of motion is (5.25) RDiagRRRRRR ABCfdq s r s f r D r Q r = ⎡ ⎣ ⎤ ⎦ ,,, , , IR V d ABCfDQ ABCfDQ ABCfDQ ABCfD ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ + ⎡ ⎣ ⎤ ⎦ = −Ψ QQ ABCfDQ er ABCfDQ ABCf dt L d dt I L = − () ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ − ∂ θ DDQ er er ABCfDQ d dt I ⎡ ⎣ ⎤ ⎦ ∂ ⎡ ⎣ ⎤ ⎦ θ θ VVVVV d dt ABCfDQ A B C f T er r =+ + + − ⎡ ⎣ ⎤ ⎦ =,,,,,;00 θ ω ΨΨΨΨΨΨΨ ABCfDQ A B C f r D r Q r T = ⎡ ⎣ ⎤ ⎦ ,,,,, IV I L ABCfDQ T ABCfDQ ABCfDQ T AB ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ =− ⎡ ⎣ ⎤ ⎦ ∂ 1 2 CCfDQ er er ABCfDQ r ABC I d dt I θ θ ω () ⎡ ⎣ ⎤ ⎦ ∂ ⎡ ⎣ ⎤ ⎦ ⋅− − 1 2 ffDQ T ABCfDQ er ABCfDQ A LII ⎡ ⎣ ⎤ ⎦ ⋅ () ⋅ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ −θ BBCfDQ T ABCfDQ ABCfDQ IR ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ PI L I elm ABCfDQ T er ABCfDQ er =− ⎡ ⎣ ⎤ ⎦ ⋅ ∂ ∂ () ⎡ ⎣ ⎤ ⎦ 1 2 θ θ AABCfDQ r ⎡ ⎣ ⎤ ⎦ ω T P p p I L e e r ABCfDQ T ABCfDQ er = + () =− ⎡ ⎣ ⎤ ⎦ ∂ () ω θ / 1 1 2 ⎡⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ δθ er ABCfDQ I J p d dt TT d dt r shaft e er r 1 ωθ ω=− =; © 2006 by Taylor & Francis Group, LLC 5-8 Synchronous Generators The phase-variable equations constitute an eighth-order model with time-variable coefficients (induc- tances). Such a system may be solved as it is either with flux linkages vector as the variable or with the current vector as the variable, together with speed ω r and rotor position θ er as motion variables. Numerical methods such as Runge–Kutta–Gill or predictor-corrector may be used to solve the system for various transient or steady-state regimes, once the initial values of all variables are given. Also, the time variations of voltages and of shaft torque have to be known. Inverting the matrix of time-dependent inductances at every time integration step is, however, a tedious job. Moreover, as it is, the phase- variable model offers little in terms of interpreting the various phenomena and operation modes in an intuitive manner. This is how the d–q model was born — out of the necessity to quickly solve various transient operation modes of SGs connected to the power grid (or in parallel). 5.3 The d–q Model The main aim of the d–q model is to eliminate the dependence of inductances on rotor position. To do so, the system of coordinates should be attached to the machine part that has magnetic saliency — the rotor for SGs. The d–q model should express both stator and rotor equations in rotor coordinates, aligned to rotor d and q axes because, at least in the absence of magnetic saturation, there is no coupling between the two axes. The rotor windings f, D, Q are already aligned along d and q axes. The rotor circuit voltage equations were written in rotor coordinates in Equation 5.1. It is only the stator voltages, V A , V B , V C , currents I A , I B , I C , and flux linkages Ψ A , Ψ B , Ψ C that have to be transformed to rotor orthogonal coordinates. The transformation of coordinates ABC to d–q0, known also as the Park transform, valid for voltages, currents, and flux linkages as well, is as follows: (5.26) So, (5.27) (5.28) (5.29) P er er er θ θθ π () ⎡ ⎣ ⎤ ⎦ = − () −+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 2 3 cos cos cos θθ π θθ π er er er − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − () −+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 2 3 sin sin sin −−− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ θ π er 2 3 1 2 1 2 1 2 V V V P V V V d qer A B C0 = () ⋅θ I I I P I I I d qer A B C0 = () ⋅θ Ψ Ψ Ψ Ψ Ψ Ψ d qer A B C P 0 = () ⋅θ © 2006 by Taylor & Francis Group, LLC Synchronous Generators: Modeling for (and) Transients 5-9 The inverse transformation that conserves power is (5.30) The expressions of Ψ A , Ψ B , Ψ C from the flux/current matrix are as follows: (5.31) The phase currents I A , I B , I C are recovered from I d , I q , I 0 by (5.32) An alternative Park transform uses instead of 2/3 for direct and inverse transform. This one is fully orthogonal (power direct conservation). The rather short and elegant expressions of Ψ d , Ψ q , Ψ 0 are obtained as follows: (5.33) From Equation 5.16, (5.34) are exactly the “cyclic” magnetization inductances along axes d and q as defined in Chapter 4. So, Equation 5.33 becomes (5.35) (5.36) (5.37) PP er er T θθ () ⎡ ⎣ ⎤ ⎦ = () ⎡ ⎣ ⎤ ⎦ −1 3 2 Ψ ABCfDQ ABCfDQ er ABCfDQ LI= ()θ I I I P I I I A B C er T d q = () ⎡ ⎣ ⎤ ⎦ ⋅ 3 2 0 θ 2 3 Ψ Ψ dsl AB dff r DD r q LLL LIMIMI L =+− + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ++ = 002 3 2 ssl AB q Q q r sl A LL LIMI LL L +− − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + =++ 002 00 3 2 2Ψ BBAB IL L 00 0 0 2 () ≈−;/ LLL LLL dm qm =+ () =− () 3 2 3 2 02 02 ; Ψ ddd ff r DD r dsldm LI M I M I LLL =+ + =+ ; Ψ qqq QQ r qslqm LI M I LLL =+ =+ ; Ψ 00 ≈ LI sl © 2006 by Taylor & Francis Group, LLC 5-10 Synchronous Generators In a similar way for the rotor, (5.38) As seen in Equation 5.37, the zero components of stator flux and current Ψ 0 , I 0 are related simply by the stator phase leakage inductance L sl ; thus, they do not participate in the energy conversion through the fundamental components of mmfs and fields in the SGs. Thus, it is acceptable to consider it separately. Consequently, the d–q transformation may be visualized as representing a fictitious SG with orthogonal stator axes fixed magnetically to the rotor d–q axes. The magnetic field axes of the respective stator windings are fixed to the rotor d–q axes, but their conductors (coils) are at standstill (Figure 5.3) — fixed to the stator. The d–q model equations may be derived directly through the equivalent fictitious orthogonal axis machine (Figure 5.3): (5.39) The rotor equations are then added: FIGURE 5.3 The d–q model of synchronous generators. I d I D I f V f I Q V q I q V d ω r ω r Ψ Ψ f r fl r fm f r fd fDD r D r Dl r Dm LLI MIMI LL =+ () ++ =+ 3 2 (() ++ =+ () + IMIMI LLI MI D r Dd fD f r Q r Ql r Qm Q r Q 3 2 3 2 Ψ qq IR V d dt IR V d dt ds d d rq qs q q rd +=− + +=− − Ψ Ψ Ψ Ψ ω ω [...]... Q 0 = lqm I q0 ; lq = lqm + lsl © 2006 by Taylor & Francis Group, LLC (5.72) 5-18 Synchronous Generators jq ω1 = ω r ωr positive − ⎞ ⎛3π − δv0 = θ0 ⎠ ⎝2 (lsl + ldm)Id0 Id0 ω1 = ωr 1dmIf0 IS0 ϕ1 ψS0 jIq0 j(lsl + lqm)Iq0 δV0 −jωrψs0 Generator torque VS0 − rsIso FIGURE 5.5 The space-phasor (vector) diagram of synchronous generators We may now introduce space phasors for the stator quantities: Ψ s 0 = Ψ... 2006 by Taylor & Francis Group, LLC 5-22 Synchronous Generators 2 s 1 ωb fDl 1 s ωb lsl iD1 iD s l ωb fl s l ωb Dll s l ωb Dl rf rD1 rD −ωrψq rs A Id Idm = Id + If + ID s l ωb dm Vd s ψ ωb d Vf B (a) Iq ωrψd s 1 ωb sl rs C s 1 ωb ql Iqm = Iq + IQ Vq s ψ ωb q IQ s 1 ωb q11 s 1 ωb q12 s 1 ωb qm rQ rQ1 rQ2 D (b) FIGURE 5.6 General equivalent circuits of synchronous generators: (a) along axis d and (b) along... suggested in previous © 2006 by Taylor & Francis Group, LLC 5-30 Synchronous Generators TABLE 5.1 Typical Synchronous Generator Parameter Values Parameter Two-Pole Turbogenerator ld (P.U.) lq (P.U.) ld′ (P.U.) ld″ (P.U.) lfDl (P.U.) l0 (P.U.) lp (P.U.) rs (P.U.) Td0′ (sec) Td′ (sec) Td″ (sec) Td0″ (sec) Tq″ (sec) Tq0″ (sec) lq″ (P.U.) Hydrogenerators 0.9–1.5 0.85–1.45 0.12–0.2 0.07–0.14 0.05–+0.05 0.02–0.08... lq = lim lq ( s ) = lq s →0 t →∞ where ld″, ld′, ld = the d axis subtransient, transient, and synchronous inductances lq″, lq = the q axis subtransient and synchronous inductances lp = the Potier inductance in P.U (lp ≥ lsl) Typical values of the time constants (in seconds) and subtransient and transient and synchronous inductances (in P.U.) are shown in Table 5.1 As Table 5.1 suggests, various inductances... space fundamental is concerned, this condition holds Once heavy local magnetic saturation conditions occur (Equation 5.57), there is a departure from reality © 2006 by Taylor & Francis Group, LLC Synchronous Generators: Modeling for (and) Transients 5-15 • No leakage flux coupling between the d axis damper cage and the field winding (LfDl = 0) was considered so far, though in salient-pole rotors, LfDl... — base flux linkage ωb (5.62) Zb = Vb Vn — base impedance (valid also for resistances and reactances) = Ib In (5.63) Lb = Zb — base inductance ωb (5.64) © 2006 by Taylor & Francis Group, LLC 5-16 Synchronous Generators Inductances and reactances are the same in P.U values Though in some instances time is also provided with a base quantity tb = 1/ωb , we chose here to leave time in seconds, as it seems... are in P.U measurements (Time t and inertia H are given in seconds, and ωb is given in rad/sec.) Equation 5.67 represents the d–q model of a three-phase © 2006 by Taylor & Francis Group, LLC 5-17 Synchronous Generators: Modeling for (and) Transients SG with single damper circuits along rotor orthogonal axes d and q Also, the coupling of windings along axes d and q, respectively, is taking place only...5-11 Synchronous Generators: Modeling for (and) Transients I f Rf − Vf = − iD RD = − dt dΨ D dt iQ RQ = − dΨ f dΨ Q dt (5.40) In Equation 5.39, we assumed that dΨ d = −Ψq dθer (5.41) dΨ q = Ψd dθer The assumptions... (d axis) axis and the voltage vector angle It may be seen from Figure 5.5 that axis d is behind Vs0, which explains why ⎛ 3π ⎞ − δV 0 ⎟ θ0 = − ⎜ ⎝ 2 ⎠ © 2006 by Taylor & Francis Group, LLC (5.75) Synchronous Generators: Modeling for (and) Transients 5-19 Making use of Equation 5.74 in Equation 5.70, we obtain the following: Vd 0 = −V 2 sin δ V 0 < 0 Vq 0 = −V 2 cos δ V 0 < 0 I d 0 = − I 2 sin ( δ V 0... Amperes Solution 1 The vector diagram is simplified as cosϕ1 = 1 (ϕ1 = 0), but it is worth deriving a formula to directly calculate the power angle δV0 © 2006 by Taylor & Francis Group, LLC 5-20 Synchronous Generators Using Equation 5.70 and Equation 5.71 in Equation 5.72 yields the following: ⎛ ω1lq I cos ϕ1 − rs I sin ϕ1 ⎞ δV 0 = tan −1 ⎜ ⎟ ⎝ V + rs I cos ϕ1 + ω1lq I sinϕ1 ⎠ with ϕ1 = 0 and ω1 = 1, . & Francis Group, LLC 5-2 Synchronous Generators 5.1 Introduction The previous chapter dealt with the principles of synchronous generators (SGs) and steady. © 2006 by Taylor & Francis Group, LLC 5-1 5 Synchronous Generators: Modeling for (and) Transients 5.1 Introduction 5-2 5.2 The