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Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
299 8.1. INTRODUCTION There are alternative formulations of induction and synchronous machine equations that warrant consideration since each has a specifi c useful purpose. In particular, (1) linearized or small-displacement formulation for operating point stability issues; (2) neglecting stator electric transients for large-excursion transient stability studies; and (3) voltage-behind reactance s ( VBR s) formulation convenient for machine-converter analysis and simulation. These special formulations are considered in this chapter. Although standard computer algorithms may be used to automatically linearize machine equations, it is important to be aware of the steps necessary to perform lineariza- tion. This procedure is set forth by applying Taylor expansion about an operating point. The resulting set of linear differential equations describe the dynamic behavior during small displacements or small excursions about an operating point, whereupon basic linear system theory can be used to calculate eigenvalues. In the fi rst sections of this chapter, the nonlinear equations of induction and synchronous machines are linearized and the eigenvalues are calculated. Although these equations are valid for operation with stator voltages of any frequency, only rated frequency operation is considered in detail. Over the years, there has been considerable attention given to the development of simplifi ed models primarily for the purpose of predicting the dynamic behavior of electric machines during large excursions in some or all of the machine variables. Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek. © 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. ALTERNATIVE FORMS OF MACHINE EQUATIONS 8 300 ALTERNATIVE FORMS OF MACHINE EQUATIONS Before the 1960s, the dynamic behavior of induction machines was generally predicted using the steady-state voltage equations and the dynamic relationship between rotor speed and torque. Similarly, the large-excursion behavior of synchronous machines was predicted using a set of steady-state voltage equations with modifi cations to account for transient conditions, as presented in Chapter 5 , along with the dynamic relationship between rotor angle and torque. With the advent of the computer, these models have given way to more accurate representations. In some cases, the machine equations are programmed in detail; however, in the vast majority of cases, a reduced-order model is used in computer simulations of power systems. In particular, it is standard to neglect the electric transients in the stator voltage equations of all machines and in the voltage equations of all power system components connected to the stator (transformers, trans- mission lines, etc.). By using a static representation of the power grid, the required number of integrations is drastically reduced. Since “neglecting stator electric tran- sients” is an important aspect of machine analysis especially for the power system engineer, the theory of neglecting electric transients is established and the voltage equa- tions for induction and synchronous machines are given with the stator electric tran- sients neglected. The large-excursion behavior of these machines as predicted by these reduced-order models is compared with the behavior predicted by the complete equa- tions given in Chapter 5 and Chapter 6 . From these comparisons, not only do we become aware of the inaccuracies involved when using the reduced-order models, but we are also able to observe the infl uence that the electric transients have on the dynamic behavior of induction and synchronous machines. Finally, in an increasing number of applications, electric machines are coupled to power electronic circuits. In Chapter 4 , Chapter 5 , and Chapter 6 , a great deal of the focus was placed upon utilizing reference-frame theory to eliminate rotor-dependent inductances (or fl ux linkage in the case of the permanent magnet machine). Although reference-frame theory enables analytical evaluation of steady-state performance and provides the basis for most modern electric drive controls, it can be diffi cult to apply a transformation to some power system components, particularly power electronic converters. In such cases, one is forced to establish a coupling between a machine modeled in a reference frame and a power converter modeled in terms of physical variables. As an alternative, it can be convenient to represent a machine in terms of physical variables using a VBR model. In this chapter, the derivation of a physical variable VBR model of the synchronous machine is provided, along with explanation of its potential application and advantages over alternative model structures. In addi- tion, approximate forms of the VBR model are described in which rotor position- dependent inductances are eliminated, which greatly simplifi es the modeling of machines in physical variables. 8.2. MACHINE EQUATIONS TO BE LINEARIZED The linearized machine equations are conveniently derived from voltage equations expressed in terms of constant parameters with constant driving forces, independent of MACHINE EQUATIONS TO BE LINEARIZED 301 time. During steady-state balanced conditions, these requirements are satisfi ed, in the case of the induction machine, by the voltage equations expressed in the synchronously rotating reference frame, and by the voltage equations in the rotor reference frame in the case of the synchronous machine. Since the currents and fl ux linkages are not independent variables, the machine equations can be written using either currents or fl ux linkages, or fl ux linkages per second, as state variables. The choice is generally determined by the application. Currents are selected here. Formulating the small- displacement equations in terms of fl ux linkages per second is left as an exercise for the reader. Induction Machine The voltage equations for the induction machine with currents as state variables may be written in the synchronously rotating reference frame from (6.5-34) by setting ω = ω e as v v v v r p XX p qs e ds e qr e dr e s b ss e b ss ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = + ω ω ω ωω ω ω ω ωω ω ωω ω ω ωω b M e b M e b ss s b ss e b M b M b M e b Mr XX Xr p XX p X p XsXr p −+− ′ + bb rr e b rr e b M b M e b rr r b rr XsX sX p XsXr p X ′′ −− ′′ + ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ω ω ω ωω ω ωω ⎢⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ i i i i qs e ds e qr e dr e (8.2-1) where s is the slip defi ned by (6.9-13) and the zero quantities have been omitted since only balanced conditions are considered. The reactances X ss and ′ X rr are defi ned by (6.5-35) and (6.5-36) , respectively. Since we have selected currents as state variables, the electromagnetic torque is most conveniently expressed as TXii ii eMqs e dr e ds e qr e = ′ − ′ () (8.2-2) Here, the per unit version of (6.6-2) is selected for compactness. The per unit relation- ship between torque and speed is (6.8-10) , which is written here for convenience THp T e r b L =+2 ω ω (8.2-3) Synchronous Machine The voltage equations for the synchronous machine in the rotor reference frame may be written from (5.5-38) for balanced conditions as 302 ALTERNATIVE FORMS OF MACHINE EQUATIONS v v v v e v qs r ds r kq r kq r xfd r kd r ′ ′ ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1 2 ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = + − r p XX p X p XX X X s b q r b d b mq b mq r b md r b md r b q ω ω ωωω ω ω ω ω ω ω rr p XX X p X p X p Xr p X s b d r b mq r b mq b md b md b mq kq b kq +− − ′ + ′ ω ω ω ω ωω ω ωω 0 11 pp X p X p Xr p X X r p X b mq b mq b mq kq b kq md fd b md ω ωωω ω 00 000 0 22 ′ + ′ ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 000 000 X r r p X X r p X p X p md fd fd b fd md fd b md b md ′ ′ + ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ωω ωω bb md kd b kd qs Xr p X i ′ + ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ω rr ds r kq r kq r fd r kd r i i i i i ′ ′ ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1 2 ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (8.2-4) where positive currents are assumed into the machine and the reactances are defi ned by (5.5-39)–(5.5-44) . With the currents as state variables, the per unit electromagnetic torque positive for motor action is expressed from (5.6-2) as TXii ii Xii ii emdds r fd r kd r qs r mq qs r kq r kq r ds r =+ ′ + ′ −+ ′ + ′ ()( ) 12 (8.2-5) The per unit relationship between torque and rotor speed is given by (5.8-3) , which is THp T e r b L =+2 ω ω (8.2-6) The rotor angle is expressed from (5.7-1) as δ ωωω ω = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ br e b p (8.2-7) It is necessary, in the following analysis, to relate variables in the synchronously rotat- ing reference frame to variables in the rotor reference frame. This is accomplished by using (5.7-2) with the zero quantities omitted. Thus f f f f qs r ds r qs e ds e ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ cos sin sin cos δδ δδ (8.2-8) 8.3. LINEARIZATION OF MACHINE EQUATIONS There are two procedures that can be followed to obtain the linearized machine equa- tions. One is to employ Taylor ’ s expansion about a fi xed value or operating point. That is, any machine variable f i can be written in terms of a Taylor expansion about its fi xed value, f io , as LINEARIZATION OF MACHINE EQUATIONS 303 gf gf g f f gf f iio ioi io i () () () () ! =+ ′ + ′′ +ΔΔ 2 2 (8.3-1) where ff f iio i =+Δ (8.3-2) If only a small excursion from the fi xed point is experienced, all terms higher than the fi rst-order may be neglected, and g ( f i ) may be approximated by gf gf g f f iio ioi () () ()≈+ ′ Δ (8.3-3) Hence, the small-displacement characteristics of the system are given by the fi rst-order terms of Taylor ’ s series, ΔΔgf g f f iioi () ( )= ′ (8.3-4) For functions of two variables, the same argument applies gf f gf f f gf f f f gf f f oo oo oo (,)(,) (,) (,) 12 1 2 1 12 1 2 12 2 ≈+ ∂ ∂ + ∂ ∂ ΔΔ (8.3-5) where Δ g ( f 1 , f 2 ) is the last two terms of (8.3-5) . If, for example, we apply this method to the expression for induction machine torque, (8.2-2) , then Tiiii Tiiii T eqs e ds e qr e dr e eqso e dso e qro e dro e (, , , ) ( , , , ) ′′ ≈ ′′ + ∂ eeqso e dso e qro e dro e qs e qs e iiii i i (, , , ) . ′′ ∂ +Δ etc (8.3-6) whereupon the small-displacement expression for torque becomes ΔΔΔΔΔTXiiiiiiii e M qso e dr e dro e qs e dso e qr e qro e ds e = ′ + ′ − ′ − ′ () (8.3-7) where the added subscript o denotes steady-state quantities. An equivalent method of linearizing nonlinear equations is to write all variables in the form given by (8.3-2) . If all multiplications are then performed and the steady- state expressions cancelled from both sides of the equations and if all products of small displacement terms ( Δ f 1 Δ f 2 , for example) are neglected, the small-displacement equa- tions are obtained. It is left to the reader to obtain (8.3-7) by this technique. Induction Machine If either of the above-described methods of linearization is employed to (8.2-1)–(8.2-3) , the linear differential equations of an induction machine become 304 ALTERNATIVE FORMS OF MACHINE EQUATIONS Δ Δ Δ Δ Δ v v v v T r p qs e ds e qr e dr e L s ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = + ωω ω ωω ω ω ω ωω ω ωω ω b ss e b ss b M e b M e b ss s b ss e b M b M b XX p XX Xr p XX p X p 0 0−+− XXsXr p XsX XiXi s Mo e b Mr b rr o e b rr M dso e rr dro e o e b ω ωω ω ω ω ω ′ + ′′ −− ′′ − XX p XsXr p XXiXi Xi X M b Mo e b rr r b rr M qso e rr qro e Mdro e ω ω ωω − ′′ + ′ + ′′ ′ − MMqro e Mdso e Mqso e q iXiXi Hp i ′ −− ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 Δ ss e ds e qr e dr e r b i i i Δ Δ Δ Δ ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ω ω (8.3-8) where s o ero e = − ωω ω (8.3-9) It is clear that with applied voltages of rated frequency, the ratio of ω e to ω b is unity. However, (8.3-8) and (8.3-9) are written with ω e included explicitly so as to accom- modate applied voltages of a constant frequency other than rated as would occur in variable-speed drive systems. The frequency of the applied stator voltages in variable- speed drive systems is varied by controlling the fi ring of the source converter. There- fore, in some applications, the frequency of the stator voltages may be a controlled variable. It is recalled from Chapter 3 that variable-frequency operation may be inves- tigated in the synchronously rotating reference frame by simply changing the speed of the reference frame corresponding to the change in frequency. Therefore, if fre- quency is a system input variable, then a small displacement in frequency may be taken into account by allowing the reference-frame speed to change by replacing ω e with ω eo + Δ ω e . It is convenient to separate out the derivative terms and write (8.3-8) in the form Ex Fx up =+ (8.3-10) where ()x T qs e ds e qr e dr e r b iiii= ′′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ΔΔΔΔ Δ ω ω (8.3-11) ()u T qs e ds e qr e dr e L vvvvT= ′′ [] ΔΔΔΔΔ (8.3-12) LINEARIZATION OF MACHINE EQUATIONS 305 E = ′ ′ − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ 1 000 00 0 000 00 0 00002 ω ω b ss M ss M Mrr Mrr b XX XX XX XX H ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (8.3-13) F =− −− ′′ rX X Xr X sX r s s e b ss e b M e b ss s e b M o e b Mr o e b ω ω ω ω ω ω ω ω ω ω ω ω 00 00 0 XXXiXi sX sX r XiX rr M dso e rr dro e o e b Mo e b rr r M qso e r −− ′′ −− ′′ + ′ ω ω ω ω 0 rrqro e Mdro e Mqro e Mdso e Mqso e i Xi Xi Xi Xi ′ ′ − ′ − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ 0 ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (8.3-14) In the analysis of linear systems, it is convenient to express the linear differential equa- tions in the form pxAxBu=+ (8.3-15) Equation (8.3-15) is the fundamental form of the linear differential equations. It is commonly referred to as the state equation. Equation (8.3-10) may be written as pxEFxEu=+ −− () () 11 (8.3-16) which is in the form of (8.3-15) with AEF= − () 1 (8.3-17) BE= − () 1 (8.3-18) Synchronous Machines Linearizing (8.2-4)–(8.2-8) yields (8.3-19) . Since the steady-state damper winding cur- rents ( ′′ ii kdo r kq o r ,, 1 and ′ i kq o r 2 ) are zero, they are not included in (8.3-19) . Since the synchro- nous machine is generally connected to an electric system, such as a power system, and since it is advantageous to linearize the system voltage equations in the synchro- nously rotating reference frame, it is convenient to include the relationship between Δ ω r and Δ δ in (8.3-19) . As in the case of linearized equations for the induction machine, ω e is included explicitly in (8.3-19) so that the equations are in a form convenient for voltages of any constant frequency. Small controlled changes in the 306 ALTERNATIVE FORMS OF MACHINE EQUATIONS frequency of the applied stator voltages, as is possible in variable-speed drive systems, may be taken into account analytically by replacing ω e with ω eo + Δ ω e in the expression for δ given by (8.2-7) . (8.3-19) In most cases, the synchronous machine is connected to a power system whereupon the voltage v qs r and v ds r , which are functions of the state variable δ , will vary as the rotor angle varies during a disturbance. It is of course necessary to account for the depen- dence of the driving forces upon the state variables before expressing the linear dif- ferential equations in fundamental form. In power system analysis, it is often assumed that in some place in the system, there is a balanced source that can be considered a constant amplitude, constant frequency, and zero impedance source (infi nite bus). This would be a balanced independent driving force that would be represented as constant voltages in the synchronously rotating reference frame. Hence, it is necessary to relate the synchronously rotating reference-frame variables, where the independent driving force exists, to the variables in the rotor reference frame. The transformation given by (8.2-8) is nonlinear. In order to incorporate it into a linear set of differential equations, it must be linearized. By employing the approximations that cos Δ δ = 1 and sin Δ δ = Δ δ , the linearized version of (8.2-8) is Δ Δ Δ Δ f f f f qs r ds r oo oo qs e ds e ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ cos sin sin cos δδ δδ ⎦⎦ ⎥ + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ f f dso r qso r Δ δ (8.3-20) Linearizing the inverse transformation yields Δ Δ Δ Δ Δ Δ Δ v v v v e v T qs r ds r kq r kq r xfd r kd r L ′ ′ ′ ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 2 0 ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = + −r p XX p X p XX XXi s b q e b d b mq b mq e b md e b md d dso ω ω ωωω ω ω ω ω rr md fdo r e b qs b d e b mq e b mq b md b md q Xi Xr p XX X p X p XX + ′ −+−− 0 ω ωω ω ω ω ωω ω ii p Xr p X p X p X p Xr p qso r b mq kq b kq b q b mq b mq kq 0 00000 0 11 2 ωωω ωωω ′ + ′ ′ + bb kq md fd b md md fd fd b fd X X r p X X r r p X ′ ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ ′ + ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 0000 000 ωω XX r p X p X p Xr p X Xi md fd b md b md b md kd b kd mq ds ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ + ′ − ω ωωω 00 000 00 oo r md dso r fdo r md qso r mq qso r mq dso r mq dso r m Xi i Xi Xi Xi Xi X+− ′ −− −() ddqso r md qso r b iXi Hp p − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 20 000000 ω ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ′ ′ ′ ′ Δ Δ Δ Δ Δ Δ Δ i i i i i i qs r ds r kq r kq r fd r kd r r b 1 2 ω ω ΔΔ δ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ LINEARIZATION OF MACHINE EQUATIONS 307 Δ Δ Δ Δ f f f f qs e ds e oo oo qs r ds r ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ cos sin sin cos δδ δδ ⎦⎦ ⎥ + − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ f f dso e qso e Δ δ (8.3-21) It is convenient to write the above equations in the form ΔΔ ΔfTfF qds r qds er =+ δ (8.3-22) ΔΔΔfTfF qds e qds re =+ − () 1 δ (8.3-23) It is instructive to view the interconnections of the above relationships as shown in Figure 8.3-1 . With the equations as shown in Figure 8.3-1 , a change in Δv qds e is refl ected through the transformation to the voltage equations in the rotor reference frame and fi nally back to the synchronously rotating reference-frame currents Δi qds e . The detail shown in Figure 8.3-1 is more than is generally necessary. If, for example, the objective is to study the small-displacement dynamics of a synchronous machine with its termi- nals connected to an infi nite bus, then Δv qds e is zero and Δv qds r changes due only to Δ δ . Also, in this case, it is unnecessary to transform the rotor reference-frame currents to the synchronously rotating reference frame since the source (infi nite bus) has zero impedance. If the machine is connected through a transmission line to a large system (infi nite bus), the small-displacement dynamics of the transmission system must be taken into account. If only the machine is connected to the transmission line and if it is not equipped with a voltage regulator, then it is convenient to transform the equations of the transmission line to the rotor reference frame. In such a case, the machine and transmission line can be considered in much the same way as a machine connected to an infi nite bus. If, however, the machine is equipped with a voltage regulator or more than one machine is connected to the same transmission line, it is generally preferable to express the dynamics of the transmission system in the synchronously rotating refer- ence frame and transform to and from the rotor reference frame of each machine as depicted in Figure 8.3-1 . Figure 8.3-1. Interconnection of small-displacement equations of a synchronous machine: Park ’ s equations. Δv e Δv r Δi r Δi e Δv k r (8.3-22) (8.3-19) (8.3-23) ′ Δi k r ′ Δv k r ′ Δi k r ′ Δi f r ′ Δe x r ′ Δv k r ′ Δi k r ′ ΔT 1 Δw r /w b Δd qds qds q1 q2 fd d qds q1 q2 d d qds 308 ALTERNATIVE FORMS OF MACHINE EQUATIONS If the machine is equipped with a voltage regulator, the dynamic behavior of the regulator will affect the dynamic characteristics of the machine. Therefore, the small- displacement dynamics of the regulator must be taken into account. When regulators are employed, the change in fi eld voltage Δ ′ e xfd r is dynamically related to the change in ter- minal voltage, which is a function of Δv qds e (or Δv qds r ), the change in fi eld current Δi fd r , and perhaps the change in rotor speed Δ ω r / ω b if the excitation system is equipped with a control to help damp rotor oscillations by means of fi eld voltage control. This type of damping control is referred to as a power system stabilizer ( PSS ). In some investigations, it is necessary to incorporate the small-displacement dynam- ics of the prime mover system. The change of input torque (negative load torque) is a function of the change in rotor speed Δ ω r / ω b , which in turn is a function of the dynamics of the masses, shafts, and damping associated with the mechanical system and, if long- term transients are of interest, the steam or hydro dynamics and associated controls. Although a more detailed discussion of the dynamics of the excitation and prime mover systems would be helpful, it is clear from the earlier discussion that the equations that describe the operation and control of a synchronous machine equipped with a volt- age regulator and a prime mover system are very involved. This becomes readily apparent when it is necessary to arrange the small-displacement equations of the com- plete system into the fundamental form. Rather than performing this task by hand, it is preferable to take advantage of analytical techniques, which involves formulating the equations of each component (machine, excitation system, prime mover system, etc.) in fundamental form. A computer routine can be used to arrange the small-displacement equations along with the interconnecting transformations of the complete system into the fundamental form. 8.4. SMALL-DISPLACEMENT STABILITY: EIGENVALUES With the linear differential equations written in state variable form, the u vector repre- sents the forcing functions. If u is set equal to zero, the general solution of the homo- geneous or force-free linear differential equations becomes xK A = e t (8.4-1) where K is a vector formed by an arbitrary set of initial conditions. The exponential e A t represents the unforced response of the system. It is called the state transition matrix. Small-displacement stability is assured if all elements of the transition matrix approach zero asymptotically as time approaches infi nity. Asymptotic behavior of all elements of the matrix occurs whenever all of the roots of the characteristic equation of A have negative real parts where the characteristic equation of A is defi ned det( )AI−= λ 0 (8.4-2) In (8.4-2) , I is the identity matrix and λ are the roots of the characteristic equation of A referred to as characteristic roots, latent roots, or eigenvalues. Herein, we will use [...]... j 18. 2 −17.0 −41 .8 ± j374 −14.3 ± j42 .8 −29.6 −24.6 ± j376 −9.05 ± j42.5 − 18. 5 310 ALTERNATIVE FORMS OF MACHINE EQUATIONS Real part 40 Positive imaginary part Real eigenvalue 0 400 –40 360 “Stator” eigenvalue 320 –120 280 –160 240 –200 200 “Rotor” eigenvalue –240 Imaginary part, rad/s Real part, rad/s 80 160 – 280 120 –320 80 –360 40 –400 0 0.2 0.4 0.6 0 .8 0 1.0 wr wb Figure 8. 5-1 Plot of eigenvalues... Induction Machine The voltage equations written in the arbitrary reference frame for an induction machine with the electric transients of the stator voltage equations neglected may be written from (6.5-22)–(6.5-33), with the zero quantities eliminated and (8. 7-7) and (8. 7 -8) appropriately taken into account vqs = rs iqs + ωe ψ ds ωb (8. 7-11) vds = rsids − ωe ψ qs ωb (8. 7-12) 316 ALTERNATIVE FORMS OF MACHINE. .. ⎟ ⎥ ⎝ 3 ⎠⎦ (8. 10-30) where the entries are defined as follows rS′′(⋅) = rs + ra′′− rb′′cos(⋅) rM (⋅) = ′′ −ra′′ − rb′′cos(⋅) 2 (8. 10-31) (8. 10-32) 330 ALTERNATIVE FORMS OF MACHINE EQUATIONS ra′′= rd′′+ rq′′ 2 − rs 3 3 (8. 10-33) rb′′= rd′′− rq′′ 3 (8. 10-34) The stator voltage equations given in (8. 10-24), along with the rotor state equations pλ kqj = − rkqj (λ kqj − λ mq ); Llkqj j = 1, M (8. 10-35) pλ... Figure 8. 10-5 Operational impedance of PVVBR model and PVVBR model with auxiliary damper winding to force Lb = 0 ′′ r r vqs = −rsiqs − ω e Ld ids + eq ′′ r ′′ (8. 11-1) v = −r i + ω e L ′′i + ed ′′ (8. 11-2) r ds r s ds r q qs where eq = ω e λ d ′′ ′′ (8. 11-3) ed = −ω e λq ′′ ′′ (8. 11-4) Equations (8. 11-1) and (8. 11-2), together with the state equations of the rotor flux linkages of (8. 10-35)– (8. 10-37)... ELECTRIC TRANSIENTS NEGLECTED 323 Figure 8. 9-1 Dynamic performance of a steam turbine generator during a three-phase fault at the terminals predicted with stator electric transients neglected Te,106N•m 8. 88 4.44 0 90 d, electrical degrees 180 –4.44 Figure 8. 9-2 Torque versus rotor angle characteristics for the study shown in Figure 8. 9-1 324 ALTERNATIVE FORMS OF MACHINE EQUATIONS From all outward appearances,... (PVCC) form, that is, machine variables The PVCC model of the induction machine is provided in (6.2-19) and (6.3-7) The PVCC model of the synchronous machine is given by (5.2-1), (5.2-2), (5.2-7), and (5.3-4) An example of the use of the PVCC of the synchronous machine is shown in Figure 8. 10-2, wherein a machine is coupled to a diode rectifier using the branch elements of Figure 8. 10-1 Both the stator... j =1 lkqj ∑L 1 1 + + Lmd Llfd (8. 10 -8) N 1 j =1 lkdj ∑L (8. 10-9) Using, (8. 10-6) and (8. 10-7), the stator flux linkages are then expressed as r λqs = Lq iqs + λq ′′ r ′′ (8. 10-10) λ = L ′′i + λ d ′′ (8. 10-11) r ds r d ds where Lq = Lls + Lmq and Ld = Lls + Lmd are the dynamic inductances The dynamic flux ′′ ′′ ′′ ′′ linkages are given by 3 28 ALTERNATIVE FORMS OF MACHINE EQUATIONS ⎛ λq = Lmq ⎜ ′′ ′′ ⎝... TABLE 8. 5-1 Induction Machine Eigenvalues Rating, hp 3 50 500 2250 Stall Rated Speed No Load −4.57 ± j377 −313 ± j377 1.46 −2.02 ± j377 −1 98 ± j377 1. 18 −0 .87 2 ± j377 −70.3 ± j377 0.397 −0.4 28 ± j377 −42.6 ± j377 0.241 85 .6 ± j313 −223 ± j83.9 −16 .8 −49.4 ± j356 −142 ± j42.5 −14.4 −41 .8 ± j374 −15.4 ± j41.5 −27.5 −24.5 ± j376 −9.36 ± j41.7 −17.9 89 .2 ± j316 −2 18 ± j60.3 −19.5 −50.1 ± j357 −140 ± j 18. 2... (8. 7-29) r kq 2 r lkq 2 kq 2 r qs r kq1 r kq 2 r r ψ ′fd = Xlfd i ′ r + X md (−ids + i ′ r + ikd ) ′ fd ′r fd (8. 7-30) ψ ′ = X ′ i ′ + X md (−i + i ′ + i ′ ) (8. 7-31) r kd r lkd kd r ds r fd r kd As in the case of the induction machine, the voltage equations for the synchronous machine may be written in terms of the currents Hence, (8. 7-32) results from the substitution of (8. 7-26)– (8. 7-31) into (8. 7-20)– (8. 7-25)... is positive over the positive-slope region of the torque-speed curve, becoming negative after maximum steady-state torque 8. 6 EIGENVALUES OF TYPICAL SYNCHRONOUS MACHINES The linearized transformations, (8. 3-22) and (8. 3-23), and the machine equations (8. 319) may each be considered as components as shown in Figure 8. 3-1 The eigenvalues of the two synchronous machines, each connected to an infinite bus, . () 11 (8. 3-16) which is in the form of (8. 3-15) with AEF= − () 1 (8. 3-17) BE= − () 1 (8. 3- 18) Synchronous Machines Linearizing (8. 2-4)– (8. 2 -8) . frame of each machine as depicted in Figure 8. 3-1 . Figure 8. 3-1. Interconnection of small-displacement equations of a synchronous machine: Park ’ s equations.