1. Electromechanical equations. Electromechanical equations are to model the effect of input-output imbal ance on the rotor speed (and therefore on the operating frequency ). The rotor of each machine can be described by the so-called swing equation d-e where e denotes the rotor position relative to a certain rotating frame, M the inertia of rotor, and d damping The term de/dt represents the angular velocity and d20/dr is the angular acceleration of the rotor. The preceding differential equation is derived from Newtons law for rotational motions and, in some respects, resembles the namical equation of a swinging pendulum(with Pin -driving torque, and Pout"restoring torque). The term Pin, which drives the rotor shaft, can be considered constant in many cases. The term Pout, the power sent out to the system, may behave in a very complicated way. Qualitatively, Pout tends to increase(respectively, decrease, as the rotor position moves forward(respectively, backward) relative to the synchronous rotating frame However, such a stable operation can take place only when the system is capable of absorbing(respectively, providing) the extra power. In a multimachine system, conflict might arise when various machines compete with each other in sending out more(or sending out less)electrical power; as a result, the stabilizing effect 2. Electromagnetic equations. The(nonlinear)electromagnetic equations are derived from Faraday s law of electromagnetic induction--induced emfs are proportional to the rate of change of the magnetic fluxes. A general form is as follows λ,+λ- dt (66.1) λ,+λ6 d = g(s)if- xd(s)id (s)i The true terminal voltage, e.g., e, for phase a, can be obtained by ombining the direct-axis and quadrature-axis components ea and respectively, which are given in Eq(66.1). On each line of Eq (66.1), P-model>SYSTEM Machin the induced emf is the combination of two sources the first is the rate of change of the flux on the same axis [(d/dt)nd on the first line, (d/dr)%a on the second ] the second comes into effect only when a Igiven by(d/dn)e]. The third term in the voltage equation represents a qualitative relationship among various the ohmic loss associated with the stator winding ectrical and mechanical quantities of a e Equation(66. 2) expresses the fluxes in terms of relevant currents: synchronous machine.e,,6,,e,are phase x is equal to inductance times current, with inductances G(s),X(s), voltages; ia,ib, i phase currents; iF rotor X,(s)given in an operational form(s denotes the derivative operator). field current; 0 relative position of rotor; Figure 66.6 gives a general view of the input-output state descri o deviation of rotor speed from synchro. tion of machine's dynamic model, the state variables of which appear The state variables appear in Eqs. (66.1) nous speed; Pin mechanical power input. in eqs.(66.1)and(662) d(66.2) e 2000 by CRC Press LLC© 2000 by CRC Press LLC 1. Electromechanical equations. Electromechanical equations are to model the effect of input–output imbal￾ance on the rotor speed (and therefore on the operating frequency). The rotor of each machine can be described by the so-called swing equation, where q denotes the rotor position relative to a certain rotating frame, M the inertia of rotor, and D damping. The term dq/dt represents the angular velocity and d2q/dt2 is the angular acceleration of the rotor. The preceding differential equation is derived from Newton’s law for rotational motions and, in some respects, resembles the dynamical equation of a swinging pendulum (with Pin ~ driving torque, and Pout ~ restoring torque). The term Pin, which drives the rotor shaft, can be considered constant in many cases. The term Pout, the power sent out to the system, may behave in a very complicated way. Qualitatively, Pout tends to increase (respectively, decrease) as the rotor position moves forward (respectively, backward) relative to the synchronous rotating frame. However, such a stable operation can take place only when the system is capable of absorbing (respectively, providing) the extra power. In a multimachine system, conflict might arise when various machines compete with each other in sending out more (or sending out less) electrical power; as a result, the stabilizing effect might be reduced or even lost. 2. Electromagnetic equations. The (nonlinear) electromagnetic equations are derived from Faraday’s law of electromagnetic induction—induced emf’s are proportional to the rate of change of the magnetic fluxes. A general form is as follows: (66.1) where (66.2) The true terminal voltage, e.g., ea for phase a, can be obtained by combining the direct-axis and quadrature-axis components ed and eq, respectively, which are given in Eq. (66.1). On each line of Eq. (66.1), the induced emf is the combination of two sources: the first is the rate of change of the flux on the same axis [(d/dt)ld on the first line, (d/dt)lq on the second]; the second comes into effect only when a disturbance makes the rotor and stator fields depart from each other [given by (d/dt)q]. The third term in the voltage equation represents the ohmic loss associated with the stator winding. Equation (66.2) expresses the fluxes in terms of relevant currents: flux is equal to inductance times current, with inductances G(s), Xd(s), Xq(s) given in an operational form (s denotes the derivative operator). Figure 66.6 gives a general view of the input–output state descrip￾tion of machine’s dynamic model, the state variables of which appear in Eqs. (66.1) and (66.2). M d dt D d dt P P 2 2 q q + = in – out e d dt d dt ri e d dt d dt ri d d q d q q d q = + = + Ï Ì Ô Ô Ó Ô Ô l l q l l q – – l l d F d d q q q Gsi X s i X s i = = Ï Ì Ô Ó Ô ( ) – ( ) – ( ) FIGURE 66.6 A block diagram depicting a qualitative relationship among various electrical and mechanical quantities of a synchronous machine. ea , eb , ec are phase voltages; ia , ib , ic phase currents; iF rotor field current; q relative position of rotor; w deviation of rotor speed from synchro￾nous speed; Pin mechanical power input. The state variables appear in Eqs. (66.1) and (66.2)