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International Journal of Science, Engineering and Technology Research (IJSETR) Volume 1, Issue 1, July 2012 Analysis of Transient Power-angle Characteristic for Synchronous Generator Naing Naing Htun Abstract— Transient condition may lead to instability where the machines in the power system fall out of synchronism. The electrical air-gap power depends on the generator loading and varies depending on the generator parameters and the power angle. It also depends on the operating state of the generator. Mechanical movement of the generator rotor is influenced by electromagnetic effects and this movement varies depending on the operating state of the generator. An important advantage of the classical model is that the generator reactance may be treated in a similar way to the reactance of the transmission lines and other network elements. the d-axis. Generators operating at lower speed, such as driven by hydro turbines, need many magnetic poles in order to operate 50 or 60 Hz. As the centrifugal forces experienced by the rotors of these low-speed machines are lower than in the corresponding turbo generators, salient-poles can be used and the rotor diameter increased. Normally salient-pole rotors have more than two poles and the angles and speed expressed in electrical units. The main problem with modeling a salient-pole machine is that the width of the air gap varies around the generator with the narrowest gap being along the d-axis and the widest along the q-axis. Index Terms—Synchronous Generator, Steady State Stability, Transient Stability, Constant Flux Linkage Model, Classical Model I. INTRODUCTION An electrical power system consists of many individual elements connected together to form a large, complex and dynamic system capable of generating, transmitting and distributing electrical energy over a large geographical area. Based on their physical character, the different power system dynamics may be divided into four groups such as wave, electromagnetic, electromechanical and thermodynamic. The electromagnetic phenomena mainly involve the generator armature and damper windings and partly the network. These electromechanical phenomena, namely the rotor oscillations and accompanying network power swings, mainly involve the rotor field and damper windings and the rotor inertia. As the power system network connects the generators together, this enables interactions between swinging generator rotors to take place. II. SYNCHRONOUS GENERATOR AND ITS EQUIVALENT NETWORK A. Salient-Pole Machine A synchronous generator consists of a stator, on which the three-phase armature winding is normally wound, and a rotor, on which the DC field winding is wound. The stator has three axes A, B and C each corresponding to one of the phase windings. The rotor has two axes such as the direct axis (d-axis), which is the main magnetic axis of the field winding, and the quadrature axis (q-axis), π/2 electrical radians behind Manuscript received Oct 15, 2011. Ma Naing Naing Htun, Electrical Power Engineering Department, Mandalay Technological University, (e-mail: [email protected]). Mandalay, Myanmar, 09401573078. Fig.1 A simplified salient-pole generator B. Stability Problem of Synchronous Machine The stability problem is concerned with the behavior of the synchronous machines after they have been perturbed. If the perturbation does not involve any net change in power, the machines should return to their original state. If an unbalance between the supply and demand is created by a change in load, in generation, or in network conditions, a new operating state is necessary. In any case all the interconnected synchronous machines should remain in synchronism if the system is stable. The transient following a system perturbation is oscillatory in nature, but if the system is stable, these oscillations will be damped toward a new quiescent operation condition. These oscillations are reflected as fluctuations in the power flow over the transmission lines. If a certain line connecting two groups of machines undergoes excessive power fluctuations, it may be tripped out by its protective equipment thereby disconnecting the two groups of machines. This problem is termed the stability of the tie line, even though in reality it reflects the stability of the two groups of machines. A statement declaring a power system to be stable is rather ambiguous unless the conditions under which this stability has been examined are clearly stated. This includes the operating conditions as well as the type of perturbation given to the system. The same thing can be said 1 All Rights Reserved © 2012 IJSETR International Journal of Science, Engineering and Technology Research (IJSETR) Volume 1, Issue 1, July 2012 about tie-line stability. Since, there is concerned with the tripping of the line, the power fluctuation that can be tolerated depends on the initial operating condition of the system including the line loading and the nature of the impacts to which it is subjected. C. The Equivalent Network Impedance Synchronous generators are rarely used to supply individual loads but are connected to a power system that consists of a large number of other synchronous generators and loads linked by the transmission network. The power ratings of all remaining generators in the system can be treated as one equivalent, very large, generating unit with an infinite power rating. This equivalent generating unit is referred to as the infinite busbar and can be represented on a circuit diagram as an ideal voltage source behind equivalent system impedance. The infinite busbar maintains a constant terminal voltage and is capable of absorbing all the active and reactive power output of the generator. This concept is illustrated in figure (2) where the generator is connected to the system via a step-up transformer represented by the series impedance. It is assumed that the ideal transformer has been eliminated from the circuit diagram by either using per units or recalculating all the quantities using a common voltage level. The rest of power system is represented by the infinite busbar that is by the ideal voltage source Vs behind the equivalent system impedance. The infinite busbar is assumed to have constant voltage and frequency, neither of which is influenced by the action of an individual generator. This means that the voltage Vs can be used as reference and the phase angles of all the other voltages and currents in the circuit measured with respect to it. The impedance of the power angle δ is defined as the phase shift between Er and Vs. As all the angles in the phasor diagram have a dual time/space meaning, δ is also the spatial angle between the two synchronously rotating rotors. This spatial angle is referred to as the rotor angle and has the same numerical value in electrical radians as the power angle. As the rotor of the infinite busbar is not affected by an individual generator, this rotor also provides a synchronously rotating reference axis with respect to which the space position of all the rotors may be defined. The elements of the equivalent circuit may be combined to give the total parameters as x d =Xd +XT +Xs (1) x q =X q +X T +Xs (2) r=R+R T +R s (3) The salient-pole generator can be described as E f = Vs + rI + jx d Id + jx q Iq (4) Fig.2 Equivalent circuit of the generator operating on an infinite busbar III. ARMATURE FLUX PATHS AND THE EQUIVALENT REACTANCE Figure (3) shows three characteristic states that correspond to three different stages of rotor screening. Immediately after the fault, the current induced in both the rotor field and damper windings forces the armature reaction flux completely out of the rotor to keep the rotor flux linkages constant as shown in figure (3a), and the generator is said to be in the sub-transient state. As energy is dissipated in the resistance of Fig.3 patch of the armature flux in (a) the sub-transient state (b) the transient state (c) the steady state the rotor windings, the currents maintaining constant rotor flux linkages decay with time allowing flux to enter the windings. As the rotor damper winding resistance is the largest, the damper current is the first to decay, allowing the armature flux to enter the rotor pole face. However, it is still forced out of the field winding itself, Figure (3b), and the generator is said to be in the transient state. The field current then decays with time to its steady-state value allowing the armature reaction flux eventually to enter the whole rotor and assume the minimum reluctance path. This steady state is illustrated in Figure (3c). It is convenient to analyze the dynamics of the generator separately when it is in the sub-transient, transient and steady states. This is accomplished by assigning a different equivalent circuit to the generator when it is in each of the above states, but in order to do this it is first necessary to consider the generator reactances in each of the characteristic states. Fig.4 Three-step approximation of the generator model: (a) rms value of the AC component of the armature current (b) generator reactances If a generator is in the sub-transient state, and the armature mmf is directed along the rotor d-axis, then the armature reaction flux will be forced out of the rotor by the currents induced in the field winding, the damper winding and the rotor core. This flux path corresponds to the direct-axis sub-transient reactance Xd''. On the other hand, if the armature mmf is directed along the rotor q-axis, then the only currents forcing the armature reaction flux out of the rotor are the rotor core eddy currents and the currents in the q-axis damper winding. If a generator only has a d-axis damper winding, then the q-axis screening effect is much weaker than that for the d-axis, and the corresponding quadrature axis sub-transient reactance Xq''is greater than Xd''. This difference between Xq''and Xd'' is called sub-transient saliency. For a generator with a damper winding in both the d-axis and the q-axis, the screening effect in both axes is similar, sub-transient saliency is negligible and Xq'' ≈ Xd''. 2 All Rights Reserved © 2012 IJSETR International Journal of Science, Engineering and Technology Research (IJSETR) Volume 1, Issue 1, July 2012 When the generator is in the transient state, screening is provided by the field winding which is only in the d-axis. However, in the round-rotor generator some q-axis screening will be produced by eddy currents in the rotor iron with the effect that Xq' > Xd'. The actual value of Xq' is somewhere between Xd' and Xq with typically Xq' ≈ 2Xd'. In the salient-pole generator the laminated rotor construction prevents eddy currents flowing in the rotor body, there is no screening in the q-axis and Xq' = Xq. Because of the absence of a field winding in the q-axis, there is some degree of transient saliency in all types of generator. IV. STEADY-STATE AND TRANSIENT CHARACTERISTICS ON THE POWER-ANGLE DIAGRAM A. Steady-state Power-Angle Characteristics Figure (5) shows the phasor diagram of a generator operating under a particular load as defined by the length and direction of the phasor I. Obviously the diagram will change if the load is changed. The excitation is constant, Ef = Eq = constant, and any change in load will change the generator terminal voltage Vg , the angle δg and the power angle δ. If the resistance r is neglected the real power supplied by the generator to the system is given by the following equation. PsEq = E q Vs xd sinδ + Vs2 x d -x q sin 2δ 2 xd xq (5) Where PsEq is a function of the power angle δ and function PsEq(δ) is referred to as the power-angle characteristic of the generator operating on the busbar. The reluctance power term deforms the sinusoidal characteristic so that the maximum of PsEq(δ) occurs at δ < π/2. winding (transient period). Consequently, changes in the armature flux can penetrate the damper winding. C. Constant Flux Linkage Model Assume that the generator is connected to the infinite busbar as shown in figure (6) and that all the resistances and shunt impedances is associated with the transformer and network can be neglected. The corresponding equivalent circuit and phasor diagram of salient-pole generator in the transient state is as shown in figure (6). The fictitious rotor of the rotor of the infinite busbar serves as synchronously rotating reference axis. The reactances of the step-up transformer and the connecting network can be combined with that of the generator to give x'd = X'd +X , x'q = X'q +X , X=XT +Xs Where Xd' and Xq' are the d- and q-axis transient reactance of the generator. The voltage equations can be constructed as E'd = Vsd +x q' Iq , Eq' = Vsq +x d' Id (7) Where Vsd and Vsq are the d- and q-components of the infinite busbar voltage Vs given by Vsd = −Vs sin δ and Vsq = Vs cos δ. As all resistances are neglected, the air-gap power is Pe = Vsd Id Vsq Iq (8) Assuming constant rotor flux linkages, then the values of the emfs Ed' and Eq' are constant, implying that both E' = constant and α = constant. The generator power-angle characteristic Pe(E', δ') can be described in terms of the transient emf and the transient power angle and is valid for any type of generator. A generator with a laminated salient-pole rotor cannot produce effective screening in the q-axis with the effect that xq'= xq. Inspection of the phasor diagram in Figure (6b) shows that in this case E lies along the q-axis so that α = 0 and δ' = δ. Pe = PE' δ' = Xd’ Eq’ Ed ’ E q ' Vs xd' sinδ ' - ' ' Vs2 x q - x d sin2δ ' ' ' 2 xd xq (9) Id Vs X q’ Fig.5 Power-angle characteristic B. Transient Power-angle Characteristics The disturbance acting on a generator will produce a sudden change in armature current and flux. This flux change induces additional currents in the rotor filed and damper windings that expel the armature flux into high-reluctance paths around the rotor so as to screen the rotor and keep the rotor flux linkage constant. As the emf Eq is proportional to the filed current, the additional induced field current will cause changes in Eq so that the assumption of constant Eq used to drive the static power angle characteristic. The induced rotor currents decay with time as the armature flux penetrates first the damper windings (sub-transient period) then the field (6) q I Iq Vs Vs d Figure.6 Generator-infinite busbar system in the transient state (a) circuit diagram (b) phasor diagram of the salient-pole generator (xq' =xq) D. Classical Model In this model, all the voltages, emfs and currents are phasors in the network reference frame rather than their components resolved along the d- and q-axes. The classical model can be expressed by ignoring transient saliency, that is assuming xd' ≈ xq'. 3 All Rights Reserved © 2012 IJSETR International Journal of Science, Engineering and Technology Research (IJSETR) Volume 1, Issue 1, July 2012 Pe = PE' δ ' Eq ' Vs xd' sin δ ' (10) Consequently, when the network reactance is large the classical model, and the constant flux linkage model give very similar results even for a generator with a laminated salient-pole rotor. It is important to note that δ is the angle between Vs and E and not the angle between Vs and the q-axis. During the transient period the emfs Ed and Eq are assumed to be constant with respect to the rotor axes and α is also constant with δ= δ + α ' (11) E'd0 = Vgd +Xq ' Iq0 E 'd0 = (- 0.690) + 0.69×1.001 =0.00069 pu 0 pu E'q0 = Vgq - Id0 Xd' E'q0 = 0.737 - -1.800 0.3=1.277 pu E''d0 = Vgd +Xq '' Iq0 E ''d0 = -0.690 + (0.27) (1.001) = - 0.419pu E''q0 = Vgq - Id0 Xd'' E''q0 = 0.737 - -1.800 0.25 =1.187pu I0 = 2.06 -17.8 φg0 = 17.8 deg The total reactances are xd = 0.995pu, xq = 67.7pu, xd= 0.32pu Taking Vg as reference, the phasor of the transient emf is E' = 1.27743.13 Vs =Vg - j XT +XL I Fig.7 Classical model of the generator in the transient state (a) circuit diagram, (b) phasor diagram V. CALCULATION OF STEADY-STATE AND TRANSIENT POWER-ANGLE CHARACTERISTICS A 230MVA salient-pole generator is referred to Yeywa hydro generating station and which can be produced 790MW. The generator real and reactive power are 1.98pu and 0.64pu. The constant flux linkage and the classical generator model are used to find and plot the steady-state and transient characteristics. * S P - jQ 1.98-j0.64 I0 = = = = 2.06 -17.8 V V 1.01 g g E Q =Vg +jXq I0 Vs = 1.01 - j 0.065×2.06 -17.8 = 1.06 - 6.92 δ0 = δ0' = 43.13 + 6.92 = 50.05° δ +φ = 50.05 + 17.8 = 67.85° 0 0 The d- and q-components of the system voltage are Vsd = -Vg sin δg0 Vsd = - 1.01 sin 50.05 = -0.774 pu Vsq = Vg cos δg0 Vsq = 1.01 cos 50.05 = 0.65pu PEq (δ) = E q Vs xd sinδ + Vs2 x d -x q sin2δ 2 xd xq E Q =1.01 + j (0.69× 2.06 -17.8) = 1.9843.13 PEq (δ) = 2.57 sin δ + 0.265sin 2δ δg0 = 43.13°, φg0 = 17.8° The transient characteristic can be calculated, PE' δ = Id0 = -I0 sin φg0 + δg0 Id0 = -I0 sin φg0 + δg0 I0 = - 2.06sin 17.8 +43.13 = -1.800 pu E q ' Vs xd' ' ' Vs2 x q - x d sinδ sin2δ ' ' ' 2 xd xq ' PE' δ = 4.13 sin δ - 0.883 sin 2δ Iq0 = I0 cos φg0 + δg0 The approximated transient characteristic for the classical model can be calculated assuming xd' = xq'. The transient emf, calculated with respect to Vs is Iq0 = 2.06cos(17.8 +43.13)= 1.001pu E ' = Vs + j X d' I Vgd = -Vg sin δg0 E' = 1.06 + j0.328 × 2.06 -17.8 = 1.421 26.93 Vgd = - 1.01 sin 43.13 = -0.690 pu α = δ - δ' = 50.05 - 26.93 = 23.12° E' = 1.421V, δ' = 26.93° Vgq = Vg cos δg0 α = δ - δ' = 50.05 - 26.93 = 23.12° Vgq = 1.01 cos 43.13 = 0.737pu The approximated transient characteristic can be calculated as E q0 = Vgq -Id0 X d E q0 = 0.737- (- 1.800) (0.93) = 2.411pu PE' δ' = Eq ' Vs xd ' sinδ' 4 All Rights Reserved © 2012 IJSETR International Journal of Science, Engineering and Technology Research (IJSETR) Volume 1, Issue 1, July 2012 PE' δ' = 1.421×1.06 sin δ' = 4.952 sin δ' 0.328 Fig.8 Steady-state and transient characteristics of laminated salient-pole machine VI. CONCLUSION The classical model gives a good approximation of the constant flux linkage model. This characteristic is shifted with respect to PEq(δ) by α = 23.12°. Figure (8) shows that by neglecting transient saliency the classical model does not generally significantly distort the transient characteristics. As xd > xd', the amplitude of the transient characteristic is greater than the amplitude of the steady-state characteristic. ACKNOWLEDGMENT The author would to express her special thanks to her parents for their noble support and encouragement. The author would like to express her gratitude to Dr. Khin Thuzar Soe, Associate Professor, Head of Department of Electrical Power Engineering, Mandalay Technological University, for providing encouragement. The author especially appreciates and thanks her teachers at MTU and corresponding people who helped her directly or indirectly for this paper. The author would like to thank U Nay Zar Win, Executive Engineer from Ministry of Electric Power Engineering, for providing the required data and information. REFERENCES [1] [2] [3] [4] [5] [6] [7] P. Kundur, “Power System Stability and Control”, EPRI Power System Engineering Series (Mc Graw-Hill, New York, 1994). I. J. Nagrath and D. P. Kothari, “Power System Engineering”, Tata McGraw-Hill, New Delhi, 1994. Pranamita Basu and Aiswarya Harichandan, “POWER SYSTEM STABILITY STUDIES USING MATLAB”, National Institute of Technology Rourkela, Rourkela-769008, Orissa. Saadat Hadi, Power System Analysis, New Delhi, Tata McGraw-Hill Publishing Company Limited, 2002. [5] www.homerenergy.com Leonard L.Grisby, “Power System Stability and Control”, Second Edition. Jan Machowski, Janusz W. Bialek, James R. Bumby, “Power System Dynamics: Stability and Control”, Second Edition. 5 All Rights Reserved © 2012 IJSETR