Moelling of Three Phase Short Circuit an Measuring Parameters of a Turbo Generator for Improve Performance M. Olubiwe, S. O. E. Ogbogu, D. O. Dike, L. Uzoechi Dept of Electrical an Electronic Engineering, Feeral University of Technology Owerri, Nigeria Abstract - For use in power system stability simulations, utilities an system operators may esire to erive accurate moel parameters of generators. Ajustments may have been mae by fiel or operating personnel that have altere the response of the euipment. In these situations, there is a nee to obtain more accurate moels for simulation. Of a great extent in time, the research works irecte at stuying transients in synchronous generators have not yet provie fully sufficient comparative stuies in respect to suen short circuits of the machine. The present paper puts forwar an iea of comprehensive process moels for ynamic moelling of short circuit faults of unloae synchronous generator, using the generalize --0 mathematical moel as starting point in erivation. Also, the computational efficiency is being increase by introucing a set of auxiliary variables common to ifferent state euations. The moels erivation is carrie out without altering the structural euations of the generalize --0 mathematical moel of synchronous generators. The research has also propose ways of obtaining synchronous machine parameters through irect test or measuring techniues. A plot of open circuit an short circuit characteristics shows why an how saturation shoul be consiere to obtain the internal voltage, hence the reactance of the generator when suen short circuit occurs. Key Wors: Synchronous generator, ynamic moelling, open circuit, Short circuit, auxillary variables. INTRODUCTION Moelling of short circuit characteristics of synchronous generator is an ongoing research for many researchers. The essence of it is to have inept knowlege of the ynamic performance either when in offline moe or on line moe. Different ratings of synchronous machines have been use an these vary from a few watts to hunres of megawatts. Synchronous machine operating as a generator is of particular interest because it prouces power to the mains. Its rotating spee is proportional to the freuency of the alternating supply an is inepenent of the loa. When it elivers power, the electromagnetic torue evelope in the generator opposes the torue of the prime mover. A synchronous machine is an ac rotating machine whose spee uner steay state conition bears a constant relationship to the freuency of current in the armature wining. A synchronous machine is one of the important types of electric machines. Large ac networks operating at constant freuency of 50Hz or 60Hz rely almost exclusively on synchronous generator, also calle alternators for the supply of electrical energy an may have synchronous compensators at key points for control of reactive power. 1.1 Over view of a Synchronous Generator The three phase machines are the largest an also perhaps the most common electric machines that run at synchronous spee. The higher efficiency of synchronous machine over others is an avantage at a higher power rating. Another avantage that makes a synchronous machine ifferent from other machines is that varying its fiel excitation can vary its power factor of operation. This property makes it useful for the inustry which is always operating at low lagging power factor. Part of the loa is hanle by synchronous machine whose fiel is ajuste such that it is operating at leaing power factor to improve the overall power factor to nearly unity. There are two types of synchronous generator; the stationary fiel an the rotating c magnetic fiel. The only ifference between these two types of generator is their armature. The stationary fiel synchronous generator has salient poles mounte on the stator. The poles are magnetize either by permanent magnets or by a c current. The armature normally containing a three phase wining is mounte on the shaft, the stator winings are each space 120 0 apart from each other roun the stator. Voltages of eual magnitue are inuce so proucing a set of balance three phase supply voltages, which can be use as either an isolate supply or can be connecte in parallel with other generators to make up gri system in which all generators prouce the same freuency an are in phase synchronism with one another. The rotating magnetic fiel (also known as revolving fiel) synchronous generator has the fiel winings woun on the Rotor an the armature woun on the Stator. A c current creating a magnetic fiel that must be rotate at synchronous spee energizes the rotating fiel wining. The fiel wining can be energize through a set of slip rings an brushes (external excitation) or from a ioe brige mounte on the rotor shaft (self excite). Large ac generators usually have exciters consisting of an ac source 2264
with soli state rectifiers. The rectifier-brige is fe from a shaft-mounte alternator which is itself excite by the pilot exciter. In externally fe fiels, the source can be a shaftriven DC generator, a separately excite DC generator or a soli state rectifier. [1]. The synchronous generators are built with either salient pole or cylinrical iron rotors, epening on the spee an size of the machine. To any machine esigner an power engineer, the preiction of the machine s performance characteristics is of highest priority [2]. In orer to unerstan the characteristics of a machine, there are a number of ifferent techniues available. These techniues are able to etermine parameters efining the steay state, transient an subtransient response of a machine. Analysis of synchronous machines has been implemente using ifferent approaches such as open circuit step response test [3], time omain test using finite-elements [4] an time omain ientification of generator transfer functions [5]. Moeling of the ynamics of three-phase roun-rotor or salient-pole machine is important in using any of these methos. Synchronous generator short circuit stuies are an essential tool for the power system esigner. The task is to calculate the fault conitions an to provie protective euipment esigne to isolate the faulte generator from the remainer of the system in the appropriate time. The interrupting capacity of breakers shoul be chosen to accommoate the largest of short circuit currents an hence care must be taken not to base the protection ecision simply on the results of a balance three-phase short circuit. The circuit breakers are capable of carrying for a short time the specifie short circuit current. However, the possibility of catastrophic failure exists if the short circuit currents are not properly calculate an the breakers are subjecte to fault uties that excee their rating. The stator phase an rotor fiel currents at short circuit stressing take angerous values, thermally overloaing the installation. The critical value of electromagnetic torue, uring short circuit transient, has to be known by the generator esigner to appraise the mechanical strength of the structure. The vast majority of commercial software esigne for short circuit transient analysis are base on the empirical calculations encompasse by the accepte stanars. Dynamic simulation of short circuit faults is always an option, not expressly for valiation of the results receive from stanarize calculations but also for an accurate an effective representation of the transient behaviour. The present paper escribes various novel, comprehensive, an general process moels for moelling of synchronous generators short circuit transients. The moels erivation was carrie out without altering the structural euations of the generalize --0 mathematical moel of synchronous generator. The propose simulation techniue offers the avantage of an increase computational efficiency by introucing auxiliary variables, common to ifferent state euations. The time-consuming matrix inversion at each step of numerical integration, performe when currents are selecte as state variables an with a view to computing the currents erivatives, is eliminate by avancing the moels in a convenient split matrix form that allows symbolic processing. For practical purposes, besies the time-omain analysis, the peak values of short circuit characteristic uantities are epicte as epenencies upon the initial value (at short circuit occurrence) of the rotor lag angle. Such kin of representation consierably facilitates the examination of the ifferences among the results corresponing to ifferent short circuit types, hence permitting to stuy the synchronous generator behaviour closely. [6] 2.0 THE BASICS OF THREE PHASE FAULT A balance 3-phase fault implies that all three phases of the power system are simultaneously short-circuite to each other through a irect or "bolte" connection. * Often, a 3-phase fault prouces the largest short-circuit current magnitue; thus, this worst-case result is then use as the basis to select the short-circuit capabilities of switchgear from manufacturers' tables. * Short-circuit calculations are easier for a balance 3-phase fault because symmetry of the fault connection permits one to consier only one of the three phases. Figure 1 Balance three phase circuit The other types of unbalance short-circuit faults are important in selecting the time-current characteristics an settings of phase-overcurrent an groun-fault protective evices to provie selective coorination. This coorination assures service continuity an minimizes amage to switchgear an loa euipment. 2265
2.1 Symmetrical RMS Current versus Short-circuit Duty In general, a 3-phase synchronous generator, previously unloae, that has been subjecte to a balance, 3-phase fault across its accessible terminals is use. (momentary) network, accoring to the entries in the secon column of Tables 4-1 an 4-2 of the 1993 eition of the IEEE Re Book (or Tables 24 an 25 of the 1986 eition). 2.4 Approximately 30 cycle network. The root mean-suare (rms) value of the asymmetrical short-circuit current waveform obtaine, is the basis for the selection of the short-circuit capabilities of circuit breakers an fuses. Calculation of the precise rms value of an asymmetrical current at any time after the inception of a short-circuit may be very involve. Accurate ecrement factors to account for the DC component at any time are reuire, as well as factors for the rate of change of the apparent reactance of the generators. This precise metho may be use, if esire; however, simplifie methos have evolve whereby the DC component is accounte for by simple multiplying factors. These multiplying factors convert the rms value of the symmetrical AC component (symmetrical rms current) into rms current of the asymmetrical waveform, incluing the DC component (asymmetrical rms current or short-circuit current uty). 2.1 Types of Networks use to Calculate Symmetrical rms Current In orer to utilize AC circuit theory in calculating symmetrical rms current, three types of networks are use to represent the power system over three time intervals of the fault-on time perio. * First-cycle (momentary) network. * Contact-parting (interrupting) network. * Approximately 30 cycle network. These networks only iffer from one another by the assignments of constant reactances for the machines. 2.2 First-cycle(momentary) Network. This network is use to calculate the first-cycle (momentary) symmetrical rms current. Here, the rotating machine sources of short-circuit current are represente, for the most part, by their subtransient reactance, accoring to the entries in the first column of Tables 4-1 an 4-2 of the 1993 eition of the IEEE Re Book (or Tables 24 an 25 of the 1986 eition). 2.3 Contact-parting (interrupting) network. This network is use to calculate the contact-parting (interrupting) symmetrical rms current for circuit breaker minimum contact-parting times of 1.5 to 4 cycles after the inception of the short-circuit fault. Here, the rotating machine sources of short-circuit current are represente by ifferent constant reactances than the first-cycle This network is often a minimum-source representation to investigate whether minimum short-circuit currents are sufficient to operate current-actuate relays. Minimumsource networks might apply at night or when prouction lines are own for any reason. Some of the source circuit breakers may be open an all motor circuits may be off. Inplant generators are represente with transient reactance or a larger reactance that is relate to the magnitue of ecaying generator short-circuit current at the esire calculation time. 3.0 THE D-Q-0 MODEL OF SYNCHRONOUS GENERATOR The generalize mathematical moel of a turbo synchronous generator encompasses two istinctive sets of structural euations. These are the ifferential euations, that is, voltage an motion euations, an the algebraic correlations between flux linkages an currents (the flux euations). The voltage euations of synchronous generators are given by means of the following orinary ifferential euations. [7-11] Stator Voltage euation v = Ri + τ λ ωλ (1) v = Ri + λ + ωλ τ (2) For the Rotor winings: v f = R f i f + λ f 0 = R D i D + λ D 0 = R Q i Q + λ Q Where: v enotes voltage i enotes current λ enotes flux linkage - enotes irect axis components f, D, Q enote variables an parameters associate with fiel wining an the - axis amper winings respectively. R enotes resistance ω enotes rotor spee (3) σ enote stator an rotor leakage inuctances 3.1 The evelopment of Three Phase Short Circuit Note the following relations for the evelopment of the three phase short circuit moels L = L σ + L m L = L σ + L m 2266
L f = L fσ + L m L D = L Dσ + L m L Q = L Qσ + L m a 0 w c The substitution X which can be current, voltage or flux linkage are to be performe in euations (3.5) an (3.6). The mapping (3.5) performs the transformation of the stator wining variables to a coorinate system in which the rotor is stationary. The euivalent wining are ientifie in the irect an uaratic axis. The irect axis (-axis) wining is the euivalent one of the phase wining but aligne irectly with the fiel. The uarature axis (-axis) is situate so that its axis is perpenicular to the axis of rotor fiel wining. In orer to improve legibility in presentation an to increase computational efficiency uring numerical integration, the following set of auxiliary variables are introuce: L 1 = L cos γ, L 2 = L sin γ, L 1 = L cos γ, L 2 = L sin γ, L m1 = L m cos γ, L m2 = L m sin γ, (7) L m1 = L m cos γ, L m2 = L m sin γ. Flux euation λ = L i + L m i m = L i + L m. i f + i (8) λ = L i + L m i m = L i + L m. i f + i (9) b Figure 2 Systems of stator phase axes an the - references frame. Rotor voltage Euations to eliminate flux variables i L + L i f i m τ f + L = R τ m τ f i f + v f (10) a = phase A axis = Direct axis b = phase B axis = Quaratic axis c = phase C axis Figure 2 shows that rotor uantities are referre to stator, it also show that the lag angle of the rotor γ is measure between the stator phase A axis an the irect (-axis) an its ecrease in time, correspon to a positive rotor angular velocity The irect an converse Park-Goren transform are given by X X 2 2 4 X cos cos cos A X B X c 3 3 3 2 2 4 X sin sin sin A X B X c 3 3 3 (5) X A = X cos + X sin + X o X B = 2 cos X 3 4 cos 3 2 sin 3 X +X o 4 3 (4) Xc = X X sin +X o. (6) respectively. i L + L i f i m τ m + L = R τ τ i (11) i L + L i = R m τ τ i (12) With rotor excitation voltage v f as input variable. Also replacing the stator - axis flux linkages, in generalize electromagnetic torue, using correlation (8) an (9) the electromagnetic torue is expresse in terms of current in - axis: T em = 3 2 p L m i m i + L m i m i. (13) The restrictive conition in this case is i A + i B + i C = 0 i 0 = i A +i B +i C 3 = 0 (14) v A = v B = v C = 0 or by applying the irect Park-Gorev transformation; v = v = 0 (15) By employing the current-base expressions (8) an (9) to eliminate the flux variables from stator voltage, the following processe voltage euations result: i L + L i f τ τ (16) L i + L τ m (17) γ τ ω + L m i τ = Ri + ω. L i + L m i i τ = Ri ω. L i + L m i f + L m i = ω (as state in (4)) τ = p J T em... (18) Where, 2267
P = Generator pole pairs T em = The riving turbine torue J = Euivalent moment of inertia γ = Rotor lag angle The short circuit moel follows by coupling (15) an (16) with the processe rotor voltage euation (10-12), (13) an (18) given interms of - axis current. In this case, the electro-magnetic torue preserves the general expression given for electromagnetic torue in terms of - axis currents. The moel incorporates seven ifferential euations an the vector of state variables inclues the stator - axis currents, the rotor (fiel, amper) current, angular velocity an the rotor lag angle. At each step of numerical integration, the stator phase currents result by means of converse Park-Gorev transform (6) having in view of euation (14) that points the annulment of the zero seuence components. Thus the three phase short circuit Moel is shown below: i A = i cosγ + i sinγ (19) Figure 3 Short circiut characteristics of a turbo generator i B = 1 2 i + 3i cos γ 1 2 3i + i sin γ (20) Figure 4 The euivalent generator s circuit uring short circuit per phase. i C = 1 2 i + 3i cos γ + 1 2 3i i sin γ (21) 4.0 MEASURING PARAMETERS OF SYNCHRONOUS GENERATOR MODEL The three uantities that must be etermine in orer to escribe the generator moel are: 1. The relationship between fiel current an flux (an therefore between the fiel current I F an the internal generate voltage E A ); 2. The synchronous reactance; 3. The armature resistance. The open-circuit test on the synchronous generator is first conucte: the generator is rotate at the rate spee, all the terminals are isconnecte from loas, the fiel current is set to zero first. Next, the fiel current is increase in steps an the phase voltage (which is eual to the internal generate voltage E A since the armature current is zero) is measure. Therefore, it is possible to plot the epenence of the internal generate voltage on the fiel current the opencircuit The SCC is a straight line since, for the short-circuite terminals, the magnitue of the armature current is: I A = E A (22) R A 2 XS 2 Where I A is Armature current R A is Armature resistance X s is synchronous reactance Figure 5 The resulting phasor iagram Figure 6 The magnetic fiels uring short-circuit test Since B S almost cancels B R, the net fiel B net is very small. Where: B R is the rotor magnetic fiel B S is the stator magnetic fiel B net is the net magnetic fiel Also since the unsaturate core of the machine has a reluctance thousans times lower than the reluctance of the air-gap, the resulting flux increases linearly first. When the saturation is reache, the core reluctance greatly increases 2268
causing the flux to increase much slower with the increase of the mmf. Z s = R A 2 + X s 2 Open circuit per phase voltage Short circuit per phase current = E A I ASE = X s (23) Uner the assumptions that the synchronous reactance Xs an the inuce emf Ea have the same values in both the open an short circuit tests, an that Xs >> Ra, Figure 7 Connection for open circuit test 4.2 Conclusion Three-phase synchronous machines account for a high percentage of any country s power generation. Unerstaning the machine s ynamic characteristics an etermining its euivalent circuit an performance characteristics are of prime importance to a power engineer. The main purpose of moelling short circuit characteristics stuy is to infer the machine's reactance. From time to time it is necessary to evelop user moels for euipment which o not have representation in commercial software stanar libraries for stability assessments. Emerging technologies an new euipment are typical bases for user moels. Figure 8 Open circuit characteristics of a turbo generator We conuct next the short-circuit test on the synchronous generator: the generator is rotate at the rate spee, all the terminals are short-circuite through ammeters, the fiel current is set to zero first. Next, the fiel current is increase in steps an the armature current I A is measure as the fiel current is increase. The plot of armature current (or line current) vs. the fiel current is the short-circuit characteristic (SCC) of the generator. 4.1 An Approximate Metho to Determine the Synchronous Reactance x s at a given fiel current: 1. Get the internal generate voltage E A from the OCC at that fiel current. 2. Get the short-circuit current I A,SC at that fiel current from the SCC. 3. Fin X S from X s = E A I Asc Since the internal machine impeance is 5.0 REFERENCES 1. W. Theoore Electrical Machines Drives an Power System, Prentice Publishers, fourth Eition, PP 901, 2000. 2. S. Hai, Power System Analysis, McGraw-Hill, Secon Eition, P324, 1999. 3. A.Walton, A Systematic Metho for the Determination of the Parameters of Synchronous Machine from the Results of Freuency Response Tests, 1999. 4. M. Amaya, Ientification of synchronous Machine Parameters by the Simulation of Time Domain Test using Finite Elements Metho, Vol. 3 3004, P1 5. P. Davi, O.H. Bosgra, M.J. Hoeijmakere, Time Domain Ientification of Synchronous Generator Transfer Function, Journal of Solar Energy Engineering, November, Vol.123/419, 2002 6. L. Lupsa-Tataru, An extension of flux linkage state-space moel of synchronous generators with a view to ynamic simulation, WSEAS Transactions on Power Systems, vol. 1, no. 12, pp. 2017 2022, 2006. 7. A. A. Gorev, Transient Processes of Synchronous Machine, Nauka, Sankt-Petersburg, Russia, 1985. 8. P. Vas, Electrical Machines an Drives: A Space-Vector Theory Approach, Clarenon Press, Oxfor, UK, 1992. 9. P. C. Krause, O. Wasynczuk, an S. D. Suhoff, Analysis of Electric Machinery an Drive Systems, Wiley-IEEE Press, New York, NY, USA, 2002. 10. C. M. Ong, Dynamic Simulations of Electric Machinery, Prentice Hall PTR, Englewoo Cliffs, NJ, USA, 1997. 2269