Analysis of the induction machines sensitivity to voltage sags. F. Córcoles and J. Pedra Ll. Guasch

Transcription

1 Analyss of the nducton machnes senstvty to voltage sags F Córcoles and J Pedra Ll Guasch Dep d Eng Elèctrca ETSEIB UPC Dep d Eng Elèctrca Mec ESTE URV Dagonal, 7 88 Barcelona Span Ctra Salou, s/n Tarragona Span e-mal: corcoles@eeupces, pedra@eeupces emal: llguasch@etseurves Abstract Ths paper proposes an algorthm for the approxmate calculaton of the effects caused by a voltage sag n the nducton machne supply system The algorthm computes the current and torque peaks, and the mechancal speed loss for an extensve range of voltage sags It has a hgh calculaton speed because t makes two smplfcatons The frst supposes that speed vares nsgnfcantly durng the frst cycles after the voltage drop and recovery ponts to solve analytcally the electrcal transents n these cycles The second neglects the electrcal transent durng the sag duraton for a quckly evaluaton of the speed loss Machne senstvty to voltage sags s graphcally shown The curves can be appled, e, to protecton calbraton Keywords: voltage sag (dp, semanalytc algorthm, exponental matrx, senstvty curves Introducton A voltage sag (dp s a short-duraton (from cycles to mnute reducton between % and 9% n rms voltage [] It s generally orgnated by a short crcut or overload on the utlty system, whch causes an excessve cables voltage drop When the reducton s hgher than 9%, the fault s named short nterrupton Typcally, voltage sag duraton s from to cycles, and the magntude depends on the power system dstrbuton and proxmty to the fault ste (In ths paper, the sag magntude s the net rms voltage n percentage Power qualty has a great mportance nowadays because the suppled equpment can produce severe problems when the supply has a falure The consequences of a voltage sag n the nducton machne supply are the speed loss and the current and torque peaks that appear n the voltage drop and recovery ponts, ponts and n Fg Ths transent can produce the actuaton of the machne or system protectons The transent shape depends on many elements, as the sag magntude and duraton, the electrcal parameters of the machne, the load and the mechancal nerta Another factors, as the fault nstant, have less nfluence The computer smulaton wll be slowly When extensve ranges of voltage drop are analyzed, the total computatonal tme s unacceptable No prevous work has been found wth ths detaled approach The papers concernng the topc [,] use the steady-state equatons of the nducton machne Ths paper proposes a method for the approxmated calculaton of the transent n partcular, current and torque peak and speed loss caused by a voltage sag n the nducton machne supply system Ths so-called semanalytc method s very quck and able to compute an extensve range of voltage drops Machne senstvty to voltage sags s graphcally shown wth the current and torque peaks, and the speed loss versus the voltage dp They can be appled, e, to machne or system protecton calbraton Voltage (% and mechancal speed (rad/s % % % 777 rad/s 77 rad/s 7 A Phase c current (A - - knm -7 A Electromagnetc torque (knm - knm - - Fg Voltage sag: ms of duraton and % of magntude Transent solved wthout approxmatons

2 Phase c current (A 7 A Exact method Approxmate method Phase c current (A A - Electromagnetc torque (knm knm Fg Phase c current and electrc torque n the drop voltage pont Comparson between the exact and proposed methods Analyss Method Detaled smulatons show that lower current and torque peaks are generally obtaned n the drop voltage pont than those obtaned n the recovery voltage pont So, the frst verson of the algorthm [] only analyzed the second transent In the optmzed verson presented n ths paper, the frst transent has also been consdered No calculaton complexty s added because both transents have smlar characterstcs On the other hand, current peaks are usually obtaned n the frst cycle after the drop or the recovery voltage ponts The torque peak s also produced n the frst cycle, but n certan condtons t can be obtaned n the next or cycles Then, only t s necessary to analyze these cycles to fnd the exact or approxmate varables values, dependng on the smplfcatons The presented algorthm analyzes separately: - the electrc transent n the early cycles from ponts and n Fg to calculate the current and torque peaks, and - the mechancal transent wthn ponts and to calculate the speed loss Electrcal Transent n the Drop and Recovery Voltage Ponts Assumng the mechancal speed s constant durng a bref nstant when the voltage drops, pont n Fg, and recovers, pont, both transents can be solved analytcally by means of the algorthm presented n [], that s resumed here On applyng the Ku tranormaton [] n the synchronously reference frame to the machne equatons, a second order dfferental equatons system s obtaned supposng the homopolar component s null n complex varables, 9 knm 8 d dt dx = Ax + Bu dt Lr = L M slr M rs + jω sls j s m r + L L M where ( + j M Ls jω sm s m Lr Lr M v M Ls v ( ω ω M r + j( ω ω s r ( f = d q = forward component of current, f ( v jv v = + = forward component of d q voltage If mechancal speed, ω m, s constant, the system ( has constant coeffcents The system soluton can be expressed as the addton of the homogeneous (or free and the steady-state (or partcular solutons The homogeneous soluton nvolves the exponental matrx evaluaton The exponental matrx can be expressed by means of the matrx R, R and the egenvalues of A The order system, second order, allows to obtan analytcally the egenvalues or characterstc polynomal roots, λ λ h h dxh = Axh xh dt λ ( t t ( = { Re + R } ( t ( t Electromagnetc torque (knm Exact method Approxmate method Fg Phase c current and electrc torque n the recovery voltage pont Comparson between the exact and proposed methods A( t t ( t = e xh ( t ( t ( t λ t t p e ( t ( t As the system s lnear, the steady-state soluton s a functon wth the same shape as the exctaton In the chosen reference, synchronously, the snusodal threephase symmetrc voltages are tranormed n constant voltages So, the steady-state soluton s also constant p (

3 = Ax p + Bu p p ( t ( t = A x p = A v B v ( t ( t Bu Then, the analytc soluton of ths system s ( t ( t ( ( λ = { e t t λ t t R + Re } p ( t + p ( t The electromagnetc torque s ( t p ( t ( t ( t p ( ( rad/s Mechancal speed (rad/s rad/s 7 - Exact method Approxmate method ( t = M * Im( Γ ( When the machne has P par of poles, the torque s P tmes hgher The ntal condtons n both transents are the fnal values of the prevous transent Electrcal transent duraton Although the voltage sag perturbaton generates an electromechancal transent, the electrcal and mechancal transents can be consdered separately because ther tme constants are dfferent It s usually accepted that the transent duraton s or tmes the hghest system tme constant In the analyzed electrcal transent, assumng the mechancal speed s constant, the tme constants are the nverse of the egenvalues real parts: λ = α + jω λ = α + jω τ = α, τ = α Torque Peak and per Phase Absolute Current Peak The peak of the real currents, not agrees wth the (t maxmum The real phase current s sa jω st jω st ( t = Re{ e }, ( t = Re{ a e } sc jω st ( t = Re{ ae }, a = sb The torque, (, and current, (7, peaks are searched numercally Mechancal Transent Durng the Voltage Sag Although the voltage sags perturb the electrcal and mechancal systems, ther tme constants are usually very dfferent: the electrcal ones are shortest Neglectng the ntal electrcal transent and takng nto account only the mechancal equaton, the angular speed durng the voltage sag, ω m, can be obtaned The ntegraton of a frst order dfferental equaton s necessary In the snusodal case, ( (7 Fg Mechancal speed from pont to pont Comparson between the exact and proposed methods where dωm ωs ds Γm Γres = J = J dt P dt (8 U s s ωs ds Γres = J k + k s + k s P dt Γ m = electromagnetc torque, Γ res = load torque, J = total mechancal nerta, ω s = stator voltage pulsaton, ω m = mechancal speed, P = par of poles, s = ( ωs Pωm ωs = slp, U s = per phase stator rms voltage, k, k, k = constants dependng on the frequency and machne parameters Smulaton Results The chosen smulaton corresponds to % voltage sag wth ms of duraton The motor works actually and drves a ventlator n a cement plant: kw, V (star, Hz, 78 Nm, 7 rpm, 8 A (phase The operatng pont s close to the nomnal condtons Other characterstcs are: - Electrcal parameters referred to the stator ( Hz measured reactances: r s = 8 Ω r r = Ω X sd = 7 Ω X rd = 7 Ω X m = 799 Ω - Load characterstcs (Γ res = K + K ω m + K ω m : K = Nm K = Nm/(rad/s K = 9 Nm/(rad/s J = 8 kg m (rotor ncluded The electrcal transent egenvalues depend on the mechancal speed The electrcal tme constants for ths machne, when the mechancal speed s 77 rad/s, are λ = 9 + j τ = 98ms λ = 9 + j78 τ = 9ms Fg shows the transent obtaned wth the exact method and Fg to Fg also nclude the approxmated one Mechancal speed, phase c current n ths case s the (9

4 hghest one n the voltage dp and recovery ponts and torque are shown The proposed method error s 7% for the mechancal speed n the pont, 8% and % for the current and torque peaks respectvely n the voltage dp pont, 9% and % n the recovery pont Senstvty Curves CBEMA curve was orgnally developed to descrbe the tolerance of man frame computer equpment to the magntude and duraton of voltage varatons on the power system Ths curve has been a standard to represent the power qualty and the equpment senstvty [,7] Smlar curves can be made to dsplay the nducton machne senstvty to the voltage sags These curves are shown n Fg to Fg, whch dsplay the results of 8 voltage sags: dfferent sag duraton (from ms to s and magntudes (from % to 97% Current and torque peaks Fg 7 and Fg are the maxmum values between that of the voltage drop and recovery ponts Fg, Fg, Fg 9 and Fg Calculaton Errors: Comparson wth an Exact Method In order to evaluate the calculaton errors, the proposed method has been compared wth an exact method A total of voltage sags wth duratons rangng from ms to s and % of voltage magntude have been evaluated A good agreement n the results has been observed In the 9% of the cases, error s lower than %, % and 9% for the current and torque peaks and the speed loss respectvely Smlar errors have been obtaned for % and 8% of sag magntude Algorthm valdty Smplfcatons made n the algorthm one supposng that speed vares nsgnfcantly to solve the electrcal transents and the other neglectng the electrcal transents to calculate the speed loss can be summarzed n a sngle supposton: electrcal and mechancal tme constants are very dfferent In ths stuaton, t can be consdered the dfferental equatons are uncoupled Algorthm s applcable when ths supposton s true, whch s so n the Sag magntude (% INSTANTANEOUS CURRENT PEAK IN THE DROP VOLTAGE POINT IN pu ( _peak/i_n Sag duraton (s Fg Curves of nstantaneous current peak (pu n the drop voltage pont 8 Sag magntude (% INSTANTANEOUS CURRENT PEAK IN THE RECOVERY VOLTAGE POINT IN pu ( _peak/i_n Sag duraton (s Fg Curves of nstantaneous current peak (pu n the recovery voltage pont Sag magntude (% INSTANTANEOUS CURRENT PEAK IN pu ( _peak/i_n Current peak (A 8 INSTANTANEOUS CURRENT PEAK (A Sag duraton (s Fg 7 Curves of nstantaneous current peak (pu Sag magntude (% Sag duraton (s Fg 8 Suace of nstantaneous current peak (A

5 Sag magntude (% ELECTROMAGNETIC TORQUE PEAK IN THE DROP VOLTAGE POINT IN pu ( torque_peak/torque_n Sag duraton (s Fg 9 Curves of nstantaneous torque peak (pu n the drop voltage pont 8 Sag magntude (% VOLTAGE POINT IN pu ( torque_peak/torque_n Sag duraton (s Fg Curves of nstantaneous torque peak (pu n the recovery voltage pont 8 Sag magntude (% ELECTROMAGNETIC TORQUE PEAK IN pu ( Torque_peak/Torque_n Sag duraton (s Fg Curves of nstantaneous torque peak (pu Torque peak (Nm x 8 Sag magntude (% ELECTROMAGNETIC TORQUE PEAK (Nm Sag duraton (s Fg Suace of nstantaneous torque peak (Nm 7 MECHANICAL SPEED (rad/s MECHANICAL SPEED (rad/s Sag magntude (% 7 7 Mechancal speed (rad/s 8 - Sag duraton (s 8 Sag magntude (% Sag duraton (s Fg Curves of mechancal speed (rad/s Fg Suace of mechancal speed (rad/s

6 majorty of machnes, except for servo-machnes, where the electrcal and mechancal tme constants have a smlar order The nerta constant gves an dea of the mechancal transent It s defned by b b b Pb Jω Jω H = = ( PΓ P where ω b = base synchronous pulsaton, P = par of poles, T b = base torque, P b = Γ b ω b / P = base power The example machne nerta constant s s Algorthm has been proved wth machnes of a wde range of power Table contans the maxmum errors Inerta loads have been chosen to gve a s nerta constant, smlar to the NEMA loads The voltage sag s always s of duraton and % of magntude In machnes wth a s nerta constant, errors are smaller than % When the nerta s tmes smaller, the maxmum error s % When t s 8 tmes smaller, the error s nferor to % In larger nerta machnes, such as the ventlator of the example, errors are smaller than % Calculaton Tme Comparson wth an Exact Method Senstvty curves n ths paper dsplay the results of 8 voltage sags: dfferent sag duraton and magntudes Ther calculaton tme n a Pentum computer ( MHz s mnutes However, t s possble to have an dea of ther approxmate shape by solvng fewer voltage sags The analyss of, for example, voltage sags wth dfferent duratons and magntudes requres mnutes No exact method has been mplemented to peorm the same calculatons Nevertheless, t s supposed that the total calculaton tme would be much longer A comparson has been made to solve a sngle transent: voltage sag of % of magntude appled to the machne from the example The exact method s a th and th order Runge-Kutta Results are that, dependng on the sag duraton, total calculaton tme n the approxmate method s to tmes shorter 7 Conclusons The hgh calculaton speed of the proposed method converts t as a very nterestng alternatve to study an extensve range of voltage sags on nducton machnes Curves have been dsplayed to represent the sag machne senstvty One of the applcatons of these curves s the machne or system protecton calbraton, as has succeeded n the smulated machne of the cement plant References [] RC Dugan, MF McGranaghan and HW Beaty, Electrcal Power Systems Qualty New York: McGraw-Hll, 99 [] JC Das, Effects of momentary voltage dps on the operaton of nducton and synchronous motors, IEEE Trans Ind Appl, vol, no, pp 7-78, jul/aug 99 [] MHJ Bollen, The nfluence of motor reacceleraton on voltage sags, IEEE Trans Ind Appl, vol, no, pp 7-7, jul/aug 99 [] F Córcoles, J Pedra and Ll Guasch, Algortmo para el estudo de los huecos de tensón sobre las máqunas de nduccón, (n Spansh Actas de las as Jornadas Hspano-Lusas de Ing Eléctrca, Salamanca, jul 997, vol I, pp 9- [] F Córcoles, J Pedra and M Salchs, Analytc algorthm for real-tme nducton machnes control, Proc of the ICEM, Vgo, jul 99, vol I, pp 9-7 [] J Lesenne, F Notelet and G Seguer, Introducton a l'electrotechnque Approfonde París: Technque & Documentaton, 98 [7] IEEE Std -987, IEEE Recommended Practce for Emergency and Standby Power Systems for Industral and Commercal Applcatons [8] J Lamoree, D Mueller, P Vnett, W Jones and M Samotyj, Voltage sag analyss case studes, IEEE Trans Ind Appl, vol, no, pp 8-89, jul/aug 99 [9] ANSI/IEEE Std 9-99, IEEE Recommended Practce for Desgn of Relable Industral and Commercal Power Systems Table Error n dfferent machnes Voltage sag: % of magntude and s of duraton Nomnal load Nomnal power Inerta (kg m H (s Voltage drop Voltage recovery Max error Current Torque Current Torque peak peak peak peak Speed HP,%,77%,%,8%,8%,8% 7, HP 99,%,9%,%,7%,%,% HP 8,%,%,%,%,88%,% HP 89,%,%,%,%,9%,% HP 8,%,7%,%,78%,%,78% HP 9,%,%,%,%,%,% kw 8,8%,%,9%,%,7%,9% HP 8,%,%,%,%,9%,% HP,9%,%,7%,%,%,% HP 7,%,%,8%,79%,8%,79% Average error,%,7%,%,%,8%,%