Exploring Synchronous Machines: Models, Exciters, and Power System Stability

ECE 576 Power System Dynamics and Stability Lecture 11: Synchronous Machines Models and Exciters Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign overbye@illinois.edu 1

Announcements Homework 3 is on the website and is due on Feb 27 Read Chapters 6 and 4 Midterm exam is on March 13 in class Closed book, closed notes You may bring one 8.5 by 11" note sheet Simple calculators allowed 2

Summary of Five Book Models a) Full model with stator transients 1 qo T 0 b) Sub-transient model c) Two-axis model s = ( ) 0 do T = ( ) 0 qo = d) One-axis model T X ) e) Classical model (const. E behind d 3

Two-Axis vs Flux Decay For 4 bus system, figure compares gen 4 rotor angle for bus 3 fault, cleared at t=1.1 seconds 4

Industrial Models There are just a handful of synchronous machine models used in North America GENSAL Salient pole model GENROU Round rotor model that has X"d = X"q GENTPF Round or salient pole model that allows X"d X"q GENTPJ Just a slight variation on GENTPF We'll briefly cover each one 5

Subtransient Models The two-axis model is a transient model Essentially all commercial studies now use subtransient models First models considered are GENSAL and GENROU, which require X"d=X"q This allows the internal, subtransient voltage to be represented as = + + ( d q q d E jE j + = + ( ) E V R jX I s ) 6

Subtransient Models Usually represented by a Norton Injection with ( d q d q s R jX + ) q d + + j + E jE + = = I jI R jX s May also be shown as ( ) ( ) q + d d q + j + j j ( ) + = = = j I jI I jI d q q d + R jX R jX s s In steady-state = 1.0 7

GENSAL The GENSAL model has been widely used to model salient pole synchronous generators In the 2010 WECC cases about 1/3 of machine models were GENSAL; in 2013 essentially none are, being replaced by GENTPF or GENTPJ In salient pole models saturation is only assumed to affect the d-axis 8

GENSAL Block Diagram (PSLF) A quadratic saturation function is used. For initialization it only impacts the Efd value 9

GENSAL Initialization To initialize this model 1. Use S(1.0) and S(1.2) to solve for the saturation coefficients 2. Determine the initial value of with ( s q E V R jX I = + + ) 3. Transform current into dq reference frame, giving id and iq 4. Calculate the internal subtransient voltage as = + + ( ) E V R jX I s 5. Convert to dq reference, giving P"d+jP"q= "d+ "q 6. Determine remaining elements from block diagram by recognizing in steady-state input to integrators must be zero 10

GENSAL Example Assume same system as before, but with the generator parameters as H=3.0, D=0, Ra = 0.01, Xd = 1.1, Xq = 0.82, X'd = 0.5, X"d=X"q=0.28, Xl = 0.13, T'do = 8.2, T"do = 0.073, T"qo =0.07, S(1.0) = 0.05, and S(1.2) = 0.2. Same terminal conditions as before Current of 1.0-j0.3286 and generator terminal voltage of 1.072+j0.22 = 1.0946 11.59 Use same equation to get initial ( 1.072 0.22 (0.01 0.82)(1.0 1.35 1.037 1.70 37.5 j = + = ) + = + + E V R jX I s + q = + 0.3286) j j j 11

GENSAL Example Then I I I I sin cos cos sin d r = q i 0.609 0.793 0.793 0.609 1.0 0.3286 0.869 0.593 = = And V = = + + + + ( ) R jX j j I s + + 1.072 1.174 0.22 (0.01 0.497 0.28)(1.0 0.3286) j j 12

GENSAL Example Giving the initial fluxes (with = 1.0) q 0.609 0.793 0.793 0.609 1.174 0.497 0.321 1.233 = = d To get the remaining variables set the differential equations equal to zero, e.g., ( 0.82 0.28 0.593 1.425, 1.104 q d E = = ) ( )( ) q q = = = 0.321 X X I q q Solving the d-axis requires solving two linear equations for two unknowns 13

GENSAL Example Once E'q has been determined, the initial field current (and hence field voltage) are easily determined by recognizing in steady-state the derivative of E'q is zero ( ( ( ) ( ) A ) Saturation coefficients were determined from the two initial values q q d = + + 1 ( ) E E Sat E X X I fd d D ) ( ( 2 ) q = + + 1.1 0.5 (0.869) 1.425 1 B E ) ( ) 2 = 1.425 1 1.25 1.425 0.8 + + = 0.521 2.64 Saved as case B4_GENSAL 14

GENROU The GENROU model has been widely used to model round rotor machines Saturation is assumed to occur on both the d-axis and the q-axis, making initialization slightly more difficult 15

GENROU Block Diagram (PSLF) The d-axis is similar to that of the GENSAL; the q-axis is now similar to the d-axis. Note saturation affects both axes 16

GENROU Initialization Because saturation impacts both axes, the simple approach will no longer work Key insight for determining initial is that the magnitude of the saturation depends upon the magnitude of ", which is independent of = + + ( ) V R jX I s Solving for requires an iterative approach; first get a guess of using 3.229 from the book ( s q E V R jX = + + ) I 17

GENROU Initialization Then solve five nonlinear equations from five unknowns The five unknowns are , E'q, E'd, 'q, and 'd Five equations come from the terminal power flow constraints (giving voltage and current) and from the differential equations initially evaluating to zero Two differential equations for the q-axis, one for the d-axis (the other equation is used to set the field voltage 18

GENROU Initialization Use dq transform to express terminal current as sin cos cos sin q i I I I I These values will change during the iteration as changes d r = Get expressions for "q and "d in terms of the initial terminal voltage and Use dq transform to express terminal voltage as sin cos cos sin q i V V ( q d j V V R I = + V Recall Xd"=Xq"=X" and =1 (in steady-state) V d r = ) ( ) Then from + = + + + + ( ) jV R jX I jI d q s d q q = + X I Expressing complex equation as two real equations d s d q + V R I X I d q s a d 19

GENROU Initialization Example Extend the two-axis example For two-axis assume H = 3.0 per unit-seconds, Rs=0, Xd = 2.1, Xq = 2.0, X'd= 0.3, X'q = 0.5, T'do = 7.0, T'qo = 0.75 per unit using the 100 MVA base. For subtransient fields assume X"d=X"q=0.28, Xl = 0.13, T"do = 0.073, T"qo =0.07 for comparison we'll initially assume no saturation From two-axis get a guess of ( )( 1.0946 11.59 2.0 1.052 52.1 = ) = + = 2.814 52.1 18.2 E j 20

GENROU Initialization Example And the network current and voltage in dq reference V V 0.7889 0.6146 0.6146 1.0723 0.7889 0.7107 0.8326 d = = 0.220 q I I 0.7889 0.6146 0.6146 0.7889 1.000 0.3287 0.9909 0.3553 d = = q Which gives initial subtransient fluxes (with Rs=0), ( q d d q j V jV V R I X I = + + = ) + ( ) ( ) + = + + + + ( ) R jX I jI s d q q = = 0.7107 0.28 0.3553 + = 0.611 d s d q 0.8326 0.28 0.9909 1.110 = V R I X I d q s a d 21

GENROU Initialization Example Without saturation this is the exact solution Initial values are: = 52.1 , E'q=1.1298, E'd=0.533, 'q =0.6645, and 'd=0.9614 Efd=2.9133 Saved as case B4_GENROU_NoSat 22

Two-Axis versus GENROU Response Figure compares rotor angle for bus 3 fault, cleared at t=1.1 seconds 23

GENROU with Saturation Nonlinear approach is needed in common situation in which there is saturation Assume previous GENROU model with S(1.0) = 0.05, and S(1.2) = 0.2. Initial values are: = 49.2 , E'q=1.1591, E'd=0.4646, 'q =0.6146, and 'd=0.9940 Efd=3.2186 Saved as case B4_GENROU_Sat 24

Two-Axis versus GENROU Response 25

GENTPF and GENTPJ Models These models were introduced by PSLF in 2009 to provide a better match between simulated and actual system results for salient pole machines Desire was to duplicate functionality from old BPA TS code Allows for subtransient saliency (X"d <> X"q) Can also be used with round rotor, replacing GENSAL and GENROU Useful reference is available at below link; includes all the equations, and saturation details http://www.wecc.biz/library/WECC%20Documents/Docum ents%20for%20Generators/Generator%20Testing%20Pro gram/gentpj-typej-definition.pdf 26

GENSAL Results Chief Joseph disturbance playback GENSAL BLUE = MODEL RED = ACTUAL Image source :https://www.wecc.biz/library/WECC%20Documents/Documents%20for %20Generators/Generator%20Testing%20Program/gentpj%20and%20gensal%20morel.pdf 27

GENTPJ Results Chief Joseph disturbance playback GENTPJ BLUE = MODEL RED = ACTUAL 28

GENTPF and GENTPJ Models GENTPF/J d-axis block diagram GENTPJ allows saturation function to include a component that depends on the stator current Most of WECC machine models are now GENTPF or GENTPJ Se = 1 + fsat( ag + Kis*It) If nonzero, Kis typically ranges from 0.02 to 0.12 29

Voltage and Speed Control ( , P ) ( ) Q, V

Exciters, Including AVR Exciters are used to control the synchronous machine field voltage and current Usually modeled with automatic voltage regulator included A useful reference is IEEE Std 421.5-2005 Covers the major types of exciters used in transient stability simulations Continuation of standard designs started with "Computer Representation of Excitation Systems," IEEE Trans. Power App. and Syst., vol. pas-87, pp. 1460-1464, June 1968 Another reference is P. Kundur, Power System Stability and Control, EPRI, McGraw-Hill, 1994 Exciters are covered in Chapter 8 as are block diagram basics 31

Functional Block Diagram Image source: Fig 8.1 of Kundur, Power System Stability and Control 32

Types of Exciters None, which would be the case for a permanent magnet generator primarily used with wind turbines with ac-dc-ac converters DC: Utilize a dc generator as the source of the field voltage through slip rings AC: Use an ac generator on the generator shaft, with output rectified to produce the dc field voltage; brushless with a rotating rectifier system Static: Exciter is static, with field current supplied through slip rings 33

Brief Review of DC Machines Prior to widespread use of machine drives, dc motors had a important advantage of easy speed control On the stator a dc machine has either a permanent magnet or a single concentrated winding Rotor (armature) currents are supplied through brushes and commutator Equations are di v i R L dt di v i R L G i dt The f subscript refers to the field, the a to the armature; is the machine's speed, G is a constant. In a permanent magnet machine the field flux is constant, the field equation goes away, and the field impact is embedded in a equivalent constant to Gif f = + f f f f = + + a a a a a m f Taken mostly from ECE 330 book, M.A. Pai, Power Circuits and Electromechanics 34

Types of DC Machines If there is a field winding (i.e., not a permanent magnet machine) then the machine can be connected in the following ways Separately-excited: Field and armature windings are connected to separate power sources For an exciter, control is provided by varying the field current (which is stationary), which changes the armature voltage Series-excited: Field and armature windings are in series Shunt-excited: Field and armature windings are in parallel 35

Separately Excited DC Exciter (to sync mach) d 1 f = + e r i N 1 1 in f in f 1 1 dt 1 = 1 is coefficient of dispersion, modeling the flux leakage 1 1 a f 1 36

Separately Excited DC Exciter Relate the input voltage, ein1, to vfd f 1 = = v K K fd a1 1 a1 a1 1 1 N K f 1 1 = N v f 1 f 1 fd Assuming a constant speed 1 a1 N K 1 d dv dt dv f 1 f 1 1 fd = N f 1 dt a1 1 N K f 1 1 fd = + e i r in in f 1 dt 1 1 a1 1 37

Separately Excited DC Exciter If it was a linear magnetic circuit, then vfd would be proportional to in1; for a real system we need to account for saturation v ( )fd v fd = + in i f v sat fd 1 K 1 g Without saturation we can write K N = a1 1 K L g1 f 1us f 1 1 Where is the L f 1us unsaturated field inductance 38

Separately Excited DC Exciter d 1 f = + e r i N 1 1 1 in f in f dt 1 Can be written as r e K L dv dt ( ) 1 1 f f us K fd = + + v r f v v 1 in fd f sat fd fd 1 1 1 g g This equation is then scaled based on the synchronous machine base values v X R X R fd = = md md E V fd fd V fd fd BFD 39

Separately Excited Scaled Values r L 1 1 f f us K g K T E E K sep 1 1 g Xmd V fd BFD V e 1 R in R ) ( V R BFD fd Xmd S E r f E 1 E fd f sat fd Vr is the scaled output of the voltage regulator amplifier Thus we have ) ( dEfd dt = + + T K S E E V E E E fd fd R sep 40

The Self-Excited Exciter When the exciter is self-excited, the amplifier voltage appears in series with the exciter field Note the additional Efd term on the end ) ( dEfd dt = + + + T K S E E V E E E E fd fd R fd sep 41

Self and Separated Exciter Exciters The same model can be used for both by just modifying the value of KE ( E E E fd T K S E dt dE ) ( ) fd = + + E V fd R = = 1 typically .01 K K K E E E self sep self 42