CN 667 Power System Stability Lecture : xciter Models Prof. Tom Overbye Dept. of lectrical and Computer ngineering Texas A&M University, overbye@tamu.edu
Announcements Read Chapter 4 Homework 3 is due today Homework 4 is posted; it should be done before the first exam but need not be turned in Midterm exam is on Tuesday Oct 7 in class; closed book, closed notes, one 8.5 by inch hand written notesheet allowed; calculators allowed 2
Types of xciters 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: xciter is static, with field current supplied through slip rings 3
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 quations are di f v f i f R f Lf dt di v i R L G i dt a a a a a m f 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 Gi f Taken mostly from C 330 book, M.A. Pai, Power Circuits and lectromechanics 4
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 5
Separately xcited DC xciter (to sync mach) e in d f rf iin N f dt a f is coefficient of dispersion, modeling the flux leakage 6
Separately xcited DC xciter Relate the input voltage, e in, to v v K K a a a f v K a in in f a d f dv dt K dt e i r N K f a dv dt f Assuming a constant speed Solve above for f which was used in the previous slide 7
Separately xcited DC xciter If it was a linear magnetic circuit, then v would be proportional to i n ; for a real system we need to account for saturation v i in f sat v v K g Without saturation we can write K g K N Where L a f f us L f us is the unsaturated field inductance 8
e r i N in f in f Separately xcited DC xciter Can be written as d dt f r L dv e v r f v v f f us in f sat Kg Kg dt This equation is then scaled based on the synchronous machine base values X X v V R R V md md BFD 9
Separately xcited Scaled Values r f L f us K T sep K g K g V R R X md V BFD e in V BFD R S r f f sat X md Thus we have d T K S V dt sep R V r is the scaled output of the voltage regulator amplifier 0
The Self-xcited xciter When the exciter is self-excited, the amplifier voltage appears in series with the exciter field d T K S V R dt sep Note the additional term on the end
Self and Separated xcited xciters The same model can be used for both by just modifying the value of K d T K S V dt R K K typically K.0 self sep self 2
I T K Values xample IT Values from a large system 3
Saturation A number of different functions can be used to represent the saturation The quadratic approach is now quite common S ( ) B ( A) An alternative model is S ( ) xponential function could also be used 2 B ( A) B x S A e x 2 This is the same function used with the machine models 4
xponential Saturation K S 0.e Steady state V R 0.5. 5.e 5
xponential Saturation xample Given:.05 K S max 0.27 S.75 0.074 max V R.0 max max 4.6 Find: A x, Bx and max A x.005 x B x S A e B. 4 x 6
Voltage Regulator Model Amplifier In steady state Big K A dv T V K V dt R A R A in min max R R R V V V V t V ref V V ref t V in V K R A Modeled as a first order differential equation There is often a droop in regulation 7
Feedback This control system can often exhibit instabilities, so some type of feedback is used One approach is a stabilizing transformer Large L t2 so I t2 0 V F N N 2 L tm dit dt 8
dt d R L N N V L L R dt dv dt di L L I R t tm F tm t t F t tm t t t 2 T F K F Feedback 9
I T xciter This model was standardized in the 968 I Committee Paper with Fig shown below 20
I T volution This model has been subsequently modified over the years, called the DC in a 98 I paper (modeled as the XDC in stability packages) Note, K in the feedback is the same as the 968 approach Image Source: Fig 3 of "xcitation System Models for Power Stability Studies," I Trans. Power App. and Syst., vol. PAS-00, pp. 494-509, February 98 2
I T volution In 992 I Std 42.5-992 slightly modified it, calling it the DCA (modeled as SDCA) Same model is in 42.5-2005 Image Source: Fig 3 of I Std 42.5-992 V UL is a signal from an underexcitation limiter, which we'll cover later 22
I T volution Slightly modified in Std 42.5-206 Note the minimum limit on FD There is also the addition to the input of voltages from a stator current limiters (V SCL ) or over excitation limiters (V OL ) 23
Initialization and Coding: Block Diagram Basics To simulate a model represented as a block diagram, the equations need to be represented as a set of first order differential equations Also the initial state variable and reference values need to be determined Next several slides quickly cover the standard block diagram elements 24
Integrator Block quation for an integrator with u as an input and y as an output is dy u I dt Ku K I s y In steady-state with an initial output of y 0, the initial state is y 0 and the initial input is zero 25
u First Order Lag Block K Ts quation with u as an input and y as an output is dy Ku y dt T y In steady-state with an initial output of y 0, the initial state is y 0 and the initial input is y 0 /K Commonly used for measurement delay (e.g., T R block with I T) Input Output of Lag Block 26
Derivative Block u KDs st D y Block takes the derivative of the input, with scaling K D and a first order lag with T D Physically we can't take the derivative without some lag In steady-state the output of the block is zero State equations require a more general approach 27
State quations for More Complicated Functions There is not a unique way of obtaining state equations for more complicated functions with a general form m du d u 0u m m dt dt n n dy d y d y 0 y n n n dt dt dt To be physically realizable we need n >= m 28
General Block Diagram Approach One integration approach is illustrated in the below block diagram Image source: W.L. Brogan, Modern Control Theory, Prentice Hall, 99, Figure 3.7 29
Write in form KD s TD T s D Derivative xample Hence 0 =0, =K D /T D, 0 =/T D Define single state variable x, then dx dt y 0u0y T K D y x u x u TD D Initial value of x is found by recognizing y is zero so x = - u 30
Lead-Lag Block u st st In exciters such as the XDC the lead-lag block is used to model time constants inherent in the exciter; the values are often zero (or equivalently equal) In steady-state the input is equal to the output To get equations write in form with 0 =/T B, =T A /T B, 0 =/T B A B y input Output of Lead/Lag T s st T T st T s A A B B B B 3
The equations are with 0 =/T B, =T A /T B, 0 =/T B then Lead-Lag Block dx 0u 0y u y dt T T A y x u x u TB B The steady-state requirement that u = y is readily apparent 32