SIMULATION OF ELECTRIC MACHINE AND DRIVE SYSTEMS USING MATLAB AND SIMULINK

Transcription

1 SIMULATION OF ELECTRIC MACHINE AND DRIVE SYSTEMS USING MATLAB AND SIMULINK Introduction Thi package preent computer model of electric machine leading to the aement of the dynamic performance of open- and cloed-loop ac and dc drive. The Simulink/Matlab implementation i adopted becaue of it inherent integration of vectorized ytem repreentation in block diagram form, of numerical analyi method, of graphical portrayal of time evolution of ignal combined with the imple implementation of the functionality of controller and power electronic excitation. The development of Simulink model of drive aemblie i a relatively imple tak coniting of combining inputoutput block repreentation of the variou component making up the ytem. Thi approach provide a powerful deign tool becaue of the eae of oberving the effect of parameter modification and of change in ytem configuration and control trategie. Under the rubric Animation, a erie of movie clip portray the motion of electric machine, magnetic field, and pace vector.

2 The approach Electric machine The tarting tep in the mathematical modeling of ac machine i to decribe them a coupled tator and rotor polyphae circuit in term of o-called phae variable, namely tator current i a, i b, i c ; rotor current i ar, i br, i cr for an induction machine or i f, i kd, i kq for a ynchronou machine; the rotor peed ω m ; and the angular diplacement θ between tator and rotor winding. The magnetic coupling i expreed in term of an inductance matrix which i a function of poition θ. The matrix expreion of the machine equation are readily formulated in Matlab or Simulink language. A detailed example of thi approach i given in a later ection. The next tep i to tranform the original tator and rotor abc frame of reference into a common k or dq frame in which the new variable for voltage, current, and fluxe can be viewed a 2-D pace vector. In thi common frame the inductance become contant independent of poition. Figure 1 illutrate variou reference frame (coordinate ytem): the triplet [A B C ] denote a three-phae ytem attached to the tator while the pair [a b ] correpond to an equivalent two-phae ytem (zeroequence component can be ignored in Y-connected ac machine in which the neutral i normally iolated). Among poible choice of dq frame are the following: a) Stator frame where ω k = 0 b) Rotor frame where ω k = ω m c) Synchronou frame aociated with the frequency ω (poibly time varying) of the tator excitation. d) Rotor flux frame in which the d-axi line up with the direction of the rotor flux vector. The choice of the common dq frame i uually dictated by the ymmetry contraint impoed by the contruction and excitation of the machine. With the complete ymmetry encountered in a three-phae induction machine with balanced inuoidal excitation, any one of the five frame can be ued, although the ynchronou frame i more convenient in a much a all ignal appear a contant dc in teady tate. However, certain control trategie may require the adoption of a pecific frame, a i the cae of vector control where the reference frame i attached to the rotor flux vector. In the preence of aymmetry, the common frame i attached to the aymmetrical member: an induction motor with unbalanced excitation or aymmetrical tator winding (the cae of a capacitor motor) will be modeled in the tator frame where a a ynchronou machine i repreented in the rotor frame. In the common dq frame, the machine dynamic equation appear a differential equation with contant coefficient (independent of rotor poition) and nonlinearitie confined to product of variable aociated with peed voltage and torque component.

3 Figure 1. Reference frame in ac machine analyi ω denote the rotational peed or angular frequency of a frame (in electrical rad/) with repect to the tationary tator. The angular poition i obtained by integrating peed over time, that i θ= ωdt.

4 Excitation and controller The imulation of the input to the machine involve the mathematical repreentation of programmed time equence of event uch a the udden application or removal of mechanical load, the ramping of the magnitude and frequency of the applied voltage, or even the change in parameter value (for intance, rotor reitance). Similarly the functionality of power electronic type of excitation can generally be realized in the form of imple mathematical expreion: a an example, a PWM ignal can be decribed by the function ignum[m - tri(t)] where m i the modulation factor ( 1 m 1) and 2 1 tri(t) = in [in( ωt γ)] i a triangular waveform of unit amplitude, π frequency ω, and phae γ Thi function i readily tranlated into Simulink block form. The imulation of peed or poition controller in drive ytem i achieved by uing a relay block in a hyterei type of controller, and a imple combination of gain, ummer and integrator (incorporating limiter to include antiwindup feature) in a PI type of controller. Initial condition Initial condition are etablihed by pecifying a teady-tate operating condition. The implet cae i encountered in the imulation of the tarting of a motor for which all initial condition are zero. In ome other cae, the initial condition can imply be calculated before running the imulation: for example, under teady-tate inuoidal excitation, an induction motor running at a pecified peed can be quickly analyzed in term of a tandard phaor equivalent circuit; by uing phaor technique, one can compute the correponding load torque and initial condition. However, in mot intance, a pecified operating condition can only be obtained after running the imulation for a time that will depend on the tarting etting of thee initial condition. Thi ituation will occur if, in the above cae of the induction motor, torque intead of peed i pecified or if the mechanical load i a nonlinear function of peed. Thi i the normal ituation encountered with power electronic input ignal: a teady-tate condition i reached when an output ignal waveform i repeated every witching cycle o that the value at the beginning and end of a cycle are equal. When a teady-tate condition i attained, one can then ave thi o-called final tate and ue it later a the initial tate in a renewed imulation which now include the pecified time equenced input event.

5 Space vector model of the induction machine (SI unit) Electrical ytem equation: dλ v = Ri + +ωkmλ dt dλr vr = Rri r + + ( ωk ωm)mλr dt fd π 0 1 where the pace vector f = and the rotational operator M = fq Flux linkage-current relation: λ = L i + L i or i =Γ λ Γ λr m r m r r r λ = L i + L i i = Γ λ +Γλr r where r m L = Lm + Ll and Γ = Γ r = Γ m = Lr = Lm + Lrl = LmLl + LmLrl + LlLrl Mechanical ytem equation: dωmec Te = J + Bmω mec + TL dt where Te = k( λ i ) = k(mλ i ) = ( λdq i λqd i ) Lm = k(i r λ r) = kl m(ir i ) = k ( λr i ) = k Γm( λr λ) Lr and 2 3p ω mec = ω m k = p 22 Nomenclature: v = voltage pace vector [V] ω o = 2π fo = Bae frequency [rad/] i = current pace vector [A] ω k = peed of dq frame [rad/]] λ = flux linkage pace vector [Wb] ω m = rotor peed [rad/] R = Reitance [Ω] T e = electromagnetic torque [N.m] L = Inductance [H] T L = load torque [N.m] Γ = Invere inductance [H -1 ] J = moment of inertia [kg.m 2 ] f 0 = Bae frequency [Hz] p = number of pole operator: cro product dot product M rotation ubcript: tator r rotor d direct axi q quadrature axi L m m r L L

6 Simulink block diagram model Thee mathematical equation can be repreented a hown below in a block diagram form that preerve the one-to-one correpondence between the 2D pace vector of the equation and the vectorized ignal (of width 2) appearing in the Simulink repreentation. It i ignificant to point out that flux linkage are elected a tate variable in the imulation. Note that, apart from the peed ignal (calar) indicated in blue, all the variable are 2-element vector hown in red. M(90o) K*u ω k STATOR v 1 λ G i ωmec p/2 ωm M(90o) K*u R Gm 3/4*p* Lm T e ROTOR Gm M(90o) K*u v r 1 Gr i r λ r Rr Space vector model of an induction machine

7 Remark It i important to point out at thi tage the difference between the pace vector introduced here and thoe motly ued in the current literature (ee reference cited). Firt, a matter of notation: here a 2D pace vector i repreented by a column vector fd having 2 real element uch a f = ; the uual repreentation in the complex plane fq i f = fd + jfq j fe θ = (often referred to a a complex pace vector, omewhat of a minomer ince it appear a a complex calar). While the manipulation of expreion and equation uing complex quantitie i perhap eaier to perform, the proce mak the underlying phyical and geometric undertanding achievable with real vector in a real 2D plane. A geometric vector ha a meaning irrepective of the coordinate ytem in which it may be expreed; it can be caled, rotated, added to or projected on another vector, or multiplied with another vector a a cro product. In mechanic, the work done by a force f moving a body along a path i expreed a the dot (calar) product f and the torque exerted by thi force relative to an origin pecified by a radiu vector r i the cro (vector) product f r. In the preent context, electromagnetic torque i the cro product of 2 pace vector uch a * λ i, where a in the complex notation it will appear a Im( λ i ) or in term of the dq component a ( λdiq λ qi d). Similarly, power i neatly the dot product v i and energy i imply λ i. Finally, for computer imulation, it become neceary to recat the differential equation from their complex expreion into real form by aembling real and imaginary part. In the cae of an induction machine, the proce reult in four equation (2 for the tator in d and q, 2 for the rotor in d and q). Thee equation can then be repreented within Simulink in a cumberome calar form; or preferably the dq component can be recombined o a to recover a vectorized model (a long way around!). Thee obervation hould not detract from the fact that direct acce to the component of ome of the vector (via Demux block) may often be required for control purpoe a i certainly the cae of the vector control of an induction machine.

8 Block diagram model of the ynchronou machine The armature and damper circuit are hown in vectorized form with the variable indicated in red. The calar ignal are indicated in blue. M K*u M K*u ω m Flux bu Current bu T e v a wo 1 λ a i a ARMATURE λ k L_1* u Demux i k invere inductance i f i f Ra i a vf wo 1 λf FIELD Rf -wo 1 DAMPERS damper reitance K*u Space vector model of the ynchronou machine (per-unit formulation)

9 Convention & formulation: Becaue the emphai in thi work i focued on drive, the o-called motor convention are adopted through out. Thi mean that electric power in and mechanical power out are conidered poitive for motoring action. Furthermore, the current are aumed to be poitive when directed into the input tator and rotor port of the electric machine. Poitive rotation i counter clockwie from d axi to q axi. The model are formulated in both the SI and per-unit ytem. The per-unit ytem ue the rating of the machine a a bai, normally horepower output for induction machine and KVA input for ynchronou machine; the bae voltage i rated peak line-to-neutral voltage [Volt]; the bae peed i ynchronou peed at bae frequency. Although both ytem utilize practically the ame notation, they can readily be identified by context. In particular, the time differential dt (with t in econd) become ω odt in a dimenionle expreion. Furthermore, uch factor a 3/2 or p/2 do not appear in a per-unit formulation.

10 A dc motor peed drive The mathematical model of dc motor (permanent magnet type) can be expreed by thee equation dia va = Raia + La + ea where ea = Kωm dt dωm Te = J + TLign( ω m) + Bmωm where Te = Kia dt The block diagram of a cacade cloed-loop peed control of the dc motor i hown below. PI with antiwindup Kp ic we Ki 1 cycle average peed current ig ave voltage peed command peed controller current controller va Clock wc we va we ic ie va va DC MOTOR ia ia torque command Sign Tl wm wm current loop peed loop va 1/La 1 ia K Te 1/J 1 wm 1/La Kt 1/J Ra -K- TL Dc motor block Tf Bm Bm K Kv CASCADE SPEED CONTROL OF A DC MOTOR DRIVE

11 Typical dynamic repone are alo hown. The motor i initially at tandtill and at no load when a tep command in peed i applied; when teady-tate condition are reached, a reveral of peed i commanded followed by a tep load application. The ytem i highly nonlinear due to the introduction of aturation needed to limit both the current delivered and the voltage applied to the motor. The ytem i in the aturation mode when the error are large; a a conequence, the controller function a a contant current ource, that i torque, reulting in the ramping of the peed ince the load in thi example i a pure inertia. The incluion of aturation limit on the PI integrator i therefore neceary to provide antiwindup action. The preence of the ignum function in the torque expreion i required in order to inure that the load i paive whether the peed i poitive or negative (a i the cae here).

12 MATLAB MATRIX FORMULATON OF INDUCTION MACHINE STATE EQUATIONS Phae variable model INDUCTION MOTOR RUNNING UNDER PULSED LOADS The purpoe of thi Matlab peudo-cript file i to how how the tate equation of a three-phae induction machine are aembled in the context of a imple mode of operation: the motor i running in teady tate at a pecified loading when a pulating load i applied. The dynamic equation are expreed in canonical firt-order matrix form uing the abc phae variable, pecifically the tate variable are the tator current ia, ib, ic; the rotor current iar, ibr, icr; the rotor peed ωm; and the angular diplacement θ between tator a_axi and rotor ar_axi. The equation are olved uing ODE45. To perform the preent imulation, imply run the m_file named IMabcrun0. A Simulink rendition i portrayed below and i given in the mdl_file named IMabcim0. MACHINE PARAMETERS [SI] Vll=220 Line-to-line rm voltage rating [V] R=0.531 Stator reitance [Ohm] Rr=0.408 Rotor reitance [Ohm] Ll=2.5e-3 Stator leakage inductance [H] Lrl=2.5e-3 Rotor leakage inductance [H] Lm=84.7e-3 Magnetizing inductance [H] freq0=60 Bae frequency [Hz] wo=2*pi*freq0 Bae frequency [rad/] p=4 Number of pole J=0.02 Moment of inertia [kg.m^2] TLi=10 Initial load torque [N.m] TLf=2 Final load torque [N.m] Bm=.01 Frictional coefficient Rotor-tator turn ratio = unity V=Vll*qrt(2/3) Peak per-phae voltage [V] ODE xo=[ wo 0]' Approximate initial condition tf=8.0 Final time tpan=[0 tf] option=odeet('reltol',1e-3,'maxtep',1e-3) [t,x]=ode45(@imabcf0,tpan,xo,option,r,rr,lm,ll,lrl,p,v,wo,j,bm,tli,tlf) OUTPUT VARIABLES pd=x(:,7)*60/(p*pi) ia=x(:,1) ; ib=x(:,2) ; ic=x(:,3) I=qrt(ia.^2+(ib-ic).^2/3) iar=x(:,4) ; ibr=x(:,5) ; icr=x(:,6) gam=2*pi/3 va=v*co(wo*t) vb=v*co(wo*t-gam) vc=v*co(wo*t+gam) Speed [rpm] Stator current [A] Stator current magnitude [A] Rotor current [A] Three-phae inuoidal tator voltage (Balanced) [V]

13 State variable vector: STATE EQUATIONS x=[ia ib ic iar ibr icr wm theta] function dxdt=imabcf0(t,x,r,rr,lm, Ll,Lrl, p,v,wo,j,bm,tli,tlf) Lm1=2/3*Lm Maximum mutal inductance between any two phae xy M xy = Lm1*co(angle between the 2 phae xy) L=Ll+Lm1; Lr=Lrl+Lm1; gam=2*pi/3 Input: Load time profile: if t<1.5, TL=TLi; eleif t>=1.5 & t<5, TL=TLf; eleif t>5, TL=TLi; end Stator voltage time profile: va=v*co(wo*t); vb=v*co(wo*t-gam); vc=v*co(wo*t+gam) Variable: V=[va vb vc 0 0 0]' I=x(1:6) wm=x(7) theta=x(8) Parameter: Reitance matrix : R=diag([R R R Rr Rr Rr]) Inductance ubmatrice : L=[L -Lm1/2 Lm1/2 -Lm1/2 L -Lm1/2 -Lm1/2 Lm1/2 L] Voltage vector [V] Current vector [A] Speed [electrical rad/] Angular diplacement between tator a_axi and rotor ar_axi [rad] [Ω] [H] Lrr=[Lr -Lm1/2 Lm1/2 -Lm1/2 Lr -Lm1/2 -Lm1/2 Lm1/2 Lr] Lr=Lm1*[co(theta) co(theta+gam) co(theta-gam) co(theta-gam) co(theta) co(theta+gam) co(theta+gam) co(theta-gam) co(theta)] dlr=lm1*[-in(theta) -in(theta+gam) -in(theta-gam) -in(theta-gam) -in(theta) -in(theta+gam) -in(theta+gam) -in(theta-gam) -in(theta)] Inductance matrix L : L=[L Lr Lr' Lrr] dl/dθ : dldtheta=[zero(3) dlr dlr' zero(3)]

14 Matrix formulation of induction machine equation in phae variable: Contitutive flux-current relation: λ = L*I or I = L\λ Torque: Te= (Wm)/ θm =p/2* (0.5*I *L*I)/ θ =p/4*i *dl/dθ*i Electrical ytem equation: V= R*I + dλ/dt = R*I + d(l*i)/dt = (R + ωm*dl/dθ)*i + L*dI/dt Mechanical ytem equation: Te = J*dωm/dt + TL + Bm*ωm ωm= dθ/dt Wm= magnetic co-energy Te=p/4*I'*dLdtheta*I didt=l\(v-(r+wm*dldtheta)*i dwmdt=p/2*(te-tl)/j-bm*wm/j ; dthetadt = wm dxdt=[didt;dwmdt;dthetadt] Electromagnetic torque [N.m] Electrical ytem equation Mechanical ytem equation State equation arrived at by concatenation Note In the phae model formulation of the machine equation, the variable V, I, λ appear a 6- element column vector (in the matrix analyi connotation); o that, for intance, the current vector i I = [ia ib ic iar ibr icr], repreenting tator and rotor current expreed in their repective tator and rotor frame. The (mutual) inductance parameter are explicitly dependent upon rotor poition θ. With the tranformation of thee original variable into a common DQ reference frame, they become pace vector (in a 2_D geometric interpretation); now current are defined a i=[id iq] and ir=[idr iqr]. Furthermore, the inductance parameter become contant, independent of poition. Circuit repreentation Three-phae coupled circuit repreentation of an induction machine

15 Simulink model

16 Concatenation ued to aemble the inductance matrix L a a function of θ Concatenation ued to aemble the matrix dl/dθ a a function of θ

17 Graphical diplay Torque veru time Speed veru time Torque-peed curve One cycle of tator voltage and current Stator current i a Three-phae rotor current