Benchmark Demonstration

Carles Batlle

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• Publié par: Nasheg
• Nombre de lectures: 24
• Langue: English

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GEOPLEX: Benchmark Demonstration


C. Batlle, A. Dòria, E. Fossas



IOC-DT-P-2004-23
Octubre 2004





















GeoplexBenchmarkDemonstration.
1,2,4 4 3,4C.Batlle ,A.Do`ria andE.Fossas
1 2 3 5EPSEVG, MAIV, ESAII,and IOC
TechnicalUniversityofCatalonia
October,2003
1 Electromechanicalbenchmark
In this section, we describe the development of a complex multi-domain electromechanical sys-
tem as an interconnection of simpler subsystems. We first give a global overview of the total
system to be modelled, then describe the subsystems of the model, and conclude with final re-
marksonhownetworkmodellingwasusedinthisproblem,andtowhatbenefit.
Our electromechanical system exchanges energy between the power grid, a local mechanical
sourceandalocalgeneralload,whichmaycontainsubsystemsfromanydomain.
1.1 Systemoverview
AgeneraldescriptionofoursystemappearsinFigure1.
Thecoreofourmodelisadoubly-fedinductionmachine(DFIM)togetherwithitscontroller,
a back-to-back 3phase converter (B2B). The DFIM is coupled to the power grid directly through
the stator, while the rotor receives power from the B2B, which in turn takes it from the power
grid.
The control objective, which does not form part of this Deliverable, is to effect the flow of
power from the grid and the local source to the local load, by means of Hamiltonian and port
relatedideas.
The 20sim model of the whole system, in bond graph notation, appears in Figure 2. We
have suppressed the transformers and the dynamics of the flywheel’s beam, but they can be
incorporatedeasilyfromthe20simlibrary.
We will describe the DFIM with some detail since it is the most complex of the subsystems
andtheonewithmoreroomformodellingimprovement.
1.2 Thedoubly-fedinductionmachine
ThedoublyfedinductionmachineappearsinFigure1.2
It contains 6 energy storage elements with their associated dissipations and 6 inputs (the 3
stator and the 3 rotor voltages). The dynamical equations are ([8][10], but see [4][2] for a discus-
sion)
dλ
v =Ri+ (1)
dt
whereR = diag(r ,r ,r ,r ,r ,r ) and the linking fluxes are related to the currents through ans s s r r r
angle-dependentinductancematrix
˜λ =L(θ)i. (2)
Weassumethat
• themachineissymmetric(allwindingsareequal)
1







local load
ve
utility grid i ie l flywheel
transformer transformer
power plants
vs
is
i statorr
converter
v rotorr ω
doubly−fed
induction machine
(DFIM)
wind turbine
Figure1: Systemoverview.
Local_load0Power_grid
B2B
I
Inertia
DFIM_bg
Control
Figure2: Systemoverviewinbondgraphnotation.
2irc
isb
+
rr
+ rr ss + v irb rbvsbNs rr Nr
isc + NrNsvsc vrc
vv N rasa s Nr
irarrisa
rs+
+
θ
Figure3: Basicschemeofthedoublyfedinductionmachine
• stator-rotor cross inductances are smooth, sinusoidal functions of θ, with just the funda-
mentalterm.
Tosimplifythenotation,wetakeN =N =N,sothatwedonothavetoreferrotorvariablestor s
statorwindings. Then
µ ¶
˜ ˜L L (θ)s sr˜L(θ) = T˜ ˜L (θ) Lrsr   
1 1 1 1L +L − L − L L +L − L − Lls ms ms ms lr mr mr mr2 2 2 2
1 1 1 1˜   ˜  − L L +L − L − L L +L − LL = L = .s ms ls ms ms r mr lr mr mr2 2 2 2
1 1 1 1− L − L L +L − L − L L +Lms ms ls ms mr mr lr mr2 2 2 2
HereL andL are leakage terms, whileL andL are magnetizing terms that can be com-ls lr ms mr
putedfromthecorereluctanceR asm
2N
L =L = .ms mr
Rm
Thecross-inductanceis  
2 2cosθ cos(θ + π) cos(θ− π)3 3
2 2˜  L (θ) =L cos(θ− π) cosθ cos(θ + π)sr sr 3 3
2 2cos(θ + π) cos(θ− π) cosθ3 3
2NwhereagainL = =L . Hence(1)isahighlynonlinearsetofODE.sr msRm
Fora2-polemachine,thetorqueisgivenintermsofthecoenergyby
∂W (i,θ)c
T (i,θ) =e
∂θ
andsinceweareassumingalinearmagneticsystem,energyandcoenergyareequal: W =W =c f
1 T ˆi L(θ)i.2
3Before proceeding to a θ-and-time-dependent transformation which eliminates most of the
nonlinearities in (1), it is better to perform a constant transformation which reduces an effective
degree of freedom for both the stator and the rotor. From the original (i,λ,v) quantities we
′ ′ ′compute (i,λ,v )bymeansof
′y =Ay (3)
wherey isanyofi,λ,v andAisa6by6block-diagonalmatrix
µ ¶
A 0sA =
0 Ar
where  √ 
2 1 1√ √ √− −
3 6 6 1 1√ √A =A = 0 − . s r 2 2
1 1 1√ √ √
3 3 3
T −1Noticethat,sinceA =A ,thisisapower-preservingtransformation:
′ ′hi,vi =hi,vi.
′ ′Thecomponentsofy areusuallydenotedbyy = (y ,y ,y ).d q 0
Underthistransformation,relation(2)becomes
′ ′ ′λ =L (θ)i
where  
L 0 0 M cosθ −M sinθ 0ss 0 L 0 M sinθ M cosθ 0ss  µ ¶′ ′T 0 0 L 0 0 0 L L (θ)ls′ s m L (θ) = = ′ ′ M cosθ M sinθ 0 L 0 0 L (θ) Lrr m r  −M sinθ M cosθ 0 0 L 0rr
0 0 0 0 0 Llr
3andM = L ,L =L +M,L =L +M.ms ss ls rr lr2
Inthenew(prime)variables,equation(1)becomes
d′ ′ ′ ′v = (L (θ)i )+Ri. (4)
dt
′It can be seen from the form of L that the homopolar components y decouple from the rest,0
yieldinganindependentlineardynamics,andfromnowonwewilldropthemfromthecompu-
tations,althoughwewillkeepthesamenotation:
d′ ′ ′ ′v = (L (θ)i )+Ri, (5)
dt
wherenoweverythingis 4-dimensional.
One can try to eliminate the complicate, θ-dependent terms in (5) by means of a change of
variables (Blondel-Park transformation). There is a whole family of transformations, depending
onanarbitrarytime-dependentparameterx(t):
′′ ′f =K(x,θ)f
with µ ¶
K (x) 0sK(x,θ) =
0 K (x,θ)r
and
−Jx −J(x−θ)K (x)≡e =B(x), K (x,θ)≡e =B(x−θ), (6)s r
4where µ ¶
cosz sinz
B(z) = , (7)
−sinz cosz
and µ ¶
0 −1
J = . (8)
1 0
NoticethatbothK andK belongtoSO(2)andhencethissecondtransformationisalsopowers r
preserving.
Underthistransformation(5)becomes
′′di′′ ′ ′ −1 −1 ′′ ′ −1 −1 ′′ ′ −1 ′′v =ωK(∂ L −LK ∂ K)K i −x˙KLK ∂ KK i +KLK +Ri (9)θ θ x
dt
where [K,R] = 0 has been used. Taking into account that B(x +y) = B(x)B(y) and B(−x) =
−1B (x)andusing(6),onegets
µ ¶
L I MI′′ ′ −1 ssL ≡KLK = . (10)
MI L Irr
′′ExploitingthefactthatthisL isindependentofbothxandθ,itiseasytoderivetheidentities
′ −1 −1 ′ −1KLK ∂ KK = ∂ KLK ,x x
′ ′ −1 −1 ′ −1K(∂ L −LK ∂ K)K = −∂ KLK ,θ θ θ
whereupon(9)becomes
′′di′′ ′′ ′′ ′′ ′′
v =L +ωΩi +x˙Ω i +Ri , (11)xdt
with µ ¶
0 0′ −1
Ω = −∂ KLK = , (12)θ −MJ −L Jrrµ ¶
L J MJ′ −1 ssΩ = ∂ KLK = . (13)x x MJ L Jrr
′′The prize for a constant inductance matrix is a nonlinear term involving ω and i . In what
′′followswewillrefertotheindividualcomponentsofaf 4−vectorasf ,f ,f ,andf .sd sq rd rq
Theco-energyinthetransformedcoordinatesisgivenby
1 1T ′′T ′′ ′′H (θ,i) = i L(θ)i = i L i +homopolarcontributions.c
2 2
However, the expression for the electrical torque T must be computed using the physical cur-e
rentsi. Hence
1 TT = i ∂ L(θ)ie θ
2
1 ′′T T T ′′ T T ′′= i KA∂ (A K L KA)A K iθ
2
1 ′′T T ′′ T ′′= i K∂ (K L K)K iθ
2
1 ′′T ′′ T ′′= i L ∂ KK i +transposeθ
2
′′T ′′
= i Ti , (14)
where µ ¶
M0 J
2T = . (15)M− J 0
2
5Themechanicalequationis(themechanicalpartistransformation-independent)
¨ ˙Jθ =T −Bθ +Te m
whereJ is the total inertia moment of the rotor,B is a friction coefficient and (T ,ω) is the me-m
chanical port to which any flywheel or rotating machinery can be coupled . Taking into account
theformofT ,thiscanbewrittenase
˙θ = ω (16)
′′T ′′Jω˙ = i Ti −Bω +T . (17)m
TheexplicitPCHformisgivenby
Tz˙ = (J(z)−R(z))(∇H(z)) +g(z)u,
n mwherez ∈ R ,J is antisymmetric,R is symmetric and positive semi-definite andu∈ R is the
control. The function H(z) is the hamiltonian, or energy, of the system. The natural outputs in
thisformulationare
T Ty =g (z)(∇H(z)) .
′′Equations (11) and (17) can be cast in this formulation with variablesz = (λ ,p = Jω), hamilto-
nianfunction
1 1′′ T ′′ −1 ′′ 2H = (λ ) (L ) λ + p , (18)
2 2J
structurematrix  
−x˙L J −x˙M Oss 2×1
′′ J = −x˙MJ −(x˙ −ω)L J MJi , (19)rr s′′TO Mi J 01×2 s
anddissipationmatrix  
R I O Os 2 2 2×1 R = O R I O , (20)2 r 2 2×1
O O B1×2 1×2 r
′′ ′′ ′′ ′′while the coupling is given by g = I with the controls u = (v ,v ,v ,v ,T ). The bond5 msq rqsd rd
graph corresponding to this description in the synchronous frame (x˙ = ω ) is shown in Figures
4. The stator and rotor resistances can be varied arbitrarily to include the effects of temperature.
Thisdq model is embedded into a 3-phase model which includes theA andK transformations,
asshownin5.
1.3 Theback-to-backconverter
Theiconicdiagramforourthree-phaseconverterappearsinFigure6.
TheB2BisaVariableStructureSystem(VSS)whichtakesitspowerfromthegridanddelivers
itinappropriateformtotherotoroftheDFIM.Itscontrolisimplementedby6pairs(3phases×
2 sides) of complementary switches. The main modelling challenge of this subsystem is the de-
taileddescriptionoftheswitches. Forth