Aitken-Schwarz heterogeneous Domain Decomposition for EMT-TS Simulation
AAitken-Schwarz heterogeneous DomainDecomposition for EMT-TSSimulation
H. Shourick , D. Tromeur-Dervout and L. Chedot Abstract
In this paper, a Schwarz heterogeneous domain decompositionmethod (DDM) is used to co-simulate an RLC electrical circuit where a partof the domain is modeled with Electro-Magnetic Transients (EMT) modelingand the other part with dynamic phasor (TS) modeling. Domain partition-ing is not based on cutting at transmission lines which introduces a physicaldelay on the dynamics of the solution, as is usually done, but only on connec-tivity considerations. We show the convergence property of the homogeneousDDM EMT-EMT and TS-TS and of the heterogeneous DDM TS-EMT, withand without overlap and we use the pure linear divergence/convergence ofthe method to accelerate it toward the searched solution with the Aitken’sacceleration of the convergence technique.
Key words: co-simulation, heterogeneous Schwarz domain decomposition,Aitken acceleration of the convergence
The introduction of renewable energies into the power grid leads to the useof more components based on power electronics which have to be well dimen-sioned in order not to be damaged by electrical disturbances. These com-ponents imply faster dynamics, for power system safety simulations, whichcannot be handled by traditional Transient Simulations (TS) with dynamicphasors. Nevertheless, for large power grids, it can be expected that the needof high level details requiring Electro-Magnetic Transient (EMT) modelingwill be localized close to disturbances, as other parts of the network still useTS modeling. This paper deals with a proof of concept to develop hetero-geneous Schwarz domain decomposition with different modeling (EMT-TS)between the sub-domains. Hybrid (Jacobi type) EMT-TS co-simulation hasto face several locks [3]: EMT and TS do not use the same time step size,the transmission of values is also a problem as the solutions do not have thesame representation and are subject to some information loss. Our approach University of Lyon, UMR5208 U.Lyon1-CNRS, Institut Camille Jordan,e-mail: [email protected] Supergrid-Institute, 14 rue Cyprien, 69200 Villeurbanne.e-mail: h.shourick,[email protected] a r X i v : . [ m a t h . NA ] F e b H. Shourick, D. Tromeur-Dervout and L. Chedot don’t use waveform relaxation [4], and the domain partitioning is not basedon cutting the transmission lines [1, 5] as we want to be able to define anoverlap between the two representations. On the contrary, we want to usethe traditional Schwarz DDM but also where the transmission conditions canlead to divergent DDM. The pure linear convergence/divergence of the lin-earized problems is then used to accelerate the convergence to the solutionby the Aitken’s technique. In Section 2, we describe the EMT and TS mod-eling and perform homogeneous Schwarz DDM accelerated by the Aitken’sacceleration of the convergence technique. Section 3 gives behavior resultsobtained for each modeling. Section 4 describes the heterogeneous EMT-TSDDM and gives first results obtained before concluding in section 5
Simulation of power grid consists in solving a system of differential algebraicequations (DAE) where the unknowns are currents and voltages. This systemis built using the Modified Augmented Nodal Analysis where each componentof the grid contributes through relations between currents and voltages andthe Kirshoff’s laws give the algebraical constraints. Let x (respectively y ) bethe differential (respectively algebraical) unknowns. For the EMT modeling,we have to solve the DAE: F ( t, x ( t ) , ˙ x ( t ) , y ( t )) = 0 , with Initial Conditions (1)The linearized BDF time discretization of (1) (Backward Euler here) leadsto solve the linear system (2) to integrate the state space representation ofthe DAE from time step t n to time step t n +1 : (cid:18) I − ∆tA BC D (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) H ∆t (cid:18) x n +1 y n +1 (cid:19) = (cid:18) I
00 0 (cid:19) (cid:18) x n y n (cid:19) (2)For TS modeling the variables are supposed to oscillate with a specific fre-quency ω and its selected harmonics taken in a subset I = { . . . , − , , , . . . } : z ( t ) = (cid:88) k ∈ I z k ( t ) e ikω t , z = { x, y } . (3)Introducing (3) into (1) leads after simplification to an another DAE systemthat takes into account the differential property of the dynamic phasor. Theresulting DAE system has smoother dynamics. The number of TS variablesis then multiplied by the number of harmonics chosen, and the number ofequations must be multiplied accordingly. For example, below is the structureof the matrix H T S by choosing two harmonics k = a and k = b and by solving itken-Schwarz heterogeneous Domain Decomposition for EMT-TS Simulation 3 the imaginary and real part separately and with S the matrix taking intoaccount the differential property of the dynamic phasor modeling. H T S = H ∆T − a ω S a ω S H ∆T H ∆T − c ω S c ω S H ∆T Let x n +1 T (respectively x n +1 E ) be the algebraic and differential unknowns of TS(respectively EMT) modeling associated to the linear system H T S x n +1 T = b nT (respectively H E x n +1 E = b nE ) We consider a linear RLC circuit of Figure 1 to develop the proof of conceptof the the Schwarz DDM on TS and EMT models. Ω C C R R E cos ωt = βL L v = 0 (4) v − v − E − Z s i = 0 (5) v − v − L di dt = 0 (6) v − v − R i = 0 (7) C ( dv dt − dv dt ) − i = 0 (8) v − v − R i = 0 (9) v − v − L di dt = 0 (10) C ( dv dt − dv dt ) − i = 0 (11) i − i = 0 (12) i − i = 0 (13) i − i = 0 (14) i − i = 0 (15) i − i = 0 (16) i − i = 0 (17) Fig. 1
Linear RLC circuit and its associated EMT modeling DAE system with x = { v , i , v , v , i , v } and y = { v , i , v , i , i , i , v , i } . L L . C C . − , R R Zs = 1 . − , ω = 2 π One Restrictive Additive Schwarz (RAS) iterate to solve Hx m +1 , ∞ = b m ∈ R n writes on subdomain Ω i : x m +1 ,k +1 i = A − i (cid:16) b mi − E i x m +1 ,ki,e (cid:17) , with R i ∈ R n i × n the operator that restricts the global vector to the subdomain Ω i , including the overlap, (cid:101) R i ∈ R n i × n the operator that restricts the globalvector to the subdomain Ω i , with setting to 0 the components of the vector H. Shourick, D. Tromeur-Dervout and L. Chedot that correspond to the overlap. W i is the global index set of the unknownsbelonging to the subdomain Ω i . A i is the part of the operator H associatedto the subdomain Ω i : A i = R i HR Ti . x m +1 i = R i x m +1 and b mi = R i b m arethe restriction to the subdomain Ω i of the solution and the right hand siderespectively. x m +1 i,e represents the external data dependencies of the subdo-main Ω i : x m +1 i,e is composed of the x m +1 j such that H kj (cid:54) = 0 with k ∈ W i and j / ∈ W i . R i,e is the restriction operator such that R i,e x m +1 = x m +1 i,e . E i is the part of the matrix H that represents the effect of the unknownsexternal to the subdomain Ω i on the unknowns belonging to the subdomain Ω i : E i = R i,e HR Ti,e .The small linear system associated with the RLC circuit is partitionedinto two subdomains using graph partitioning without overlaping (Figure 2top) and with an overlap of 1 (Figure 2 bottom). Each subdomain needs twovalues from the other to solve its equations. Ω = Ω ∪ Ω Ω Ω C C R R E cos ωt = βL L i , v i , v Ω = Ω ∪ Ω Ω Ω C C R R E cos ωt = βL L i , v i , v Fig. 2
Graph partitionning of the RLC circuit in two subdomains and the associatedmatrix partioning without overlap (top) and with overlap of 1 (bottom).
The RAS applied to each time step has a pure linear convergence i.e. theerror operator P does not depend on the RAS iterate. x m +1 ,p +1 − x m +1 , ∞ = P ( x m +1 ,p − x m +1 , ∞ ) (18)Thus it can be accelerated if it does not stagnate to obtain the searchedsolution regardless of its convergence or divergence [2]. itken-Schwarz heterogeneous Domain Decomposition for EMT-TS Simulation 5 λ ( P ) withoutoverlap withoverlap Schwarz ∆t EMT ± ± . − TS k=1 -36.6318 ± ± . − TS k=0 -36.77 ±
0i -36.77 ± ±
0i RMS 2 . − TS k=1 -1.28888 ± ± . − TS k=0 -1.427 ±
0i -1.427 ±
0i RMS 2 . − Table 1
Larger eigenvalue for P error operator for RAS and EMT modeling ( ∆t =2 . − ), and for RMS and TS k = 0 , ∆T = 2 . − , ∆T = 2 . − ) modeling. x m +1 , ∞ = ( I d − P ) − ( x m +1 , − P x m +1 , ) (19) P can be compute numerically from the values of the iterated transmissionconditions. For this small problem it can be directly computed working onthe matrix partitioning. P = − [( ˜ R ) t A − E ,e R ,e + ( ˜ R ) t A − E ,e R ,e ] (20)Table 1 gives the larger eigenvalue in modulus for the P RAS error operatorfor the EMT modeling and for the P RMS(Restricted Multiplicative Schwarz)error operator for the TS modeling main harmonic k = 1 applied to theRLC circuit. In both cases EMT and TS modeling the eigenvalue modulus isgreater than one, so the method diverges. We can observe that the overlapdoes not impact the divergence of the method. The time step increasing from ∆t = 2 . − to ∆T = 2 . − has a beneficial effect on the TS-TS DDMdivergence. Nevertheless, the divergence is purely linear and the Aitken’sacceleration (19) can be performed after the first iterate. v1 DDM v2 DAE14 DDM BDF
EMTDDM v3 DAE14 DDM BDF
EMTDDM v4 DAE14 DDM BDF
EMTDDM v5 DAE14 DDM BDF
EMTDDM v6 DAE14 DDM BDF
EMTDDM v7 DAE14 DDM BDF
EMTDDM -3 i12 DAE14 DDM BDF EMTDDM -3 i56 DAE14 DDM BDF EMTDDM k=1 real part v1 DDM
DDM combine v2 DDM
TSDDM combine v3 DDM
TSDDM combine v4 DDM
TSDDM combine v5 DDM
TSDDM combine v6 DDM
TSDDM combine v7 DDM
TSDDM -3 combine i12 DDM TSDDM -3 combine i56 DDM TSDDM
Fig. 3
Homogeneous DDM results comparison with DAE monodomain: (Left) RAS forEMT modeling with ∆t E = 1 . − and (right) RMS for TS modeling with ∆t T = 2 . − . H. Shourick, D. Tromeur-Dervout and L. Chedot Our goal is to simulate, using heterogeneous RAS DDM, the electrical net-work with one part with a TS modeling which can use large time steps ∆T and the other part with the EMT modeling which requires smaller time steps ∆t as the high oscillations remain.These two representations TS and EMT of the solution imply having someoperators E T Semt (respectively E emtT S ) to transfer the solution from the subdo-main EMT (respectively TS) to the other TS (respectively EMT). The E emtT S operator needs to compute the fundamental harmonic and other harmonicschosen of the solution from the history of the EMT solution. The history timelength is one period. This is performed by the FFT of the solution over thetime period and keeping the mode corresponding to the chosen harmonics.The E T Semt operator is more simple as it consists in recombining the TSmodes of the solution with the appropriate Fourier basis modes.Let us consider a linear electrical network with the TS modeling. Thetime discretisation of the DAE to integrate from T N to T N +1 , assuming that ∆T = m∆t can be witten as: (cid:18) I − ∆T A T S B T S C T S D T S (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) H TS (cid:18) x N +1 T S y N +1 T S (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) w N +1 TS = (cid:18) I
00 0 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Θ emt (cid:18) x NT S y NT S (cid:19) + (cid:18) E AT S E BT S E CT S E DT S (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) E emtTS (cid:18) x memt y memt (cid:19) (21)Similarly one time step for the EMT side to integrate from t n to t n +1 can bewitten as: (cid:18) I − ∆tA emt B emt C emt D emt (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) H emt (cid:18) x n +1 emt y n +1 emt (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) w n +1 = (cid:18) I
00 0 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Θ emt (cid:18) x nemt y nemt (cid:19) + (cid:18) E Aemt E Bemt E Cemt E Demt (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) E TSemt (cid:18) x N +1 T S ( t n +1) y N +1 T S ( t n +1 ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) W N +1 ( t n +1 ) (22)The m time steps can be gathered in one larger system considering t = T N : itken-Schwarz heterogeneous Domain Decomposition for EMT-TS Simulation 7 I − Θ emt H emt − Θ emt H emt . . . . . . − Θ emt H emt − Θ emt H emt (cid:124) (cid:123)(cid:122) (cid:125) H emt w emt w emt w emt ... w m − emt w memt (cid:124) (cid:123)(cid:122) (cid:125) W emt = I E
T Semt E T Semt . . . . . . E T Semt E T Semt (cid:124) (cid:123)(cid:122) (cid:125) E TSemt ( x , y ) t W N +1 ( t ) W N +1 ( t )... W N +1 ( t p − ) W N +1 ( t p ) (cid:124) (cid:123)(cid:122) (cid:125) W N +1 TS (23)This system needs the values that the TS solution connected to the EMTpart taken on the small time steps.The two domains are connected via the connected or flowing variables.Since these variables should be the solution at time T N +1 , we need theSchwarz iterative algorithm to obtain the exact values. We then iterate theiteration p + 1 by taking the connected values, at the iteration p , from theother subdomain. We can used the multiplicative form or the additive formas follows: (cid:26) H T S w N +1 , p + T S = Θ T S w NT S + E emtT S w m, p emt H emt W N +1 , p + emt = E T Semt W N +1 , p T S (24)Figure 4 (left) show the solutions v EMT et i TS of heterogeneousDDM EMT( ∆t = 2 . − )-TS( ∆T = 2 . − ) with comparison with the DAEsolution on monodomain. We proceed to a jump in amplitude at t = 0 . log of the error betweentwo consecutive RAS iterates at t = 0 .
02. It shows a linear convergencebehavior and can therefore be accelerated by the Aitken’s accelerating of theconvergence technique after 9 iterates needed to numerically construct theerror operator P . A Schwarz heterogeneous DDM was used to co-simulate an RLC electricalcircuit where a part of the domain is modeled with EMT modeling and the
H. Shourick, D. Tromeur-Dervout and L. Chedot v4 EMT/TS
EMTDDM -3 i71 EMT/TS EMTDDM
RAS iterates -14-12-10-8-6-4-20 l og || ( x n + , p + - x n + , p || EMT-TS DDM convergence on Aitken acceleration
TsEMTAccelerateEMTAccelerateTS
Fig. 4
Heterogeneous EMT ( ∆t = 2 . − )-TS( ∆T = 2 . − ) DDM results comparisonwith DAE monodomain (Left) and RAS convergence error for each subdomain at t = 0 . P computed numerically from 9 iterates (right). other part with TS modeling. We showed the convergence/divergence prop-erty of the homogeneous DDM EMT-EMT and TS-TS and of the heteroge-neous DDM TS-EMT, with or without overlap and we use the pure lineardivergence/convergence of the method to accelerate it toward the searchedsolution with the Aitken’s acceleration of the convergence technique. The do-main partitioning is only based on connectivity considerations since we want,in the long term, for the electrical network, to take advantage of the two TSand EMT representations on the overlap in order to identify the loss of in-formation between the two models. We would like then to use this knowledgeto work on other transmission conditions than Dirichlet to conserve someinvariants such as electrical power. References
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