Conference article

Improving Newtonand#8217;s method for Initialization of Modelica models

Johan Ylikiiskilä
Modelon AB, Lund, Sweden

Johan Åkesson
Departement of Automatic Control, Lund University, Sweden

Claus Führer
Departement of Numerical Analysis, Lund University, Sweden

Download articlehttp://dx.doi.org/10.3384/ecp1106397

Published in: Proceedings of the 8th International Modelica Conference; March 20th-22nd; Technical Univeristy; Dresden; Germany

Linköping Electronic Conference Proceedings 63:12, p. 97-104

Show more +

Published: 2011-06-30

ISBN: 978-91-7393-096-3

ISSN: 1650-3686 (print), 1650-3740 (online)

Abstract

Initializing a model written in Modelica translates to nding consistent initial values to the underlyinDAE. Adding initial equations and conditions creates a system of non-linear equations that can be solved for the initial conguration. This pap reports an implementation of Newton’s method to solve the non-linear initialization system. This implementation also uses a regularization method to deal with singular Jacobians as well as sparse solvers to exploit the sparsity structure of the Jacobian. The implementation is based on the opensource projects Jmodelica.org and Assimulo; KINSOL from the SUNDIALS suite and SuperLU.

Keywords

Initialization; Newton’s method; regularization; Jmodelica.org; Assimulo; KINSOL; SuperLU

References

[1] The assimulo homepage. http://www.jmodelica.org/page/199 .

[2] The coconut benchmark: Library 3 constraint satisfaction test problems. http://www.mat.univie.ac.at/~neum/glopt/coconut/Benchmark/Library3_new_v1.html .

[3] The cppad webpage. http://www.coin-or.org/CppAD/ .

[4] Stefan Behnel; Robert Bradshaw; Dag Sverre Seljebotn; and other contributors. Cython cextensions for python - homepage. http://www.cython.org/.

[5] Aaron M. Collier; Alan C. Hindmarshand Radu Serban; and Carol S. Woodward. User Documentation for kinsol v2.6.0. Center for Applied Scientific Computing; Lawrence Livermore National Laboratory; May 2009.

[6] The SciPy community. SciPy Reference Guide; 0.9.0.dev6665 edition; October 2010.

[7] J.E. Dennis and R.B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall; 1983.

[8] Hilding Elmqvist. A Structured Model Language for Large Continuous Systems. PhD thesis; Department of Automatic Control; Lund University; Sweden; may 1978. TFRT-1015.

[9] Jan Eriksson. A note on the decomposition of systems of sparse non-linear equations. BIT Numerical Mathematics; 16(4):462-465; 1976. 10.1007/BF01932730.

[10] Jinyan Fan and Jianyu Pan. A note on the levenberg-marquardt parameter. Applied Mathematics and Computation; 207:351-359; 2009.

[11] John E. Hopcroft and Richard M. Karp. An 2=5 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing; 2(4):225-231; 1973.

[12] X. S. L. W. Demmel James W. Demmel; John R. Gilbert; Stanley C. Eisenstat; John R. Gilbert; Xiaoye S. Li; and Joseph W. H. Liu. A supernodal approach to sparse partial pivoting. SIAM J. MATRIX ANAL. APPL; 20(3):720-755; 1999.

[13] Xiaoye S. Li James W. Demmel; John R. Gilbert. SuperLU Users Guide.

[14] Johan Åkesson; Karl-Erik Årzén; Magnus Gäfvert; Tove Bergdahl; and Hubertus Tummescheit. Modeling and optimization with optimica and jmodelica.org - languages and tools for solving large-scale dynamic optimization problem. Computers and Chemical Engineering; 34(11):1737-1749; nov 2010.

[15] K. Levenberg. A method for the solution of certain nonlinear problems in least squares. Quart. Appl. Math.; 2:164-166; 1944.

[16] D.W. Marquardt. An algorithm for leastsquares estimation of nonlinear inequalities. SIAM J. Appl. Math.; 11:431-441; 1963.

[17] S.E Mattson; H. Elmqvist; M. Otter; and H. Olsson. Initialization of hybrid differential-algebraic equations in modelica 2.0. In Second International Modelica Conferencem; Proceedings; pages 9-15. The Modelica Association; March 2002.

[18] Sven Erik Mattsson and Gustaf Söderlind. Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput; 14(3):677-692; May 1993.

[19] Arnold Neumaier. Solving ill-conditioned and singular linear systems: A tutorial on regularization. SIAM Review; 40(3):636-666; September 1998.

[20] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer; 2006.

Citations in Crossref