Abstract
Electromagnetic descriptor models are models which lead to differential algebraic equations (DAEs). Some of these models mostly arise from electric circuit and power networks. The most frequently used modeling technique in the electric network design is the modified nodal analysis (MNA) which leads to differential algebraic equations in descriptor form. DAEs are known to be very difficult to solve numerically due to the sensitivity of their solutions to perturbations. We use the tractability index to measure this sensitivity since it can be computed numerically. Simulation of DAEs is a very difficult task especially for those with index greater than one. To solve higher-index DAEs, one needs to use multistep methods such as Backward difference formulas (BDFs). In this paper, we present an easier method of solving DAEs numerically using special projectors. This is done by first splitting the DAE system into differential and algebraic parts. We then use the existing numerical integration methods to approximate the solutions of the differential part and the solutions of the algebraic parts are computed explicitly. The desired solution of the DAE system is obtained by taking the linear combination of the solutions of the differential and algebraic parts. Our method is robust and efficient, and can be used on both small and very large systems.
Highlights
Consider a linear Resistor-Inductor-Capacitor (RLC) electric network which connects linear capacitors, inductors and resistors, and independent voltage v(t) ∈ RnV and current sources ı(t) ∈ RnI
Using the commonly used Modified Nodal Analysis (MNA) [ ], we introduce the incidence matrices AC ∈ Rne,nC, AL ∈ Rne,nL and AR ∈ Rne,nG, which describe the branch-node relationships for capacitors, inductors and resistors
We introduce the incidence matrices AV ∈ Rn,nV and AI ∈ Rne,nI, which describe this relationship for voltage and current sources, respectively
Summary
Consider a linear Resistor-Inductor-Capacitor (RLC) electric network which connects linear capacitors, inductors and resistors, and independent voltage v(t) ∈ RnV and current sources ı(t) ∈ RnI. This motivated us to do some modifications in the März decomposition, using special basis vectors, which leads to a modified decoupled system of dimension n This decoupling preserves the spectrum of the matrix pencil (E, A) of the DAE. März decomposition leads to decoupled system of larger dimension ( + μ)n than the dimension n of ( ) and it doesn’t preserve the spectrum of the matrix pencil This motivated us to modify her decoupled system using special basis vectors introduced in [ ] and [ ] for index- and - , respectively. This modification leads to a decoupled system which preserves the dimension and the stability of the DAE system.
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