Simulation of electromagnetic descriptor models using projectors
© Banagaaya and Schilders; licensee Springer 2013
Received: 26 June 2012
Accepted: 6 February 2013
Published: 11 February 2013
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.
where , and are the capacitance, inductance and conductance matrices, respectively which are usually assumed to be symmetric and positive-definite. We need to find the unknowns of the system (1). If we let and , then , and . Equation (1) can be solved depending to the elements in the electric networks. For example if we consider a resistor network, (1) simplifies to a system of algebraic equations which implies that . Thus we need to solve a linear system of the form . But if the network contains a combination of resistors and other elements, this leads to differential algebraic equations (DAEs), i.e., . In this paper, we consider electric networks which leads to DAEs.
DAEs are very difficult to solve numerically due to the sensitivity of their solutions to perturbations. This sensitivity is measured by the index concepts such as differentiation index, perturbation index and tractability index. We use the tractability index because it can be obtained numerically. According to , a DAE-index from electric networks cannot be greater than 2. Thus we assume (1) has maximum index of 2. In order to solve DAEs we assume the following: (i) the matrix pencil must be nonsingular for some . (ii) the input vector u must be smooth enough. Further more consistent initial values must be applied.
A lot of work has been done to solve DAE system using multistep methods such BDFs, NDFs and Runge–Kutta methods . Although these methods are accurate they need a lot of computational effort depending on the index of DAE system, for example index-2 systems, BDF is convergent and globally accurate to but these require tight Newton solutions accurate to . The implicit Euler method loses order of accuracy with each increase in index, thus cannot be used in practice to solve DAE systems numerically .
Thus, we need to make sure the physical properties of the DAE system (1) such as stability to be inherited in its decoupled system. 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. Moreover, this decoupling preserves the spectrum of the matrix pencil of the DAE.
This paper is organized as follows: In Section 2, we discuss the decoupling of LTI DAE system using März decomposition and its limitation. In Section 3, we propose the modification of the März decomposition using bases of special projectors. In this section we further discuss the modified decoupling of index-1 and -2 electric systems. We observed that higher index DAEs can decoupled into two ways depending on the spectrum of the matrix pencil . In Section 4, we check whether the decoupled system preserves the physical properties of the DAE system. In Section 4.1, we compare the numerical accuracy of the decoupled and undecoupled system. In Section 5, we test the proposed method on both simple and industrial problems. The industrial examples show the feasibility of this method on real-life applications. This paper is concluded by some final remarks in Section 6.
2 Decoupling of LTI DAE systems using projectors
As a consequence, (2) can be decoupled into 1 differential part and μ algebraic parts. Unfortunately, März decomposition leads to decoupled system of larger dimension than the dimension n of (1) and also 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-1 and -2, respectively. This modification leads to a decoupled system which preserves the dimension and the stability of the DAE system. In fact, the dimension of the differential part is equal to the dimension of the finite spectrum of the matrix pencil . Thus, the stability of the solutions of the decoupled system is guaranteed. We discuss this modification for index-1 and -2 systems in the next section.
3 Modification of the März decomposition
In this section, we modify the März decomposition using special basis vectors. This decomposition leads to a decouple system that preserves the dimension and the spectrum of the matrix pencil of the DAE system.
3.1 Decoupling of index-1 electric networks
where , , and . Observe that (3a) and (3b) can be solved in a hierarchical way and the desired solutions of the DAE system can be obtained using (4). We observe that the differential part (3a) can be solved using ODE numerical methods and algebraic part (3b) can solved explicitly using the solutions from the differential part. This decoupled system always preserves the dimension since and the stability of (1) since it can easily be proved that . The proof can be found in . and is the number of differential and algebraic equations, respectively.
3.2 Decoupling of index-2 electric networks
We now assume (1) is of index-2 which implies in Equation (2). We first construct basis vectors in with their inversion for the projectors and , where , . For this case we have two possibilities depending on the spectrum of the matrix pencil .
3.2.1 Matrix pencil with at least one finite eigenvalue
Theorem 1 Let , . Then are projectors in provided the constraint condition holds.
is a strictly lower triangular nilpotent matrix of index 2. Also the decoupled system (7a)-(7c) can be solved in a hierarchical way and the desired solutions of the DAE system can be obtained using (7c).
3.2.2 Matrix pencil with no finite eigenvalues
where , , and is also a strictly lower triangular nilpotent matrix of index 2 which is defined as (8). We can observe that we do not need initial conditions to solve system (11a) and (11b), in fact the solutions are computed explicitly. We note that the input vector u must be at least times differentiable.
We have discussed that index-1 and -2 electric networks can be decoupled using special basis vectors. These special basis vectors can be computed numerically in an efficient way using the LUQ routine discussed in . Hence this procedure can be applied even on very large electric networks as illustrated in Section 5.2. We call this method the Split-DAE method since it involves splitting the DAE system into differential and algebraic variables.
4 Analysis of decoupled electric networks
where , and is the projected state space. We note that the solutions of (1) and (12a) and (12b) coincide since the two systems are equivalent to each other. This implies the spectrum of matrix pair and must also coincide. System (12a) and (12b) is much easier to solve than its counter part and moreover it reveals the interconnection or structure of the DAE system. Assume the matrix pencil of (1) has at least one finite eigenvalue then following theorem below holds:
This implies that .
Thus the stability of the decoupled system (12a) and (12b) depends on the stability of the differential part. Hence if Equation (1) is stable then also (12a) and (12b) must be stable.
Index 1 electric networks
Index 2 electric networks
Matrix pencil with at least one finite eigenvalue
Matrix pencil with no finite eigenvalues
We observe that the descriptor form with differential part takes the same form for the case of index-1 and -2 system though for index-1 systems. We use this form only for analysis of the solutions of the DAE system (1) but not for solving. For solving one need to solve the decoupled systems derived in the previous section.
4.1 Numerical accuracy of the decoupled system
Index 1 systems
Index 2 systems
Matrix pencil with at least one finite eigenvalue
Matrix pencil with no finite eigenvalue
Hence implicit Euler method loses accuracy for each increase in the index if used on the DAE system directly. Thus first decoupling the DAE system (1) into differential and algebraic parts is a robust and effective way of solving DAEs. Since on the differential part one can use any numerical integration method and then the solutions of the algebraic part can be obtained explicitly. Hence the Split-DAE method is a very accurate method and very easy to implement as compared to its counterparts discussed in .
5 Numerical experiments
In this section, we test our proposed method in Section 3 using problems from electromagnetic community since is the main focus of this paper but it can also be applied to other applications.
5.1 Simple examples
Here, we explicitly discuss how to decouple DAEs into differential and algebraic parts using projectors. We illustrate this using examples below. Example 1 is an index-1 electric network while Example 2 is an index-2 electric network whose decoupled system has a differential part. In Example 3, we illustrate the decoupling of index-2 system without differential part.
We can easily check that is non-singular. Thus Equation (27) is of index-2.
We observe that , , and . Equations (34a) and (34b) can be solved if we apply . Thus the DAE system (27) is decoupled into 1 differential and 3 algebraic equations. The solutions of system (27) and (34a) and (34b), (35) coincides though the later is easier to solve than the former.
5.2 Industrial applications
In this section, we test the Split-DAE method on large scale descriptor electromagnetic models from industries. Examples 4 and 5 are index-1 and -2 systems respectively. These examples are multiple-input multiple-output (MIMO) type, i.e. we are interested in a few solutions of the DAE system.
In conclusion, splitting DAEs using special projectors leads to decoupled systems which are easier to solve than DAE systems directly. We have tested the Split-DAE method on both large and small problems and proved to be a robust and efficient method. We can now use any numerical integration method to solve DAE systems. Hence this method can be used to simulate electromagnetic descriptor models.
This work was supported by The Netherlands Organization for Scientific Research (NWO).
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