# Forced inactivity in electric elements by modal perturbations

FIELD: electricity.

SUBSTANCE: electric element having ports and linear electric properties characterised in matrix, which is impedance matrix, admittance matrix or dissipation matrix of electric element and connecting voltage applied to ports with current passing through these ports. Electric element has inactivity determined by means of parameters perturbation up to perturbated set of parameters provided that this perturbated set of parameters corresponds to function in Boolean values.

EFFECT: increase of forecast accuracy for technically relevant linear electric properties of electric elements.

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The technical field to which the invention relates.

The invention relates to a method for modeling the linear properties of an electrical component with the enforcement of passivity.

The level of technology

The residual disturbance (OB) (RP) is often used as a means to enforce passivity in the models describing the linear properties of an electrical component. One known approach S uses a quadratic programming (CP) (QP) for solving the problem by the method of least squares with constraints.

As an example, consider a model with deductions in the poles for the full matrix Y conductivities

where s is the angular frequency, R_{m}for m=1 to N are matrices independent of s (N represents the number of poles or resonances taken into account), D is the matrix that is independent of s, and a_{m}for m=1 to N are complex angular frequencies of the poles or resonances.

The model parameters should be outraged so that the perturbed model met the criteria of passivity that the real part of the eigenvalues of Y is positive for all frequencies, ie,

The outrage should be done so as to minimize the change in the original model, ie,

img src="https://img.russianpatents.com/1107/11079887-s.jpg" height="12" width="107" />

The traditional way of handling equation (2b) is to minimize changes for ΔY in the sense of least squares.

The invention

The problem solved by the present invention is to provide a method with higher precision.

This problem is solved by the method according to claim 1 of the claims. This invention is based on the understanding that the weakness of the approach in the prior art is that the small eigenvalues of Y can easily be distorted due to the perturbation (ΔY). The invention overcomes this problem by "modal perturbations, i.e. by finding an approximate solution for this problem:

where F is the function describing the dependence of the matrix Y from the independent variable s, while p_{1},...,p_{K}are parameters (which must be resent) model. t_{i}there are a number of independent ports electrical component (device).

For the model with deductions in the poles of the function F is expressed by the equation (1), and the parameters of the p_{1},..., p_{K}can, for example, correspond to the elements of matrix R_{m}and D.

In addition to equation (3) requires the restriction that guarantees the passivity of the matrix Y, analogous to equation (2A). According to the present invention is a generalized version of equation (2A) can be expressed by requiring that stabilizamainly set of parameters p_{
1}+Δp_{1},..., p_{K}+Δp_{K}match suitable function C in Boolean values:

An approximate solution for n vector equations (3) predominantly found by minimizing the sum of squares of each coordinate vectors of each of the equations under the condition of equation (4).

The constraint expressed by a function of the conditions may be, for example, a constraint according to equation (2A). But it may also be suitable restriction such, for example, what is obtained using the eigenvalues of the Hamiltonian matrix, as, for example, described in S.Grivet-Talocia, "Passivity enforcement via perturbation of Hamiltonian matrices" ("enforcing passivity via perturbation of Hamiltonian matrices", IEEE Trans. Circuit and Systems I, vol.51, no. 9, pp.1755-1769, Sept. 2004.

Further options, advantages and applications are dependent claims and the following detailed description.

Embodiments of the inventions

Device simulation

As mentioned, the present invention relates to the modeling of linear electrical properties electrical component with n ports.

The term "electrical component" should be understood in a broader sense, it can refer to a particular device, such as a transformer, or the site from multiple devices such as a system of transformers, motors, etc. are mutually connected by power lines.

Linear electrical properties of such devices can be expressed by a matrix Y of size n×n, which in General refers to the voltage applied to the port for flowing current through it. The matrix Y can be a full matrix conductivity, as described in the introduction, but it can be, for example, also the matrix of impedances (usually called Z) or the scattering matrix (usually called S) of the device. Therefore, even though the matrix Y is predominantly a full matrix conductivity, it can also be described in any other type of linear response of the device.

The model describes the dependence of the matrix Y from the independent variable s, which can be frequency, but it can also be, for example, time or discrete z-domain. Therefore, even though this independent variable s is mainly frequency, it can also be any other independent variable, the dependence on which is described by the model.

The dependence of the matrix Y from the independent variable s can be, for example, the described model, with deductions in the poles in equation (1). This model has several parameters that need to be resent to ensure passivity. In the example of equation (1) these parameters are matrix elem is now matrices R_{
m}and D. Alternatively, these parameters can also be, for example, the eigenvalues of the matrix R_{m}and D. in Addition, it is also possible to perturb the pole frequency and_{i}.

It should be noted, however, that equation (1) is not the only model that can be used to describe the matrix Y in the context of the present invention. In particular, equation (1) can be refined by adding additional expressions, namely s·E matrix E of size n×n, which describes the linear dependence of the matrix Y from the independent variable s.

In more General terms the dependence of the matrix Y from s can be described by the matrix-function F, defined above, ie,

p_{1},...,p_{K}which are the model parameters that need to be resent to enforce passivity.

The function F is usually a polynomial function, rational function or polynomial amount and (or) rational functions.

The function F is mainly a rational function, primarily given as a ratio between two polynomials in s, the model with deductions in the poles model state space, or any combination thereof.

Enforce passivity

The options are subject to perturbation so that the matrix Y has become passive. The outrage" in this context means,
the parameters of the p_{1},...,p_{K}(slightly) adjusted to become indignant set of parameters p_{1}+Δp_{1},..., p_{K}+Δp_{K}.

If, for example, the matrix Y is a matrix of full resistance, passivity can be achieved for the perturbed set of parameters, if the following conditions are met:

where eig_{i}() is the operator issuing its own value of i from its matrionnogo argument. If the function F has a model with deductions in the poles of equation (1) and if the perturbation modifies only the matrix R_{m}and D, this gives:

where ΔR_{m}and ∆ D are the changes introduced in matrix R and D in consequence of the disturbance.

In the case of equation (1) is equivalent to the condition given by equation (2A). It should be noted, however, that there are other conditions that ensure the passivity of the matrix Y, such as restrictions derived from the eigenvalues of the Hamiltonian matrix, as mentioned above. Thus, in a more General form of the condition that the matrix Y perturbed set of parameters p_{1}+Δp_{1},..., p_{K}+Δp_{K}passive, can be expressed a function of conditions in the Boolean values that depend on the perturbed set of parameters p_{1}+Δp_{1},..., p_{K}+Δp_{K}. Namely, when a suitable definition of the function C passivity is achieved,
if:

The perturbation algorithm

The goal of the algorithm described here is to find the perturbed set of parameters p_{1}+Δp_{1},..., p_{K}+Δp_{K}that satisfies equation (6) or, in more General terms, equation (8) under the condition that the perturbation is maintained "as low as possible".

Adopted in the present invention approach is motivated by the fact that the matrix Y can be made diagonal by converting it into the matrix of eigenvectors of T. namely:

where Λ is a diagonal matrix with the eigenvalues of Y as its nonzero elements, and T is a matrix of size n×n formed by placing n of eigenvectors of t_{i}matrix Y in its columns. The multiplication on the right of equation (9) with T and taking the derivative of the first order while ignoring the expressions that includes ΔT gives for each pair (λ_{i}, t_{i}):

In other words, the perturbation matrix Y leads to the corresponding linear perturbation of each fashion and own space.

The present invention is based on the understanding that the outrage should be maintained "as low as possible" in the sense that the perturbation of each mode is weighted by inverse transformation of the elements of its sobstvennosti.

For the case of models with deductions in the poles in equation (7) this means that it is possible to minimize the error in the following equations:

In a more General case, equation (5) this corresponds to equation

Therefore, the purpose of this algorithm is to nd an approximate solution to equations (12) or, for example, for the model with deductions in the poles of the solutions to equations (11) for all i=1 to n. Since for each i there is a vector-valued equation, this means that you need to approximate the entire n×n scalar equations in the case that one of the conditions(6)-(8).

This approximation, as a rule, is performed by minimizing the sum of squared errors of all equations using quadratic programming algorithms.

Many of these algorithms minimize suggest that the subject of approximation equations are linear in the parameters, which should be outrage. This is already the case for equation (11). For the General case of equation (12) it may not necessarily be needed. For example, if the model is used with deductions in the poles for the equation (1), but change and frequency a_{m}poles, equation (11) becomes nonlinear in vozmushchaemym parameters Δa_{m}. In this case, the equations should be linearizability before those who,
as they will be put into the standard quadratic programming algorithms. For the General case of equation (12) this linearization can be expressed as:

Before the introduction of data in the quadratic programming algorithm can calculate the derivatives in equation (13). In addition, the values of eigenvectors of t_{i}and eigenvalues λ_{i}that relate to the unperturbed matrix, Y, is calculated to optimize.

Instead of minimizing the error of equation (12) in mean-square sense, you can use any appropriate measure (norm) of each vector element equations (13). Such measures are known to experts.

The method according to the present invention substantially reduces the problem of disturbances that distort the behaviour of the model used in the simulation with arbitrary boundary conditions, in particular, if the matrix Y has a large spread of eigenvalues. This is achieved by formulating the part of the least squares problem limited optimization, so the size of the perturbation of the eigenvalues of the full conductance is inversely proportional to the size of the eigenvalues. This allows you to get around the fact that small eigenvalues become distorted. Application to models with a large violation of inactivity, dormancy is shows the new approach preserves the behavior of the original model, whereas large deviations lead to alternative approaches. The approach of modal perturbation is computationally more expensive than alternative methods, and is mainly used sparse solvers for quadratic programming.

1. An electrical component having n > 1 ports and having a linear electrical properties, as described in the matrix Y, which is the impedance matrix, the matrix of the full conductance or the scattering matrix of the electrical component and connecting the voltage applied to the ports, with the current passing through these ports, and the dependence of Y on the independent variable s, which is the frequency, time or discrete z-region is approximated by the model

where R_{1},..., p_{K}are the model parameters, a F is metricheskuyu function describing the dependence of Y on the variable s,

in this case the electric component has the passivity determined by perturbation of the mentioned parameters R_{1},..., R_{K}to the perturbed set of parameters p_{1}+Δ_{1},..., p_{K}+Δ_{K}while ensuring that this perturbed set of parameters corresponds to the conditions function in the Boolean values

and by finding approximate solutions to equations

for i=1...n, and t_{i}and λ_{i}are self-vectors and eigenvalues of the matrix Y.

2. An electrical component according to claim 1, and equation 1.3 linearized through expressions

3. An electrical component according to claim 1, and function With the condition is an expression

where eig_{i}() is the operator that returns the private value of i his matrionnogo argument.

4. An electrical component according to claim 2, and function With the condition is an expression

where eig_{i}() is the operator that returns the private value of i his matrionnogo argument.

5. An electrical component according to any one of claims 1 to 4, and the above function F is a function from the group of rational functions, polynomials, functions deduction in pole models in the state space or their combinations.

6. An electrical component according to claim 4, and equation 1.1 represents

and R_{m}when m=1...N are matrices independent of s (where N is the number of poles or resonances taken into account), D is a matrix that is not dependent on s, and a_{m}when m=1...N depict ablaut a complex angular frequency poles or resonances,
moreover, the above-mentioned matrix R_{m}and/or D and/or the pole and_{m}depend on these parameters p_{1},..., R_{K}.

7. An electrical component according to claim 5, where each element in the above matrix R_{m}and D is one of the mentioned parameters R_{1},...,R_{K}.

8. An electrical component according to claim 5, each eigenvalue of the above matrix R_{m}and D is one of the mentioned parameters p_{1},...,R_{K}.

9. An electrical component according to claim 5, whereby the above-mentioned equation 1.3 is a

10. An electrical component according to any one of claims 1 to 4, and the approximate solution of equation 1.3 is found by minimizing the measures of each vector element of each of the equations 1.3.

11. An electrical component according to claim 7, with the approximate solution of equation 1.3 is found by minimizing the measures of each vector element of each of the equations 1.3.

12. The electrical component of claim 8, with the approximate solution of equation 1.3 is found by minimizing the measures of each vector element of each of the equations 1.3.

13. An electrical component according to claim 9, with the approximate solution of equation 1.3 is found by minimizing the measures of each vector element of each of the equations 1.3.

14. An electrical component according to any one of claims 1 to 4, and an approximate solution is equation 1.3 is found by minimizing the sum of squares of each vector element of each of the equations 1.3.

15. An electrical component according to claim 9, with the approximate solution of equation 1.3 is found by minimizing the sum of squares of each vector element of each of the equations 1.3.

16. An electrical component according to any one of claims 1 to 4, and the electrical component is a separate device, in particular a transformer, or a site from multiple devices, in particular a system of transformers or motors, are mutually connected by power lines.

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