Максимова Наталия Николаевна, Астапович Юрий Михайлович
Саратовский государственный технический университет
имени Гагарина Ю. А., Саратов, Россия
МОДЕЛИРОВАНИЕ ДИНАМИКИ МНОГОМОСТОВЫХ ИНВЕРТОРОВ
Митяшин Никита Петрович, д.т.н., профессор каф. «Системотехника
Maksimova N.N., Astapovich Yu.M.
Yuri Gagarin State Technical University of Saratov, Saratov, Russia
MODELING OF MULTIBRIDGE INVERTERS DYNAMICS
Mityashin N.P., Doctor of Engineering Science, Professor
Рассматриваются различные виды моделей динамики многомостовых силовых
преобразователей постоянного напряжения в переменное заданной частоты. Задача
исследования динамики преобразователей может быть решена с помощью стандартных
пакетов моделирования. Однако такая методика, позволяющая отобразить переходные и
установившиеся режимы при любом наборе значений параметров системы, не обладает
общностью результатов, характерной для аналитических методов. В частности указанная
методика не позволяет рассчитывать передаточные функции преобразователя,
необходимые для исследования динамики систем.
Предложенная
методика
позволяет
получить
модели
многомостовых
преобразователей в форме передаточных функций, что является важным ее
преимуществом.
The purpose of this work is receiving transfer functions of inverter modules. The
solution of this problem will allow to research dynamics of conversion systems with
use of methods of the theory of automatic control. At the solution of this problem was
used methodology of representation of inverter modules in the form of a linear
continuous link, which is described in detail in [1]. This methodology is based on
structural symmetry of schemes of converting equipment and on the relevant laws of
following of management impulses by its valves. It is possible to select interval of
symmetry for the most schemes of inverters.
Interval of symmetry is one of m isomorphic structural conditions of the inverter
relative to some linear conversion K, which it accept on the period of frequency of
inverting. The duration of stay inverter module in each of these conditions is
  T m , where T – period of the output frequency of the inverter. During work the
inverter module passes from one interval into another in connection with changes of a
set of working valves. The matrix of linear conversion of K is forming cyclic group
of an order of m [2], so K m  E . This corresponds to the fact that in m switching
inverter module reverts to the original condition. Value m is determined by number of
phases and feature of structure of the scheme.
At theoretical substantiation of used methodology it is necessary to use four
spaces of conditions of inverter modules of same dimension n:
 original
space of the physical variables of condition Kf, which coordinates of

vectors X (t ) are real continuous functions of time – currents and voltages of inertial
elements of the scheme of inverter modules;

 spaces of settlement variables of condition Kc, which vectors X * (t ) are

connected with vectors X (t ) on i interval the following ratios:


(1)
X * (t )  K i X (t ) ,
at that coordinates of this vector, generally speaking, are the discontinuous functions,
which breakpoints coincide with the moments of switching working valves of
inverter modules;

 spaces of discrete variables of condition Kd, which coordinates of vectors x k are

values of the corresponding coordinates of vector X * (t ) at the moments of switching
of k


xk  X * (k ), k  0,1,2,...
 spaces of continuous variables of a condition Ke, which coordinates of vectors


y (t ) are envelope around the corresponding discrete coordinates of vectors xk , that is


(2)
y (k )  xk , k  0,1,2,...
The algorithm of representation of the inverter module in the form of a linear
continuous link consists of the following stages:
1. Construction of system of differential equations for variables of conditions of
inverter on the first interval of symmetry in space Kf as



X (t )  AX (t )  Gu.

Here X (t ) is a vector of variables of condition of inverter module in space Kf,

A and G – matrixes coefficients of equation, t – time, u – vector of voltage of DC
link. Formulas written down will be correct if to accept that values of voltage of DC
link don’t experience considerable changes during an interval. In case such change is
considerable, for value u take an average value of voltage for an interval, which
supplies on input of inverter or input filter of inverter.
2. Transition into space Kc by application of formula (1).
3. Construction of the equation in space Kd according to methodology, described
in [1]:



xk 1  Dxk  Hu k .
Matrixes of this equation are calculated by the formulas:
(3)
D  Ke A ; H  K (e A  E) A1G.


Here eAτ – matrix exponent, u k . – value of vector u on k interval.
4. Transition to continuous model of inverter in the form of Cauchy in space Ke,

based on envelope of transient of vector xk in space Kd. Introduce continuous vector

functions y (t ) , for which formula (2) is correctly, and  (t ) , for which is


 (k )  u k .
Differential equation for this enveloping vector can be written as



(4)
y  Sy  F ,
where
1
S  ln( D); F  S (e S  E) 1 (e S  K ) A1G.
(5)

Here ln(D) is matrix logarithm, which calculation is associated with specific
computational difficulties. Authors have developed program, which implements the
algorithm for computing the matrix logarithm with formula of Lagrange-Sylvester
[3]:
n
n
(D   j E)
ln( D)   ln( i ) 
,
i 1
j 1, j i (i   j )
where λi (i=1,…,n) – eigenvalues of matrix D.
The equation (4) taking into account the equation (5) represents linear
continuous dynamic model of the inverter module, which can be called the
enveloping model. After receive this model we can pass to determination of the

transfer functions of components of vector y (t ) about voltages of DC link, which are
control variables in many problems of regulation by converters.
According to known methodology it is necessary to pass to operator form of
equation (4) and to solve it about vector of operator representations of variables of

conditions y( p) :


y( p)  (S  pE) 1 Bu ( p).

Matrix W ( p)  ( S  pE) 1 B is transfer matrix, so its components
y ( p)
Wij ( p )  i
u j ( p)
are represented unknown transfer functions of inverter modules.
Algorithm of construction of transfer functions of inverter modules, given
above, was applied for receive transfer functions of double-bridge autonomous
current inverter. This inverter module is included in converter complexes of various
purposes.
The algorithm proposed in this article is computationally heterogeneous and
various methods of computational mathematics, used in it, are very difficult to unite
into single computational process in the existing mathematical packages. So authors
have developed user program, which implements algorithm, described above, and
solves above-noted difficulties.
At the first stage of program is executed computation of matrix exponent eAτ as a
result of numerical solution of Cauchy problem of homogeneous system of
differential equations for first interval of symmetry


x  Ax.
Because in this case differential order is equal to three, then integration is

performed three times with vectors of original conditions x i (0) , i=1,2,…, its
coordinates are equal
0, если j  i
x ij (0)  
, j  1,2,...
1, если j  i
Determination of matrix D is performed by multiplication of the matrix
exponent eAτ left by the matrix K, whereupon it is necessary to compute characteristic
polynomial of this matrix. In the developed program computation of characteristic
polynomial of this matrix D is performed by method of Krylov. Further characteristic
equation
det D  E  0,
is solved by method of parabolas, with result that eigenvalues of matrix D are
determined.
The expression for continuous model of inverter is included matrix S, for which
computation it is required to find a matrix logarithm of ln(D). In implementable
algorithm for computation matrix logarithm is applied formula of LagrangeSylvester.
The final stage of construction of transfer function of inverter is determination
the matrix inverse to matrix S  pE . Because this matrix contains the parameter p
interpolation method is the best suited to solution this problem. The interpolation
method consists in sequential execution procedures of selection of interpolation grid,
application of Gauss method for inversing of numerical matrix and construction of
interpolation polynomial of Newton method [4].
The main result of work is development of algorithm and the program for
computation of transfer functions of the double-bridge inverter, it is necessary for the
solution of problems of the analysis and synthesis of automatic control systems of
converting complexes.
References
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