The mathematical model realization algorithm of high voltage cable

Размещено на http://www.allbest.ru//

Размещено на http://www.allbest.ru//

The mathematical model realization algorithm of high voltage cable

M.Tirshu

Аннотация

При реализации алгоритма математической модели очень важно знать порядок расчета необходимых выражений и их вид представления. В зависимости от подключенных нагрузок или источников сигнала в определенных точках модели очень важно знать, как составить уравнения в этой точке для того, чтобы можно было определить все неизвестные. Количество уравнений, которые описывают эту точку, должно совпадать с количеством неизвестных, а матрица, описывающая коэффициенты при неизвестных, должна быть квадратная и иметь размерность, равную количеству неизвестных. В данной работе описывается метод и последовательность программирования рассматриваемой математической модели.

Ключевые слова: математическое моделирование, высоковольтный кабель, процессы распространения волн тока и напряжения

Introduction

Simultaneously with growth of the power systems unification tendency in parallel grows also requirements to increase of power safety. In result, it will lead to requirements growth of functioning reliability of all power system components. The high voltage cables constitute one the most important components of power electric distribution system. Therefore, predictive maintenance or cable defects localization in short time will conduct to power safety increase. But, in order to do predictive maintenance it's necessary to make different measurements at workings high voltage cables. At present, is well known that partial discharges are source of information concerning current technical state of high voltage cable (or other high voltage equipment). The partial discharges really are short-time voltage pulses (ns or µs), which measurement is very difficult. But, due to the technical progress, today, practically, there no problems to make so kind of measurements. The problem consists in analysis of measured data and formulation of adequate conclusion about really technical state of analyzed object. This problem is caused by reflection, refraction and energy dissipation phenomena, because high voltage cable can be represented as a long line with distributed parameters, which include unhomogeneities. Just from these unhomogeneities take place pulses reflection produced by partial discharges. As result, at the measurement points (usually cable ends) we have received pulses parameters of which are very different from original parameters. In order to have possibility to research propagation phenomena of short-time pulses in high voltage cable was elaborated mathematical model which describe adequate it physical structure. This mathematical model give possibility to simulate different really situation (different kind of defects placed in arbitrary points of cable). The purpose of these researches is to establish in future some propagation particularities of these pulses and increase correctness of conclusions about current technical state of cable. But, is very important to be able to program this mathematical model and do needed changes. In this paper will be described the model realization algorithm in any programming system.

1. The high voltage cable mathematical model base relations

As is known from the literature distribution of waves in cable is described with the help of the equations so-called cable equations:

(1)

The cable equations, which describe wave propagation processes in cable, produced by partial discharges (PD) in insulation are differential equations of hyperbolic type. The solution of this hyperbolic equations system is useful to find with characteristics method [1, 2, 3].

Calculations according to this method are made with the help of a characteristic grid which is generated constantly during calculation (fig.1).

Размещено на http://www.allbest.ru//

Размещено на http://www.allbest.ru//

In order to received numerical solution let introduce a non-uniform grid on segment of discontinuous change of argument with value of a step The voltage and current unknown functions will be determined in semi-nodes . The division is made so that for any the condition satisfied, and on boundary division of line parts with distributed parameter which differ both values and boundary conditions must coincided with points.

The calculation is made by consecutive steps on time. Knowing the function values and on the previous time interval (for examples, initial conditions), we can determine unknown function values on next time interval , were - time variation step. For approximation of the central differences first derivatives the system equation (1) is replaced with the analogue in the form of final differences, receiving so the equations:

(2)

or in another mode:

(3)

Indexes in top correspond to time interval “top”. The additional values determined in points are calculated from relation:

from were we determine:

(4)

For i and u values computation in and points the boundary conditions and corresponding relations on characteristics are used:

(5)

If the condition (6) satisfies:

, , (6)

then equation system (5) have one solution only.

The necessary and sufficient stability condition of numerical computation method is obtained with a priory estimation method of energy balance from relation [2, 3]:

. (6)

2. Mathematical model studying cable

On the fig.2 is presented high voltage equivalent scheme, which have connected a voltage source on input, a circuit in each odd point connected at ground and a circuit in each even point connected in series. These circuits will allow to modeling any type defect. For this scheme will describe the mode of computation algorithm realization.

Fig.2 The equivalent high voltage cable scheme with input connected to voltage source and circuits connected in parallel in all odd points and in series in all even points.

So, we have N+1 particular points, were must to determine parameters. We will start with first point .

Here we have 4 unknown variables: , were - left-side current, - current in circuit, - right -side current, - voltage in this point. We will paste these unknown variables in vector of unknown parameters in follow mode: . As we have 4 unknown variables, we must have 4 equations and write its in follow mode:

According to the previous description the first will be the characteristic equation:

;

; ;

From these equations we obtained values, which are known:

;

.

The matrix B of factors at unknown variables will be:

;;

As result, we can determine the values of unknown variables: .

The unknown variables in odd digitization points we will determine in follow mode:

Let's note unknown functions: , were - the current in point i from left side, - the current in point i from right side, - the current in circuit from point i, - voltage in point i, were . We will paste these unknown variables in vector of unknown parameters as follow: . As we have 4 unknown variables, we must have 4 equations:

;

;

.

From these equations we determine known values:

;

; .

The matrix of factors at unknown variables will be:

;;

As result, we can determine the values of unknown variables: .

The unknown variables in even digitization points we will determine in follow mode:

Let's note unknown functions: , were - the voltage in point p from the left, - the voltage from the right side after the circuit from point p, - the voltage on the circuit from point p, - the current in point p, were . We will paste these unknown variables in vector of unknown parameters as follow: . As we have 4 unknown variables, we must have 4 equations:

;

;

.

From these equations we determine known values:

;

; .

The matrix of factors at unknown variables will be:

;;

As result, we can determine the values of unknown variables: .

The unknown variables at the end of cable (point N) we will determine in follow mode:

Here we have 2 unknown variables only:. We will paste these unknown variables in vector of unknown parameters as follow: . As we have 2 unknown variables, we must have 2 equations:

;

.

From these equations we determine known values:

;

.

The matrix of factors at unknown variables will be:

;.

As result, we can determine the values of unknown variables: .

From this moment we can do the computation program. The functioning scheme is submitted below.

3. The programming algorithm of mathematical model

For programming procedure simplification of mathematical model we will introduce some modifications:

D - current in semi-nodes; U - voltage in semi-nodes; Dx - current in nodes; Ux - voltage in nodes.

; ; ; .

To exclude Gibbs oscillations at cable with lose we will change the values of and as follow:

for, for , for and for . So, we received:

, , , .

As result equations (3) can be writing in follow mode:

The equations for current and voltage in hole node will look as follow:

model voltage cable current

Or in another form:

Now, we can present the functional diagram of mathematical model computation from fig.2, which represent a part of high voltage cable with distributed parameters and circuits with concentrated parameters connected at the ends of cable and in odd points in parallel, but in even points in series. Also, at the input of cable we have connected a voltage source for given period of time.

Conclusions

The proposed algorithm is easy to do and allow to provide a large parametrical investigations of transients, which is very important for correct define of working and realization mode of technical system with distributed and concentrated unhomogeneities without a big idealization of respective objects.

Reception of similar results like above mentioned with help of experimental methods is very difficult and frequently is impossible to find the satisfying solution.

The mathematical model allows simulating any type of defect and pasting it in any place of cable. At the same time, the source allows applying needed voltage shape for given time.

Acknowledgments

The researches are carried out with support of INTAS - International Association for the promotion of cooperation with scientists from the New Independent States of the former Soviet Union, an international non-profit association organised under the laws of Belgium in frame of project “Young Scientist Fellowship”, INTAS Ref.Nr 05-115-5129.

Bibliography

Naval I.C., Rвbachin B.P., Ceban V.G. Matematicescoe modelirovanie ecologhiceschih proюesov. Chiєinгu, "EVRIKA", 1998, 246 p.

Muzвcenco V.P., Rimschi V.X. Matematiceschie metodо issledovania rasprostranenia voln. Dangavpils: MO SSSR, DVVAIU, 1986. - 200 p.

Rimschi V., Berzan V., Tоrєu M. Volnovвie iavlenia v neodnorodnвh liniah, Tom.I. Teoria rasprostranenia voln potenюiala i toca. Pod redacюiei V.K.Rimscogo.-Kiєinev: Tipografia Academii Nauk R.Moldova, 1997.-295 p.

Размещено на Allbest.ru