Modeling Dynamic Propagation of Characteristic Gases in Power Transformers Oil-Paper Insulation
Modeling Dynamic Propagation of Characteristic Gases in Power Transformers Oil-Paper Insulation
A. Shahsiah, R.C. Degeneff and J.K. Nelson
Rensselaer Polytechnic Institute
110 8th Street
Troy, NY 12180, USA
ABSTRACT
This paper presents and verifies a new mathematical model to explain the dynamic behavior of characteristic gases in the oil-cellulose insulation of high voltage devices like power transformers. The model is based on the diffusion process. Parameters of the model are already quantified from experiments and presented in previous publications. The mathematical model and assumptions are presented here. The model solution is obtained analytically and concentration change inside the paper insulation as a function of time is simulated. The results are converted to the concentration change in the oil using the principle of conservation of mass and validated with experimental measurements. The model presented can be used to reduce the error of Dissolved Gas Analysis due to the migration of characteristic gases inside a healthy power transformer.
Index Terms — Transformers, diffusion, oil-paper system, Dissolved Gas Analysis.
1 Introduction
Faults in power transformers will cause decomposition of the transformer liquid dielectric and generate gases. By inspecting of the transformer liquid dielectric (oil), the type and severity of faults can be detected using the type and amount of the gases generated inside the transformer. The detection method is called Dissolved Gas Analysis (DGA). The general diagnostic methods using DGA are summarized in ANSI/IEEE and IEC standards [1, 2]. Obviously, the detection method is not able to measure the amount of the gases that are inside the solid insulation. However, temperature variations can cause the generated gases to migrate into the solid insulation or more gases come out from the solid insulation into the liquid. This could generate error in DGA measurements or trigger a false alarm. A mathematical model can be used to convert the DGA results to the real amount of gas present in the system based on the current gas concentration in the oil and the system temperature.
The results of previous research show that the migration phenomenon in the oil-paper insulation of power transformers can be explained by diffusion equations and the relevant parameters have been obtained [3, 4, 5]. Findings of the previous research include diffusion coefficients and steady-state information on the characteristic gases in transformer insulation systems. The mathematical model presented here is an attempt to explain a portion of the dynamic behavior of characteristic gases in oil-paper insulation systems that is due to the temperature change. The proposed model is limited to the migration of the gases that already exist in the insulation system. Generation of new gases, either inside the paper insulation or in the oil, is not considered. The migration phenomenon inside the solid insulation is explained by the diffusion process.
Assumptions are made to limit the effects of the migration phenomenon to the diffusion process inside a single oil duct of power transformers. The mathematical model uses steady-state information to predict the mass distribution in the paper insulation during the transient conditions as a function of time and distance inside the paper insulation. The boundary conditions are found based on the physical facts. The mathematical model is transformed into per-unit form to simplify solving the equations. The model is then solved and the results are compared with experiments. The model sensitivity to the oil-paper boundary layer is investigated.
Experiments are designed to test the derived model. The experimental setup and method are explained in this paper. The experiments are based on measurements in the oil. The mathematical model, however, solves for the amount of gas inside the solid insulation. The principle of conservation of mass is used to convert the amount of gas inside the paper insulation to that inside the oil. Measurements inside the oil are then used to compare the simulation results against experiments.
2 Theory
2.1 The diffusion model
Assuming a constant diffusion coefficient (at a certain temperature), transfer of mass through a volume as a function of time can be expressed by the following general equation:
(1)
in which c is the mass concentration in the oil, t is time, D is the diffusion coefficient, is the material velocity vector, T is temperature, and is the thermal diffusion factor. The last term in Equation 1 is due to a thermal diffusion phenomenon known as the Soret effect. This effect is normally negligible compared to the diffusion caused by the difference in partial pressures of the species. It is assumed in this work that there is no generation of mass due to chemical reactions.
Inside the oil-impregnated paper insulation there is no transport of mass, therefore; movement of gases in this medium is defined by diffusion only, ignoring the Soret effect. Equation 1, therefore, reduces to Equation 2 inside the paper insulation.
(2)
in which cp is the gas concentration inside oil-impregnated paper insulation, Dp is the diffusion coefficient of gas in this medium, t is time and x is distance.
In the oil region it is assumed that the gases are quickly mixed with the bulk of the oil, therefore; gas concentration is evenly distributed in this region. Equation 2 defines the dynamics of the oil-paper system in terms of the dissolved characteristic gases. The dynamics of the system is a function of temperature. In other words, the system is at equilibrium at a certain temperature; when temperature is changed, the system goes toward another steady-state condition, the dynamics of which is defined by Equation 2. Dependency of Equation 2 on temperature is reflected through Dp which is a function of system temperature. In the suggested diffusion model the average of the diffusion coefficients at the two equilibrium temperatures are used in Equation 2 to obtain the dynamic behavior of the system during a step change from one temperature to another.
Figure 1. Steady-state concentrations in an oil-paper system at two temperatures.
Figure 2. Dynamic distribution of concentrations in oil-paper system in transition between the steady-states at the two temperatures.
Figure 1 illustrates the oil-paper system in equilibrium at two temperatures, T1 and T2. Figure 2 illustrates the same system under dynamic conditions in transition between the two temperatures.
Equation 2 is solved (along with the boundary conditions that will be defined later) to find the concentration of gases inside the oil-impregnated paper insulation as a function of time and distance inside the insulation thickness. To verify the model, however, gas concentration is measured inside the oil (using the DGA method). The principle of concentration of mass is used to correlate the measured gas concentration inside the oil to that inside the paper insulation. Equation 3 shows this correlation.
(3)
In Equation 3 is the average concentration change in the paper insulation, is the paper insulation thickness, cp is the apparent gas concentration in the paper insulation, cps is the gas concentration in the paper insulation at the steady state, t is time and x is distance.
2.2 Boundary conditions
Boundary conditions for Equation 2 are defined for gases inside the paper insulation in reference to Figure 2. The first two boundary conditions are obtained based on the fact that the initial and final steady-state concentrations inside the paper insulation are equal to these values at the time zero and time infinity (cpi and cps).
(4)
(5)
The third boundary condition assumes that the paper insulation is semi-infinite. In other words, gases inside the paper insulation will not diffuse all the way to the other end. This assumption is justifiable based on the diffusion time-constant of gases inside the paper insulation. In the case of carbon dioxide, for example, it takes about 550 hours for 90 percent of the total gas to diffuse all the way through a 1 mm piece of oil-impregnated pressboard (Hi-Val) at 23oC based on the experimental data [3, 4]. Comparing this time to the time that it takes for the oil and paper insulation at the boundary to exchange gases when there is a temperature change (almost immediate), the paper insulation can be assumed semi-infinite. Equation 6 defines the third boundary condition based on this assumption.
(6)
The fourth boundary condition is associated with the diffusion of mass at the oil-paper boundary and is derived based on the fact that there is no accumulation of mass at the boundary. In other word, the rates of mass diffusion into the paper insulation and out of the oil (or vise versa) are equal at the boundary. This is shown in Equation 7.
(7)
in which cp and co are concentrations inside the paper insulation and the oil, respectively; cw is the concentration of gas at the oil-paper surface (cw=cp|x=0) as illustrated in Figure 2. Parameter δd is the thickness of the boundary layer, Dp and Do are the diffusion coefficients in the paper insulation and in the oil, ρp and ρo are mass densities of the paper insulation and the oil, respectively, and x is distance inside the paper insulation.
The right-hand side of Equation 7 is derived based on the assumption that the concentration change along the boundary layer changes linearly. The reason is that the equilibrium time constant of gases in the boundary layer is much smaller than that in the paper insulation [3] and, therefore, the system can be assumed at steady-state in this region. Based on this assumption, the concentration distribution in the boundary layer will be linear, since in Equation 2 the term will be equal to zero (steady state). Hence, the term is also zero which means a linear distribution of c with respect to x.
Rearranging Equation 7 results in Equation 8 which is the last boundary condition of the problem.
(8)
2.3 equations in per-unit form
In order to make it easier to solve the proposed diffusion model, the equations and boundary conditions can be transformed into a per-unit form. Table 1 shows the variables in per-unit which are underlined. The variable Ks is the steady-state distribution coefficient which is the ratio of the steady-state concentration of gases in the oil to the steady-state concentration of gases in the oil-impregnated paper insulation at a certain temperature (Ks=cos/cps). Basically the mass concentrations are scaled between zero and one based on their initial and steady-state values. Therefore, the per-unit concentrations will define the percentage of mass with respect to the final steady-state value minus the initial value.
Substituting the normalized variables in Equations 2, 3, 4, 5, 6 and 8, results in the following set of equations. These equations are to be solved for the concentration of gas inside the paper as a function of normalized time and normalized paper insulation thickness. It should be noted that the two new variables, rv and rk, are defined to make the equations more compact.
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
2.4 The analytical solution
The solution to the boundary value problem cannot be found using conventional analytical methods because of the last boundary condition stated in Equation 13. Guggenberg and Melcher, however, solved the boundary value problem of Equations 9 to 13, analytically, for the moisture diffusion [6]. The general solution to this partial differential equation can be found using the method of separation of variables in the following form [7]:
(17)
the eigen values, γn, and constant coefficients, Bn, can be found from the boundary conditions. Substituting Equation 17 (for a particular n) and Equation 14 into the boundary condition of Equation 13 (using the fact that cw=cp at x=0) yields:
(18)
integrating and rearranging Equation 18 results in the following relation.
(19)
Solving Equation 19 gives the solution eigen values, γn, that can be used in Eqution17 for the general solution. Figure 3 illustrates the solutions of Equation 19.
Figure 3. Illustration of the solution eigen values.
In order to find the constant coefficients, Bn, the first boundary condition can be used. Using Equation 10 and the general solution of Equation 17 the following relation can be obtained:
(20)
Multiplying both sides of Equation 20 by and integrating over the paper insulation thickness results in Equation 21 (using the substitute variable u=1-x).
(21)
Since the problem eigen values are not orthogonal (as illustrated in Figure 3), none of the integration terms in Equation 21 will be equal to zero. Therefore, the coefficients, Bn, should be calculated for all n. Using the first 50 eigen values calculated from Equation 19, the integrations in Equation 21 are calculated (for n=050 and m=050 ) and therefore, the first 50 coefficients are obtained (B0, …, B50). Substituting these values into Equation 17 provides the solution to the boundary value problem (the infinite series approximated with the first 50 elements).
3 Simulation results
A MATLAB™ code has been written to solve the boundary value problem shown in Equations 9 to 16 and to obtain and plot the gas concentration inside the oil-impregnated paper insulation as a function of the insulation thickness and time. The parameters to be used to solve the equations are obtained from experiments for some of the characteristic gases and documented in previous publications [3, 4]. Some of these parameters are repeated here in Tables 2 and 3 (averaged over the values at two temperatures, 23oC and 70oC). In Table 2, D0 is the average diffusion coefficient of gas in the oil and Dp is that in the paper insulation.
It should be noted that the parameters rv and rk shown in Table 2 are different from those shown in the reference [3]. In Table 2 the parameters rv and rk are not calculated from Equations 15 and 16, rather they are the values obtained from comparing the simulation and experimental results as will be discussed later.
The simulation results inside the paper insulation show the trends of the concentration change in normalized form. In order to convert these values to parts per million (ppm), the values of Ks shown in Table 3 and the knowledge of the present gas concentrations in the oil are required. The simulation result of concentration change inside the paper insulation for carbon dioxide is calculated and plotted in Figure 4.
Figure 4. The simulation results of carbon dioxide inside the paper insulation thickness (all parameters are normalized to steady-state values).
4 Experiments
Experiments have been conducted to test the validity of the simulation results. Experiments are based on measurements in the oil since measurements in the paper insulation were not possible with the test equipment available. Equation 14 is used to convert the simulation results in the paper insulation to that in the oil. Oil samples are taken from the experimental setup and gas concentrations are measured using the DGA method.
4.1 Experimental setup
Figure 5 shows the experimental setup designed to validate the simulation results of the suggested diffusion model. The setup utilizes the same basic design and method used in the previous experiments to obtain the model parameters [3, 4], while altered to meet the new objective.
Figure 5. The experimental setup to validate the simulation results
The main container consists of one section, stacked with layers of paper insulation (Hi-Val). The insulation inside the main container is arranged in a way that ensures maximum surface contact with the oil, as oil is circulated between the container and the relaxation chamber. The main container is wrapped with a heater tape the temperature of which is precisely controlled by a PID controller. The edge of paper insulation inside the main container is covered with silicone encapsulating compound to prevent the gases from being adsorbed or desorbed from the edge. This is because the suggested model to be validated with these experiments is one dimensional and considers gas migration only through the surface of paper insulation. The relaxation chamber is filled with the oil to eliminate the effect of gas partitioning between the oil and the headspace. The reservoir compensates the oil samples withdrawn from the main container for DGA measurements. The oil flow rate between the main container and the relaxation chamber is 100 ml/min and the oil-paper volumetric ratio is about 25:1. Figure 6 shows a schematic of the main container with paper insulation (pressboard) and the counter-flow oil paths.
Figure 6. The main container in the setup shown in Figure 5.
4.2 Experimental method
The system is heated up to about 70oC and kept under vacuum for about a week before the oil is introduced into the system. In this way the gases and moisture already trapped in the dry paper insulation is purged as much as possible. Clean oil is then introduced into the system and the pump is left running for another week to ensure the equilibrium is reached between the oil and paper insulation.
The main container is then sealed off and its oil content is emptied into another container. The headspace area of the new container is filled with a mixture of standard characteristic gases with high concentrations and the container is left in this condition over night with a magnetic stirrer running inside the oil. Using this method, a high concentration of characteristic gases is dissolved into the oil. This inoculated oil is then returned back to the main container of the experimental setup and circulated through the system while the system is heated up to 70oC. Oil sampling and DGA measurements are started at this point and the time is marked at zero.
The system is left at 70oC with the pump running for about two weeks. Then the system is cooled down to 23oC and left running for another two weeks. This made sure that gases inside the oil are in equilibrium with the paper insulation inside the main container. After about 850 hours, the temperature is stepped up back to 70oC and kept at this temperature for about 300 hours. As soon as the system temperature is increased a jump in the gas content of the oil is observed through DGA measurements. Since there is no source of gas generation (at this relatively low temperature), the only conclusion is that these gases come from the paper insulation as the oil and paper insulation solubility change due to a temperature increase. The most obvious change is in the concentrations of carbon dioxide and carbon monoxide.