The correlation of the thermal mathematical models (TMMs) of spacecrafts with the results of the thermal test is a demanding task in terms of time and effort. Theoretically, it can be automatized by means of optimization techniques, although this is a challenging task. Previous studies have shown the ability of genetic algorithms to perform this task in several cases, although some limitations have been detected. In addition, gradient-based methods, although also presenting some limitations, have provided good solutions in other technical fields. For this reason, the performance of genetic algorithms and gradient-based methods in the correlation of TMMs is discussed in this paper to compare the pros and cons of them. The case of study used in the comparison is a real space instrument flown aboard the International Space Station.
Thermal control of spacecrafts and experiments is one of the key technologies needed to ensure the success of any space mission. This technology tries to make sure that the temperature of any component of the spacecraft or experiment is always inside the range foreseen during the design. Very high or very low temperatures can damage the different components of the spacecrafts, so these extreme temperatures must be avoided. Also, heat transfer between different components must be maintained under control. The design of the thermal control system is usually done following the hot and cold cases technique. This approach (consult the book of Karam [
Prior to any space mission, the thermal mathematical models of the spacecraft are created and different sets of temperatures are numerically calculated for particular mission phases or events. The use of the Thermal Lumped Parameters (TLP) method is very frequent in the space industry. This method divides the instrument in isothermal nodes with associated thermal inertias (MCs). These nodes are connected by means of conductive conductances (GLs) and radiative conductances (GRs). Reference [
These computational models must be verified by comparison with the results of the thermal balance test. The spacecraft or the experiment is placed inside a vacuum chamber where different thermal conditions are applied and it is tested under different operation modes, reproducing the situations expected in orbit. These boundary conditions, the operation modes, and the different thermal properties of the materials produce a range of thermal gradients inside the spacecraft. The temperatures of the different parts of the spacecraft are measured with thermocouples that are located in strategic positions, mainly in places where the thermal engineers need to know the temperatures with some degree of accuracy. As a result, various distributions of temperatures are obtained when the thermal test is done. For more detailed information about the thermal control testing, the book of Meseguer et al. [
In parallel, the TMMs of the spacecraft have predicted sets of temperatures for the situations tested in the thermal balance test. In an ideal world, both sets of temperatures would be the same, if the TMMs were constructed with care and the geometry and material properties of the components were known exactly. However, as it could be expected, this is not the case. There are always differences between the measured temperatures and the predicted ones, even for TMMs done with extreme care. Although several well-established methods exist to calculate the TMM parameters, these methods have some limitations, so this type of modeling is always approximated. For example, Garmendia et al. explain in [
The main objective of the correlation task is to minimize the differences between the calculated and measured temperatures. Therefore, the correlation problem can be formulated as an optimization problem. This optimization is performed modifying the parameters of the TMM (mainly the GLs, GRs, and MCs) in such a way that the temperatures calculated by the thermal solver are as near as possible to the temperatures measured in the test. If this target is achieved, the thermal engineers will have a reliable TMM that can be used to explore thermal scenarios that could hardly (or not at all) be studied by thermal tests.
This type of optimization problem has the added difficulty that there is not a unique solution; that is, different sets of parameters of the TMM can produce almost the same temperature results. As an extreme situation, it could be possible to have a set of thermal parameters that produce a set of temperatures which fits exactly the measured temperatures but has no physical meaning. As a consequence, the changes in the TMM parameters should be done taking into account the physics behind the calculation of the parameters (geometry of the different parts of the spacecraft, material properties, etc.). This is possible when the correlation is tackled manually but it is difficult when automatic correlation is performed by means of optimization methods.
Different mathematical approaches can be used to handle the optimization problem already described, and a very complete description of them can be found somewhere else [
All in all, we decided to evaluate the performance of both types of algorithms, basing the comparison on the reduced thermal mathematical model of an experiment that was flown aboard the International Space Station (ISS). This experiment, called TriboLab, executed several tribology experiments during its mission. More information about it can be found in [
The detailed mathematical description of the different optimization methods employed in the correlation of thermal tests is outside the scope of this paper. However, we feel that a succinct description of the employed algorithms and the software that implements them is a necessary approach to the study we are doing.
The two types of algorithms studied require the use of one thermal analysis software to calculate the fitness function value. An in-house developed software able to solve steady-state and transient thermal problems including the use of thermostats and time-dependent boundary conditions has been used. It solves the set of
We have used four different available FORTRAN subroutines to generate four programs to run correlations. These subroutines are public domain and can be used subjected to some conditions (see [
Genetic algorithms (GAs) are optimization algorithms of general purpose inspired in Darwin’s theory of evolution. The main advantage of these types of algorithms is to be able to find the global optimum independently of the characteristics of the function. Although they are approximated algorithms, that is, do not guarantee the finding of the exact optimum, they are able to provide a good approximation in a reasonable time.
The basic idea consists in generating a random population of individuals, where each of them represents one possible solution to the problem. The population progresses by means of crossover and mutation operators that generate new individuals, the children, from two previous individuals, the parents. The parents are randomly selected from the population but better individuals (those that represent better solutions to the problem) have more probabilities to be selected. The children from better parents are expected to be better individuals, so new better individuals are generated with the successive generations.
Genetic algorithms are a well-established method of optimization. Although there are many possible ways of implementation, one of them must be selected depending on the problem. For this reason, only a brief description of the configuration of the GA used in this case is included here. References [
The genetic algorithm used in this case is an in-house development similar to the GA used in [
Values of the GA definition parameters.
Parameters | Target |
Iteration |
Population in steady-state cases | Population in transient cases | Crossover probability |
Mutation probability |
Variation of |
---|---|---|---|---|---|---|---|
Value | 0.5 | 10,000 | 10 | 20 | 0.2 | 0.8 | 1.1 |
The GA is an approximated algorithm where the final results are influenced by some random values. For this reason, for each case of study subjected to correlation, the GA has been run 5 times obtaining 5 correlated models. The reason of this procedure is twofold. On one side, it makes it possible to evaluate the percentage of success of the algorithm, and on the other side it permits evaluating the variability between the different results obtained.
The thermal mathematical model used to evaluate the performance of the studied algorithms corresponds to the reduced model of the TriboLab experiment previously introduced. The design of the thermal control system of this experiment was performed following the hot and cold cases methodology and the thermal mathematical model is based on the Thermal Lumped Parameters method. As a result, two different models were obtained. A detailed model formed by 47 nodes and the reduced model used in this work. The latter is shown in Figure
Thermal network of the TriboLab reduced model (reprinted from Acta Astronautica, 108 (2015), Eva Anglada and Iñaki Garmendia, Correlation of thermal mathematical models for thermal control of space vehicles by means of genetic algorithms, 1–17, Copyright (2014), with permission from Elsevier).
In order to evaluate the performance of the compared algorithms, it is interesting to have a model where reference temperatures for all nodes of the model and the theoretical values of the variables are known. In this way, the results of the correlation can be better evaluated not only in terms of error level but also checking that temperature values are good for all nodes and that reasonable values have been obtained for the correlated variables. For this reason, instead of using as reference temperatures the values measured during thermal tests, the following procedure has been employed.
The TriboLab reduced model has been assumed as the reference model. That is, the values of its parameters (GLs, GRs, and MCs) are considered the real ones and the temperatures obtained from this model are taken as reference temperatures substituting the measured temperatures.
This model has been transformed modifying the value of its parameters (GLs, GRs, and MCs) randomly. As a result, a new model, called base model hereafter, has been obtained. This base model has been subjected to correlation to evaluate the performance of the compared algorithms. The four GLs, the two GRs, and the four MCs that form the model have been modified in a percentage up to 75%. This implies big differences between the theoretical values of the parameters of the reference model and the parameters used in the base model. These differences are higher than the expected ones in real cases, but this way the capacity of the algorithms to succeed in the correlation is tested with a challenging case. This model is the same as that previously used in the work presented in [
The number of variables included in the correlation is 6 in the steady-state cases (the 4 GLs and the 2 GRs) and 10 in the transient cases (the 4 GLs, the 2 GRs, and the 4 MCs).
Results have been evaluated considering the temperature values, the values assigned to the correlated variables, and the CPU times. The accuracy of the temperature results has been quantified by means of the error expression shown in the following equation:
The error levels existing in the base model subjected to correlation, that is, the initial situation, are collected in Table
Temperature errors of the base models (°C).
Case | Steady-state |
Steady-state |
Transient |
Transient |
---|---|---|---|---|
Error (°C) | 8.074 | 9.308 | 7.928 | 5.778 |
A first approximation to evaluate the capacity of the algorithms to correlate TMMs has been done with the steady-state cases. Table
Error values achieved with Powell’s algorithms for steady-state cases (°C).
TOLMIN | NEWUOA | BOBYQA | LINCOA | |
---|---|---|---|---|
Hot case (°C) | 0.000 | 0.000 | 0.000 | 0.000 |
Cold case (°C) | 0.351 | 0.000 | 0.442 | 0.351 |
Error values achieved with the GA for steady-state cases (°C).
GA |
GA |
GA |
GA |
GA |
GA | |
---|---|---|---|---|---|---|
Hot case (°C) | 0.190 | 0.216 | 0.237 | 0.392 | 0.301 | 0.267 |
Cold case (°C) | 0.407 | 0.310 | 0.439 | 0.318 | 0.293 | 0.353 |
The analysis of the temperature values obtained in all cases in each node (see Figures
Steady-state hot case. Temperatures obtained from correlations.
Steady-state cold case. Temperatures obtained from correlations.
As it has been introduced previously, one of the difficulties of the TMM correlation is the fact that the solution is not unique. Therefore, different combinations of the involved parameters (GLs, GRs, and MCs) may give solutions which fulfil the error criteria used to evaluate the correlation but assign values with no physical sense for some variables. For this reason, it is interesting to evaluate the values obtained for the different variables in order to check whether some algorithms provide better values than others.
Figure
Steady-state, classical methods. Variable values obtained from correlations.
The values obtained in the correlation of the cold case present higher variability compared to the hot case. In some cases, for example, for the GRs, the values are closer to the reference values than to the initial ones. But, in other cases as
Figure
Steady-state GA. Variable values obtained from correlations.
After performing independently the correlations of the hot and cold cases, it could be expected that the values of each variable should be the same in the hot and cold cases, but, as can be observed, the values obtained in the correlation of the hot case do not match with those obtained in the cold case correlations. In fact, if the values obtained in the correlation of the hot case, for example, with NEWUOA algorithm, are used in the cold case, the temperatures obtained present an error equal to 3.2°C, lower than the initial error (9.3°C) but not very accurate. In the alternative case, when the values obtained in the correlation of the cold case (NEWUOA) are used with the hot case, the error is equal to 9.6°C, higher indeed than the error of the base model (8.07°C). This fact was also studied in a previous work devoted to the TMM correlations with GAs [
As it has been observed, the independent correlation of the hot and cold cases is not valid to obtain a set of variables useful for both cases. Therefore, it is needed to tackle the correlation considering the hot and cold cases simultaneously.
In the case of the GA, this is not a difficulty and both cases can be simultaneously correlated without problems. In the case of the present implementation of Powell’s algorithms, this is not possible in a straightforward way, so the following workaround has been depicted. The cases have been combined consecutively: first the hot case and next the cold case. In this way, a new case to be used in the correlation has been created, combining the boundary conditions of each case sequentially. Figure
Transient hot and cold cases combined sequentially.
Table
Error values achieved with Powell’s algorithms for transient cases (°C).
TOLMIN | NEWUOA | BOBYQA | LINCOA | |
---|---|---|---|---|
Hot and cold cases combined (°C) | — | 0.400 | 1.965 | 0.885 |
In the case of the GA, the results are collected in Table
Error values achieved with the GA for transient cases (°C).
GA |
GA |
GA |
GA |
GA |
GA | |
---|---|---|---|---|---|---|
Hot and cold cases combined (°C) | 0.202 | 0.118 | 0.191 | 0.163 | 0.216 | 0.178 |
Figures
Transient hot case. Temperatures obtained from correlations with NEWUOA.
Transient hot case. Temperatures obtained from correlations with GA.
Transient cold case. Temperatures obtained from correlations with NEWUOA.
Transient cold case. Temperatures obtained from correlations with GA.
If we consider the variable values (see Figure
Transient. Variable values obtained from correlations.
Table
CPU times (in seconds).
TOLMIN | NEWUOA | BOBYQA | LINCOA | GA |
|
---|---|---|---|---|---|
Steady-state, hot case | 10 | 3 | 2 | 5 | 1.5 |
Steady-state, cold case | 13 | 13 | 5 | 5 | 15 |
Transient, hot and cold case | — | 1020 | 89 | 104 | 179 |
Numerical values of the elapsed CPU times presented in Table
Although one of the drawbacks of the genetic algorithms is usually the high number of iterations needed to perform the correlation, as can be observed in this case, this does not imply long CPU times.
It is very convenient to study the convergence of the results obtained with the different correlation methods explained so far. In fact, this is a critical point that could make the difference between a method that is interesting and a method that could be used for practical applications. There are two main aspects that are to be studied: first the temperature distribution (those measured in the tests versus those predicted in the models) and second the value of the thermal parameters of the model (GLs, GRs, and MCs).
The methods already presented (gradient-based and genetic algorithms-based) are, in general, capable of doing a good job in reproducing the temperatures measured in the tests. To do so, they change the values of the thermal parameters and reach a reasonable solution with moderate CPU times, as has been shown before. For bigger models, the CPU times increase significantly, but values can be affordable, at least with the examples studied with genetic algorithms (see [
To reach the reference parameters when starting with the base parameters through the optimization of the differences of temperatures is a difficult task. The first and most important difficulty is the fact that the inverse problem does not have a unique solution; that is, different sets of thermal parameters can produce the same temperature distribution. If this is so, how could the optimization method distinguish between a set of “correct” thermal parameters and a set of “incorrect” ones if both produce a minimum in the temperature optimization method? There is not a simple answer to this question, but it is interesting to note that better results for the thermal parameters are obtained if both hot and cold cases are used simultaneously to do the temperature correlation. As a consequence, a complementary study has been developed in order to understand this fact.
GLs values for case 1 and case 2 [W/°C].
Parameter |
|
|
|
|
|
|
---|---|---|---|---|---|---|
Case 1 [W/°C] | 8.00 | 6.00 | 5.00 | 4.00 | 8.00 | 3.00 |
Case 2 [W/°C] | 6.00 | 7.00 | 5.57 | 8.00 | 6.94 | 2.97 |
Complementary model.
The GLs values of case 1 have been defined arbitrarily. The GLs of case 2 have been selected to be different from the values of case 1 but to produce the same temperature distribution. The procedure followed to obtain the GLs values of case 2 has been the following: the temperature results for case 1 have been calculated with the help of the TK software, obtaining the values shown in Table
Temperature results (°C).
Node | 1 | 2 | 3 | 4 |
---|---|---|---|---|
Case 1 ( |
30.84 | 25.69 | 26.75 | 20.00 |
Case 2 ( |
30.84 | 25.69 | 26.75 | 20.00 |
Case 2 produces the same nodal temperatures as case 1 (see Table
Heat flows (W).
Parameter |
|
|
|
|
|
|
---|---|---|---|---|---|---|
Case 1 | 41.24 | 24.53 | 54.22 | −4.267 | 45.51 | 20.27 |
Case 2 | 30.93 | 28.62 | 60.44 | −8.53 | 39.47 | 20.09 |
Moreover, if we change the power applied in node 1 and set it to 180 W (
Temperature results (°C).
Node | Case 1 |
Case 2 |
Case 1 |
Case 2 |
---|---|---|---|---|
1 | 36.27 | 36.27 | 25.69 | 25.69 |
2 | 28.53 | 28.53 | 29.38 | 29.89 |
3 | 30.13 | 30.13 | 25.51 | 26.62 |
4 | 20.00 | 20.00 | 20.00 | 20.00 |
However, if we modify original cases 1 and 2 applying 120 W in node 2 rather than node 1, that is,
This fact implies that two sets of thermal parameters that produce the same temperature distribution with a particular set of boundary conditions will produce a different distribution of temperatures if a different set of boundary conditions is applied. This is a mechanism to discriminate between two sets of thermal parameters that produce the same result for one set of boundary conditions.
When the hot and cold cases are used at the same time for the correlation of temperatures, we are somehow applying different sets of boundary conditions. In doing so, we are “discriminating” between solutions that produce a bigger error than others and we are conducted by the optimization algorithms towards a solution closer to the reference values. The hot and cold cases “activate” different aspects of the model and the algorithms have more information in order to get the reference answer. Somehow, we are providing the optimization algorithms with more useful information. Finally, using transient hot and cold cases for the correlations means that more information is added in the intermediate steps, facilitating the finding of the reference values of the thermal parameters. If in these transient cases the boundary conditions change with time (which is quite usual, for example, heaters switch on or off depending on the temperatures, space sink temperatures change with time, etc.), we are even adding more useful information to the algorithm.
TriboLab model definition.
Parameter | TriboLab model | Alternative model |
---|---|---|
|
0.033270 | 0.02567 |
|
0.48830 | 0.2 |
|
3.2190 | 5.0 |
|
4.4309 | 2.0014 |
|
0.038300 | 0.0529 |
|
0.061200 | 0.0420 |
The change applied has been to reduce the input powers to the half in these two cases. The new distributions of temperatures obtained for the two models (TriboLab and alternative) are different, as can be seen in Table
Temperature results [°C].
Node | TriboLab model |
Alternative case |
---|---|---|
85040 | 21.8251 | 24.1327 |
85041 | 19.4234 | 21.5761 |
85070 | 1.7115 | 1.5587 |
85071 | −0.6405 | −0.9970 |
All these initial steps about the convergence of the thermal parameters towards the reference parameters need more in-depth investigation, trying to formulate in a consistent way the advantages of using hot and cold cases simultaneously. We believe that the presence of GRs in the TMMs facilitates the finding of the reference parameters when optimizing the correlation. This is so because of the nonlinearity implied in the radiation heat transfer process. If two sets of thermal parameters have the same temperature distribution and the applied power changes, the new distributions of temperatures for the two sets will probably be somehow different. We also believe that the addition of other thermal scenarios of the mission (not only the hot and cold cases but also the nominal functioning, for instance) will provide more chances to obtain better results.
Results obtained for steady-state cases with both types of algorithms, classical and GA, are good in terms of temperatures. The error levels have been reduced from 8.1 and 9.3°C to values below 0.44°C, and Powell’s algorithms indeed have reached the exact value in some cases. The nodal temperature distributions obtained are also appropriate and the CPU times are extremely short, less than 15 seconds.
Unfortunately, the values assigned to the variables are not good. The main problem is the difference between the values obtained in the correlation of the hot and the cold cases. The values obtained in one case, hot case, for example, fail when used in the other case, cold case, and vice versa. To obtain a set of values appropriate for both cases, performing the simultaneous correlation of both cases is needed. This is not a problem for the GA but unfortunately it is not possible with the present implementation of Powell’s algorithms.
In the transient case combining the hot and cold cases, the results are good in terms of temperature with the exception of TOLMIN algorithm which does not converge. Results obtained with BOBYQA and LINCOA are not exceptional but get a reduction of the initial error (6.8°C) to 1.97 and 0.88°C, respectively. NEWUOA algorithm reaches an error level equal to 0.4°C, which is a good value, and the GAs succeed in reaching values below 0.22°C. The nodal temperature distribution is also appropriate and the CPU times are very short with the GA, less than 3 minutes. The time needed by NEWUOA algorithm is longer, 17 minutes, but is a good value compared with the time needed for manual correlation.
Referring to the variable values, in this case, the sets of values obtained are valid for the hot and cold cases. With the exception of BOBYQA, which clearly fails in the calculation of some values, the rest provide reasonable values. It is remarkable that, in this combined case, the values obtained tend to converge to values closer to the reference values than to the initial values. A complementary model has been presented to explain the better convergence in this situation.
As a general conclusion, it can be said that the studied gradient-based solutions are not an appropriate tool for the steady-state correlation of TMMs of spacecrafts, with the use of a GA being a better solution. In the case of the correlation of transient TMMs, the use of NEWUOA algorithm is a valid option but the GA seems to be a better solution. The GA has been faster and the values provided for the thermal parameters tend to converge a little better to the reference values.
This work has been performed as an independent research and no financial help has been received for it.
The authors declare that there are no conflicts of interest regarding the publication of this paper.