International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 7683457, 12 pages

https://doi.org/10.1155/2017/7683457

## Performance of Gradient-Based Solutions versus Genetic Algorithms in the Correlation of Thermal Mathematical Models of Spacecrafts

^{1}Industry and Transport Division, TECNALIA, Mikeletegi Pasealekua 2, 20009 San Sebastián (Donostia), Spain^{2}Mechanical Engineering Department, Engineering School of Gipuzkoa, University of the Basque Country (UPV/EHU), Plaza de Europa 1, 20018 San Sebastián (Donostia), Spain

Correspondence should be addressed to Eva Anglada

Received 30 January 2017; Revised 11 April 2017; Accepted 26 April 2017; Published 24 May 2017

Academic Editor: Paolo Tortora

Copyright © 2017 Eva Anglada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1] for information) bases the design on the coldest and the hottest situations expected during the mission. It is performed by means of two thermal mathematical models (TMMs) that represent the hot and the cold cases. The books of Gilmore and Redor [2, 3] can be also consulted for more information about the spacecraft thermal control.

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 [4] explains in detail this methodology.

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. [5] can be consulted.

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 [6] the difficulties associated with the calculation of the conductive conductances. Moreover, there are parameters, for example, the contact conductances, whose exact calculation is almost impossible.

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 [7]. In our previous works [8, 9], our attention was focused on genetic algorithms (GAs), which are good candidates to solve the global optimization problem present in the TMM correlation. At the same time, we felt that classical optimization methods (mainly, gradient-based methods) deserved a deeper attention, as they provide good and operative solutions in other technical fields [10]. Although this type of methods tends to find the local optimum, we think that thermal parameters (GLs, GRs, and MCs) of TMMs produced carefully by thermal engineers are, in many cases, close enough to the real values and, as a consequence, the solution provided by the thermal solvers is a good starting point for using classical optimization methods. In a previous work [11], the use of several algorithms of Broyden class in TMM correlation was compared with the results obtained by means of a GA. Although some limitations were detected for the use of Broyden class algorithms, the results obtained were promising. For this reason, it has been considered of interest to evaluate more in deep the application of classical methods to the TMM correlations. In particular, the extensive work done by Powell, presented in [12–16], and the availability of FORTRAN subroutines that implemented this work were crucial to the decision to make a comparison between classical optimization methods and genetic algorithms.

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 [17]. The thermal control system of this instrument was formed by the radiator, the heaters, the thermostats, and the multilayer insulation (MLI).

#### 2. Optimization Methods in TMM Correlations

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 nonlinear transient algebraic equations (see (1)) that are obtained using the Thermal Lumped Parameters method, which is briefly explained in Section 3. The set of equations is solved by means of the Newton-Raphson linearization technique, using the iterative methods included in the public domain subroutines of the ITPACK 2C package developed by Kincaid et al. [18].

##### 2.1. Classical Optimization Methods (Powell Approach)

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 [19]). They are based on an early work of Professor Powell in 1969, the subroutine VA05, available in [12]. This subroutine minimizes the sum of squares of given functions, , each of variables, , without the use of any partial derivatives, that is, to find to minimize. In our case, is the number of nodal temperature differences multiplied by the number of time instants where temperatures are calculated (in steady state, 1 time instant). Also, is the number of thermal parameters (GLs, GRs, and MCs) whose real values are looked for. The method that VA05 uses is a compromise between three different algorithms for minimizing a sum of squares, namely, Newton-Raphson, Steepest Descent, and Marquardt. The subroutine automatically obtains an approximation to the first derivative matrix following the ideas of Broyden:(i)*Subroutine TOLMIN*: this subroutine was developed later, in 1989. The subroutine minimizes a general differentiable objective function subject to linear constraints. The same definition of the objective function as that in the subroutine VA05 applies here but now linear constraints can be applied. This is a very interesting point, as the values of the thermal parameters must always be positive to have physical sense. This subroutine needs the user to supply the partial derivatives of the function with respect to the thermal parameters (gradient-based method)(ii)*Subroutine NEWUOA*: the year of publishing of this software is 2004. The software seeks the least value of a function (in our case, the sum of the temperature differences again) but this time without constraints and without derivatives. By this definition, it is clear that we are using a subroutine similar to the VA05 software but with a different approach. In fact, a quadratic approximation of the function is used to obtain a fast convergence ratio and successive values of the approximation are used as the calculations progress(iii)*Subroutine BOBYQA*: this piece of software was presented in 2009. The subroutine finds a minimum of a function subject to bounds in the variables. The user does not supply derivatives. In our case, this particular approach seems to be very interesting, as extreme values for the thermal parameters can be imposed. These extreme values are, in fact, related to physical values of the problem; for instance, values of one of the thermal inertia must be between 2000 J/°C and 3000 J/°C. As can be seen by this definition, BOBYQA is related with TOLMIN software(iv)*Subroutine LINCOA*: this is the final work of Professor Powel, made public in 2014, before his passing away in April 2015. LINCOA stands for linearly constrained optimization. The software can be used even if the derivatives of the function are not available. Linear constraints are possible with this software and emphasis is put on the successive calculations of the quadratic function that approximates function

##### 2.2. Genetic Algorithms

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 [20–22] can be consulted for more information about GAs.

The genetic algorithm used in this case is an in-house development similar to the GA used in [8] where the detailed description can be found. The fitness function used is defined in (2) and (3), where corresponds to the number of temperature points or nodes,* K* corresponds to the number of instants, and corresponds to the number of cases. The parents are selected by proportional selection based on a weighted fitness, following (4), by means of stochastic sampling with replacement. The crossover operator used corresponds to the simple arithmetic crossover. The range of the mutation operator is modified when the fitness has not been improved in last 20 iterations. The new population is created using an elitist strategy, so it is formed by the children and by the best individual of the previous population. The values of the definition parameters used are collected in Table 1.