An ILP-Assisted Two-Stage Layout Optimization Method for Satellite Payload Placement

In the satellite overall design phase, it is a crucial step to perform satellite layout design to guarantee that the aggregation of electronic components can operate normally and stably in an appropriate temperature environment. In order to handle the satellite payload placement problem of the DongFangHong 4 (DFH-4) platform, the heat pipe-constrained component layout optimization (HCLO) problem is proposed with the HCLO model formulated. Through careful investigation, it can be divided into two optimization subproblems that can be solved subsequently. Based on the divide-and-conquer strategy, an integer linear programming- (ILP-) assisted two-stage layout optimization method is proposed. In stage one, component-heat pipe distribution optimization is performed using the ILP technique so that speci ﬁ c heat pipes occupied by each component can be determined and the horizontal movement range of components can be reduced. In stage two, the detailed component layout optimization is investigated to obtain the ﬁ nal positions of components. First, the sequence layout sampling (SeqLS) method is used to generate one nonoverlap initial layout. Next, swap operation between components is incorporated to reduce the centroid deviation. Finally, sequential quadratic programming (SQP) search is conducted based on the generated promising initial layout solutions. Therefore, the SeqLS-based heuristic layout search algorithm is proposed in the second stage. Two layout test cases, including 15 components and 90 components, respectively, are investigated to demonstrate the validity and e ﬃ cacy of the proposed layout design method. Experimental results show that it is promising to apply such a two-stage approach for satellite payload placement in engineering.


Introduction
During the satellite overall design phase, satellite layout design plays a crucial role in determining its on-orbit performance and functionality and avoiding design failure. The aim of satellite layout design is to arrange proper positions of the electronic components or equipment inside or outside the satellite to satisfy various system performance requirements (mass characteristics, thermal control, etc.). In engineering practice, it usually takes months for engineers to manually place components by trial and error and figure out one feasible layout design scheme that can meet all complex design constraints. When the number of components that need to be placed increases, the layout design task will become more difficult and time-consuming. In order to accelerate the design process and shorten the manufacturing cycle, satellite layout optimization design (SLOD) methods have been developed to tackle the satellite layout design problem so that optimization techniques can be incorporated to alleviate the design complexity [1,2].
In general, the SLOD methodology can be roughly divided into two key stages: modeling and optimization. In the modeling stage, problem formulation and geometry modeling are two essential basic topics for conducting the subsequent optimization. For problem formulation, there are two common types of basic SLOD models concerning mass characteristics and dynamics performance in the literature [3,4]. The first kind of SLOD model was built to minimize the dynamic nonequilibrium performance for one self-rotating reentry vessel in 2001 by Teng et al. [3]. After that, Sun and Teng [4] took the layout design of an international commercial communication satellite as an example and established the commonly used SLOD model, where the moments of inertia of the satellite were optimized due to the requirement of attitude control subsystem. It is a classical three-axis stabilized satellite layout design model, which has been widely adopted in most researches. Apart from the problem formulation, geometric modeling of layout components is the other essential part of constructing the entire mathematical model. The purpose of geometric modeling is to deal with the nonoverlapping constraint between components or between components and the layout domain. It demands that not only collision detection can be realized but also the volume of overlap that might happen during optimization can be precisely calculated. In earlier research [2][3][4], the nonfit polygon method was utilized to detect the overlap. However, it is difficult to realize the overlap calculation between rotating rectangular components. Hence, an enhanced finite-circle method [1] and the phi-function method [5] were proposed, which can easily tackle the nonoverlapping constraint between more complex geometries. With the aforementioned mathematical modeling, more researches have been conducted in developing layout optimization algorithms so that the global optimal placement of components can be identified effectively and efficiently. Actually, as a kind of three-dimensional (3D) packing problem with complex performance constraints, the SLOD problem is too hard to be solved in polynomial time, which is known as NP-hard [6].
To handle this problem, many heuristic and metaheuristic algorithms have been proposed to avoid being stuck in the local optimum and search for the optimal layout solution, such as quasiphysical quasihuman algorithm [7], nonlinear optimization formulation [8], particle swarm optimization [1,6], differential evolution [9,10], and artificial bee colony algorithm [11]. However, these layout search algorithms are problem-dependent, and their performance in different layout cases is not consistently good.
In addition to research based on the basic SLOD model, other vital performance requirements should be incorporated into the optimization design, such as the thermal performance [1,[12][13][14][15]. In terms of the layout optimization driven by thermal performance, the temperature simulation analysis was directly integrated into the optimization loop for assessing the temperature field of different layout schemes [12]. Despite the precise evaluation, it is a time-consuming process [14,15]. Hence, two approximate description methods [13,14] have been developed to represent the uniformity of temperature distribution based on the dissipated power of components. However, both approaches are qualitatively assessing thermal performance, which cannot realize the accurate temperature control with the layout changing. Consequently, to evaluate the temperature field precisely without increasing the computational burden, Chen et al. [16] proposed a deep learningbased surrogate modeling method to construct one cheap mapping between the layout and the resulting temperature field and replace the numerical simulation during optimization. Experiments demonstrated the effectiveness of the proposed approach and showed a great potential of the deep neural network surrogate in dealing with the ultra-highdimensional field prediction problem.
In this paper, considering the deployment of heat pipes used in satellite thermal control subsystem, we propose a new SLOD formulation for satellite payload placement in the DongFan-gHong 4 (DFH-4) bus, which is summarized as the heat pipeconstrained component layout optimization (HCLO) problem. Compared with the former classical SLOD model, more com-plicated design constraints, including the component-heat pipe overlapping constraint and heat pipe dissipation capacity constraint, have been combined in this model to restrict components to be placed on top of heat pipes so that the generated heat of components can be dissipated outside to the cold space. In addition, to solve this new complex layout design task, we employ a divide-and-conquer strategy and propose an integer linear programming-(ILP-) assisted two-stage layout optimization method. In the first stage, an ILP formulation for component-heat pipe distribution optimization is established to determine which heat pipe(s) components should be placed on. Next, the detailed component layout optimization model is built based on the distribution result of the previous stage, and the sequence layout sampling-(SeqLS-) based heuristic layout search method is developed to identify the final layout design efficiently. It becomes easier to figure out optimal layout solutions by tackling two layout subproblems one by one.
The remainder of this paper is organized as follows. In Section 2, the problem statement is presented as well as the mathematical model of the HCLO problem. Then, the proposed ILP-assisted two-stage layout optimization approach is elaborated by introducing the solution for each stage clearly in Section 3. In Section 4, two layout cases with different complexity are investigated to demonstrate the feasibility and efficacy of the proposed method. Finally, conclusion is draw in Section 5.

Problem Description and HCLO Model
In this section, the HCLO problem is firstly introduced, and then, the HCLO model is formulated with defining proper design variables, layout constraints, and the optimization objective.
2.1. Problem Description. Driven by the engineering application of determining the optimal satellite payload placement in the DFH-4 bus, the HCLO problem is proposed, illustrating a new kind of component layout optimization problem. DFH-4 bus is an international advanced large telecommunication satellite platform for the high capacity broadcast, regional mobile communication, etc., aiming for international and domestic commercial communication satellite markets (Krebs, Gunter D. "DFH-4 Bus". Gunter's Space Page. Retrieved December 22, 2021, from https://space .skyrocket.de/doc_sat/ch__dfh-4.htm). With some simplification, a two-dimensional (2D) view of bearing boards of DFH-4 satellite payload bay is displayed in Figure 1. There are three main boards, including one satellite-towards-Earth (STE) board and two side boards, in this payload bay. The plotted Cartesian coordinate system O − XYZ illustrates their spatial relationship. The STE board means that this plane always points to Earth when the satellite is flying in its space orbit. On this board, antennas are usually installed on the outer surface of the STE board while signal receivers or transmitters on its inner surface, as well as some components with low heat-generating power. Two side boards, namely, the north and south boards, are utilized to arrange some components with high heat power since most of the generated heat by components is dissipated through these 2 Space: Science & Technology two boards outside the satellite. To be more specific, heat is firstly conducted by heat pipes embedded in two side boards to their outer surfaces and then radiated to the cold deep space (constant at 4K). It is assumed that layout components on the STE board have been placed in their right position in advance. Therefore, the task layout problem is established to address how to properly arrange the positions of components on the north and south boards with meeting satellite performance requirements on the heat dissipation and system static stability. As shown in Figure 1, this 3D layout task can be simplified to a 2D layout optimization problem identifying X and Z coordinates of components on two side boards without considering their spatial intrusion with other satellite parts in the Y-axis direction. Each board is divided into two feasible layout domains, denoted by red rectangles, where electronic devices (components), represented by blue shaded rectangles, are restricted to be placed without any protrusion. The thin green bar denotes the heat pipe, which efficiently absorbs the heat generated by components and quickly reaches a balanced heat dissipation state. The design requirements on thermal performance and mass characteristic for satellite payload placement of DFH-4 are mainly discussed in this paper, including the following: (i) The system centroid of satellite payload bay should approach the expected one or satisfy the given specific centroid range (ii) Components are required to overlay on the surface of heat pipes to guarantee that the generated heat can be directly dissipated by heat conduction (iii) The total heat power dissipated by every single heat pipe, which depends on components covering its surface, cannot exceed its maximum load capacity (iv) The dissipated power over different heat pipes in one board should be balanced, that is, as close as possible such that the uniformity of temperature distribution over the entire board can be maximized and heat concentration can be avoided Following the aforementioned assumptions, the HCLO problem is defined as one typical layout design scenario where the thermal performance based on heat pipes is maximized by optimizing the placement of components, simultaneously meeting the static stability constraint.

The HCLO
Model. The mathematical model of the HCLO problem is constructed in this subsection. Notice that three assumptions have been made in advance. First, components are assumed to be rigid cuboids with a uniform mass distribution, which means their centroids are coincident with their geometric centers. Second, components that are required to be placed on these two side boards have been a priori allocated to different layout domains according to some engineering practice. It means that swap operation of components between different layout domains is not allowed during optimization. Third, components are restricted to be placed orthogonally, and rotation of 90 degrees is also not allowed during the optimization. The last two assumptions reduce the design space and thus lower the optimization difficulty to some extent. Therefore, this HCLO problem can be established as a 2D layout optimization model, which is simply illustrated using Figure 2 without loss of generality.

Design Variables.
For computational convenience, a 2D Cartesian coordinate system O − xy is defined, where x -axis is aligned with the horizontal direction and y-axis with the vertical direction. Heat pipes are evenly distributed along the x-axis direction to realize better heat transfer and spread heat over the domain uniformly. Taking geometric centers as reference points, the placement of components can be easily determined using their 2D position coordinates ðx i , y i Þ. On the basis of this, design variables of the HCLO model where i denotes the i th component, N c means the number of components to be placed, and X represents the component layout scheme, which can uniquely define the placement in their allocated layout domain. It can be easily concluded that the number of design variables is 2N c . As the number of components N increases to the magnitude of hundreds, the HCLO problem will become high dimensional, greatly increasing the complexity of the search space and thus the difficulty of identifying its optimal solution.

Layout Constraints.
Four types of layout constraints should be satisfied to accommodate satellite thermal performance and system mass characteristics: the nonoverlapping constraint, system centroid constraint, component-heat pipe overlapping constraint, and heat dissipation capacity constraint.
(1) Nonoverlapping Constraint. The nonoverlapping constraint, as a basic spatial geometry constraint, demands no overlap between different components and between components and the layout domain. The overlap volume should be strictly equal to zero to guarantee the feasibility of one layout scheme. Hence, the nonoverlapping constraint can be expressed as where ΔV ij ði, j > 0, i ≠ jÞ refers to the amount of intersection area between components i and j; when i = 0, object i denotes the layout domain, and ΔV ij means the amount of protrusion area of component j out of the layout domain.
The overlap area between rectangles placed orthogonally can be easily calculated analytically in this case. For more complex geometries, it is an efficient approach to analytically and explicitly measure the amount of overlap by distance using the phi-function method [5].
(2) Static Stability Constraint. Static stability constraint here represents the system centroid constraint, where the mass center of satellite payloads along the y-axis direction in the O − xy system (i.e., the X-axis centroid in the O − XYZ system) is required to be controlled within its permissible range. In other words, the system centroid deviation should be limited in its predefined offset. Note that this constraint can also be modeled as the objective that the real y-axis system centroid y c is expected to be as close as the expected one. It means that the system centroid error should be minimized. Following its definition, we can formulate this constraint as where y c and y e are the real and expected y-axis system centroid, respectively, and δy e is the maximum allowable centroid deviation.
(3) Component-Heat Pipe Overlapping Constraint. The component-heat pipe overlapping constraint is defined as another geometry constraint, requiring that the foot area of components must be placed on top of any one or several heat pipes horizontally to ensure effective heat conduction. Therefore, in a 2D projection view, components should intersect with several bar-shaped heat pipes geometrically. To guarantee a safe assembly operation and an efficient heat transfer, the width that is occupied on any one heat pipe by each component should be controlled as a larger value than the threshold. The threshold width d hp com is set as the width of the heat pipe in this case, which means that only the situation where the component cross over the entire heat pipe can be regarded as one valid heat pipe occupation. Thus, this constraint can be represented as   where d i denotes the distance away from its nearest heat pipe when component i does not intersect with any one heat pipe under the assumption of the valid heat pipe occupation. This distance is analytically calculated using the adjusted phi-function approach as introduced in [5]. As long as the component can validly cross over at least one heat pipe, this distance value would maintain zero. For example, component 5 in Figure 2, which has a large horizontal length, can readily satisfy this constraint under arbitrary placement. Following their mutual relationship, the number of heat pipes occupied by one component can only switch between two possible options. However, its value depends on its specific position.
(4) Heat Dissipation Capacity Constraint. Each heat pipe has its maximum heat dissipation capability, defined as the maximum load power P hp max . Heat dissipation capacity constraint describes that the actual total power dissipated by each heat pipe cannot exceed its maximum load capacity. The total dissipated power by one heat pipe is determined by accumulating the heat power of components that crosses over this heat pipe. When the heat generated by one component is dissipated through several heat pipes, that is, one component occupies multiple heat pipes simultaneously, it is assumed that the component power is averaged and then added to the real load calculation of each of its occupied heat pipes. Hence, this constraint can be written as where P hp j is the real load power of the j th heat pipe, H j represents the set of components that occupy the j th heat pipe, P i and n hp i denote the heat power of component i and the number of heat pipes occupied by component i, respectively, and N hp is the total number of heat pipes. Note that n hp i varies with the position of component i and should be updated after its movement according to the rule of valid heat pipe occupation. When investigating DFH-4-like cases, heat pipes cross over the entire board along the X-axis, that is, two layout domains. The calculation of real heat pipe load power should be performed based on the placement of components in these two layout domains simultaneously.

Optimization Objective.
Since the layout design goal, in this case, is to improve the uniformity of the temperature field based on heat pipes, the optimization objective is set to minimize the maximum real load power of heat pipes in one layout board. Thus, the objective can be formulated as .
When considering the DFH-4 case involving two side boards, this objective is extended as the summation of their respective maximum real load power of heat pipes in differ- Let us investigate this problem more deeply from the view of optimization. First, the dimension of design variables varies twice with the number of components, easily making the HCLO problem a large-scale one. Second, despite a few constraint types, the number of constraints that need to be satisfied behind is huge. For example, the nonoverlapping constraint requires that any two objects need to be prevented from the intersection, thus including N c ðN c + 1Þ/2 specific nonintersection constraints if N c components are placed in one layout domain. Besides, the feasible region of design space is quite small as constraints can scarcely be satisfied at the same time in most randomly generated layout schemes. When one component translates in some direction for the nonintersection purpose with another one, it is possible to cause intrusion with the third one or violate the valid heat pipe occupation rule. The complexity of constraints further makes this problem intractable. Third, by seeing the objective regardless of design variables and constraints, the HCLO problem is a kind of combinatorial optimization. There may exist two or more distinct component layout schemes with the same objective value, displaying the feature of multimodal optimization. For example, exchanging positions of components 1 and 2 in Figure 2 will not change the objective. In a word, the proposed HCLO case is a continuous constrained multimodal single-objective optimization problem with great complexity.

An ILP-Assisted Two-Stage Layout Optimization Method
To deal with the complex HCLO problem, there are two classes of optimization approaches that can possibly work in general. One is global optimization based on population-based evolutionary algorithms, and the other is local optimization represented by gradient algorithms. Population-based 5 Space: Science & Technology evolutionary algorithms show their unique advantages in searching for global optimal component placement than local optimization techniques. Therefore, many works on efficient layout optimization algorithms [1,2,6,17] have been conducted to improve the satellite layout design. However, when faced with this new layout case, current layout optimization algorithms do not perform well in either efficacy or efficiency or both due to problem complexity. In addition, the high dimension that may be encountered in engineering cases brings more challenges to design efficiency and result diversity. Overall, the optimization randomness and the low convergence rate greatly hamper the application of evolutionary algorithms in engineering practice unless more sophisticated and robust algorithms are proposed.
Through more careful analysis, it is surprisingly found that the HCLO problem can be divided into two optimization subproblems which can be figured out subsequently. The first one is to determine the component-heat pipe distribution relationship without taking the nonoverlapping constraint into account, thereby leading to a componentheat pipe distribution optimization task. By arranging components in proper heat pipes, the maximum real load power of heat pipes can be minimized first, which is the HCLO objective. The other one is to perform the detailed component layout optimization with minimizing system centroid deviation based on the resolved component-heat pipe allocation, thus generating the final layout design scheme. It should be noticed that heat pipes actually define the component distribution in the x-axis direction (horizontally) while the static stability constraint requires a proper vertical distribution to obtain a satisfied y-axis system centroid. Based on this divide-and-conquer idea, an ILPassisted two-stage layout optimization method is proposed as illustrated in Figure 3.

Stage One: ILP-Based Component-Heat Pipe Distribution
Optimization. In this section, an integer linear programming (ILP) model is implemented to perform the component-heat pipe distribution optimization. The task is to determine which heat pipe(s) one component should be placed on top of to minimize heat pipes' maximum total dissipated power. We represent the occupation state of each heat pipe by one component using binary variables c ij ∈ f0, 1g, where i denotes the component index and j means the heat pipe index. When component i crosses over heat pipe j, c ij = 1. Otherwise, c ij = 0.
When one component occupies n hp i heat pipes simultaneously, variables should satisfy that c ik = c iðk+1Þ = ⋯ = c iðk+n hp i −1Þ = 1 while c ij = 0ððj < kÞ ∪ j > ðk + n hp i − 1ÞÞ. However, it is very hard to define this constraint using a linear mathematical expression, which is not acceptable for ILP formulation. To handle this problem, we further define the reference heat pipe that one component occupies as the minimum index of all heat pipes that it occupies, and thus, the rest of the heat pipes can be inferred easily by its occupation number n hp i . Based on this, another kind of auxiliary binary variable (0/1) e ij is introduced to describe whether one heat pipe is chosen as the reference one. Then, the linear constraint ∑ j e ij = 1 can uniquely determine which heat pipe component i is placed on top of as long as n hp i is given. Therefore, the component-heat pipe distribution optimization can be formulated as To avoid using maximum operation in ILP formulation, the original objective function f ðXÞ = max k P hp k is equivalently modified as Z (8a) with adding constraint (8f) at the same time. It is because that if we have constraint (8f) satisfied, it can be easily verified that f ðXÞ ≤ Z. When Z gets its optimum, it must be Z min = max k P hp k . Apart from (8h), there are another three kinds of constraints that need to be satisfied. Constraint (8b) defines that each component can only be placed in one reference heat pipe. After introducing the reference heat pipe representation, it is a necessary step to construct a linear transformation between e ij and c ij . Therefore, linear constraint (8c) is formulated to realize the purpose by introducing a transformation matrix M i with size of ðN hp − n In this example, if we denote e i = ðe i1 , e i2 , e i3 , e i4 Þ and should be predetermined properly to ensure the linearity of constraints. The possible values of n hp i of one component, which usually includes two candidates at most, depend on the width of the component and the distribution of heat pipe. To minimize the objective for better heat dissipation, it is desirable that the generated heat of one component should be distributed on as many heat pipes as possible.

Stage Two: SeqLS-Based Heuristic Layout Search Method.
In this stage, the nonoverlapping constraint and the centroid deviation constraint are two major considerations to be balanced. From this perspective, we propose an efficient SeqLS-based heuristic layout search method. It includes three main parts, which are the sequence layout sampling (SeqLS) procedure, the heuristic component swap operation, and the gradient-based layout search. The nonoverlapping constraint is taken into account first, and the SeqLS method is utilized to actively generate constraint-satisfying layout samples. Then, to approach the zero centroid deviation as much as possible, a heuristic component swap operation is designed. After performing the above two steps, the layout candidate maintains a small constraint violation to a maximum extent and can be seen as a promising initial guess to conduct gradient-based search for the final layout solution in the last step. In the following, each part will be introduced in detail.

Sequence Layout Sampling (SeqLS) Method.
The main idea of the SeqLS method [18] is to sequentially add compo-nents in a layout area in a certain order and place each component randomly within its currently feasible layout area.
However, when applying the SeqLS method to the HCLO problem, two points need to be adjusted: (1) When determining the component placement area from the layout area, not only the boundary of the layout area but also the intersection relationship between the component and the heat pipe determined by stage one needs to be considered (2) The order in which the components are placed needs to be defined according to the problem rather than a random order. For example, the placement order can be determined according to the height or area of the components, increasing the success rate of generating a feasible layout The advantage of the SeqLS method is that a large number of valid samples can be collected efficiently. It can aggressively combine nonoverlapping constraints with the layout sampling process. Moreover, the method can satisfy the randomness of sampling in the feasible layout space to ensure that any feasible layout can be obtained.

Heuristic Component Swap
Operation. The goal of the entire stage two is to obtain valid samples that satisfy both the centroid constraint and the nonoverlapping constraint based on the relationship between components and heat pipes determined in stage one. After the nonoverlapping samples are obtained by the SeqLS method, these samples can be directly put into the optimization algorithm aiming at the expected centroid to obtain a layout that meets the requirements. However, as the initial value of the optimization algorithm, these generated samples have strong randomness, which will increase the difficulty of optimization and even fail to find qualified results. Consider that the initial value input into the optimization algorithm seriously affects the performance of the optimization. So we propose a heuristic component swap operation to act on the layouts generated by the SeqLS method. The purpose is to obtain the layout samples with the centroid closer to the expected centroid as the initial  The core of the heuristic component swap operation consists of two steps. First, randomly select a heat pipe in an area to determine whether there is room for adjustment of the order of components on this heat pipe. Then, if there is room for adjustment, swap the positions of the components with higher mass and those with lower mass in the expected centroid direction. For example, as shown in Figure 4, if the real centroid is smaller than the expected centroid, then the actual centroid is expected to move up. So the swap operation is that a component with a larger mass needs to swap positions with a component with a lower mass located above it on the same heat pipe. If the components on the randomly selected heat pipe are arranged in descending order of mass, it means that there is no room for adjustment of this heat pipe. When the centroid of the layout after the swap operation crosses the range of the expected centroid, the swap operation is stopped.
It is worth noting that the layout obtained after adopting the component swap operation may not satisfy the nonoverlapping constraint. However, this is not the focus of component swap operation. As long as a layout that matches the expected centroids as much as possible is found, the nonoverlapping constraint can be solved by the local gradient search algorithm.

Gradient-Based Layout Search Method.
Compared with the layout generated by the SeqLS method, the layout obtained using the component swap operation may not satisfy the nonoverlapping constraint, but it tends to meet the centroid constraint. The nonoverlapping constraint is a problem that gradient-based search algorithms are good at. So using the component swap operation to get a layout can be seen as a better initial value for the search method. Moreover, this initial layout is conducive to finding the final layout more efficiently.
According to the characteristics of the problem, this paper chooses to use sequential quadratic programming (SQP) as the gradient-based layout search algorithm. The optimization variables are the coordinates of components, and the minimization objective is the centroid deviation. The range of optimization variables is determined by the positional relationship between components and heat pipes determined by stage one. At the same time, the nonoverlapping constraints between components must be satisfied.
The entire algorithm flow of stage two is shown in Algorithm 1.

Case Study
In this section, two layout design cases, including one simple layout example and one DFH-4 bus-like layout case, are studied to demonstrate the validity and efficacy of the proposed ILP-assisted two-stage layout optimization method.

Case 1: A 15-Component Layout Example.
In this numerical example, a simple layout system involving 15 heatgenerating components is investigated to demonstrate the effectiveness of the proposed layout design method. As shown in Figure 5, one layout design scheme is prepared manually for reference, and 15 components are required to be placed within one layout domain, which is denoted by the red rectangle. The location range of this layout domain is [100,900] in the x-axis direction and [25,475] in the y-axis direction. Hence, the size of this domain is 800 × 450. There are six heat pipes, of which the widths are 30 mm, embedded in this layout area to spread the heat generated by components. Each heat pipe has a maximum load power of 60 W; that is, P hp max = 60 W. The center position coordinate of the first heat pipe in the x-axis direction is 203.57 mm, and the center interval distance between two adjacent heat pipes is 118.57 mm. The size (width w and height h), mass m, and heat power P of 15 layout components to be arranged are displayed in Table 1. The last column in this table lists the number of heat pipes that one component should occupy n hp i . As can be seen, component 11 can occupy two or three heat pipes simultaneously according to its width and the positions of heat pipes. To dissipate heat more evenly and minimize the objective, we set n hp 11 = 3. Components 6, 7, 9, and 13 are predetermined to be placed on top of two heat pipes simultaneously, while the rest of components can only occupy one heat pipe according to the valid heat pipe occupation rule. The expected y-axis centroid coordinate is set as y e = 325 mm, and the permissible maximum deviation is set as δy e = 5 mm.
Based on the setups mentioned above, an ILP-assisted two-stage layout optimization method can be performed to search for the optimal layout solution. In the first stage, the ILP model for this case can be easily established, where H max = 450 mm and P hp max = 60 W. To lower the difficulty of stage two, add one minimum distance margin 10 mm for   Space: Science & Technology each two components in applying vertical length capacity constraint. Then, OR-Tools [19], a portable open-source software suite for combinatorial optimization developed by Google, is utilized to construct the ILP problem with Python, and the open-source linear programming solver SCIP is selected. In the second stage, the termination condition of SQP is that the iteration number reaches the predefined maximum iteration limit G max = 300. The optimization is conducted using a laptop computer with Intel(R) Core(TM) i7-10710U CPU @ 1.10 GHz and 16.00 GB RAM. By performing ILP with SCIP, the theoretical optimal component-heat pipe distribution scheme with Z min = 49:3 is obtained in a short running time (around 35 seconds in our environment). The optimal distribution scheme is presented in Table 2, where P hp j denotes the total load power of the j th heat pipe. The maximum total load power is 49.3 W in the 6 th heat pipe. Note that there are actually 32 optimal distribution solutions that maintain the same minimum objective value (49.3 W), which demonstrates the multimodal property of the HCLO problem. Only one is displayed for validation and then provided to stage two for detailed component layout optimization to get the final layout design.
In stage two, the optimization is modeled as minimizing the centroid error to satisfy the nonoverlapping constraint and component-heat pipe overlapping constraint simultaneously. The component-heat pipe distribution scheme determines which heat pipes one component should be placed on, thereby restricting the movement range of one component in the x-axis direction. By incorporating the resolved distribution result in stage one, the location range of components is updated first. Following the algorithm process of stage two, we then generate one feasible layout scheme with no overlap existing as shown in Figure 6(a), of which its y-axis centroid is y c = 244:4 mm. Notice that the desired value of layout y e is 325 mm. Hence, swap operation needs to be used to vertically change the relative position relationship to approach the expected centroid as much as possible. During this step, swap operation stops until no exchange of any two components may improve the objective, and the layout scheme displayed in Figure 6(b) is obtained with its centroid y c = 266:6 mm. Finally, the optimal layout design shown in Figure 7 is realized by performing SQP-based local gradient search with y c = y e = 325:0 mm. This layout example clearly validates the feasibility and effectiveness of the proposed two-stage layout design method.
Furthermore, to illustrate the superiority of the SeqLSbased layout search algorithm, an ablation study is investigated. For convenient comparison, the same number of initial layout solutions is provided for gradient search, and final successful layout designs are counted for getting the layout success rate. There are three main algorithm modules: SeqLS generation, swap operation, and SQP search. Two methods are prepared according to whether to apply the first two modules. The first method for providing initial solutions is based on pure random generation without considering the nonoverlapping constraint. It means that the SeqLS method and swap operation are not utilized. The second method is taken as the SeqLS generation, whereas the swap operation Input: Component-heat pipe distribution scheme obtained in stage one, the expected centroid y e Output: Final layout design scheme 1 Update the coordinate range of components based on the result of stage one 2 Generate one non-overlapping layout sample using the SeqLS method 3 f lag =1 4 whilef lagdo 5 ifThere is room for centroid improvementthen 6 Randomly select one heat pipe for swap operation 7 Swap positions of components for smaller centroid deviation 8 ifReal centroid spans expected centroid rangethen  Space: Science & Technology is not performed. In our experiment, each method is allowed to apply the SQP search method 50 times, and three independent runs are performed to avoid contingency.
The statistic results are given in Table 3. It can be observed that gradient search based on random initial solutions can rarely generate target layout designs whereas our method with the SeqLS method and swap operation can locate final optimal layout designs at the highest success rate. It can improve the algorithm performance only to apply the SeqLS generation for nonoverlapping initial layout solutions to some extent. However, swapping components for a more close centroid to the expected one is also indispensable for a more robust local search, which is clearly illustrated from the figures in Table 3. All in all, the experimental results demonstrate the high robustness of the SeqLS-based layout search algorithm in stage two, which further verifies the feasibility and effectiveness of the two-stage optimization design method.  Table 4. Each board is divided into two layout domains, including the lower and upper part, which is represented using 1 and 2 for the north board and 3 and 4 for the south board in Table 4. The size of the upper layout domain is 1800 × 900 mm, and its center position coordinate is ð1000 , 500Þ mm. The other domain has the same size while its location is symmetric about the axis y = 0 mm. It can be found that 22 components are restricted in the lower domain of each side board while 23 components are in the upper domain of each side board. Each side board maintains 16 uniformly distributed heat pipes with the width w hp = 30 mm and the maximum dissipation power P hp max = 120 W.    (3,9,13) 48.0 2 (6,9,10,13) 49.0 3 (2,6,12) 49.0 4 (1,4,11) 46.8 5 (7,8,11,14) 48.8 6 (5,7,11,15) 49.3 * P hp j denotes the total load power of the j th heat pipe. The center x-axis coordinate of the first heat pipe is 192.65 mm, and the interval between centers of two adjacent heat pipes is 107.647 mm. The target y-axis centroid coordinate is set as y e = 0 mm, and the allowable maximum error is set as δy e = 5 mm. Based on the aforementioned setups, ILP based on SCIP solver using OR-Tools is performed for the first stage optimization. The maximum time limit of running SCIP is set as 60 seconds for an immediate solution to save the computational resource. In this stage, two boards are optimized separately since their thermal performance is independent of each other. However, two layout domains in one side board are considered simultaneously as heat pipes cross the entire board in the y-axis direction. Taking the north board for example, there are 45 components involving 1411 design variables and 829 linear constraints in the ILP formulation. It is a more difficult ILP problem than the       Table 5.
In the next stage, the component-heat pipe distribution scheme is incorporated firstly in the movement range of components, and then, the SeqLS-based heuristic layout search algorithm is performed. As shown in Figure 8, the nonoverlapping layout scheme is generated as the seed of the initial solution using the SeqLS method. Considering that its centroid y c = 4:7 mm >y e = 0 mm, swap operation is utilized to exchange positions of some components to approach the target centroid. The layout scheme obtained by swap operation is plotted in Figure 9, where components 50, 59, and 65 are selected to make an exchange, and the system centroid becomes y c = −0:9 mm. Taking the layout shown in Figure 9 as the initial solution for SQP search, we can obtain the final  13 Space: Science & Technology best layout design displayed in Figure 10, of which the centroid y c = 0:0 mm is coincident to the expected one. From this experiment, it can be demonstrated again that the proposed ILP-assisted two-stage layout optimization method is feasible and effective.

Conclusion
Originating from the layout design driven by the thermal performance of the DFH-4 satellite payload, the HCLO problem, as a kind of special component layout optimization, is proposed and formulated in this paper. By taking heat pipe-related performance and system mass characteristics into account, a mathematical model for the HCLO problem is established with the objective of minimizing the maximum total load power of heat pipes. With careful analysis, an ILP-assisted two-stage layout optimization method is proposed to deal with the HCLO problem for better satellite payload placement. In the first stage, the component-heat pipe distribution optimization task is constructed to minimize the objective of the HCLO problem and simultaneously satisfy heat dissipation capacity constraint and vertical length capacity constraint. Then, ILP technique is used to perform the quick search. In the next stage, the detailed component layout optimization model is built to minimize system centroid deviation with meeting the nonoverlapping constraint and component-heat pipe overlapping constraint simultaneously. Then, the SeqLS-based heuristic layout search algorithm, including SeqLS generation, swap operation, and SQP search, is proposed in stage two to conduct the optimization based on the result of stage one. Two layout test cases are studied, and results have verified the validity and efficacy of the proposed two-stage layout design method. It is very promising to apply such an effective approach to generate satellite payload placement solutions for reference in engineering practice.

Data Availability
The data used to support the findings of this study are available upon reasonable request by contacting chenxianqi12@nudt.edu.cn.  14 Space: Science & Technology