A Simulation and TOPSIS Approach to the Satellite Constellation Design Problem

Abstract

The design of satellite constellations is a complex optimization problem interdependent with other decision problems and multiple competing, user-specific criteria. Consequently, it is very difficult to make a final decision on the constellation design. This study proposes a full simulation and evaluation framework for designing a satellite constellation. Firstly, constructing a solution space by constraining orbital parameters and varying satellite count and plane configuration. Secondly, employing six evaluation metrics—covering both cost and coverage—that are weighted via the case company, Sternula’s setting, with the TOPSIS approach for ranking the candidate constellations. A subsequent sensitivity analysis evaluates robustness to shifts in criterion weights and per-satellite cost. The study indicates that a Walker Star constellation with 97.5° inclination, 105 satellites in 15 planes (phasing 7) achieves the best cost–coverage balance for the case company and remains stable under weight and cost variations.

1. Introduction

Satellites play a major role across multiple fields and industries, including communication, broadcasting, and observation of the Earth [1]. Additionally, with current technological advancements, launching and operating satellites is becoming increasingly affordable and accessible [2]. Concurrently, the space sector received more than 122 billion euros in funding throughout 2024, and the same period saw the launch of 2877 satellites [3], making it a large and economically significant industry. Furthermore, satellites are crucial in other fields such as navigation and location tracking, especially in the maritime sector. Today, over 200 thousand large vessels and passenger ships are required to use the Automatic Identification System (AIS) to broadcast their location and other information [4,5]. However, a new technology known as VHF Data Exchange System (VDES) or AIS 2.0 offers greater security and capacity for this data transmission [6]. A front-runner within this field is Sternula A/S, who previously operated a single Low Earth Orbit (LEO) satellite in a sun-synchronous polar orbit (the Sternula-1). This type of orbit is useful for several reasons, such as the availability of ride-sharing, robust solar exposure, and frequent connection to established ground stations near the poles [7]. Additionally, due to the shape of the ships’ receivers, the satellite antenna is pointed towards the horizon rather than straight downwards. The mounting angle also yields a larger, more elliptical coverage area.

In order to offer great connectivity, Sternula would need multiple satellites communicating with each other, also known as a satellite constellation. Given the success of Sternula-1 and the operational knowledge gathered from this mission, alongside the polar location of the ground stations, and the possibility of ride-sharing, a Walker Star constellation [8] is the preferred option for this case-study. This constellation pattern was also indicated by Sternula, and will be the only pattern considered. As satellites are relatively expensive in the development and deployment phase, an increase in the number of satellites, and thereby coverage, often comes at a great expense, creating a trade-off between the two. At the same time, other criteria, such as the size of the coverage gaps, might be of importance as well. Sternula focuses on increased connectivity in the Arctic due to the hazards of icebergs. Therefore, the coverage in this area might be more important than the coverage elsewhere. The case problem of this research paper underlines the general need for decision-making methods to aid in the final selection of satellite constellations. Given the current exponential growth in the space industry and the continued investment in both upstream and downstream services, we expect this trend to make the decision even more difficult, as more considerations and trade-offs arise.

All of this leads to a complex optimization problem, where multiple objectives and company-specific preferences must be taken into account to select a final constellation design. This is an extremely difficult decision problem, and the decision maker is often overwhelmed by the complexity and the number of trade-offs. Given the case-specific scoping, the immediate focus of this paper is the integration of the sensitivity of preferences with the evaluation model’s ability to support the design phase. We can therefore omit a great deal of the exploration of different constellation patterns. Thus, this paper investigates the utilization of a multi-criteria decision-making (MCDM) method, specifically TOPSIS, to evaluate different designs of simulated Walker-Star constellations based on performance metrics, such as the number of satellites, costs, maximum coverage gap, and other objectives, which are weighted by importance from Sternula to fit their specific use case.

Literature Review

Designing satellite constellations requires knowledge of several fields, including selecting appropriate constellation geometry, understanding orbital mechanics, and evaluating the economic aspects of these constellations. All these considerations create a complex optimization problem, specifically within what is known as a satellite management value chain optimization problem, which involves considering multiple aspects, as seen in Figure 1.

Although the problems illustrated in Figure 1 are interdependent, this paper concentrates solely on the design of LEO satellite constellations, a topic which has already received attention in the literature. The authors Cakaj et al. [9] investigated the coverage of LEO satellites, however, only for satellites pointing directly down towards the sub-satellite point, and discovered that a constellation of LEO satellites is convenient for achieving high coverage. In Pachler et al. [10], the authors focused on LEO satellites currently in operation and compared four broadband providing constellations based on their throughput. The authors found that the performance was not always improved when increasing the number of satellites, indicating the significance of constellation design.

Over the years, different constellation patterns have been proposed, and the Walker Star pattern is one of the best known. This pattern was first proposed in 1971 by Walker [8] and has since been explored extensively in the literature. While some sources like Wertz [11] introduce them in general, others focus on specific parameters, such as the phasing parameter analyzed in Liang et al. [12]. Additionally, Su et al. [13] investigate both the use of Walker Star and Delta constellation patterns in connection to broadband communication, while also reviewing the surrounding value chain and key technologies. In addition to constellations covering the Earth, Conti and Circi [14] have proposed a design for a lunar global positioning–navigation–timing (PNT) and communication system using halo orbit constellations, thereby establishing a continuous line-of-sight satellite constellation design for Earth–Moon communications and allowing ground stations on Earth to participate in lunar mission operations even if direct visibility of the Moon is not possible.

Regarding optimizing the constellation design, analytical methods have previously been used to optimize different aspects of the constellation as explained by Cornara et al. [15]. However, in more recent literature, the use of metaheuristic algorithms has gotten attention. The work of Savitri et al. [16] propose the use of a genetic algorithm (GA) to optimize the coverage and minimize revisit time. Similarly, Huang et al. [17] and Qin et al. [18] use a genetic algorithm to optimize LEO satellite constellations for navigational purposes, with the former focusing on the number of satellites and the latter on visibility. Both obtained constellations that improved performance on their respective objectives. Additionally, Liu et al. [19] proposes a Particle Swarm Optimization (PSO) algorithm combined with analytical methods to optimize a LEO satellite constellation to achieve continuous coverage in specific regions. Wang et al. [20] have extended this work by proposing a hybrid-resampling PSO (HRPSO) algorithm, in which multiple resampling strategies are incorporated to enhance population diversity. This approach improves global search efficiency and mitigates premature convergence, a well-known limitation of standard PSO. Others propose multi-objective optimization when designing satellite constellations. Imoto et al. [21] formulate the problem as a multi-objective mixed-integer programming problem and then propose a meta-heuristic solution approach. The study bases the constellations design process on the ground tracks, and while these authors are not restricted to a specific type of constellation, others, such as Huang et al. [22], focus specifically on the Walker Star constellation when incorporating multiple criteria in the constellation design process. Finally, Tan et al. [23] use a GA to optimize the spatial coverage density at different latitudes simultaneously and achieves an increase in performance when compared to an existing constellation.

However, none of these include more advanced preferences or rankings of the criteria in the optimization, which is the purpose of multi-criteria decision-making (MCDM) methods. At the same time, MCDM methods are often more appropriate when dealing with small discrete solution spaces, where large degrees of exploration are unnecessary, which is the case for constellation design with case-specific limitations. Moreover, multi-objective version of GAs can be computationally expensive when dealing with many complex objective functions that must be evaluated repeatedly, thereby limiting the problem domain to smaller scales. This is indeed the case, for both Tan et al. [23] and Imoto et al. [21], who only investigate coverage of smaller regions, while only considering constellations with 30 satellites at most. Such limitations are unnecessary for most MCDM methods, in which the objective values need to be computed only once.

Over the years, a variety of MCDM methods have been introduced as explained by Thakkar [24] and Ishizaka and Nemery [25]. In recent literature, these methods have been expanded in different ways. The fuzzy environment, first introduced in 1970 by Bellman and Zadeh [26], has led to methods such as Fuzzy TOPSIS, reviewed by Sorin et al. [27], to better include uncertainties and incomplete information in the decision-making. Others propose combinations of different methods into hybrid MCDM methods. Tyagi et al. [28] proposes a hybrid between AHP and TOPSIS, while Trivedi et al. [29] uses a combination of AHP and PROMETHEE. Both use AHP to construct the weights in order to incorporate multiple expert inputs.

Sahoo and Goswami [30] review the advantages and limitations of a variety of the best-known MCDM methods, and while methods like ELECTRE and AHP allow for more complex preference structures, TOPSIS uses a simple, intuitive weighting scheme, yielding straightforward results. The simplicity of the TOPSIS algorithm might explain its extensive use in the literature. Indeed, Wang and Chang [31] uses the method in the aircraft industry due to its simple computations and easy interpretation. Rehman et al. [32] applies TOPSIS to wind turbine selection and compares it to other MCDM methods. They found that it was more robust than other methods for the problem at hand. More theoretically, Madi et al. [33] explores some of the limitations of TOPSIS, while the papers by Butler et al. [34] and Bana E Costa [35] focus on the sensitivity analysis when using the algorithm. TOPSIS has also been applied to satellite-related problems in both Alp et al. [36] and Vasegaard et al. [37], though none of these involve constellation design.

Thus, motivated by the gap in the literature and the previously mentioned benefits of using MCDM methods when dealing with small discrete solution spaces, this paper proposes the use of a simulation-based multi-criteria decision-making framework to solve the satellite constellation design problem. Here, a sensitivity analysis is employed to better understand the robustness of the final recommendation and the relation to the case company’s preferences.

The rest of the paper will be structured as follows: First, Section 2 briefly states some preliminary theory on constellation design and analysis, as well as the TOPSIS algorithm in detail. Afterwards, Section 3 presents the proposed solution approach with the resulting analysis included in Section 4. Finally, Section 5 and Section 6 discuss and conclude on the results.

2. Background

2.1. Constellation Pattern

Sternula has previous experience in using polar orbits, i.e., orbits with an inclination of ~97.5°. The inclination is one of the six Keplerian Elements and explains the angle between the orbital plane and a reference plane (usually the equator) [11]. Continuing on Sternula’s approach, the type of constellation considered in this paper will be Walker Star Constellations (also known as Streets of Coverage Constellations) [11,12,13].

Walker Star constellations are a collection of orbital planes, with each plane being in a polar orbit and with a predetermined number of planes evenly distributed across the equator (see Figure 2).

Such a constellation can be uniquely defined by four parameters, denoted by $i:T/P/F$, where i is the inclination of the orbits, T is the total number of satellites in the constellation, P is the number of orbital planes, and F is the phasing parameter taking integer values from 0 to $P-1$. From these parameters, the number of satellites in each plane, S, is easily calculated as $S=T/P$, while the ascending nodes of the planes are evenly distributed at intervals of $180/P$ around the equator. This imposes restrictions on the relationship between T and P, since the number of satellites in each plane must be an integer and the same across all planes. Additionally, satellites in the same orbit are uniformly distributed at intervals of $360/S$. Finally, the phasing parameter, F, can be used to calculate the phase difference, $\delta$ as

$$\delta =F·360°/T.$$

The phase difference describes the angle to the corresponding satellite in the adjacent plane [12]. In other words, F controls how much the satellites in the next orbit are shifted relative to the ones in the current orbit.

2.2. Coverage Analysis

After designing the constellation, it is useful to analyze how effective the coverage is. There are multiple ways to perform this analysis as explained by Wertz [11]. One is to simply visualize the constellation (see Figure 2) or by ground track plots. However, since multiple constellations with a large number of satellites will be considered, visualizing each one is impractical and can be subject to bias. Other methods, such as approximating the coverage area analytically or calculating figures of merit from a simulation, are more easily summarized for multiple constellations. The latter will be the primary method of evaluating coverage in this paper.

In order to analyze the coverage through simulation, multiple observation points are evenly distributed across the globe, such that coverage analysis can be performed on each of these instead of a “continuous” globe. Considering each of these points in turn, multiple figures of merit can be calculated, all of which are related to the coverage histogram illustrated in Figure 3.

The coverage histograms are constructed such that a value of 1 on the vertical axis indicates that the point is covered by at least one satellite, while 0 means no coverage at the time. The simulation is run in minutes, which is displayed on the horizontal axis. The data used for the histogram can be collected in a vector of ones and zeroes, which will be referred to as the coverage vector, s, having a length of N time steps equal to the duration of the simulation.

The first figure of merit, Percent Coverage, describes the percentage of the observed duration where a single point received coverage from a satellite and has been used in previous literature by HE and LI [38]. It can be calculated using this binary effect from the histogram and is defined as the number of times that a point is covered, divided by the total number of time steps. This is equal to the average of all components in the coverage vector as described by:

$$\begin{array}{c}\hfill \mathit{Percent}\mathit{Coverage}=\mathrm{Mean}(\mathrm{s})={\displaystyle \frac{{\sum }_{i=1}^N{s}_i}{N}}\end{array}$$

(1)

Following the Percent Coverage, the next figure of merit is the Maximum Coverage Gap, defined as a point’s longest observed period with no coverage. In addition to the Maximum Coverage Gap, the Mean Coverage Gap can also be calculated by averaging all gaps. The Mean Coverage Gap can give a skewed interpretation of the constellation design, since some points might rarely be covered even though the Mean Coverage Gap is low. Therefore, the Maximum Coverage Gap will be the focus in this paper, since it clearly explains the worst case scenarios, while also having received more attention in the literature, e.g., in the works by Zong and Kohani [39] and Singh et al. [40]. These figures of merit can be calculated by the cumulative sum operator, the max operator, and the count operator, which identifies the number of instances every observation occurs. Additionally, we utilize the function NonZeroMean(.), which computes the average of the non-zero elements in the vector. Note, we subtract one as we are interested in the number of time instances where it is not covered.

$$\begin{array}{cc}\hfill \mathit{Maximum}\mathit{Coverage}\mathit{Gap}& =\mathrm{Max}(\mathrm{Count}(\mathrm{CumSum}(s)))-1\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \mathit{Mean}\mathit{Coverage}\mathit{Gap}& =\mathrm{NonZeroMean}(\mathrm{Count}(\mathrm{CumSum}(s))-1)\hfill \end{array}$$

(3)

Finally, the last figure is the Mean Response Time, favored by Wertz [11], which is defined as the expected time for a satellite to arrive at any given observation point. It is calculated by running through the coverage vector and for each step denoting the time distance to the last 1, i.e., the last time the point was covered. Finally, the mean time distance is calculated across all time steps. This might seem to be the opposite of the definition, but it gives the desired result because of symmetry, at least for the coverage gaps in the main part of the coverage vector. If the coverage vector begins and ends with a 1, indicating that the point is covered at the beginning and the end, the method also holds. However, if either the beginning or end is a 0, the method assumes that the value immediately before or after would be 1. This might not be true, but the effect of this can be minimized by running the simulation long enough, such that the beginning and end are of minimal influence. We calculate this by using the function RestartCumSum(., $r=0$), which computes the cumulative sum in a vector, but restarts from zero every time it encounters the value $r=0$. Note, the ${\mathbf{1}}_N$ is the 1-vector with dimensions N.

$$\begin{array}{cc}\hfill \mathit{Mean}\mathit{Response}\mathit{Time}& =\mathrm{Mean}(\mathrm{RestartCumSum}({\mathbf{1}}_N-s,r=0)))\hfill \end{array}$$

(4)

2.3. Evaluation Method

The evaluation method known as the Technique for Order Preference and Similarity to Ideal Solution (TOPSIS) will be utilized in this paper and utilizes an $n\times m$decision matrix, where the n rows each represent a possible solution, and the m columns represent the criteria on which the final decision is based. TOPSIS is based on minimizing the distance to the ideal solution, while maximizing the distance to the anti-ideal solution. The decision maker’s preferences are incorporated in the form of weights for each of the criteria, and the relatively simple procedure can be divided into the following five steps [24,25].

1.

The decision matrix, X, is normalized per criteria, such that the scale of the criteria has no effect on the outcome. Each element in X is normalized as

$${\tilde{x}}_{ij}={\displaystyle \frac{x_{ij}}{\sqrt{{\sum }_{u=1}^n{x}_{uj}^2}}}.$$

The normalized decision matrix will be denoted $\tilde{X}$.

2.

The normalized decision matrix is scaled by the corresponding weights, such that:

$$v_{ij}=w_i·{\tilde{x}}_{ij}.$$

3.

The ideal solution, $a^{+}$, and anti-ideal solution, $a^{-}$, are defined. Although, multiple methods exist, see Ishizaka and Nemery [25], however in this paper they are defined as

$$\begin{array}{cc}\hfill a^{+}& =\left\{(\underset{i}{max}{v}_{ij}|j\in J_{+}),(\underset{i}{min}{v}_{ij}|j\in J_{-})\right\}\hfill \\ \hfill a^{-}& =\left\{(\underset{i}{min}{v}_{ij}|j\in J_{+}),(\underset{i}{max}{v}_{ij}|j\in J_{-})\right\},\hfill \end{array}$$

where $J_{+}$ and $J_{-}$ include the indices associated with maximization and minimization criteria, respectively.

4.

The euclidean distance to the ideal, $d^{+}$, and anti-ideal, $d^{-}$, are calculated for each solution i:

$$\begin{array}{cc}\hfill d_i^{+}& =\sqrt{\sum _{j=1}^m{(a_j^{+}-v_{ij})}^2},i=1,2,\dots n.\hfill \\ \hfill d_i^{-}& =\sqrt{\sum _{j=1}^m{(a_j^{-}-v_{ij})}^2},i=1,2,\dots n.\hfill \end{array}$$

5.

The relative closeness for solution i can then be calculated as

$$C_i={\displaystyle \frac{d_i^{-}}{d_i^{+}+d_i^{-}}},i=1,2,\dots n.$$

The value of $C_i\in [0,1]$, with 1 being the preferable outcome since this is equivalent to a small distance to $d^{+}$ and a large distance to $d^{-}$. The different solutions can now be ranked based on the relative closeness, with the best solution being the one with the highest value.

The intuition behind using relative closeness rather than only the distance to the ideal is to better account for the trade-off between different criteria. Extreme values in one criterion can affect $d^{+}$, such that a worse (based on the other criteria) solution is ranked higher.

The algorithm above results in one optimal solution, based on the specific weighting of the criteria. However, since determining the specific weighting is very subjective, it would be beneficial to analyze how sensitive the results are with respect to changes in the weights. This is the basis of the sensitivity analysis. Additionally, robustness of the solution has been emphasized in previous literature on constellation design, where Deccia et al. [41] optimizes satellite constellation design with respect to multiple weighted objectives and argues the importance of properly calibrated weights. Similarly, Gorr et al. [42] investigates the uncertainty of the parameters for constructing the design space when designing satellite constellations.

Inspired by Butler et al. [34] and Bana E Costa [35], the new weights will be generated by sampling from a probability distribution. In this paper, the Dirichlet distribution [43] will be used, since it is a multi-dimensional distribution, which gives values between 0 and 1, while these also sum to 1. Thus, sampling from this returns an interpretable weighting without any need for further transformation.

First of all, let $W=(w_1,\dots ,w_m)$ be the vector consisting of the original weights. By letting new weights $V\sim \mathrm{Dir}({\alpha }_1,\dots ,{\alpha }_m)$ with ${\alpha }_i=c·w_i,c\in \mathbb{R}$, the marginal means and variances can be calculated as

$$\mathbb{E}\left[V_i\right]={\displaystyle \frac{{\alpha }_i}{{\alpha }_0}}={\displaystyle \frac{c{w}_i}{c{w}_0}}=w_i\mathrm{and}\mathrm{Var}\left[V_i\right]={\displaystyle \frac{{\alpha }_i({\alpha }_0-{\alpha }_i)}{{\alpha }_0^2({\alpha }_0+1)}}$$

where ${\alpha }_0$ and $w_0$ are the sum of the $\alpha$- and w-values, respectively. This leads to the mean $\mathbb{E}\left[V_i\right]$ being equal to the original weight $w_i$ for the given criteria. Additionally, the variance $\mathrm{Var}\left[V_i\right]$ can be controlled by the scaling of the original weights, with an increase in c equaling a decreasing variance.

An important consideration about this procedure is that it will not necessarily result in the same ranking of the criteria. However, this might not be desirable if the weights are relatively close, suggesting that the criteria are equally important.

Sampling from this distribution creates new weights, which can then be used in the TOPSIS algorithm to calculate new relative closeness scores and rankings of the alternatives. Inference can then be made from these new results to evaluate the robustness of the original solution.

3. Methodology

In this section, the applied solution approach will be introduced. This includes defining the solution space, such that a finite number of possible constellations or solutions can be constructed, thereby enabling the inclusion of Sternula’s preferences and the use of TOPSIS. Secondly, the criteria by which the alternatives are evaluated are defined and weighed by taking Sternula’s preferences into account. The final part of the section is dedicated to explaining how the simulation will be implemented.

3.1. Solution Space

When building a satellite constellation, multiple parameters can be tweaked in order to gain a wide variety of constellations. First of all, there are parameters related to the satellites themselves:

• The altitude of the satellites in the constellation.
• The beam width of the antenna.
• The mounting angle of the antenna, which decides the orientation of the beam.

According to Sternula, their satellites use an antenna with a 77° beam width, and the antenna is directed such that the edge of the beam reaches the horizon behind the satellite, meaning that the footprint is “dragged” behind the satellite. Thus, these two parameters will be assumed fixed in this paper. Regarding the altitude of the satellites, Sternula already has a single satellite at an altitude of 550 km, which works satisfactorily. Therefore, this altitude will be considered fixed as well.

Other parameters are related to the constellation pattern, and since Walker Star constellations are used, these include:

• The inclination, i.
• The total number of satellites, T.
• The number of orbital planes, P.
• The phasing between adjacent planes, F.

As mentioned in the introduction, Sternula uses sun-synchronous orbits, hence the inclination will be determined by the inclination formula of Wertz [11] [p. 838], which results in approximately 97.5° inclination.

Finally, with respect to the parameters influencing the Earth’s motion, we assume an ideal two-body dynamics. Under this assumption, the Earth is modeled as a perfect sphere with a uniform mass distribution, and the satellites are treated as point masses influenced solely by the Earth’s central gravitational attraction. This also means, that perturbative factors such as Earth’s oblateness, atmospheric drag, solar radiation and third-body influences (e.g., the Moon) are neglected in the design and simulation process. This simplification allows for the design of the constellation to be completely determined from the Keplerian elements.

In relation to the total number of satellites in the constellation, Sternula reported that they are looking at a range of 60 to 120 satellites. For simplicity and to ensure that the number can be divided into several orbits, this range is discretized such that

$$T\in \{60,65,70,75,80,85,90,95,100,105,110,115,120\}.$$

The number of satellites must be divisible by the number of planes, which puts restrictions on the possible values for this parameter. Additionally, if one uses the formulas derived in Appendix A and the assumed altitude, inclination, and beam width, the footprint area is approximately $2.6$ million km². Since the surface of the Earth is about 510 million km², this means that 197 satellites are needed to cover the Earth entirely at a single point in time, assuming that these are arranged perfectly. Since this amount of satellites is not available, overlap between the footprints should be kept to a minimum. By this argument, the maximum number of planes will be determined as $T/\lfloor T/16\rfloor$, where 16 is the number of footprint widths needed to cover half of the circumference of the earth. In the same way, the length of the footprint is used to determine the minimal number of orbital planes as $\lfloor T/15\rfloor$, since 15 footprint lengths would approximately cover the entire circumference. The graphical intuition behind these arguments can be seen in Figure 4.

After defining the boundaries for the number of planes, the different combinations of the T and P are made by keeping T fixed and evaluating each possible value of P. Only the combinations where $T/P$ results in an integer are kept.

By the same argument as before, and to reduce computation time, the phasing parameter will be assumed fixed in this paper. It will be determined as $\lfloor P/2\rfloor$, since this results in a shift of about half the distance between 2 adjacent satellites in the same orbit.

This procedure results in 36 different combinations, which can be seen in

Table 1. Solution space given by of number of satellites, T, and orbital planes, P.

T	P	T	P	T	P	T	P
60	4	70	5	80	16	100	10
60	5	70	7	85	5	105	7
60	6	70	10	85	17	105	15
60	10	70	14	90	6	110	10
60	12	75	5	90	9	110	11
60	15	75	15	90	10	120	8
60	20	80	5	90	15	120	10
65	5	80	8	90	18	120	12
65	13	80	10	95	19	120	15

.

3.2. Criteria

To evaluate the different combinations in the solution space, multiple criteria will be used. They can be divided into two sections: cost-related and coverage-related. Each of these will be explained individually below.

The total cost of a constellation, C, of course, depends on the number of satellites. Thus, the price of a satellite is needed to calculate the cost, but since this information is not available, it will be described as the variable $C_S$.

Additionally, the total cost also depends on the number of orbital planes, P, since a single launch can only supply a single orbit. Since Sternula uses the Rideshare Program from SpaceX, the launch cost is calculated using the information from [44]. The price, given in million USD, depends on the weight you want to launch, and four sizes are available:

$$\mathrm{Launch}\mathrm{Cos}\mathrm{t}=\left\{\begin{array}{cc}0.3\hfill & \mathrm{for}50\mathrm{kg}.\hfill \\ 0.6\hfill & \mathrm{for}100\mathrm{kg}.\hfill \\ 1.2\hfill & \mathrm{for}200\mathrm{kg}.\hfill \\ 1.8\hfill & \mathrm{for}300\mathrm{kg}.\hfill \end{array}\right.$$

This means that the launch cost for a single orbit depends on the number of satellites in each orbit, S, and the weight of the satellites. The weight is approximately 10 kg, and this is used to summarize the total cost as

$$C=T·C_S+P\left({\chi }_{\{S\le 5\}}·0.3+{\chi }_{\{5<S\le 10\}}·0.6+{\chi }_{\{10<S\le 20\}}·1.2\right),$$

(5)

where $\chi$ is the indicator function and it is used that $max(S)=20$ in the solution space described above.

To include the coverage as criteria for evaluation, the figures of merit explained in Section 2.2 are used. These figures were only calculated for individual points, and thus they must be summarized into a single number that represents the entire globe. This means that some details will be lost and that the summaries might not be the best representatives. It is, however, a necessary simplification.

Firstly, the percent coverage is used as a criterion by taking the mean across all points, in order to get a single figure for the whole constellation. In this connection, Sternula also expressed a desire to include the coverage in the Arctic and the equator as these are specific areas of interest. Thus, the mean coverage percentage for all points within the Arctic Circle is included as a criterion. This corresponds to all points above a latitude 66°, see [45]. Additionally, all points within 5° from the equator are used to represent this area of interest, and the mean coverage percentage across these points is included as a criterion as well.

Secondly, the maximum coverage gap is used. Again, these were only calculated for individual points, and thus, the maximum of all these values will be used as a criterion. This decision is based on the fact that this will be the worst case.

Lastly, the mean response time is used as a criterion as well, by taking the mean across all points. This criterion is included based on the preference shown for this metric by Wertz [11] and as a supplement to the maximum coverage gap.

3.3. Weighing of Criteria

According to Sternula, the most important criteria among the six are percent coverage across the entire planet and the maximum coverage gap. Next is the cost of the constellation, followed by the coverage of the Arctic. These are followed by the mean response time, which means that the coverage of the equator is the least important. The criteria and their weights are summarized in

Table 2. The evaluation criteria and their corresponding weights.

Criteria	Weight
Cost	0.2
Coverage Percent (Earth)	0.3
Coverage Percent (North)	0.1
Coverage Percent (Equator)	0.04
Maximum Coverage Gap	0.3
Mean Response Time	0.06

. Using the notation from Section 2.3, these weights correspond to the vector W, and it can also be seen that the weights sum to 1, making them applicable for TOPSIS.

3.4. Simulation

In order to use TOPSIS, the different criteria must be calculated for each of the possible solutions in the solution space. The cost can be calculated directly using (5). The other criteria need to be simulated, as explained in Section 2.2. This will be done by generating 1000 points equally distributed across the Earth using the Fibonacci Grid Method, see [46]. A Visualization of these points can be seen in Figure 5. This method includes both the exact north and south poles. However, due to the inclination of the orbits, the altitude of the satellites, and the size of the footprint, these two points do not receive any coverage. A graphical explanation is included in Figure 6.

This means that the maximum coverage gap would be the same for all constellations, while the other criteria would be greatly affected as well. Thus, these two points are removed from the simulation, which is reasonable considering that these points are either covered by ice or on land, making them less relevant from a maritime point of view.

With the points generated, the simulation is run using the Space Mission Analysis part of the Aerospace Toolkit in MATLAB [47]. The simulation period is 24 h since the orbits are sun-synchronous and would simply repeat after this period. Additionally, it is simulated in time steps corresponding to one minute, meaning that the data for each point is gathered every minute for 24 h. This time-resolution is both more than sufficient for Sternula’s case-study, and for determining the long-term behavior. This data is then used to calculate the criteria values in R, after which the TOPSIS function from the MCDA package in R [48] is used.

4. Results

In the following section, the previously explained solution approach will be applied, after which the results will be analyzed. This analysis first includes a look at how each of the solutions in the solution space performs according to the different criteria. Next, TOPSIS is used to decide the best solution according to the weights from Sternula. And lastly, the sensitivity of the solution is investigated through the procedure explained in Section 2.3.

Simulating the 36 solutions resulted in data that could be used to evaluate the criteria for each of the solutions. Here it should be mentioned that even though the cost of a satellite is unknown, Sternula estimates it to be around a million euros. Since the Euro and USD are relatively close in value (1 Euro = 1.05 USD per 17 December 2024), the cost will initially be assumed fixed at 1 million USD, to have it in the same currency as the rest of the costs. The entire result of this process can be seen in Appendix B, where the criteria values are collected in a table. These results are summarized in the graphs below.

From Figure 7 it appears that the criteria behave much as expected. First of all, the cost increases with the number of satellites and orbital planes, and it seems that the number of satellites is the most influential.

Additionally, all three coverage measures generally increase when the number of satellites rises. Interestingly, for the coverage around the north pole, it seems that a lower number of orbital planes leads to a higher coverage percentage in most cases. This can be explained by the increase in satellites per plane when the number of planes decreases and the fact that all planes pass over the area. Thus, more satellites in each plane would result in a higher frequency of visits around the North Pole. This also explains why the coverage percentage is much higher in this area compared to the others.

Another interesting aspect of the percent coverage in the north is that it does not seem to be linear with respect to the number of satellites, as opposed to the coverage in the other areas. It seems to be more concave, indicating that it might reach a saturation point, which in the case of coverage would be full coverage of 100%. The same arguments could be made about the mean response time, simply the other way around. The progression seems to be convex rather than linear when the number of satellites increases, and it looks like it could reach a saturation point at zero, which would again mean full coverage.

The decreasing mean response time is again the expected behavior, and the same applies to the maximum coverage gap. However, the number of orbital planes seems to be more influential on these criteria than the others, since a high number of orbital planes, indicated by the light-blue color, are generally placed at a lower criteria value.

4.1. Application of Evaluation Method

The TOPSIS algorithm explained in Section 2.3 is now applied to the decision matrix constructed from

Table A1. Performance of the 36 possible solutions on different criteria.

Satellites	Orbital Planes	Cost, Million USD	Percent Coverage (Earth)	Percent Coverage (North)	Percent Coverage (Equator)	Maximum Gap, Minutes	Mean Response Time, Min:Sec
60	4	64.8	26.38	63.23	17.15	405	43:41
60	5	66.0	26.28	62.63	17.20	372	30:11
60	6	63.6	26.27	63.33	17.22	359	21:58
60	10	66.0	25.99	60.69	17.28	311	9:49
60	12	63.6	25.61	56.27	17.19	315	8:28
60	15	64.5	25.72	57.85	17.20	276	8:15
60	20	66.0	23.83	41.52	17.24	271	8:49
65	5	71.0	28.27	65.52	18.64	380	29:50
65	13	68.9	27.92	63.55	18.64	297	7:22
70	5	76.0	30.21	68.72	20.07	373	29:32
70	7	74.2	30.03	67.76	20.07	331	15:54
70	10	76.0	30.02	68.42	20.06	308	8:40
70	14	74.2	29.19	59.01	20.07	296	6:41
75	5	81.0	32.19	72.03	21.42	374	29:17
75	15	79.5	31.47	66.24	21.44	297	6:4
80	5	86.0	33.76	73.92	22.89	374	29:2
80	8	84.8	34.01	73.96	22.84	321	11:43
80	10	86.0	34.01	74.34	22.93	305	7:47
80	16	84.8	32.66	60.86	22.93	296	5:32
85	5	91.0	35.09	75.52	24.06	369	28:50
85	17	90.1	34.93	67.96	24.33	297	5:2
90	6	97.2	37.92	79.52	25.77	354	20:12
90	9	95.4	37.61	76.40	25.77	312	8:43
90	10	96.0	37.85	78.65	25.75	303	7:6
90	15	99.0	37.30	74.16	25.73	295	4:39
90	18	95.4	35.95	62.18	25.76	296	4:40
95	19	100.7	38.24	69.28	27.13	278	4:14
100	10	106.0	41.49	81.64	28.69	311	6:34
105	7	113.4	43.26	82.95	30.05	342	14:13
105	15	114.0	43.07	81.13	30.07	281	3:42
110	10	122.0	44.89	83.61	31.50	300	6:12
110	11	116.6	44.81	83.21	31.50	302	5:5
120	8	129.6	48.49	86.10	34.27	322	10:8
120	10	132.0	48.27	85.04	34.36	299	5:53
120	12	127.2	48.31	85.64	34.37	292	3:56
120	15	129.0	48.30	84.69	34.36	282	3:2

, with the weights from Table 2. This results in the relative closeness scores shown in Figure 8, where the different solutions on the horizontal axis are denoted as (total number of satellites)_(number of orbital planes).

From this figure, it is apparent that the best solution is the one with 105 satellites distributed across 15 orbital planes, and this has a relative closeness of $0.684$. From the figure, it is also clear that, in general, large parameter values lead to better solutions, while many of the solutions are close in performance, especially around the top. The next best solution has a relative closeness of $0.676$, which is close to that of the best solution. All of this motivates the performance of a sensitivity analysis.

4.2. Sensitivity Analysis

In order to analyze the sensitivity of the result, 1000 alternative weight vectors are sampled from a Dirichlet distribution, as introduced in Section 2.3. The original weight vector W is used as the concentration parameters, with the variance of the parameters being controlled by scaling these parameters. This scaling factor directly controls the concentration of the distribution: increasing the scaling factor reduces variance and concentrates samples around the mean W, while decreasing the scaling factor increases dispersion and allows for greater variability in the sampled weights. This effect is illustrated in Figure 9, where histograms of the sampled weights for the cost criterion are presented to demonstrate how different scaling factors influence the shape and dispersion of the resulting distributions.

From Figure 9, it can be seen that applying no scaling or a small scaling factor, such as 10, results in distributions that are highly skewed with long right tails. Such high dispersion implies a large probability mass far from the original weight W, which may exaggerate the degree of uncertainty, particularly given that only the original weight (one sample) is available. Conversely, a very large scaling factor (e.g., scale = 1000) produces a distribution with extremely small variance. This means that sampled weights group close to the original weight vector W, thereby nearly removing the uncertainty and undermining the purpose of the sensitivity analysis. We found that a scaling factor of 100 provides a great balance between these two extremes. This means it ensures that the expected value of the sampled weights aligns with the original weight vector W, while also introducing a reasonable level of variability. The same applies to the remaining criteria, and this scaling is therefore chosen for the remaining analysis.

Now, new weightings $V_k$, for $k=1\dots 1000$, are samples from the Dirichlet distribution with parameter $100·W$, and the TOPSIS algorithm is run with each new weighting. For each of the 1000 repetitions, the solutions are ranked from 1 to 36, with 1 being the best solution. The results are summarized as a box plot in Figure 10 inspired by Butler et al. [34], where the rankings are used on the vertical axis, instead of the relative closeness as previously.

Looking at the medians in Figure 10, they are very similar to the result in Figure 8, and again, the best ranking solution according to the median is 105 satellites and 15 planes. However, the best ranking solution depends entirely on how “best” is defined. Before it was defined by the highest median, but one could also be interested in having the best-ranked worst case. According to the box plot, this would be the solution with 90 satellites across 10 planes. Furthermore, one could also define it as the solution with the highest achieved performance across all repetitions. For this, one would need to look at the relative closeness scores rather than the rankings.

A last, and perhaps more intuitive, way of defining the best solution is simply the one that ranked number 1 in the most repetitions. This is again the 105_15 solution, which ranked the best 503 times, corresponding to 50.3% of the cases. None of the other solutions came close to this value as seen in the following

Table 3. The number of times different solutions ranked number 1.

060_15	065_13	075_15	080_10	085_17	090_10	095_19	100_10	105_15	110_11	120_12	120_15
2	3	21	2	11	51	156	18	503	56	116	61

, which summarizes the number of times different solutions have ranked number 1. Again, the solutions are named (number of satellites)_(number of orbital planes).

These results only take into account the best solution. One way to look at the full picture is to use the ranking based on the original weights, W, from Table 2. By using this ranking as a base case and comparing it to one of the rankings based on the sampled weights, say $V_1$, it is possible to calculate how many of the solutions ranked the same in the two and how many did not. This procedure can then be done for each of the 1000 repetitions. Figure 11 shows the result of this procedure as a histogram, where the horizontal axis describes the number of solutions that changed rankings when compared to the base case.

From Figure 11 it can be seen that many of the cases have a relatively high number of changes. In fact, the mean is 25, which is a lot considering only 36 is possible. Thus, in conclusion, the original ranking is quite sensitive as a whole, but the best solution remains the same in most of the cases and is therefore more robust than the rest of the rankings.

Effect of Satellite Cost

In the previous part, the weights were considered variable, and everything else was fixed. This last statement is true for most of the parameters, due to the assumptions made in Section 3. However, here it was mentioned that the cost of a satellite is unknown, and hence, this section is dedicated to analyzing the sensitivity of the previous result with respect to the cost.

The analysis is made by constructing a sequence of equidistant satellite costs ranging from 0.1 to 10 million USD. The criteria are then calculated anew, and TOPSIS is used to evaluate the solutions for each of the cases. The solutions can now be ranked and the best solution for each of the different costs can be seen in Figure 12.

Even though it is not clear from Figure 12 alone, there are only two solutions that rank the highest, namely 110_11 and 105_15, and it is therefore clear that the optimal solution is quite robust to changes in cost. Only when the cost of a satellite is very low does the optimal solution change. This might be explained by the fact that constellation cost is not the highest ranking criterion and thus is not influential enough to change the optimal solution.

4.3. Final Recommendation

Based on the initial result of TOPSIS and the sensitivity analysis, it is clear that the best solution is the Walker Star constellation 97.5:105/15/7. Thus, this would be the recommended constellation, based on the analysis made in this paper. Additionally, the second and third-best solutions could be other alternatives. These are chosen based on the sensitivity analysis and Table 3. A summary of these solutions and their performance can be seen in

Table 4. Performance of the three best solutions.

Rank	1	2	3
Satellites	105	95	120
Orbital Planes	15	19	12
Cost, Million USD	114.0	100.7	127.2
Percent Coverage (Earth)	43.07	38.24	48.31
Percent Coverage (North)	81.13	69.28	85.64
Percent Coverage (Equator)	30.07	27.13	34.37
Maximum Gap, Minutes	281	278	292
Mean Response Time, Min:Sec	3:34	4:14	3:56

, while a visualization of the three constellations is included in Figure 13.

From Table 4, it appears that the three recommendations all perform similarly in the different criteria, and none of them is significantly better. In this connection, it is important to note that the results and recommendations are biased towards the weighting of the criteria from Sternula as well as the use of the TOPSIS algorithm. The latter creates a bias concerning the use of relative closeness as the measure for comparison, while also being restricted to linear weightings, in the sense that the weight of a criterion will be the same, regardless of the value. The latter might be unrealistic since a criterion such as cost could become more important if it reaches a certain threshold.

4.4. Exploration of Best Solution

In the last part of the analysis, the best solution is explored through the figures of merit. This is done to get a deeper understanding of how well the constellation performs across the entire Earth, as opposed to the previously used summaries of the figures of merit. Thus, the following plots show the figures of merit, for each of the 1000 simulated points spanning the globe. Since these points are constructed with the Fibonacci grid method, the indexing starts at the north pole and moves southwards, ending at the south pole. This means that the indexing can be thought of as latitude, with index 500 being approximately the equator.

In Figure 14, the first thing to notice is that the percent coverage is much higher around the poles, which was the expected outcome, due to the Walker Star pattern of the constellation. Secondly, it can be seen that the maximum coverage gap is also higher around the poles, which seems counter-intuitive when considering the coverage in the area. However, it might be explained by the inclination not being exactly 90°, which means there will be a gap where the neighboring orbits run in opposite directions, leading to a few very large gaps. This argument can somewhat be seen visually in the previous Figure 6. The few large coverage gaps at the poles are confirmed by the plot of the mean coverage gaps, since this figure of merit is not unusually large in this area. It can also be seen that there are sweet spots closer to the poles, which have both low mean and maximum coverage gaps. From the lower right plot in Figure 14, it can be concluded that the mean response time is relatively constant across all points, with a few exceptions at the poles.

In the figure above, it is possible to make out the distributions of the different figures, but for the sake of completeness, histograms are included in Figure 15.

The histograms above tell much the same story as the previous plots. However, a noteworthy result is seen from the histogram of maximum coverage gaps. Here, it can be concluded that very few points have a gap of about 350 min, and most of the other values are not even half that size. This raises the question of whether or not the maximum across all points is a good representative of the figure when using it as the criterion for evaluation. Another possibility would have been to use metrics, such as a percentile gap or inter-quartile range, to represent the variance in the coverage gaps instead of the worst case. Such a criterion might also be added as a supplement rather than replacing the maximum coverage gap to capture both aspects.

5. Discussion

The choice of a simulation and TOPSIS-based approach was primarily based on the discrete solution space and the need to incorporate company-specific preferences. The simulation provided a simple yet time-demanding way to calculate a variety of different coverage objectives for all of the 36 solutions. The subsequent use of TOPSIS made for an easy and interpretable ranking of the solutions. As mentioned in the introduction, other optimization methods have been used in the previous literature, such as a genetic algorithm. Such methods would, however, require evaluation of a large number of solutions across multiple generations, which would make it more computationally demanding than our approach. On the other hand, if the analysis was not restricted to a small discrete solution space, such a solution would offer better possibilities for exploring more possible solutions, though this might require a simplification of the objective space. At the same time, these metaheuristic methods are not guaranteed to yield a single optimal solution, but instead lead to a Pareto front, and might therefore still require inclusion of preferences from the DM post-optimization. Additionally, a meta-heuristic is not guaranteed to result in the optimal solution, but merely an approximation, which can change between two runs of the algorithm. In this aspect, TOPSIS is more robust since it will give the same conclusion each time, assuming that the parameters stay the same. Another advantage is the transparency of the TOPSIS results, which might be easier to comprehend for the decision maker than a metaheuristic black box would be.

During the work process, major assumptions and simplifications were made, affecting the realism of the simulation and results. Most of these were made in Section 3 during the construction of the solution space, and they were motivated by simplifying the solution space and reducing computation time. However, it would have been interesting more representative to analyze the effect of having other variable parameters, such as the altitude, inclination, or mounting angle. In this regard, the exploration of other constellation patterns or even a combination of different ones would also have given a more thorough exploration of the actual solution space. On the other hand, different patterns or inclinations would require additional analysis into the placement of ground stations, since the current stations might not be sufficient, and at the same time, an expansion of the solutions space would add to the already long computation time. Additionally, many of the assumptions were based on the Sternula’s specific operations and thus, the framework could be argued to be tailored to the specific scenario and thereby be less generalizable. However, the general framework consisting of simulation and TOPSIS could be adapted to other varieties of satellite missions. This might require a change in evaluation criteria, since other aspects, such as communication speed in the case of broadband, or elevation angle in the case of Earth observation, could be more relevant.

To evaluate the solution space, multiple criteria were introduced, and specifically, the cost function was a simplification of reality, since it ignores additional costs related to the launch, such as insurance, while also disregarding the dimensions of the rented space and the satellites. Even though the weight limit is satisfied, the number of satellites might not fit in the available space aboard the launch rocket. Thus, the inclusion of these aspects could have made the cost function more realistic. At the same time, the other used criteria were summaries of the simulated figures of merit. However, as mentioned in Section 4.4, these summaries were not necessarily good representatives, and it would therefore have been beneficial to include other criteria as well. These would not have to be related to the figures of merit and might evaluate aspects such as collision possibilities, but on the other hand, the addition of more criteria would make the weighing more difficult for Sternula.

Another aspect that was left out of the analysis was the time aspect, both in the form of the lifetime of a satellite and the time it takes to plan and launch new satellites. The work only considered complete constellations, but in reality, the satellites are not all launched at the same time. Thus, it would have been more realistic to do the simulation in steps, such that new satellites were added with time according to a given launch plan. Another criterion could then be included in the form of deviance from this launch plan. This would, however, increase the computation time as well as the solution space significantly, which is one of the reasons this was not included. Another reason was that no such launch plan existed.

The last part of the analysis involved the use of TOPSIS and the sensitivity analysis of the results. TOPSIS was chosen based on the varying units of measure and its simplicity regarding the weighing of the criteria. Nevertheless, as mentioned earlier, this method comes with a bias towards the weights and their linearity, along with the relative closeness as the ranking measure. At the same time, the sensitivity analysis showed that the main portion of the rankings was very sensitive to changes in the weights, which might suggest that the weight-structure is too simple and that TOPSIS might be unsuitable for this problem. In this respect, it might have been better to use other MCDM methods, such as ELECTRE, which can incorporate non-linear weights. Another possibility is AHP, but this method, as well as ELECTRE, would require a much more detailed preference structure, which would take much longer to define and might not even be available.

6. Conclusions

In this paper, we present a method for identifying a fitting satellite constellation given multiple criteria, while incorporating the case company Sternula’s preferences. This was done by constructing a solution space that included 36 possible constellations, and introducing six different criteria for evaluation representing both cost and coverage of the constellations. These criteria were then weighed and ranked, while the performance of the different solutions was assessed through simulation.

By using the given weights from Sternula and the TOPSIS algorithm, it was concluded that the best-performing constellation consisted of 105 satellites distributed across 15 planes, i.e., a 97.5:105/15/7 Walker Star constellation. The robustness of this result was explored in the sensitivity analysis, where the same solution also ranked number 1 in 503 out of 1000 cases. This was by far the largest number of top rankings, with the second-best only ranking number 1 in 156 cases.

Future Work

During the discussion in Section 5, several limitations of the current study were identified, all of which decreased the realism of the solved problem. Therefore, any further work should prioritize addressing these aspects to improve the practical relevance and applicability of the results. This would involve the inclusion of additional evaluation criteria to better capture the complexity of satellite constellation design, as well as the development of a more accurate and comprehensive cost function. The latter, however, would require access to more detailed operational and economic data, such as launch costs, maintenance schedules, or satellite-specific constraints. Potential new criteria could include collision avoidance to ensure safe orbital separation, deviance from a specified launch or deployment plan, and robustness measures that evaluate system performance under partial satellite failure or unexpected operational disruptions. Additionally, expanding the design space would also be an important focus for future work, allowing the exploration of alternative orbital patterns, multiple altitude shells, varied phasing strategies, and different payload mounting angles. Collectively, these enhancements would enable a more realistic, resilient, and flexible optimization framework capable of capturing the full range of trade-offs (e.g., the potential trade-off between altitudes and coverage gap) involved in practical constellation design.

Finally, other MCDM methods could be applied, such as ELECTRE, which would allow for non-linear outranking relations, such that the weight is not simply a scaling of the performance. This would be more realistic since the importance of a criterion might increase if the value crosses a certain threshold. E.g the cost might weigh more if it gets too large. However, it would also be more complicated to elicit the preferences needed for these methods.