e-mail: [email protected] Dale Borja is a principal engineer of Reliability Engineering at. SSL. He received his B.S. (1980) degree in Ele...

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satellite fails in service, a spare can be brought on line to restore lost coverage [3-5]. With such a large quantity, launching all satellites of the constellation fleet at once to support the mission may prove to be difficult and quite often infeasible due to manufacturing, cost and launch schedule limitations. This results in most of the constellations being designed for multi-stage launches [4, 5]. With multi-stage launches, only a fraction of the entire constellation fleet is sent into orbit at each stage. To ensure that reliability and/or availability requirements are met throughout the entire mission, each stage needs to include sufficient satellites so that spares are available to replace failures between launches. With more satellites on orbit, higher constellation reliability is achieved. Thus, reliability plays a significant role in constellation configuration designs and launch planning [6, 7]. Although reliability analysis can prescribe the minimum required number of satellites for each stage, simply using the results for multi-stage launch planning may not be cost effective, especially when manufacturing and launch costs vary at different stages. Let’s consider a situation where the costs at next stage is expected to be much higher than the current one. In order to reduce total cost, operators may want to launch more satellites than necessary at the current stage so that less satellites are required at the next stage. Clearly there is a need for an optimization strategy for multi-stage-launched constellations that harmonizes reliability requirements with overall cost. In view of the lack of reports on these considerations from existing publications, this paper presents a study on an optimal two-stage launch plan for a satellite constellation with an arbitrary number of satellites, to minimize the overall cost while maintaining compliance to the reliability requirements throughout the mission. 2 SATELLITE CONSTELLATION WITH TWO-STAGE LAUNCHES Let’s consider a satellite constellation with a two-stage launch plan. The mission life of the constellation is denoted

as tM . To ensure reliable services, the constellation is required to have a reliability (probability of success) at least RG throughout the entire mission. Also denote the reliability of a single satellite as Rs (tM ) by the end of the mission life. Assuming the lifetime of a single satellite follows an exponential distribution (for a distribution other than exponential, the optimization models to be developed in the paper remain unchanged), the equivalent failure rate (denoted as λs ) of the single satellite can be determined from

λs = −

ln [ Rs (tM ) ] tM

.(1)

Furthermore, assume the constellation is required to have at least M satellites to provide the required services during the mission. The two-stage launch plan would send a fraction of the whole constellation fleet (denoted as N1 ) into orbit in the first stage at time zero, and the remaining ones (denoted as N 2 ) in the second stage at a later time ( denoted as t2L ). Note that the reliability is a monotonically decreasing function of time t, for the time period from time 0 to t2L , if the constellation reliability meets its requirement at time t2L , it will certainly meet the requirement at any time prior to t2L . So one simple approach to determine N1 is to calculate the minimum required N1 from the requirement at t2L . Then based on the calculated N1 from the first stage, N 2 can be determined with the same approach from the requirement at tM . The drawback of this approach is that it does not take into account the overall cost. If the expenses in manufacturing and launching satellites are expected to increase dramatically from the first stage to the second, one may want to launch more satellites than the minimum required N1 in the first stage. By doing so, the minimum required N 2 to meet the reliability requirement at tM can be lower than the original calculated minimum number from the simple approach, potentially saving money with fewer satellites for the second stage. Balancing N1 and N 2 to achieve the maximum cost saving is an optimization problem. 3 OPTIMIZATION PROBLEM Denoting the overall cost of a satellite constellation as C. and the reliability of the constellation as a function of time t as R(t ) , the optimization problem is minimize: C, subject to: R(t ) ≥ RG , ∀t ∈ [0, t ] . Assume the cost associated with manufacturing and launching of a single satellite in the two stages are C1 and C2 respectively. Then, the overall cost C can be written as C = C1 ⋅ N1 + C2 ⋅ N 2 .

(2)

The optimization problem is to minimize C with N1 and N 2 being the optimization variables to be solved.

Furthermore, due to the monotonic characteristics of the reliability function, the inequality constraints of the optimization problem are reduced to R(t2 L ) ≥ RG , R(tM ) ≥ RG .

(3) (4)

Since R(t ) is a non-convex function of N1 and N 2 , this optimization problem is a non-convex optimization, and its global optimal solution needs to be found from comparison of local optimal solutions. 3.1

Objective Function

The objective function is to minimize the overall cost defined by Eq. (2). To study how the cost variation at different stages could affect the two-stage launch planning, two cost models are considered in this paper: (i) C2 remains unchanged from C1 , and (ii) C2 is changed from C1 with a rate r = ( r (C2 − C1 ) / C1 )). The cost model (i) is a special case of the model (ii) with r = 0 . In addition, the model (ii) allows study of a broad range of cost variation from the second stage costing less than the first stage ( r < 0 ) to costing more than the first stage ( r > 0 ). From Eq. (2), with normalizing the overall cost C to C1 , the objective function of the optimization problem becomes minimize: N1 + (1 + r ) ⋅ N 2 . 3.2

(5)

Inequality Constraints

The inequality constraints are governed by the constellation reliability requirement of Eqs. (3) and (4). The left-hand side reliability functions can be developed in different stages as follows: (1) For 0 ≤ t ≤ t2 L , The satellite constellation is equivalent to the M -out-ofN1 : G system, and the reliability function is given by [8] R(t ) (denoted GN1 / M ( λs , t )) R(t ) = Pr{at least M out of N1 satellites survive by time t}

=

N1 i N1 −i ⋅ Rs ( ≈ s , t ) ⋅ 1 − R ( ≈ s , t ) i=M i N1

∑

(6)

N where 1 is the combination of i chosen from N1 , and i R ( λs , t ) the reliability of a single satellite as a function of

time t; that is, R ( λs , t = ) exp ( − λs ⋅ t ) .

According to the inequality constraint Eq. (3), Eq. (6) needs to meet N1

N1 i N1 −i ≥ RG . ⋅ Rs ( λs , t2 L ) ⋅ 1 − R ( λs , t2 L ) i=M i

∑

(7)

From Eq. (7), a minimum required N1 can be solved. In general, N1 > M . (2) For t2 L < t ≤ tM ,

Due to satellites being added from the second stage the first batch exceeding N1 − M . To maintain the required launch, the constellation reliability is “replenished” at t2L . coverage, the failed satellites need to be replaced by those in At the same time, however, the first-stage launched batch of the second-stage launch batch. Hence the constellation reliability can be developed from N1 satellites could have additional failures in the second stage, resulting in an accumulative number of failures from R(t ) = Pr{at least M out of the first batch N1 satellites survive by time t} + Pr{exact M − 1 out of the first batch N1 satellites survive by time t} × Pr{at least one of the second batch N 2 satellites survives from time t2 L to t} + Pr{exact M − 2 out of the first batch N1 satellites survive by time t} × Pr{at least two of the second batch N 2 satellites survive from time t2 L to t} + Pr{exact M − K out of the first batch N1 satellites survive by time t} × Pr{at least K of the second batch N 2 satellites survive from time t2 L to t}

where K = min{N 2 , M } . Note that for any j = 1, 2, , or K , Pr{exact M − j out of the first batch N1 satellites survive by time t} N N −M + j ， = 1 ⋅ RsM − j ( λs , t ) ⋅ 1 − R ( λs , t ) 1 − M j the constellation reliability in the second stage becomes

= R(t ) GN1

M

K N1 (λs , t ) + ∑ j =1 M −

M−j Rs (λs , t ) j

× [1 − R(λs , t ) ]

N1 − M + j

(8)

GN2 j (λs , t − t2 L )

where

(

)

− t 2L GN 2 / j λs , t =

N2 − j N2 j ⋅ Rs λs , t − t 2 L ⋅ 1 − R λs , t − t 2 L j i=M N1

∑

(

)

(

)

(denoted as min{N1} ) is governed by Eq. (7) without involving C and N 2 , start by calculating min{N1} and increase N1 incrementally. Then, for each N1 , calculate the minimum required number of N 2 given N1 (denoted as min{N 2 | N1} ). Finally, compare the objective function value of each pair of ( N1 , N 2 ) to find the pair producing the minimum objective function value, which is the desired optimal solution ( N1* , N 2* ) . (1) For the cost model (i), With r = 0 , the objective function Eq. (5) is independent of the cost, and the optimization problem becomes finding the minimum numbers N1 and N 2 only. In this case, the optimal solution is N1* = min{N1} , and

According to the inequality constraint Eq. (4), Eq. = (8) N 2* min{ = N 2 | N1 N1*} . This can be proven in the needs to meet following: K N1 M − j N1 − M + j Assume there exists another optimal solution ( N1* , N 2* ) GN1 M (λs , t ) + ∑ Rs (λs , t ) [1 − R(λs , t ) ] M − j j =1 (9) that is different from (min{N1}, min{N 2 | N1 = min{N1}}) . N1 N 2 j N2 − j × ∑ Rs (λs , tM − t2 L ) [1 − R(λs , tM − t2 L ) ] ≥ RG Then N1* + N 2* < min{N1} + min{N 2 | N1 = min{N1}} . j i=M * Note that N1 ≥ min{N1} (otherwise, Eq. (7) is violated), Combining Eqs. (5), (7) and (9) yields the consolidated (a) If N1* = min{N1} , objective function and inequality constraints of the optimization problem. To satisfy the inequality constraint Eq. (9), 3.3 Solving for Optimal Solution of N 1 and N 2

N 2* ≥ min{N 2 | N1 = min{N1}} . This contracts to the

Solving for a non-convex optimization problem starts with solving for local optimization problems, which can be treated as convex functions locally. Solving a convex optimization could be complicated. Many methods have been proposed for convex optimization. The most commonly used method is the interior point (or barrier) method [9]. For most reliability optimization problems, however, evaluating the gradient of the constraint functions is difficult. A simple but efficient heuristic approach is typically used [10, 11]. Noticing that the minimum required number of N1

assumption implying N 2* < min{N 2 | N1 = min{N1}} . Hence there is no another optimal solution different than * = N 2 | N1 min{N1}} . N1* = min{N= 1} , and N 2 min{

(b) If N1* > min{N1} , Define = L N1* − min{N1} . Clearly L is a positive integer. In the second stage, each satellite surviving the first stage from the first batch N1* satellites would have a higher failure probability than a satellite from the second stage

launched N 2* satellites, due to their longer time in operation. This means that, with the same total number of satellites combined from the two batches, launching more satellites in the first stage would worsen the constellation reliability than launching more satellites in the second stage. Therefore, from a reliability stand point, a plan with a

constellation reliability requirement up to tM .

* = = min{N1} + L and N 2 = N 2* combination of N 1 N 1 would not be better than a combination of N1 = min{N1}

and N= N 2* + L . 2

With the latter, N1 = min{N1} , (a)

already proves that the optimal solution is N1* = min{N1} , and N 2* min{ = = N 2 | N1 min{N1}} . (2) For the cost model (ii), With r ≠ 0 , the objective function Eq. (5) involves the cost by the rate r. The optimal solution ( N1* , N 2* ) may not be equal to ( min{N1} , min{N 2 | N1 = min{N1}} ). The heuristic approach described above is used to find the optimal solution. 4 EXAMPLE Following is a hypothetical example to demonstrate how the proposed optimization strategies influence the two-stage launch plan of a satellite constellation. Assume the constellation is required to have at least M = 100 satellites to provide services during a mission life of tM = 15 years. The second launch is scheduled at the half way point of the mission life; that is, t2 L = 7.5 years. The constellation reliability is required to be RG ≥ 0.8 throughout the entire mission. Also assume that each satellite has a reliability Rs (tM ) = 0.6 by the end of the mission life. This implies that the failure rate of the satellite is λs = 3,887.6 FITs (1 FIT = 1 failure per billion hours). (1) With the cost model (i), By = N 2*

solving

min{ = N 2 | N1

for N1*}

N1* = min{N1}

(2) With the cost model (ii), To study the impact of a wide range of cost variations, let’s consider the rate r varies from r = -0.2 to 0.7 in 0.1 increments. Table 1 lists the optimal solutions. As can be seen from Table 1, when r < 0 , the optimal solution is identical to the solution with r = 0 . Intuitively this is correct since r < 0 means the cost of manufacturing and launching a single satellite in the second stage is lower than the cost in the first stage, and hence launching fewer satellites in the first stage would be more cost effective. On the other hand, to comply with the reliability requirement in the first stage requires a minimum number of N1* = 134 to be launched in the first stage, so that the optimal solution would be kept as N1* = 134 and N 2* = 32 for all r < 0 cases. Table 2 – List of optimal solutions for r = -0.2 to 0.7 r N 1* N 2* Total

-0.2 134 32 166

-0.1 134 32 166

0.0 134 32 166

0.1 136 30 166

0.2 136 30 166

0.3 141 26 167

0.4 171 3 174

0.5 171 3 174

0.6 174 1 175

0.7 174 1 175

and

from Eqs. (7) and (9) separately, the

optimal solution is calculated to be = 32. = 134, and For comparison, if the entire constellation fleet is launched only once at the beginning of the mission, the minimum required number of satellites would be 175 to meet the reliability requirement, 9 more satellites needed than the twostage launch plan. Figure 1 shows the plot of the constellation reliability as a function of mission time. As can be seen from Fig. 1, in both stages, the constellation reliability is a decreasing function of time t. With the minimum required number of N1 satellites being launched in the first stage, the constellation reliability meets the requirement of 0.8 up to t2L . Beyond that, the constellation reliability may drop below 0.8. Launch of N 2 satellites from the second stage launch improves the constellation reliability at t2L , maintaining compliance to the N1*

Figure 1 – Plot of constellation reliability as a function of time

N 2*

When r > 0 , the cost of manufacturing and launching a single satellite in the second stage is higher than the cost in the first stage. To reduce the overall cost of the constellation, the preference would be to launch more satellites in the first stage, so that the number of satellites to be launched in the second stage could be reduced (although the total required number of satellites in the constellation fleet may increase). This is demonstrated by the optimal solutions with r ≥ 0.1 . Once r > 0.5 , however, the two-stage launch plan would be better off having as many satellites as possible launched in the first stage, and the optimal solution is to launch just one satellite in the second stage! Also noted from Table 1, when r > 0 , even with incremental change of r, the optimal solutions of N1* and N 2* do not change incrementally. Instead, the change could be dramatic at some points (for example, when r changes from 0.3 to 0.4). This is because when N1 increases incrementally,

the min{N 2 | N1} either decreases with the same increment or remains unchanged, resulting in the total number of the constellation fleet N1 + min{N 2 | N1} to be an increasing step function, while within each step, the total number of satellites remains unchanged, as shown in Fig. 2.

an increasing step function, the global optimal solution would lie at the lowest point of the first step; for a decreasing step function, it would lie at the lowest point of the last step; and for a flat step function, it would lie at the lowest points of multiple steps. Figure 3 shows two plots of the objective functions with r = 0.1 and 0.7 respectively. This result suggests that if the cost associated with a single satellite is expected to be considerably higher in the future ( r > 0.2 in this example), a multi-stage (more than two) launch plan may be preferred. The calculation results presented above are based on an exponential distribution assumption for a satellite. If the distribution is not exponential, the reliability value of the satellite at t2L could be different resulting in different optimal solutions. However the methodology to solve for the optimal solutions is still valid. 5 CONCLUSIONS

Figure 2 – Plot of total number of constellation fleet as a function of N1

(a) r = 0.1

(b) r = 0.6 Figure 3 – Plots of objective function with r = 0.1 and 0.6 respectively Depending on the value of rate r, the objective function can be an increasing, a decreasing or a flat step function, while within each step, the objective function decreases. Hence, for

This paper provides an optimization strategy for a twostage launch plan of satellites for a constellation with the objective of minimizing the overall cost. Inequality constraints of the optimization problem are developed from the constellation reliability and meeting the specified reliability requirement. Two cost models are considered in the objective function with one model assuming the cost of a single satellite remains unchanged at both stages, and the other model assuming that the cost changes from the first stage to the second. The methodology to solve for the best solutions for the optimization problem is discussed. A hypothetical example is provided to demonstrate the benefits of using an optimal two-stage launch plan compared to a single launch, together with the pros and cons of the proposed two-stage launch plans when the cost change rate varies. Application of the optimization strategy in real problem solving may require extension of the two-stage launch plan to a multi-stage (more than two) plan, which remains to be done. Also, the presented optimization strategy assumes all satellites have identical lifetime distributions. For a constellation with different satellites that have different lifetime distributions, the presented inequality constraints from the constellation reliability would be invalid and need to be redeveloped. In addition, this optimization strategy is intended for constellation planning prior to the start of a mission. If the minimum required number of satellites for the second stage could be determined after completion of the first stage, the success/failure status of each satellite in the first launched batch would be known. Then the probability of success (or failure) of each surviving one in the second stage may be evaluated by using conditional probability. That would, in general, make the inequality constraint function of the second stage even harder to solve. For an exponentially distributed satellite, however, due to the property of memoryless, the conditional probability in the second stage would be the same as the ordinary probability by resetting the clock at the beginning of the second stage. But such simple clock resetting does not apply to other distributions.

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J. S. Abel, and J. W. Chaffee, “Existence and uniqueness of GPS solutions”, IEEE Transactions on Aerospace and Electronic Systems, Vol. 27, No. 6, November 1991, pp. 952-958. 2. C. E. Fossa, R. A. Raines, G. H. Gunsch, and M. A. Temple, “An overview of the IRIDIUM (R) low Earth orbit (LEO) satellite system”. Proceedings of the IEEE 1998 National Aerospace and Electronics Conference (NAECON), Dayton, Ohio, July 1998. pp. 152–159. 3. Z. Peng, Y. Lu, A. Miller, T. Zhao, and C, Johnson, “Formal Specification and Quantitative Analysis of a Constellation of Navigation Satellites”, Quality & Reliability Engineering International, Vol. 32, No. 2, March 2016, pp. 345-361. 4. S. Cornara, T. W. Beech,, M. Belló-Mora, and A. Martinez de Aragon, “Satellite Constellation Launch, Deployment, Replacement, and End-of-Life Strategies”, Proceedings of the 13th AIAA/USU Conference on Small Satellites, Logan, Utah, August, 1999, SSC99-X-1. 5. S. Cornara, T. W. Beech, M. Belló-Mora, “Satellite constellation mission analysis and design”, Acta Astronautica, Vol. 48, Issue 5, March 2001, pp. 681-691. 6. J. F. Ereau, and M. Saleman, “Modeling and simulation of a satellite constellation based on Petri nets”, Proceedings of 1996 Annual Reliability and Maintainability Symposium, Las Vegas, NV, August 2002, pp. 66-72. 7. S. Zhou, J. Jiao, and Q. Sun. “The Modeling and Simulation of Constellation Availability Based on Satellite Reliability”, Applied Mechanics & Materials, 2014, Vol. 522-524, February 2014, pp. 1215-1219. 8. W. Kuo, V. R. Prasad, F. A. Tillman, and C.-L. Hwang. Optimal Reliability Design: Fundamentals and Applications, Cambridge University Press, Cambridge, U.K., 2001, pp. 16-17. 9. S. Boyd, and L. Vandenberghe. Convex Optimization. Cambridge University Press, Cambridge, U.K., 2004, pp. 127-273. 10. K. .B. Misra, and M. D. Ljubojevic, “Optimal Reliability Design of a System: A New Look”, IEEE Transactions on Reliability, Vol. R-22, No. 5, December 1973, pp. 255258. 11. A. K. Sheikh, M. Younas, and A. Raouf A. “Reliability Based Spare Parts Forecasting and Procurement Strategies”. In: M. Ben-Daya., S. O. Duffuaa, A. Raouf (ed) Maintenance, Modeling and Optimization, Springer, Boston, MA, 2000, pp. 81-110. BIOGRAPHIES Wei Huang, PhD Reliability Engineering Space Systems/Loral, LLC 3825 Fabian Way Palo Alto, CA 94303 USA e-mail: [email protected]

Wei Huang is a principal engineer of Reliability Engineering at SSL. He received his M.S. (1998) degree in Reliability & Quality Engineering, and Ph.D. (2002) degree in Systems & Industrial Engineering from University of Arizona; and B.S. (1982), M.S. (1985), and Ph.D. (1988) degrees in Mechanical Engineering from Nanjing University of Science and Technology, P. R. China. His current research interests center on reliability modeling and analysis of complex systems, reliability life test planning and data analysis, and failure modes, effects, and criticality analysis for satellite equipment. James Loman, PhD Engineering Technologies Space Systems/Loral, LLC 3825 Fabian Way Palo Alto, CA 94303 USA e-mail: [email protected] James Loman is the executive director of Engineering Technologies at SSL. He received his B.S. (1975) degree from Villanova University, M.S. (1977) degree from University of Notre Dame, and Ph.D. (1980) degree from University of Delaware all in Physics. He is the author of over 20 publications and three invention disclosures. He was a leader of GEs "Design for Reliability" initiative which has been an integral part of the well-known GE Six Sigma initiative. Roy Andrada Reliability Engineering Space Systems/Loral, LLC 3825 Fabian Way Palo Alto, CA 94303 USA e-mail: [email protected] Roy Andrada is the manager of Reliability Engineering at SSL. Prior to SSL, he managed the Reliability Engineering Department at Sun Microsystems, Inc. He received his B.S.E.E. (1985) degree from the University of California at Davis. He is a certified Six Sigma Green Belt and completed three Six Sigma projects while at Sun Microsystems, Inc. Mark L. Hanson, PhD Product Strategy & Development Space Systems/Loral, LLC 3825 Fabian Way Palo Alto, CA 94303 USA e-mail: [email protected] Mark L. Hanson is a Project Manager at SSL. He received his B.S. degree from Massachusetts Institute of Technology, M.S. degree from University of Michigan, and Ph.D. degree from George Washington University. He has over 20 years of experience in automated system development for space applications, including autonomous navigation, autonomous fault management (particularly in RPO missions), electric propulsion, and advanced payload accommodation.

Dale Borja Reliability Engineering Space Systems/Loral, LLC 3825 Fabian Way Palo Alto, CA 94303 USA e-mail: [email protected] Dale Borja is a principal engineer of Reliability Engineering at

SSL. He received his B.S. (1980) degree in Electrical Engineering and Computer Sciences from the University of California at Berkeley, and M.S. (2001) degree in Computer Science from Loyola Marymount University. He served as Chairman for the Subcommittee on Software Reliability under the SAE Ground Vehicle Reliability Committee from 2009 through 2014.