## A recourse-based solution approach to the design of fuel cell aeropropulsion systems

##### Abstract

The past few decades have witnessed a growing interest in the engineering communities to approach the handling of imperfect information from a quantitatively justifiable angle. In the aerospace engineering domain, the movement to develop creative avenues to nondeterministically solving engineering problems has emerged in the field of aerospace systems design. Inspired by statistical data modeling and numerical analysis techniques that used to be relatively foreign to the designers of aerospace systems, a variety of strategies leveraging upon the probabilistic treatment of uncertainty has been, and continue to be, reported. Although each method differs in the sequence in which probabilistic analysis and numerical optimization are performed, a common motif in all of them is the lack of any built-in provisions to compensate for infeasibilities that occur during optimization. Constraint violations are either strictly prohibited or striven to be held to an acceptable probability threshold, implying that most hitherto developed probabilistic design methods promote an avoid-failure approach to developing aerospace systems under uncertainty.
It is the premise of this dissertation that such a dichotomous structure of addressing imperfections is hardly a realistic model of how product development unfolds in practice. From a time-phased view of engineering design, it is often observed that previously unknown parameters become known with the passing of each design milestone, and their effects on the system are realized. Should these impacts happen to be detrimental to critical system-level metrics, then a compensatory action is taken to remedy any unwanted deviations from the target or required bounds, rather than starting the process completely anew. Anecdotal accounts of numerous real-world design projects confirm that such remedial actions are commonly practiced means to ensure the successful fielding of aerospace systems. Therefore, formalizing the remedial aspect of engineering design into a new methodological capability would be the next logical step towards making uncertainty handling more pragmatic for this generation of engineers.
In order to formulate a nondeterministic solution approach that capitalizes on the practice of compensatory design, this research introduces the notion of recourse. Within the context of engineering an aerospace system, recourse is defined as a set of corrective actions that can be implemented in stages later than the current design phase to keep critical system-level figures of merit within the desired target ranges, albeit at some penalty. The terminology is inspired by the concept of the same name in the field of statistical decision analysis, where it refers to an action taken by a decision maker to mitigate the unfavorable consequences caused by uncertainty realizations. Recourse programs also introduce the concept of stages to optimization formulations, and allow each stage to encompass as many sequences or events as determined necessary to solve the problem at hand. Together, these two major premises of classical stochastic programming provide a natural way to embody not only the remedial, but also the temporal and nondeterministic aspects of aerospace systems design.
A two-part strategy, which partitions the design activities into stages, is proposed to model the bi-phasal nature of recourse. The first stage is defined as the time period in which an a priori design is identified before the exact values of the uncertain parameters are known. In contrast, the second stage is a period occurring some time after the first stage, when an a posteriori correction can be made to the first-stage design, should the realization of uncertainties impart infeasibilities. Penalizing costs are attached to the second-stage corrections to reflect the reality that getting it done right the first time is almost always less costly than fixing it after the fact. Consequently, the goal of the second stage becomes identifying an optimal solution with respect to the second-stage penalty, given the first-stage design, as well as a particular realization of the random parameters. This two-stage model is intended as an analogue of the traditional practice of monitoring and managing key Technical Performance Measures (TPMs) in aerospace systems development settings. Whenever an alarmingly significant discrepancy between the demonstrated and target TPM values is noted, it is generally the case that the most cost-effective recourse option is selected, given the available resources at the time, as well as scheduling and budget constraints.
One obvious weakness of the two-stage strategy as presented above is its limited applicability as a forecasting tool. Not only cannot the second stage be invoked without a first-stage starting point, but also the second-stage solution differs from one specific outcome of uncertainties to another. On the contrary, what would be more valuable given the time-phased nature of engineering design is the capability to perform an anticipatory identification of an optimum that is also expected to incur the least costly recourse option in the future. It is argued that such a solution is in fact a more balanced alternative than robust, probabilistically maximized, or chance-constrained solutions, because it represents trading the design optimality in the present with the potential costs of future recourse. Therefore, it is further proposed that the original two-stage model be embedded inside a larger design loop, so that the realization of numerous recourse scenarios can be simulated for a given first-stage design. The repetitive procedure at the second stage is necessary for computing the expected cost of recourse, which is equivalent to its mathematical expectation as per the strong law of large numbers. The feedback loop then communicates this information to the aggregate-level optimizer, whose objective is to minimize the sum total of the first-stage metric and the expected cost of future corrective actions. The resulting stochastic solution is a design that is well-hedged against the uncertain consequences of later design phases, while at the same time being less conservative than a solution designed to more traditional deterministic standards.
As a proof-of-concept demonstration, the recourse-based solution approach is presented as applied to a contemporary aerospace engineering problem of interest - the integration of fuel cell technology into uninhabited aerial systems. The creation of a simulation environment capable of designing three system alternatives based on Proton Exchange Membrane Fuel Cell (PEMFC) technology and another three systems leveraging upon Solid Oxide Fuel Cell (SOFC) technology is presented as the means to notionally emulate the development process of this revolutionary aeropropulsion method. Notable findings from the deterministic trade studies and algorithmic investigation include the incompatibility of the SOFC based architectures with the conceived maritime border patrol mission, as well as the thermodynamic scalability of the PEMFC based alternatives. It is the latter finding which justifies the usage of the more practical specific-parameter based approach in synthesizing the design results at the propulsion level into the overall aircraft sizing framework. The ensuing presentation on the stochastic portion of the implementation outlines how the selective applications of certain Design of Experiments, constrained optimization, Surrogate Modeling, and Monte Carlo sampling techniques enable the visualization of the objective function space. The particular formulations of the design stages, recourse, and uncertainties proposed in this research are shown to result in solutions that are well compromised between unfounded optimism and unwarranted conservatism. In all stochastic optimization cases, the Value of Stochastic Solution (VSS) proves to be an intuitively appealing measure of accounting for recourse-causing uncertainties in an aerospace systems design environment.