## 2 A Brief Review of the Literature  
Typical classifications of fuel cells are based on the type of electrolyte used in the cell, the operating temperature range, the fuel used, and other factors [9; 10]:* Direct methanol fuel cell (DFMC), about 60 \({}^{\circ}\)C
* Proton exchange membrane (or polymer electrolyte membrane) fuel cell (PEMFC), less than 120 \({}^{\circ}\)C, typically about 80 \({}^{\circ}\)C
* Alkaline fuel cell (AFC), about 100 \({}^{\circ}\)C or less
* Phosphoric acid fuel cell (PAFC), about 150-200 \({}^{\circ}\)C
* Molten carbonate fuel cell (MCFC), about 600-700 \({}^{\circ}\)C
* Solid oxide fuel cell (SOFC), about 600-1000 \({}^{\circ}\)C  
Reversible fuel cells [11] represent another class of fuel cells that, in addition to producing electricity from hydrogen and oxygen, can use electricity to produce, by electrolysis, hydrogen and oxygen from water. Microbial fuel cells [12] are yet another category of fuel cells where biological organisms (bacteria, for example) act as the electrocatalyst. The literature on the modeling of fuel cells is expansive (see, for example, [9; 13; 14; 15; 16; 17; 18; 19; 20; 21]). The fuel cell is a complex, non-linear system, and not all aspects of the physics, chemistry and engineering of the system are completely understood yet. Given the enormity of the problem, a single model typically focuses on one particular aspect (e.g., liquid water removal from a PEMFC [22] or real-time simulation [23]) of the system. Fuel cell models are usually characterized by the number of physical dimensions (number of different geometrical axes) they consider. Clearly, this number can vary from zero to three. Zero-dimensional (0-D) models (e.g., [24; 25; 26]) employ a concentrated (as opposed to distributed) or lumped representation of the parameters or metrics (e.g., voltage, current, temperature or species partial pressures) under consideration and therefore make more use of mathematical abstraction than the spatial geometry of the cell. In other words, the variation of parameters along/within the cell is ignored, that is, the \"local\" and \"global\" values are the same in such models. This type of modeling is particularly useful when the interaction of the cell with other components of the (integrated) system is to be investigated (e.g., the modeling of SOFC-based hybrid power generation systems) [27]. A simple, lumped, dynamic model of an integrated SOFC plant is developed for power system simulation in [28], building upon which a large number of fuel cell models have been proposed in the literature (see, e.g., [6; 7; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55]). One-dimensional (1-D) models (e.g., [23; 34; 35; 56; 57]) take into account the variation of the relevant metrics (electrochemical, physical, or thermal) along only one dimension of the cell, ignoring the other two dimensions. For instance, a 1-D model can compute local changes in current density in axial directions of tubular fuel cells [58]. Zero- and one-dimensional models are often quite effective for control purposes. Two-dimensional (2-D) models (e.g., [59; 60; 61]) consider a two-dimensional cross-section of the cell geometry, ignoring the variations along the third dimension. Three-dimensional (3-D) models (e.g., [62; 63; 64; 65; 66; 67]) use distributed (not lumped) metrics or parameters and are the closest to the physical geometry of the cell, describing spatial properties in each of the three coordinates. Because of their high computational costs, 3-D models are somewhat more esoteric than 2-D or 1-D models; in Reference [68], we find an example of the reduction of a 3-D model to 2-D without significant compromises with the quality of results. Zero-dimensional or lumped models are also referred to as macro-models [27]. Because models in two or three dimensions look at finer levels of granularity, they are sometimes called micro-models [69].