# Control-Oriented Models for SO Fuel Cells from the Angle of V&V: Analysis, Simplification Possibilities, Performance

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models for SOFC and their Verification Degree

#### 2.1. Modelling of the Temperature of an SOFC Stack

#### 2.2. A Short Overview of Methods with Result Verification

#### 2.3. Verification and Validation Assessment Verification Degree

- C4
- The CSM uses standard floating point or fixed point arithmetic (the lowest verification degree).
- C3
- The CSM is meaningfully subdivided into its constituent parts; the numerical implementation uses at least standardized IEEE 754 floating point arithmetic; and sensitivity analysis is carried out for uncertain parameters or uncertainty is propagated through the subsystems using traditional methods (e.g., Monte-Carlo).
- C2
- Relevant subsystems of a CSM are implemented using tools with result verification or delivering reliable error bounds; and the convergence of numerical algorithms is proven via analytical solutions, computer-aided proofs, or fixed point theorems.
- C1
- The whole CSM is verified using tools and algorithms with result verification or using real number algorithms, analytical solutions, or computer-aided existence proofs; software and hardware comply with the IEEE754 and follow the interval standard P1788; uncertainty is quantified and propagated throughout the CSM using verified or stochastic (or both) approaches; and sensitivity analysis is carried out.

#### 2.4. Verification Degree of ODE Based SOFC Models

- (i)
- the way the simulated solution $y({t}_{k},p)$ of the IVP is obtained in Equation (2):
- (i.a)
- $y({t}_{k},p)$ is computed analytically,
- (i.b)
- $y({t}_{k},p)$ is approximated by an analytic expression (e.g., using the Euler method) and the approximation error is neglected, which is justified in cases in which the integration step size is smaller by several orders of magnitude than the smallest time constant of an asymptotically stable ODE system (cf. Equation (4)), and
- (i.c)
- $y({t}_{k},p)$ is computed using a “black box” numerical solver (no explicit expression for the solution); and

- (ii)
- the kind of used arithmetic:
- (ii.a)
- traditional floating point arithmetic,
- (ii.b)
- interval arithmetic, and
- (ii.c)
- affine arithmetic, Taylor models, etc.

## 3. Simplifying the Models

#### 3.1. Simplified Model MPC1: Constant Heat Capacities

#### 3.2. Simplified Model MPL1: Linear Heat Capacities

## 4. Numerical Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Rauh, A.; Senkel, L.; Auer, E.; Aschemann, H. Interval Methods for the Implementation of Real-Time Capable Robust Controllers for Solid Oxide Fuel Cell Systems. Math. Comput. Sci.
**2014**, 8, 525–542. [Google Scholar] [CrossRef] - Rauh, A.; Senkel, L.; Aschemann, H. Interval-Based Sliding Mode Control Design for Solid Oxide Fuel Cells with State and Actuator Constraints. IEEE Trans. Ind. Electron.
**2015**, 62, 5208–5217. [Google Scholar] [CrossRef] - Rauh, A.; Senkel, L.; Kersten, J.; Aschemann, H. Reliable Control of High-Temperature Fuel Cell Systems using Interval-Based Sliding Mode Techniques. IMA J. Math. Control. Inf.
**2016**, 33, 457–484. [Google Scholar] [CrossRef] - Rauh, A.; Senkel, L.; Aschemann, H. Reliable Sliding Mode Approaches for the Temperature Control of Solid Oxide Fuel Cells with Input and Input Rate Constraints. In Proceedings of the 1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, Saint Petersburg, Russia, 24–26 June 2015; pp. 390–395. [Google Scholar]
- Oberkampf, W.L.; Trucano, T.G.; Hirsch, C. Verification, Validation, and Predictive Capability in Computational Engineering and Physics. Appl. Mech. Rev.
**2004**, 57, 345–384. [Google Scholar] [CrossRef] - Auer, E.; Luther, W. Numerical Verification Assessment in Computational Biomechanics. In Dagstuhl Seminar 08021: Numerical Validation in Current Hardware Architectures — From Embedded Systems to High-End Computational Grids; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2009; Volume 5492, pp. 145–160. [Google Scholar]
- Kiel, S.; Auer, E.; Rauh, A. Uses of GPU Powered Interval Optimization for Parameter Identificationin the Context of SO Fuel Cells. IFAC Proc. Vol.
**2013**, 46, 558–563. [Google Scholar] [CrossRef] - Moore, R.E.; Kearfott, R.B.; Cloud, M.J. Introduction to Interval Analysis; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 2009. [Google Scholar]
- Lohner, R. On the Ubiquity of the Wrapping Effect in the Computation of Error Bounds; Perspectives on Enclosure Methods; Springer: Berlin/Heidelberg, Germany, 2001; pp. 201–218. [Google Scholar]
- Auer, E.; Senkel, L.; Kiel, S.; Rauh, A. Performance of Simplified Interval Models for Simulation and Control of Solid Oxide Fuel Cells. In Proceedings of the Fourth International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering; Tsompanakis, Y., Kruis, J., Topping, B.H.V., Eds.; Civil-Comp Press: Stirlingshire, UK, 2015. [Google Scholar] [CrossRef]
- Huang, B.; Qi, Y.; Murshed, A. Dynamic Modeling and Predictive Control in Solid Oxide Fuel Cells; John Wiley & Sons, Inc.: New York, NY, USA, 2013. [Google Scholar]
- Bove, R.; Ubertini, S. (Eds.) Modeling Solid Oxide Fuel Cells; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Yu, S.; Fernando, T.; Iu, H.H.C. A Comparison Study for the Estimation of SOFC Internal Dynamic States in Complex Power Systems Using Filtering Algorithms. IEEE Trans. Ind. Inform.
**2017**, 13, 1027–1035. [Google Scholar] [CrossRef] - Ma, R.; Gao, F.; Breaz, E.; Huangfu, Y.; Pascal, B. Multi-Dimensional Reversible Solid Oxide Fuel Cell Modeling for Embedded Applications. IEEE Trans. Energy Convers.
**2017**. [Google Scholar] [CrossRef] - Auer, E.; Cuypers, R.; Luther, W. Process-oriented approach to verification in engineering. In Proceedings of the ICINCO 2012, Rome, Italy, 28–31 July 2012; SciTePress: Setubal, Portugal, 2012; pp. 513–518. [Google Scholar]
- Auer, E. Result Verification and Uncertainty Management in Engineering Applications; Habilitation.Verlag Dr. Hut: Munich, German, 2014. [Google Scholar]
- Henninger, H.; Reese, S.; Anderson, A.; Weiss, J. Validation of Computational Models in Biomechanics. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine. SAGE J.
**2010**, 224, 801–812. [Google Scholar] - De Figueiredo, L.H.; Stolfi, J. Affine Arithmetic: Concepts and Applications. Numer. Algorithm
**2004**, 34, 147–158. [Google Scholar] [CrossRef] - Berz, M. Modern Map Methods for Charged Particle Optics. Nucl. Instrum. Methods
**1995**, 363, 100–104. [Google Scholar] [CrossRef] - Zepper, J.; Aragon, K.; Ellis, M.; Byle, K.; Eaton, D. ASCI Applications Software Quality Engineering Practices; Technical Report SAND2002-0121; Sandia National Laboratories: Albuquerque, NM, USA, 2002.
- Knuth, D.E. Literate Programming. Comput. J.
**1984**, 27, 97–111. [Google Scholar] [CrossRef] - Rump, S.; Ogita, T.; Morikura, Y.; Oishi, S. Interval arithmetic with fixed rounding mode. Nonlinear Theory Appl. (IEICE)
**2016**, 7, 362–373. [Google Scholar] [CrossRef] - Hofschuster, W.; Krämer, W.; Neher, M. C-XSC and Closely Related Software Packages. In Numerical Validation in Current Hardware Architectures LNCS 5492; Cuyt, A.M., Krämer, W., Luther, W., Markstein, P.W., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; Volume 5492, pp. 68–102. [Google Scholar]
- Nedialkov, N.S. The Design and Implementation of an Object-Oriented Validated ODE Solver; Kluwer Academic Publishers: Norwell, MA, USA, 2002. [Google Scholar]
- Pyzdek, T.; Keller, P. Quality Engineering Handbook; Marcel Dekker: New York, NY, USA, 2003. [Google Scholar]
- Auer, E.; Kiel, S.; Pusch, T.; Luther, W. A Flexible Environment for Accurate Simulation, Optimization, and Verification of SOFC Models. In Proceedings of the Second International Conference on Vulnerability and Risk Analysis and Management (ICVRAM) and the Sixth International Symposium on Uncertainty, Modeling, and Analysis (ISUMA), Liverpool, UK, 13–16 July 2014. [Google Scholar]
- Rauh, A.; Senkel, L.; Aschemann, H. Sensitivity-Based State and Parameter Estimation for Fuel Cell Systems. In Proceedings of the 7th IFAC Symposium on Robust Control Design, Aalborg, Denmark, 20–22 June 2012; pp. 57–62. [Google Scholar]
- Auer, E.; Kiel, S.; Rauh, A. Verified Parameter Identification for Solid Oxide Fuel Cells. In Proceedings of the 5th International Conference on Reliable Engineering Computing, Brno, Czech Republic, 13–15 June 2012; pp. 41–55. [Google Scholar]
- Pusch, T. Umsetzung einer Verifizierten Parameteridentifikation mit GUI Realisierung für Festoxidbrennstoffzellen. Master’s Thesis, Universität Duisburg-Essen, Duisburg, German, 2013. [Google Scholar]
- Wächter, A.; Biegler, L.T. On the Implementation of an Interior-Point Filter Line-search Algorithm for Large-scale Nonlinear Programming. Math. Program.
**2006**, 106, 25–57. [Google Scholar] [CrossRef] - Auer, E.; Kiel, S. Uses of Methods with Result Verification for Simplified Control-Oriented Solid Oxide Fuel Cell Models. In Proceedings of the 7th International Workshop on Reliable Engineering Computing, Bochum, Germany, 15–17 June 2016; pp. 299–317. [Google Scholar]
- Kiel, S. UniVerMec – A Framework for Development, Assessment, Interoperable Use of Verfied Techniques; with Applications in Distance Computation, Global Optimization, and Comparison Systematics. Ph.D. Thesis, University of Duisburg-Essen, Duisburg, German, 2014. [Google Scholar]
- Hansen, E.; Walster, G.W. Global Optimization Using Interval Analysis; Marcel Dekker: New York, NY, USA, 2004. [Google Scholar]
- Brown, P.; Byrne, G.; Hindmarsh, A. VODE: A Variable-Coefficient ODE Solver. SIAM J. Sci. Stat. Comput.
**1989**, 10, 1038–1051. [Google Scholar] [CrossRef] - Rauh, A.; Auer, E. Verified Simulation of ODEs and DAEs in ValEncIA-IVP. Reliab. Comput.
**2011**, 5, 370–381. [Google Scholar] - Lagarias, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E. Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions. SIAM J. Optim.
**1998**, 9, 112–147. [Google Scholar] [CrossRef]

**Figure 2.**Results for models MQ1, MQ3. MQ1 corresponds to $1\times 1\times 1$ spatial configuration, MQ3 to $1\times 3\times 1$. “Fminsearch” denotes results obtained using MATLAB, “IpOPT” the ones using the C++ tool from [30], and “IA Filtering” the global optimisation.

**Figure 3.**Differences in the quality between MPC1 (in red), MPL1 (in green), and MPQ1 (in blue) under S-MATLAB, “c” stands for heat capacities.

**Figure 4.**Simulated temperatures for different parameter sets (cf. Table 3) in comparison to measurements in case of MPC1 (left) and MPL1 (right). Since the curves are very similar for MPL1, we show only a plot close-up in the Figure. “Cf” means that the closed-form solution was used, otherwise VNODE-LP. A MATLAB solver was employed to identify the parameters from Sets 1 and 4. For parameters in Sets 2, 3 and 5, the solver IPOPT was used. The parameters for Set 6 are the same as for Set 5.

**Figure 5.**Simulations with uncertainty in the temperature parameters for MPL1. For Set 5, the uncertainty of $1\%$ in all parameters can be handled very well (green curves). For Set 6 (here with parameters identified for Set 4), only the uncertainty of $5\times {10}^{-6}\%$ leads to meaningful results (red curves). “Cf” means that the closed-form solution was used, otherwise VNODE-LP.

Model | (i.a) | (i.b) | (i.c) | ||||||
---|---|---|---|---|---|---|---|---|---|

(ii.a) | (ii.b) | (ii.c) | (ii.a) | (ii.b) | (ii.c) | (ii.a) | (ii.b) | (ii.c) | |

Models without preheaters, with quadratic $c\left(\theta \right)$, 1, 3 and 9 volume elements, resp. | |||||||||

MQ1 | not possible | [27] | [28] | [26,29] | |||||

MQ3 | not possible | [27] | [7] | [29] | too slow | too slow | |||

MQ9 | not possible | [27] | future work | too slow | |||||

Model without preheaters, with linear $c\left(\theta \right)$, 1 volume element | |||||||||

ML1 | [26] | not interesting | not interesting | ||||||

Models with preheaters, quadratic $c\left(\theta \right)$, 1 and 3 volume elements, resp. | |||||||||

MPQ1 | not possible | [4] | future work | future work | |||||

MPQ3 | not possible | [4] | future work | future work | |||||

Model with piecewise constant preheaters, linear $c\left(\theta \right)$, 1 volume element | |||||||||

MPL1 | this paper | this paper | not interesting | ||||||

Models with preheaters, constant $c\left(\theta \right)$, 1, 3 and 9 volume elements, resp. | |||||||||

MPC1 | this paper | not interesting | |||||||

MPC3 | future work | not interesting | |||||||

MPC9 | too complex | future work | too slow |

Parameters (to Identify) | |

${\alpha}_{i}$, ${\alpha}_{j}$, ${\alpha}_{k}$ | coefficients of heat convection in $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

${c}_{\mathrm{N}2,0}$, ${c}_{\mathrm{N}2,1}$ | heat capacity of nitrogen as ${c}_{\mathrm{N}2}\left(\theta \right)={c}_{\mathrm{N}2,0}+{c}_{\mathrm{N}2,1}\xb7\theta $ in $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${c}_{\mathrm{CG},0}$, ${c}_{\mathrm{CG},1}$ | heat capacity of the cathode gas as ${c}_{\mathrm{CG}}\left(\theta \right)={c}_{\mathrm{CG},0}+{c}_{\mathrm{CG},1}\xb7\theta $ in $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${T}_{\mathrm{AG}}^{\mathrm{inv}}$ | inverse time constant of the anode gas preheater in s${}^{-1}$ |

${T}_{\mathrm{CG}}^{\mathrm{inv}}$ | inverse time constant of the cathode gas preheater in s${}^{-1}$ |

${T}_{\mathrm{SL},\mathrm{AG}}^{\mathrm{inv}}$ | inverse time constant of the anode gas supply line in s${}^{-1}$ |

${T}_{\mathrm{SL},\mathrm{CG}}^{\mathrm{inv}}$ | inverse time constant of the cathode gas supply line in s${}^{-1}$ |

c | specific heat capacity of the stack module in $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

m | mass of the stack module in kg |

${c}_{m}^{\mathrm{inv}}$ | $=1/(c\xb7m)$ |

Variables | |

${y}_{0}={v}_{\mathrm{N}2}$ | product of the mass flow and the temperature ${\dot{m}}_{\mathrm{N}2}^{\mathrm{in}}\xb7{\theta}_{\mathrm{N}2}$ of the anode gas (nitrogen) at the preheater outlet in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

${y}_{1}={v}_{\mathrm{N}2}^{\mathrm{in}}$ | product ${\dot{m}}_{\mathrm{N}2}^{\mathrm{in}}\xb7{\theta}_{\mathrm{N}2}^{\mathrm{in}}$ for the nitrogen at the stack inlet in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

${y}_{2}={v}_{\mathrm{CG}}$ | product ${\dot{m}}_{\mathrm{CG}}^{\mathrm{in}}\xb7{\theta}_{\mathrm{CG}}$ for the cathode gas at the preheater outlet in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

${y}_{3}={v}_{\mathrm{CG}}^{\mathrm{in}}$ | product ${\dot{m}}_{\mathrm{CG}}^{\mathrm{in}}\xb7{\theta}_{\mathrm{CG}}^{\mathrm{in}}$ for the cathode gas at the stack inlet in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

$\theta $ | temperature of the stack in K |

Control Inputs | |

${\dot{m}}_{\mathrm{N}2}^{\mathrm{in}}$ | mass flow of anode gas in $\frac{\mathrm{kg}}{\mathrm{s}}$ (recorded data) |

${\dot{m}}_{\mathrm{CG}}^{\mathrm{in}}$ | mass flow of cathode gas in $\frac{\mathrm{kg}}{\mathrm{s}}$ (recorded data) |

${\theta}_{A}$ | ambient temperature in K |

${\theta}_{\mathrm{AG}}^{\mathrm{d}}$ | desired temperature of the anode gas in K (recorded data) |

${\theta}_{\mathrm{CG}}^{\mathrm{d}}$ | desired temperature of the cathode gas in K(recorded data) |

${u}_{1}={v}_{\mathrm{N}2}^{\mathrm{d}}$ | desired ${v}_{\mathrm{N}2}={\theta}_{\mathrm{AG}}^{\mathrm{d}}\xb7{\dot{m}}_{\mathrm{N}2}^{\mathrm{in}}$ in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

${u}_{2}={v}_{\mathrm{CG}}^{\mathrm{d}}$ | desired ${v}_{\mathrm{CG}}={\theta}_{\mathrm{CG}}^{\mathrm{d}}\xb7{\dot{m}}_{\mathrm{CG}}^{\mathrm{in}}$ in $\frac{\mathrm{kg}\xb7\mathrm{K}}{\mathrm{s}}$ |

**Table 3.**Results for MPC1/MPL1 under different conditions. Results for MPQ1 as a comparison. WT1 is the wall time for the optimisation, WT2 for the simulation. The parameter identification using the closed form solution and IPOPT have not been carried out yet for MPL1 due to reasons explained in Section 3.2, therefore “n.d” for Set 6 means “not done”. Times in MATLAB are given only once as a reference.

Set | Model/Setting | Accuracy e | WT 1 | WT 2 | Degree |
---|---|---|---|---|---|

1 | MPC1/S-MATLAB | 4.8255 K | – | – | C3− |

2 | MPC1/S-VERICELL(i.b) | 35.595 K | 39 s | 348 s | C2− |

3 | MPC1/S-VERICELL(i.a) | 3.5259 K | 636 s | <1 s | C2− |

4 | MPL1/S-MATLAB | 2.1260 K | – | – | C3− |

5 | MPL1/S-VERICELL(i.b) | 0.5310 K | 95 s | 232 s | C2− |

6 | MPL1/S-VERICELL(i.a) | n.d. | n.d. | <1 s | C2− |

7 | MPQ1/S-MATLAB | 2.1641 K | ≈18 h | ≈1800 s | C3− |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Auer, E.; Senkel, L.; Kiel, S.; Rauh, A.
Control-Oriented Models for SO Fuel Cells from the Angle of V&V: Analysis, Simplification Possibilities, Performance. *Algorithms* **2017**, *10*, 140.
https://doi.org/10.3390/a10040140

**AMA Style**

Auer E, Senkel L, Kiel S, Rauh A.
Control-Oriented Models for SO Fuel Cells from the Angle of V&V: Analysis, Simplification Possibilities, Performance. *Algorithms*. 2017; 10(4):140.
https://doi.org/10.3390/a10040140

**Chicago/Turabian Style**

Auer, Ekaterina, Luise Senkel, Stefan Kiel, and Andreas Rauh.
2017. "Control-Oriented Models for SO Fuel Cells from the Angle of V&V: Analysis, Simplification Possibilities, Performance" *Algorithms* 10, no. 4: 140.
https://doi.org/10.3390/a10040140