# Optimal parameter extraction of fuel cells based on interval branch-and-bound optimization algorithm

Raphaël Chenouard, Ragab A El-Sehiemy

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## To cite this version:

Raphaël Chenouard, Ragab A El-Sehiemy. Optimal parameter extraction of fuel cells based on interval branch-and-bound optimization algorithm. Energy Reports, 2026, 15, pp.108932. 10.1016/j.egyr.2025.108932. hal-05443163

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# Graphical Abstract

## Optimal parameter extraction of fuel cells based on interval branch-and-bound optimization algorithm

Raphaël Chenouard, Ragab A. El-Sehiemy,,

<figure><img src="image_3.png" /></figure>
# Highlights

## Optimal parameter extraction of fuel cells based on interval branch-and-bound optimization algorithm

Raphaël Chenouard, Ragab A. El-Sehiemy,,

* A new application of the interval branch-and-bound optimization algorithm (IBBO) is developed for parameter extraction of four types of fuel cells.

* The optimization results are successfully compared with other methods from the literature for the tested FCs.

* A detailed analysis of the IBBO performance is provided in terms of convergence speed and success rates, and the degree of loss of data or uncertainty.

* The results simulated based on IBBO show a high closeness to the experimental data for I-V and P-V curves.

* The IBBO results consist of a feasible point and guaranteed bounds of the objective values, thus leading to a certified FC model accuracy.
Optimal parameter extraction of fuel cells based on
interval branch-and-bound optimization algorithm

Raphaël Chenouard<sup>a</sup>, Ragab A. El-Sehiemy<sup>b, c, d</sup>

<sup>a</sup>Nantes Université, Ecole Centrale de Nantes, CNRS, LS2N UMR
6004, Nantes, F-44000, France

<sup>b</sup>Kafrelsheikh University, Faculty of Engineering - Electrical Engineering
Department, El-Geish Street, Kafrelsheikh, 33516, Egypt

<sup>c</sup>Higher Institute for Engineering and Technology, Electronics Engineering
Department, Manzala, 35642, Egypt

<sup>d</sup>Széchenyi István University, Sustainability Competence Centre, Egyetem tér 1.,
Hungary, Győr, H9026, Hungary

**Abstract**

Fuel cells play an important role in reducing environmental impacts to
produce cleaner electricity. Numerical models are used to simulate their per-
formance and build efficient observers in real use. The accuracy of these
models is a major concern, as they can be parameterized by several val-
ues. Most of the previous works study the estimation of these parameters
using various metaheuristics. While these methods are stochastic and do
not provide any proof of optimality, the current paper introduces a global
optimization method to accurately bound the optimal root mean square er-
ror between the parameterized model and some experimental data. The
proposed algorithm is based on a deterministic Interval Branch-and-Bound
optimization (IBBO) framework. Interval arithmetic ensures set-based com-
putations to safely bound the objective function value. Four types of fuel
cells, with their experimental data, are used to demonstrate the efficiency
of the proposed methods. IBBO results are compared with some competing
optimization methods used in the literature. They show a better accuracy for
the computed feasible solutions (upper bounds) and a guaranteed value of the
best possible solutions (lower bounds). This last information is not possible
to obtain with metaheuristic algorithms. Compared to other Branch-and-
Bound algorithms, IBBO proposes a new mix of mechanisms (e.g. advanced
constraint propagation, specific search heuristic and feasible point finding
method). Due to the deterministic nature of IBBO, results can be repeated.
Its convergence analysis is detailed on four fuel cells from which a real test
system based on Scribner technology is used to demonstrate the accuracy
and robustness of IBBO on several usage scenarios.

*Keywords:* Fuel cell, Parameter estimation, branch-and-bound, Interval analysis

# 1. Introduction

## 1.1. Motivation

The recent trend in the energy sector is to increase the dependence on clean and renewable energy. This trend has become a necessary issue for the continuous reduction of conventional resources and the increasing awareness of environmental issues. In November 2022, Egypt hosted the 27th version of the Conference of the Parties of the UNFCCC (COP 27). One of the main concerns was the increased progress of renewable energy and its integration into electrical grids, and the way to produce cleaner energy [1], in the context of climate change. This report underlines the clean energy mix, highlighting the need to rapidly transform energy systems to make them more reliable, resilient, and secure, notably by accelerating the transition to renewable energies.

One of the most important renewable energy sources is hydrogen via fuel cell technology. It is based on the chemical transformation of energy contained in hydrogen into an electrical source. FCs are considered a reliable clean energy source when using green hydrogen [2]. They are classified into different types according to the fuel used and their electrolyte. FCs based on a proton exchange membrane (PEMFCs) are characterized by their operating performance at low temperatures and pressures without wasted material. Depending on the operating conditions, their efficiency is in the range of 30% to 60% for the nominal operating temperature (from 50°C to 100°C) [3]. Over the past few decades, the modeling of PEMFCs has attracted a lot of interest [4, 5] with applications in power grids [2].

## 1.2. Related works

Recently, several optimization techniques have been developed to solve the extraction of parameters for PEMFCs to find the optimal values of the control parameters [6]. Several methods were reported to properly find
these unknown parameters, like electrochemical impedance spectroscopy-
based techniques [7], adaptive filter-based [8], black box-based approaches [9],
and current switching methods[10]. Mann et al. proposed a semi-empirical
formulation based on the electrochemical model in [11] to emulate the steady
state operation of the polarization properties with varied operating situa-
tions. However, this model does not accurately identify the unknown. pa-
rameters that vary with the load conditions. This problem increases the
non-linearity of the model and consumes more time without an acceptable
degree of precision [12].

Several other methods have been applied to extract the PEMFC's pa-
rameters with high accuracy and a time-efficient process. Most of these
methods are based on metaheuristics, for example: WOA) [13], flower pol-
lination method [14], pathfinder algorithm [15], shark smell technique [16],
grasshopper optimization algorithm (GOA) [17], chaotic Harris hawk algo-
rithm [18], gray wolf algorithm [19], tree seed algorithm (TSA) and neural
network algorithm [20], jellyfish searching algorithm [21], manta ray forage
optimization [22], and coyote optimization algorithm [23], artificially ecosys-
tem optimization [24], salp swarm algorithm [25]. In [26], a hybrid algorithm
between a differential evolution approach and teaching learning optimizers
was developed in [25].

Reference [27] collects the main key observations concerning solution
methodologies in the last three years. Most of the previous approaches are
based on stochastic algorithms that solve a least-squares problem. However,
deterministic optimization techniques can also be used, like local search al-
gorithms based on the Gauss-Newton and its extension as the Levenberg-
Marquardt algorithm [28, 29]. However, these algorithms may diverge and
also fall to local optima depending on the initial guess. Another type of
deterministic strategy is the branch-and-bound framework, which proposes
a global optimization strategy to find efficient parameter values. Interval
branch-and-bound optimization (IBBO) uses interval analysis [30] to reli-
ably compute and converge to optimal solutions [31]. In most of the papers,
the goal to be minimized is the sum of squared errors (SSE) between the
measured voltage at the fuel cell terminal and the estimated one using the
parameterized model across current data. In most papers, the SSE or RMSE
objective function is used, and they both lead to the same optimal value. SSE
is often preferred because its formula is simpler (no square root). However,
other functions can be considered, like the Mean Absolute Error (MAE). In
Ref. [27], the authors consider the Huber loss function as an alternative cost
function.

## 1.3. Main contributions

This paper aims to introduce a new algorithm for the extraction of fuel cell parameters based on the IBBO framework. Therefore, the current paper has the following main contributions that are summarized as follows:

* A new application of the interval branch-and-bound optimization algorithm (IBBO) is developed for parameter extraction of different types of fuel cells.

* The optimization results are successfully compared with other methods from the literature on four FCs.

* A detailed analysis of the IBBO algorithm is provided in terms of convergence speed and success rates, and the degree of loss of data or uncertainty

* Optimization results and their use to simulate I-V and P-V curves show the high closeness to the experimental data.

* The IBBO results consist of a feasible point and guaranteed bounds of the objective values, thus leading to a certified FC model accuracy.

## 1.4. Paper organization

The next sections of this paper are organized as follows: Section 2 presents the mathematical representation of the fuel cell, and section 3 shows the proposed methodology that is based on the IBBO framework. The simulation results are reported with a detailed discussion in section 4. Finally, section 5 concludes the paper and identifies some future works.

# 2. Mathematical modeling of Proton Exchange Fuel Cell

## 2.1. Operation of PEM fuel cell

This section describes the operation principles of the tested PEMFCs that are composed of the following three parts: anode, proton exchange membrane, and cathode. The ionization of hydrogen gas produces protons and electrons, as presented in Figure 1.
<figure><img src="image_4.png" /><figcaption>Figure 1: Basic composition and working principle of PEMFC [32]</figcaption></figure>

A fuel cell is considered a chemical conversion element. The reactions at the anode and cathode support the energy conversion process. The anode and cathode reactions and the associated accumulated reactions are presented in Equations 1, 2, and 3, respectively:

$$2H_2 \rightleftharpoons 4H^+ + 4e^- \qquad (1)$$

$$4H^+ + 4e^- + O_2 \rightleftharpoons 2H_2O \qquad (2)$$

$$2H_2 + O_2 \rightleftharpoons 2H_2O \qquad (3)$$

## 2.2. Electrical modeling

Figure 2 shows the equivalent electrical circuit that is considered for PEM-FCs. By applying Kirchhoff’s laws, the output voltage at the terminals of a PEMFC ($V_{fc}$) is computed by Equation 4 as in [6]:

$$V_{fc} = E_{Nst} - (\text{All the fuel cell losses}) \qquad (4)$$

where $E_{Nst}$ is produced from the thermodynamic potential, the other losses resulting from the conversion process as defined in [6, 33]. This voltage is
<figure><img src="image_5.png" /><figcaption>Figure 2: Electrical Model of the Fuel Cell</figcaption></figure>

computed at temperatures T (in Kelvin) below 1000 °C. The partial voltage results from the partial pressure of hydrogen and oxygen as in Equations 5, 6, and 7 [33].

$$E_{Nst} = 1.229 - (8.5 \times 10^{-3}) \times (T - 298.15) + (4.308 \times 10^{-5}) \times T \times \log(P_{H_2}^* \sqrt{P_{O_2}^*}) \quad (5)$$

$$P_{H_2}^* = \left( \frac{RH_a \times P_{H_2O}}{2} \times \left( \frac{1}{\frac{RH_a \times P_{H_2O}}{P_a} \times e^{1.635 \times (I/A)/T^{1.334}}} - 1 \right) \right) \quad (6)$$

$$P_{O_2}^* = (RH_c \times P_{H_2O} \times \left( \frac{1}{\frac{RH_c \times P_{H_2O}}{P_c} \times e^{4.192 \times (I/A)/T^{1.334}}} - 1 \right)) \quad (7)$$

where RH_a and RH_c are the relative humidity in the anode and cathode, respectively, P_{H_2O} is the saturation pressure of water vapor in atm, which is given by [11]:
$$P_{H_2O} = 10^{2.95 \times 10^{-2} \times (T-273.15) - 9.18 \times 10^{-5} \times (T-273.15)^2 + 1.44 \times 10^{-7} \times (T-273.15)^3 - 2.18} \quad (8)$$

There are three types of losses associated with PEM fuel cells: activation, ohmic, and concentration losses, which cause FC's output voltage to vary with the load current level in Amperes [34]. Additional losses can also be considered in more complex models, like fuel crossover and internal current [35], but the modeling in this paper stays at the system level for tractability in optimization and simulation methods. The voltage losses of PEMFCs at various current levels are depicted in Figure 3. Equation 9 gives the loss activation voltage. This loss activation voltage is the major part of losses compared to other components. It refers to the amount of voltage loss that allows ions to pass between electrodes [36]. The activation loss can be calculated as follows [17, 37, 38]:

$$V_a = -(\xi_1 + \xi_2 \times T + \xi_3 \times T \times \log(CO_2^*) + \xi_4 \times T \times \log(I)) \quad (9)$$

where the factors $\xi_1$, $\xi_2$, $\xi_3$ and $\xi_4$ represent the coefficient of the activation loss voltage. It can be written as a function of the temperature, T in Kelvin, the concentration percentage of oxygen $CO_2^*$, and the current load level I in A. The actual temperature and the $P_{O_2}^*$ affects on the $CO_2^*$ value as in Equation 10:

$$CO_2^* = \frac{P_{O_2}^*}{5.08 \times 10^6} \times e^{498.15/T} \quad (10)$$

Another component of the loss of voltage is the ohmic voltage loss. It is mainly defined as two resistances, the so-called electronic and proton resistances. These resistances are the flow resistance of protons in the solid polymer membrane and the resistance of electron transfer through the cell electrodes. Mathematically, Equation 11 and 12 present the computation of this ohmic loss voltage component as:

$$V_{ohmic} = V_{ohmic}^{electronic} + V_{ohmic}^{proton} \quad (11)$$

$$= I \times R_c + I \times R_m \quad (12)$$
<figure><img src="image_6.png" /><figcaption>Figure 3: Voltage components of the Fuel Cell</figcaption></figure>

where $R_c$ is the value of electronic resistance, which is unknown. It is thought to be roughly constant throughout the cell's operating conditions. The resistance of proton membrane, $R_m$, that is identified by the membrane's specific resistivity, $\rho_m$ (ohm.cm), area A (cm²), and thickness l (cm) is expressed in Equation 13:

$$R_m = \frac{\rho_m \times l}{A} \qquad (13)$$

An empirical formula is applied to compute the value of resistivity $\rho_m$[11] as:

$$\rho_m = \frac{181.6 \times (1 + 0.03 \times (I/A) + 0.062 \times (T/303)^2 \times (I/A)^{2.5}}{\lambda - 0.634 - 3 \times (I/A) \times e^{4.18 \times \frac{T-303}{T}}} \quad (14)$$

The adjustable parameter $\lambda$ demonstrates the effective water content of the membrane. It could range from 13 in ideal conditions (100% relative humidity) to 22 and 23 in supersaturated conditions [11].
The concentration loss is the third voltage loss of a PEM fuel cell. It occurs at high limiting currents, where the concentration decreases due to the difficulty in transporting sufficient reactant to the electrode surface [17, 37, 38].

$$V_{conc} = -b \times \log(1 - \frac{I/A}{J_{max}}) \qquad (15)$$

where b denotes the unknown parametric coefficient.

Equation 16 presents the accumulated voltage that represents the PEMFC output voltage as:

$$V_{fc} = E_{Nst} - (V_a + V_{ohmic} + V_{conc}) \qquad (16)$$

If $n_{cells}$ they are connected in series to achieve a desired voltage, the resulting stack voltage is the sum of the output of all cells, as shown in [37, 39]:

$$V_{stack} = n_{cells} \times V_{fc} \qquad (17)$$

### 3. Parameter Estimation of PEMFC with IBBO

#### 3.1. Parameters estimation of PEMFC as an optimization problem

When dealing with measurement data, the most common approach is to minimize the sum of squared errors (SSE), which evaluates the sum of the differences between the simulated and measured data. In the case of PEMFCs, the SSE objective function evaluates the differences between the simulated model and the measured voltages, known as $V_{model}$ and $V_{measured}$, respectively. The target parameters to be optimized are ($\xi_1, \xi_2, \xi_3, \xi_4, R_c, \lambda$, and $b$), which are presented in detail in the mathematical formulation in section 2. Then, the following objective function is considered [37]:

$$\min_{x=\{\xi_1, \xi_2, \xi_3, \xi_4, R_c, \lambda, b\}} \quad \text{SSE}(x) \qquad (18)$$

The cost function, $SSE(x)$, is defined in Equation 19:

$$SSE(x) = \sum_{i=1}^{m} (V_{measured}(i) - V_{model}(x, i))^2 \qquad (19)$$
where *m* refers to the recorded number of measured voltages at various loading conditions.

The objective function in Equation 18 is minimized based on the vector of decision variables, which is the parameter estimation of the PEMFC that is defined as $x = \{\xi_1, \xi_2, \xi_3, \xi_4, R_c, \lambda, b\}$. These decision variables are only subjected to the following inequality constraints, related to the lower and upper boundary limitations for the control variables, and represented as follows:

$$x_i^{min} \leq x_i \leq x_i^{max} \quad \forall i \in \{1, \dots, 7\} \qquad (20)$$

## 3.2. Proposed IBBO-based parameter extraction method

### 3.2.1. IBBO structure

The optimization method used in this work is based on the IBBO algorithm. This is a deterministic global optimization algorithm based on the iterative exploration of a search tree composed of boxes of intervals. To facilitate the implementation of such algorithms, the search tree is built and managed through a list *L* of pending nodes, i.e., boxes to be processed. Intervals naturally capture the domain of continuous decision variables. A Cartesian product of intervals is called a box and represents a rectangular polytope in *n* dimensions. Figure 4 describes the main steps and mechanisms of this type of algorithm. Boxes are handled by applying set-based computations through interval analysis [30, 40]. Moreover, it also takes into account the rounding effects of floating-point number computations to make them rigorous.

As an example, computing the sum of two intervals with real bounds is first captured with an enclosing interval with floating-point numbers as bounds:

$$[a, b] + [c, d] \subseteq [\nabla(a), \Delta(b)] + [\nabla(c), \Delta(d)], \qquad (21)$$

where *a*, *b*, *c*, *d* are real numbers. $\nabla(k)$ Returns the greatest floating point number smaller than or equal to *k* and, respectively, $\Delta(k)$ returns the smallest floating point-number greater or equal to *k*. Then, any operation, like the addition, is extended to intervals to ensure that no solution on reals is lost. To be safe, it should handle the rounding at each operation to always obtain an outer approximation of the interval on real numbers as for the addition in Equation 22:

$$[\nabla(a), \Delta(b)] + [\nabla(c), \Delta(d)] = [\nabla(\nabla(a) + \nabla(c)), \Delta(\Delta(b) + \Delta(d))] \quad (22)$$
<figure><img src="image_7.png" /><figcaption>Figure 4: Global process of the interval Branch-and-Bound</figcaption></figure>
To limit the size of the search tree, at each node, the constraints are checked to cut the branches that are irrelevant and only explore the promising ones. The best-known upper bound of the objective function is used as a constraint to cut branches or to reduce boxes:

$$f(x) \leq \text{best\_upper\_bound} \qquad (23)$$

Depending on the optimization problem to solve, other constraints are also used to reduce the boxes to the possibly feasible boxes using contractors [41, 42]. In the context of parameter extraction for PEMFC, only this upper bound constraint is used to limit the size of the search tree.

When a branch is not cut, that means that the box of the node may contain a solution. The existence of a feasible solution can be proved as well for inequality constraints as for equality constraints with, for instance, the interval extension of the Newton algorithm for systems of equations [43, 44]. Then, if the box is precise enough, there is no need to split it, and it can be considered a solution. Otherwise, it is decomposed into sub-boxes to be processed in some further steps, and the IBBO iterates until the stopping criteria are triggered. The IBBO algorithm stops based on several criteria:

* a user-defined time limit;

* an absolute precision on the objective function approximation;

* a relative precision on the objective function approximation.

At any time in IBBO, an interval approximation defines the possible values of the objective function. This approximation is defined by the smallest known upper bound (UB), i.e., the value of the best feasible point found so far, and the lowest lower bound (LB) of the objective function. This lower bound is dynamically evaluated as the minimum of all pending nodes in the search tree:

$$LB = \min(\{f(\mathbf{x}) | \forall \mathbf{x} \in L\}) \qquad (24)$$

Then, at any time, the algorithm provides an interval enclosing the optimal value $f^*$:

$$f^*(\mathbf{x}) \in [LB, UB] \qquad (25)$$

The precision of the objective function approximation is defined by the width of this interval, while the relative precision is the ratio of this width
with LB (except when it is equal to zero). When the time limit is triggered, it
results in a partial search tree exploration, and the best feasible solution that
was found may not be the best one. Conversely, the two other criteria are
met only when the search tree is completely generated; remaining pending
boxes cannot improve the objective function value. In these two cases, the
objective function value is certified to be in the interval [LB, UB], having an
absolute or relative precision corresponding to the criterion that stopped the
solution.

3.2.2. *ibexopt strategy*

The IBBO algorithm is implemented by the ibexopt algorithm, which is
part of the IBEX<fnref n="1" /> open-source C++ library for constraint processing over
real numbers.

IBEX library can be used from C++, but it also provides a specific lan-
guage to define mathematical problems, namely the *Minibex* language. Then,
problem files can be parsed and solved by the default ibexopt strategy, and
it generates an output binary file with the best-known solution and the re-
maining boxes to process (in the case of a partial solving due to a timeout).

To solve the parameter estimation problem presented in this paper (see
Equation 18), we have to write a description of the optimization problem in
a minibex file (for example, see Appendix C) to run ibexopt on this file.
The main ingredients of the ibexopt optimizer are:

* A fixed-point propagation strategy of a set of contractors;

* Contractors implementing several filtering strategies like 2B [45], ACID [46], and polytope relaxations [47].

* A feasible point-finding strategy that updates the upper bound mainly based on a line search algorithm;

* A proving algorithm using the interval Newton test;

* A bisection algorithm to split boxes.

* A bi-level strategy to select the next box to process [48];

* A default relative precision of 0.001;

<footnote n="1">¹https://github.com/ibex-team/ibex-lib</footnote>
• A default absolute precision of 10⁻⁷.

The two main parameters of `ibexopt` are the relative and absolute precisions that directly influence the consideration of boxes as leaves in the search tree, that is, the search tree depth. The user can also use a time limit to stop the algorithm before the exhaustive exploration of the search tree, and it influences the quality of the objective value approximation.

# 4. Applications

## 4.1. Tested Fuel Cells

In the current paper, the applications are carried out on three standard fuel cells and are extended to one real fuel cell. These fuel cells are: Fuel Cell 250 W, Fuel Cell H-12, Fuel Cell Nedstack PS6 and Fuel Cell SCRIBNER 850e, LLC. These types are denoted by FC1, FC2, FC3 and FC4, respectively. Table 1 defines the characteristics and operational conditions used for the estimation of parameters.

## 4.2. Assessment Tools

The results obtained with `ibexopt` are compared to some famous metaheuristics that were previously used to extract FC parameters: Grey Wolf Optimization (GWO), Particle Swarm Optimization (PSO), Moth Flame Optimization (MFO), Bacterial Foraging Optimization (BFO), and Flower Pollination Optimization (FPO). All these algorithms were taken from the NiaPy library<fnref n="2" /> [49], and all results were generated on the same laptop as the results obtained with `ibexopt`. As an example, the model for H-12 FC is presented in Appendix C in the Minibex language.

The default parameters were used for the metaheuristics, but the stopping criterion is based on a maximal number of evaluations equal to 20,000. For `ibexopt`, the default stopping criteria were used without a time limit. All metaheuristics were executed 20 times, while `ibexopt` was applied a single time since it is deterministic and does not require any random initialization procedure. Table 2 defines the bounds for the decision variables.

To illustrate the results obtained from the optimization, a basic simulation of the model is conducted using a Python script (using Numpy and Matplotlib libraries) with the formula presented in section 2.

<footnote n="2">²https://github.com/NiaOrg/NiaPy [accessed 10/14/2024]</footnote>
<table><caption>Table 1: Characteristics and operational conditions for fuel cells</caption><thead><tr><th>Name</th><th>FC1</th><th>FC2</th><th>FC3</th><th>Unit</th></tr></thead><tbody><tr><td>n<sub>cells</sub></td><td>24</td><td>13</td><td>65</td><td>-</td></tr><tr><td>A</td><td>27</td><td>8.1</td><td>240</td><td>cm<sup>2</sup></td></tr><tr><td>l</td><td>0.0178</td><td>0.0025</td><td>0.0178</td><td>cm</td></tr><tr><td>J<sub>max</sub></td><td>0.86</td><td>0.86</td><td>1.2</td><td>A/cm<sup>2</sup></td></tr><tr><td>T</td><td>338.15</td><td>302.15</td><td>343</td><td>K</td></tr><tr><td>PO<sub>2</sub><sup>*</sup></td><td>1</td><td>1</td><td>1</td><td>atm</td></tr><tr><td>PH<sub>2</sub><sup>*</sup></td><td>1</td><td>0.5</td><td>1</td><td>atm</td></tr></tbody></table>

<table><caption>Table 2: Bounds used for the seven parameters to estimate</caption><thead><tr><th>Parameters</th><th>lower bounds</th><th>upper bounds</th></tr></thead><tbody><tr><td>ξ<sub>1</sub></td><td>-1.1997</td><td>-0.8532</td></tr><tr><td>ξ<sub>2</sub> × 10<sup>-3</sup></td><td>1</td><td>5</td></tr><tr><td>ξ<sub>3</sub> × 10<sup>-5</sup></td><td>3.6</td><td>9.8</td></tr><tr><td>ξ<sub>4</sub> × 10<sup>-5</sup></td><td>-26</td><td>-9.54</td></tr><tr><td>R<sub>c</sub> × 10<sup>-3</sup></td><td>0.1</td><td>0.8</td></tr><tr><td>λ</td><td>13</td><td>23</td></tr><tr><td>b</td><td>0.0136</td><td>0.5</td></tr></tbody></table>
### 4.3. Numerical simulations

#### 4.3.1. Results for FC1

The results obtained for FC 250W are summarized in Table 3. Ibexopt obtained the best SSE value in about 73 seconds. Then, the best metaheuristic is FPO, closely followed by GWO, PSO, BPO, and finally, MFO. Since ibexopt is a deterministic algorithm, the average value is the same as the best one, and the standard deviation is zero. Conversely, the metaheuristics give a different solution at each run, and while most of these algorithms give a quite good best solution, MFO has the highest standard deviation and average SSE value.

<table><caption>Table 3: Results on FC 250W and comparison to other optimization algorithms</caption><thead><tr><th></th><th>ibexopt</th><th>GWO</th><th>PSO</th><th>MFO</th><th>BFO</th><th>FPO</th></tr></thead><tbody><tr><td>ξ<sub>1</sub></td><td>-0.8532</td><td>-1.0690</td><td>-0.8532</td><td>-0.8732</td><td>-1.1589</td><td>-0.9637</td></tr><tr><td>ξ<sub>2</sub></td><td>2.2708</td><td>3.2335</td><td>2.6175</td><td>2.8211</td><td>3.4714</td><td>3.3204</td></tr><tr><td>ξ<sub>3</sub></td><td>3.6</td><td>5.9257</td><td>6.0802</td><td>7.0270</td><td>5.6814</td><td>8.7664</td></tr><tr><td>ξ<sub>4</sub></td><td>-17.4923</td><td>-17.4840</td><td>-17.3679</td><td>-17.0735</td><td>-17.6135</td><td>-17.5664</td></tr><tr><td>R<sub>c</sub></td><td>0.1</td><td>0.1407</td><td>0.1</td><td>0.7998</td><td>0.3230</td><td>0.1</td></tr><tr><td>λ</td><td>19.9531</td><td>20.3071</td><td>19.3415</td><td>20.3966</td><td>20.3754</td><td>20.0032</td></tr><tr><td>b</td><td>0.0146</td><td>0.0150</td><td>0.0136</td><td>0.0136</td><td>0.0136</td><td>0.0146</td></tr><tr><td>Best SSE</td><td><strong>0.336017</strong></td><td>0.336563</td><td>0.337184</td><td>0.392778</td><td>0.347314</td><td>0.336244</td></tr><tr><td>Worst SSE</td><td><strong>0.336017</strong></td><td>0.349930</td><td>0.343679</td><td>2.735820</td><td>0.802822</td><td>0.338913</td></tr><tr><td>Avg SSE</td><td><strong>0.336017</strong></td><td>0.340887</td><td>0.339362</td><td>1.019317</td><td>0.433958</td><td>0.337291</td></tr><tr><td>Std SSE</td><td><strong>0.0</strong></td><td>0.003341</td><td>0.002729</td><td>0.577471</td><td>0.095799</td><td>0.000762</td></tr></tbody></table>

Figure 5 shows the I-V curve and the P-V curve simulated by the parameterized model based on the solution obtained with ibexopt. It shows the good quality of the simulation model to generate smooth curves close to and between the data points. One other aspect of the result obtained with ibexopt demonstrates the robustness of the parameters' values since it stops by triggering the relative precision criterion. Then, the best possible value for the SSE objective function for this FC and these operating conditions cannot be lower than $UB=0.335681963$. The value is guaranteed thanks to the IBBO exploration process and the interval analysis computations.

#### 4.3.2. Results for FC2

For the H-12 FC, ibexopt is again the best optimization algorithm, and it took about 25 seconds to converge, while it takes a few seconds for the
<figure><img src="image_8.png" /><figcaption>Figure 5: Simulated curves using the ibexopt solution for fuel cell 250W</figcaption></figure>
metaheuristics. For this fuel cell, the results are closer than for the previous one and the standard deviations are more stable. The best metaheuristic, behind ibexopt, is PSO closely followed by FPO and GWO. Again, with ibexopt we obtain a guaranteed lower bound of 0.11779422.

<table><caption>Table 4: Results on FC H-12 and comparison to other optimization algorithms</caption><thead><tr><th></th><th>ibexopt</th><th>GWO</th><th>PSO</th><th>MFO</th><th>BFO</th><th>FPO</th></tr></thead><tbody><tr><td>ξ<sub>1</sub></td><td>-0.8532</td><td>-0.9515</td><td>-1.1997</td><td>-1.0405</td><td>-1.1347</td><td>-1.0930</td></tr><tr><td>ξ<sub>2</sub></td><td>1.5630</td><td>1.9030</td><td>3.1982</td><td>2.8941</td><td>2.6006</td><td>2.6675</td></tr><tr><td>ξ<sub>3</sub></td><td>3.6</td><td>3.7083</td><td>7.1394</td><td>8.7341</td><td>4.3894</td><td>5.8505</td></tr><tr><td>ξ<sub>4</sub></td><td>-9.54</td><td>-9.54</td><td>-9.54</td><td>-9.54</td><td>-9.8897</td><td>-9.54</td></tr><tr><td>R<sub>c</sub></td><td>0.8</td><td>0.5232</td><td>0.8</td><td>0.8</td><td>0.1</td><td>0.8</td></tr><tr><td>λ</td><td>13.625</td><td>14.4471</td><td>13.0</td><td>13.0</td><td>21.6057</td><td>13.0</td></tr><tr><td>b</td><td>0.1548</td><td>0.1583</td><td>0.1539</td><td>0.1651</td><td>0.1696</td><td>0.1538</td></tr><tr><td>Best SSE</td><td><strong>0.117943</strong></td><td>0.118177</td><td>0.118020</td><td>0.128554</td><td>0.124832</td><td>0.118032</td></tr><tr><td>Worst SSE</td><td><strong>0.117943</strong></td><td>0.309721</td><td>0.321416</td><td>0.436117</td><td>0.197903</td><td>0.118412</td></tr><tr><td>Avg SSE</td><td><strong>0.117943</strong></td><td>0.137333</td><td>0.138905</td><td>0.226089</td><td>0.153595</td><td>0.118150</td></tr><tr><td>Std SSE</td><td><strong>0.0</strong></td><td>0.0568140</td><td>0.060814</td><td>0.080402</td><td>0.021369</td><td>0.000106</td></tr></tbody></table>

### 4.3.3. Results for FC3

Considering the Nedstack Ps6, ibexopt performed well in about 93 seconds. Only MFO is far from the other algorithms and again FPO, GWO, and PSO are close to ibexopt result with a small standard deviation. The lower bound is 2.0982204, so no feasible solution can exist under this certified value for this modeling.

### 4.3.4. Results for a real fuel cell (FC4)

Ibexopt was also applied to the SCRIBNER 850e fuel cell using the experimental results from [50]. A SCRIBNER 850e fuel cell test system is used to control and monitor the fuel cell operation. The characteristics and operating conditions of the tested FC are reported in Table 6 and Table 7. This fuel cell is tested at the operating pressures of 1.0, 1.25, and 1.5 bar and a temperature of 353.15 K.

Table 8 shows the values obtained with ibexopt for the three scenarios, and Figure 6 shows the results of simulations for them as I-V and P-V curves. In all of these results, the order of magnitude of the SSE is the same, even if the solving time can vary from 3 to 13.5 seconds. Indeed, the results on
<table><caption>Table 5: Results on FC Nedstack Ps6 and comparison to other optimization algorithms</caption><thead><tr><th></th><th>ibexopt</th><th>GWO</th><th>PSO</th><th>MFO</th><th>BFO</th><th>FPO</th></tr></thead><tbody><tr><td>ξ<sub>1</sub></td><td>-0.8532</td><td>-0.9864</td><td>-1.1997</td><td>-1.1928</td><td>-1.1570</td><td>-0.9740</td></tr><tr><td>ξ<sub>2</sub></td><td>2.3976</td><td>2.8785</td><td>3.6454</td><td>4.2377</td><td>3.4080</td><td>3.2640</td></tr><tr><td>ξ<sub>3</sub></td><td>3.6</td><td>4.2654</td><td>5.2983</td><td>9.7006</td><td>4.4052</td><td>7.2759</td></tr><tr><td>ξ<sub>4</sub></td><td>-9.54</td><td>-9.54</td><td>-9.54</td><td>-9.54</td><td>-9.8479</td><td>-9.5430</td></tr><tr><td>R<sub>c</sub></td><td>0.1</td><td>0.1037</td><td>0.1018</td><td>0.1</td><td>0.1</td><td>0.1018</td></tr><tr><td>λ</td><td>13.3262</td><td>13.5480</td><td>13.6042</td><td>13.1551</td><td>13.2875</td><td>13.6042</td></tr><tr><td>b</td><td>0.0136</td><td>0.0160</td><td>0.0174</td><td>0.0136</td><td>0.0136</td><td>0.0174</td></tr><tr><td>Best SSE</td><td><b>2.100319</b></td><td>2.117981</td><td>2.124346</td><td>3.015218</td><td>2.714080</td><td>2.124346</td></tr><tr><td>Worst SSE</td><td><b>2.100319</b></td><td>3.997748</td><td>2.825146</td><td>59.403067</td><td>9.360342</td><td>2.475572</td></tr><tr><td>Avg SSE</td><td><b>2.100319</b></td><td>2.538150</td><td>2.270961</td><td>19.44090</td><td>6.098408</td><td>2.270961</td></tr><tr><td>Std SSE</td><td><b>0.0</b></td><td>0.503053</td><td>0.217359</td><td>14.92755</td><td>1.978422</td><td>0.107887</td></tr></tbody></table>

the three first FCs also show a high variability of the solving time, whereas the number of variables and their domain are the same. When solving a new problem, it's difficult to predict the time to solve it because of the IBBO algorithm's complexity. Hopefully, the heuristics used, like in the contractors' feasible point finding strategy, allow for reducing a lot of the time spent in most cases.

<table><caption>Table 6: Characteristics of SCRIBNER 850e FC</caption><thead><tr><th>Variable</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>n<sub>cells</sub></td><td>1</td><td>-</td></tr><tr><td>A</td><td>5</td><td>cm<sup>2</sup></td></tr><tr><td>l</td><td>0.0183</td><td>cm</td></tr><tr><td>J<sub>max</sub></td><td>1.1</td><td>A/cm<sup>2</sup></td></tr><tr><td>T</td><td>353.15</td><td>K</td></tr></tbody></table>

### 4.4. Discussion and Performance Analysis of Ibexopt

Since ibexopt is deterministic, several runs return the same result in about the same amount of time. Figure 7 depicts the evolution of LB and UB during the solving of the tested fuel cells. Ibexopt can quickly find good upper bounds, as shown by the quick decrease of the curves represented by solid lines. Conversely, the lower bound, i.e., the curves as dashed lines, are longer to increase. This can be explained by the IBBO framework, in
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Figure 6: Simulated curves using the ibexopt solutions for fuel cell SCRIBNER 850e on the three scenarios.</figcaption></figure>
<table><caption>Table 7: Operating conditions for the three operating scenarios</caption><thead><tr><th>Scenario #</th><th>PO<sub>2</sub><sup>*</sup> (bar)</th><th>PH<sub>2</sub><sup>*</sup> (bar)</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td></tr><tr><td>2</td><td>1.25</td><td>1.25</td></tr><tr><td>3</td><td>1.5</td><td>1.5</td></tr></tbody></table>

<table><caption>Table 8: Results on FC SCRIBNER 850e with ibexopt algorithm</caption><thead><tr><th>-</th><th>Scenario 1</th><th>Scenario 2</th><th>Scenario 3</th></tr></thead><tbody><tr><td>ξ<sub>1</sub></td><td>-0.8532</td><td>-0.8532</td><td>-0.8532</td></tr><tr><td>ξ<sub>2</sub></td><td>1.3420</td><td>1.4928</td><td>1.6654</td></tr><tr><td>ξ<sub>3</sub></td><td>3.6</td><td>3.6</td><td>3.6</td></tr><tr><td>ξ<sub>4</sub></td><td>-26</td><td>-24.6514</td><td>-16.0837</td></tr><tr><td>R<sub>c</sub></td><td>0.8</td><td>0.8</td><td>0.8</td></tr><tr><td>λ</td><td>13</td><td>13</td><td>13</td></tr><tr><td>b</td><td>0.1098</td><td>0.1126</td><td>0.1460</td></tr><tr><td>SSE</td><td>0.003635</td><td>0.004174</td><td>0.005120</td></tr><tr><td>Time (s)</td><td>3.35</td><td>9.37</td><td>13.50</td></tr></tbody></table>

which LB is computed as the lowest objective function approximation on the remaining boxes to process. Then, it is very sensitive to the approximation mechanisms that are used. Even with efficient linear relaxation, the accuracy of the values that are computed may lead to a high number of iterations before the objective function approximation, namely the interval defined by the lower bound and upper bound, meets the expected precision.

The main parameters that impact the solving process are the absolute and relative precisions used to stop the algorithm. Figure 8 shows the evolution of the solving time depending on the relative precision. We can see that until precision $10^{-7}$, the solving time is reasonable at around 50 seconds for H-12 FC, but for $10^{-8}$ the required accuracy leads to a huge number of additional nodes in the search tree almost for all FCs. In fact, before this precision, the upper bounds that are found can cut branches that are not good enough. However, in the last case, the precision is too high, and the interval contractor used for the upper bound constraint is not able to eliminate the huge number of small boxes generated in the deep branches of the search tree, thus leading to a huge number of iterations to explore them and to an important increase of the solving time. This effect may also highlight the numerical limits of the
<figure><img src="image_11.png" /><figcaption>Figure 7: Convergence and evolution of SSE lower and upper bounds for fuel cell H-12.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 8: Convergence for fuel cell H-12 with decreasing relative precisions.</figcaption></figure>

used models with the experimental data. With reasonable precision, the time and number of nodes in the search tree do not increase too much, and they can even decrease for a small amount, like for $10^{-6}$ relative precision for H-12 or 0.0001 for Scribner FCs. Moreover, the improvement of the upper bound value when increasing the precision is not always worth it. For instance, the value obtained at a relative precision of $10^{-3}$ for H-12 FC (0.117912014415) is very close to the one with precision $10^{-8}$ (0.117909545051), i.e., there is a gap lower than $2 \times 10^{-5}$. The relative precision handles the order of magnitude of the objective function value, and the default 0.001 value is enough in most cases to get accurate solutions, since the optimal SSE may be small when there is not too much noise in the measured data.

## 5. Conclusion and Future directions

In this paper, the ibexopt optimizer was presented to solve optimization problems to find the parameters of fuel cell models. It has good conver-
gence properties and performed the best in the investigated FCs compared to common metaheuristics used in similar works. Since ibexopt is a global optimization solver, the computed solutions are guaranteed using an enclosing interval. The convergence analysis shows the algorithm's good behavior, but for too high accuracies, the algorithm may take a lot of time to finish. Even in this case, if a time limit is fixed, the objective function is approximated through an interval. Moreover, the mechanism used to find feasible solutions allows decreasing the upper bound very quickly to come close to the best solution. Another limitation of ibexopt is the intrinsic nature of the optimization problems it can solve. Indeed, the complete mathematical definition of the objective function has to be defined in the optimization model, in contrast to black-box optimization algorithms that can use a simulation code to evaluate the objective function.

Four FCs were used to demonstrate the performance of the algorithm. The I-V and P-V curves show the good properties of the computed solutions when simulating fuel cell behavior. Even for the last one, based on a recent industrial platform, the algorithm provided an efficient result over the 3 scenarios that were tested.

When estimating the parameters' values, the optimal solution may be on the bounds of their initial domains. This specific case may call out the engineer about the validity of the modeling or the too noisy data. The noise in experimental data may also lead to a poor approximation of the parameters, and removing obvious outliers is a basic and efficient solution to limit the impact on the SSE minimization. Otherwise, the Mean Absolute Error (MAE) should be preferred since it is known to be more robust in this case. It is not detailed in this paper, but the MAE objective function was also tested and it gives very good results. It shares similar behavior compared to SSE, while modeling the MAE in the same manner as SSE, i.e. using intermediary variables for the absolute errors and constraints to define their values.

Ibexopt is a generic solver capable of solving a wide variety of nonlinearly constrained problems. Parameter estimation problems formulated as least squares always have the same structure, namely, a sum of squared differences between data and a model's results. Then, further work can investigate the integration of such specificities in some of the mechanisms of ibexopt. For instance, the contractors that are used do not exploit the fact that the objective function is a sum of positive values, like in the Gauss-Newton strategy. Additionally, ibexopt spends a lot of time exploring small boxes because of a
too-wide approximation of the objective function, as illustrated by the slow increase of the lower bound during the solving. Investigating several types of approximation may lead to an early rejection of most of these small boxes, thus reducing the size of the generated search tree. Other problems can be considered with a high degree of accuracy like the parameter estimation of induction machines, transformers, and the upgraded photovoltaic solar cells.

## Acknowledgments

Dr. Ragab A. El-Sehiemy thanks École Centrale de Nantes for supporting this work.

## Appendix A. Nomenclature

<table><tr><td>λ</td><td>Parameter relating to the membrane's effective water content</td><td>-</td></tr><tr><td>ρ<sub>m</sub></td><td>Membrane specific resistivity</td><td>Ω cm</td></tr><tr><td>ξ<sub>1</sub></td><td>PEMFC 1st semi-empirical coefficient</td><td>-</td></tr><tr><td>ξ<sub>2</sub></td><td>PEMFC 2nd semi-empirical coefficient</td><td>-</td></tr><tr><td>ξ<sub>3</sub></td><td>PEMFC 3rd semi-empirical coefficient</td><td>-</td></tr><tr><td>ξ<sub>4</sub></td><td>PEMFC 4th semi-empirical coefficient</td><td>-</td></tr><tr><td>A</td><td>Membrane area</td><td>cm<sup>2</sup></td></tr><tr><td>b</td><td>Concentration coefficient</td><td>V</td></tr><tr><td>CO<sub>2</sub><sup>*</sup></td><td>concentration of O2</td><td>mol cm<sup>-3</sup></td></tr><tr><td>E<sub>Nst</sub></td><td>Thermodynamic potential of the PEMFC</td><td>V</td></tr><tr><td>I</td><td>PEMFC output current</td><td>A</td></tr><tr><td>I<sub>max</sub></td><td>Maximal current</td><td>A</td></tr><tr><td>J<sub>max</sub></td><td>Maximal current density</td><td>A cm<sup>-2</sup></td></tr><tr><td>l</td><td>Membrane thickness</td><td>cm</td></tr></table>
<table><tr><td>n<sub>cells</sub></td><td>Number of cells in the PEMFC</td><td>-</td></tr><tr><td>P<sub>a</sub></td><td>Inlet pressure at anode</td><td>atm</td></tr><tr><td>P<sub>c</sub></td><td>Inlet pressure at cathode</td><td>atm</td></tr><tr><td>P<sub>H2O</sub></td><td>Saturation pressure of water vapour</td><td>atm</td></tr><tr><td>P<sup>*</sup><sub>H2</sub></td><td>Partial pressure of inlet H<sub>2</sub></td><td>atm</td></tr><tr><td>P<sup>*</sup><sub>O2</sub></td><td>Partial pressure of inlet O<sub>2</sub></td><td>atm</td></tr><tr><td>R<sub>c</sub></td><td>Electronic resistance</td><td>Ω cm<sup>-2</sup></td></tr><tr><td>R<sub>m</sub></td><td>Proton membrane resistance</td><td>Ω cm<sup>-2</sup></td></tr><tr><td>RH<sub>a</sub></td><td>Relative humidity at anode</td><td>%</td></tr><tr><td>RH<sub>c</sub></td><td>Relative humidity at cathode</td><td>%</td></tr><tr><td>T</td><td>PEMFC cell temperature</td><td>K</td></tr><tr><td>V<sub>a</sub></td><td>Activation loss voltage</td><td>V</td></tr><tr><td>V<sub>conc</sub></td><td>Voltage concentration loss</td><td>V</td></tr><tr><td>V<sub>fc</sub></td><td>Cell output voltage</td><td>V</td></tr><tr><td>V<sub>ohmic</sub></td><td>Ohmic loss voltage</td><td>V</td></tr><tr><td>V<sup>electronic</sup><sub>ohmic</sub></td><td>Electronic loss voltage</td><td>V</td></tr><tr><td>V<sup>proton</sup><sub>ohmic</sub></td><td>Proton loss voltage</td><td>V</td></tr><tr><td>V<sub>stack</sub></td><td>PEMFC output voltage</td><td>V</td></tr></table>

## Appendix B. Supplementary material

The source code for these applications is available on GitHub:

https://github.com/raphaelchenouard/optimal_fuel_cell_parameter_estimation.
# Appendix C. Minibex modeling for H-12 FC

## Constants

N=20; // Number of experimental data points

T=29+273.15; // operating cell temperature in K

N_s = 13; // Number of cells in serie

P_H_2_O = 10^(2.95e-2 * T - 9.18e-5 * T^2 + 1.44e-7 * T^3 - 2.18);

P_H_2 = 0.5;

P_O_2 = 1;

C_O_2 = (P_O_2/5.08e6)*exp(498.15/T);

E_Nernst = 1.229-0.85e-3*(T-298.15)+4.3085e-5*T*(ln(P_H_2)+0.5*ln(P_O_2));

Vm_stack[N] = (9.58;9.42;9.25;9.20;9.09;8.95;8.85;8.74;8.65;8.45;8.41;8.2;8.12;8.11;8.05;7.99;7.95;7.94;7.9;7.57);

Im_stack[N] = (0.104;0.2;.309;0.403;0.51;0.614;0.703;0.806;0.908;1.076;1.127;1.288;1.39;1.45;1.578;1.707;1.815;1.9;2.06;2.44);

A = 8.1; // membrane area in cm²

J_max = 0.860; // in A/cm²

l = 0.0025; // thickness of membrane in cm

## Variables

ksi_1 in [-1.1997, -0.8532];

ksi_2 in [1, 5];

ksi_3 in [3.6, 9.8];

ksi_4 in [-26, -9.54];

R_C in [.1, .8]; // connections resistance in mohm

lambda in [13.23];

beta in [0.0136, 0.5];

r[N];

function V_act(I, ksi_1, ksi_2, ksi_3, ksi_4)
    return -(ksi_1 + ksi_2*1e-3*T + ksi_3*1e-5*T*ln(C_O_2) + ksi_4*1e-5*T*ln(I))
end

function V_ohmic(I, R_C, lambda)
return I*(R_C*1e-3 + (1*(181.6*(1+0.03*I/A+0.062*(T/303))^2*
(I/A)^2.5)) / ((lambda-0.634-3*I/A)*exp(4.18*(T-303)/T)))
/A)

end

function V_con(I, beta)

return -beta*ln(1-(I/A)/J_max)

end

function V_fc(I, R_C, ksi_1, ksi_2, ksi_3, ksi_4, lambda, beta)

return N_s * (E_Nernst -
V_act(I, ksi_1, ksi_2, ksi_3, ksi_4) -
V_ohmic(I, R_C, lambda) - V_con(I, beta))

end

Minimize sum(i=0:N-1,(r[i])^2)

Constraints

for i=0:N-1;

r[i] = Vm_stack[i] -
V FC (Im_stack[i], R_C, ksi_1, ksi_2, ksi_3, ksi_4, lambda, beta);

end;

End

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