# Voltage and Overpotential Prediction of Vanadium Redox Flow Batteries with Artificial Neural Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{+}

_{2}and VO

^{2+}ions, while the negative electrolyte has V

^{2+}and V

^{3+}ions. Both electrolytes are recirculated by pumps into the cell. The electrodes, inside the cell, provide an active area for the electrochemical redox reactions to occur. To prevent cross-contamination and allow for protons to pass and preserve the charge conservation, an ion-selective membrane is added between both electrodes [22].

## 2. Materials and Methods

#### 2.1. Governing Equations

^{+}permeability [37]. The following assumptions were made in the numerical model:

- Stationary conditions;
- Incompressible electrolytes;
- The fluids were assumed to be completely diluted;
- Side reactions were neglected;
- Both electrodes and the membrane were considered isothermal;
- The properties of the electrodes, electrolyte, and membrane were isotropic;
- Changes in the z-direction of the cell were ignored (depth in Figure 1).

_{i}is the concentration of species i, and S

_{i}is the source term of the species (listed in Table 1).

_{i}represents the charge of species i, u

_{i}is the ionic mobility, F is the Faraday constant, and φ

_{l}is the liquid potential. In the third term (convection), $\overrightarrow{\mathrm{u}}$ represents electrolyte velocity.

_{i}was calculated using the Nernst–Einstein equation, as shown in Equation (7), where R is the universal gas constant and T is the temperature.

_{f}is the fibre diameter and k

_{ck}is the Kozeny–Carman constant.

_{4}

^{2−}:

_{R}) directly corresponds to the charges leaving the electrolyte ${\overrightarrow{\mathit{\u2373}}}_{l},$ which in turn equate to the charges entering the electrode ${\overrightarrow{\mathit{\u2373}}}_{\mathrm{s}}.$ Both the liquid and solid current densities are expressed by Equations (12) and (13).

_{s}is the electrode bulk conductivity, listed in Table 3, with other parameters related to the electrodes.

_{R}) was calculated for both electrodes, positive (“+”) and negative (“−”), as shown in Equation (15) and Equation (16), respectively.

_{+}and k

_{−}are the reaction rate constants for the positive and negative side, respectively.

_{+}and E

_{−}were obtained using the Nernst equation, as shown in Equations (21) and (22).

#### 2.2. Boundary Conditions

_{0}, the anode external boundary is set as an electrical ground, that is, the solid potential is equal to zero:

_{cell}is the cell width, and L

_{e}is the electrode thickness. Analogously, the electrodes have a pressure outlet at y = h

_{cell}, and the flux of the species caused by diffusion is neglected.

_{avg}denotes the user-defined current applied to the boundary. The sign of this parameter determines whether the cell is in charge or discharge. This leads to the application of an electrical insulation to the upper and lower boundaries of the membranes and electrodes.

#### 2.3. Numerical Model

^{−6}.

#### 2.4. Neural Network

## 3. Results

#### 3.1. Model Validation

^{2}cell in a static solution and measuring the charge-discharge curves at two different current densities: 40 mA cm

^{−2}and 80 mA cm

^{−2}. Figure 4 illustrates the excellent agreement between the numerical results of the in-house simulation and experimental data from You et al. [28]. The model demonstrated an average relative error of 1.6% when calculating the voltage, which is comparable to the level of agreement between the numerical simulations and experiments described in You et al. [28]. The relative error has been calculated by means of the following expression:

#### 3.2. Artificial Neural Network Validation

^{2}in the discharge mode for an SoC of 50%. Figure 7 shows a section view across the membrane (the xy plane in Figure 1), where the positive and negative electrode voltages across the membrane and neighbouring fluid are shown for the numerical simulations to the left and ANN2 to the centre. Observing the numerical results, one may note a near-discrete step in the electrode voltage over the membrane and a non-uniform distribution, particularly near the lower boundary. ANN2 captured a substantial fraction of the electrode voltage distribution of the section cut, as can be observed by comparing the results with the CFD results. However, there were clear discrepancies near the membrane and the lower boundary. These discrepancies are emphasised by the subtraction of the numerical results and ANN2, as illustrated on the rightmost surface in Figure 7.

#### 3.3. ANN Effiency Benchmark

## 4. Conclusions

^{−2}).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | Specific surface area [m^{2}] |

c | Concentration [mol m^{−3}] |

D | Diffusion coefficient |

d_{f} | Fiber diameter [m] |

E | Equilibrium potential [V] |

F | Faraday constant [C mol^{−1}] |

h | Length [m] |

${h}_{i}$ | Postsynaptical output (ANN) |

i | Current density [mA cm^{−2}] |

i_{0} | Exchange current density |

i_{R} | Electrochemical reaction rate |

K | Permeability [m^{2}] |

K_{ck} | Kozeny-Carman constant |

k | Reaction rate constant [m s^{−1}] |

L | Thickness [m] |

N | Charged species flux [mol m^{−3} s^{−1}] |

p | Pressure [Pa] |

Q | Volumetric flow rate [ml min^{−1}] |

R | Ideal gas constant [J mol^{−1} K^{−1}] |

S | Source term [mol m^{−3} s^{−1}] |

T | Temperature [K] |

t | Time |

u | Mobility [mol s kg^{−1}] |

u | Velocity [m s^{−1}] |

w | Width [m] |

$x$ | Input (ANN) |

$y$ | Output (ANN) |

z | Species charge |

Greek | |

α | Charge transfer coefficient |

ε | Porosity |

η | Overpotential [V] |

σ | Conductivity [S m^{−1}] |

φ | Potential [V] |

µ | Dynamic viscosity [Pa s] |

$\theta $ | Layer bias (ANN) |

$\omega $ | Layer weights (ANN) |

Superscripts and subscripts | |

+ | Positive side or cathode |

− | Negative side or anode |

‘ | Standard |

avg | Average |

e | Electrode |

eff | Effective |

i | Species |

in | Inlet |

l | Liquid |

m | Membrane |

out | Outlet |

s | Solid |

Abbreviations | |

ANN | Artificial Neural Network |

CFD | Computational Fluid Dynamics |

MLP-BP | Multi-Layer Perceptron with Backpropagation |

RFB | Redox Flow Battery |

VRFB | Vanadium Redox Flow Battery |

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**Figure 1.**Schematic of the VRFB cell employed, with the negatively charged fluid symbolised in red and the positively charged fluid denoted in green. Additional details can be found in the text.

**Figure 4.**Comparison of experimentally obtained charge-discharge curves from [42] and simulated curves in a 5 cm

^{2}cell for: (

**a**) a current density of 40 mA cm

^{−2}; (

**b**) a current density of 80 mA cm

^{−2}.

**Figure 5.**Correlation coefficient of the test set ANN for overpotential prediction: (

**a**) ANN1 for voltage prediction; (

**b**) ANN2 for overpotential prediction.

**Figure 6.**ANN1 predictions of voltage (coloured surfaces) and CFD results (black circles): (

**a**) charging; (

**b**) discharging.

**Figure 7.**Comparison between the CFD data and ANN predictions of the overpotential for a current density of 60 mA/m

^{2}in discharge mode for a State of Charge of 50%.

Source Term | Positive Electrode | Negative Electrode |
---|---|---|

S_{II} (V(II) mass conservation equation) | - | $\overrightarrow{\mathit{\u2373}}/F$ |

S_{III} (V(III) mass conservation equation) | - | −$\overrightarrow{\mathit{\u2373}}/F$ |

S_{IV} (V(IV) mass conservation equation) | $\overrightarrow{\mathit{\u2373}}/F$ | - |

S_{V} (V(V) mass conservation equation) | −$\overrightarrow{\mathit{\u2373}}/F$ | - |

S_{H}^{+} (proton concentration equation) | - | −$2\nabla \xb7\overrightarrow{\mathit{\u2373}}/F$ |

Term | Symbol | Value |
---|---|---|

V(II) diffusion coefficient | ${D}_{V2}$ | $2.4\times {10}^{-10}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [40] |

V(III) diffusion coefficient | ${D}_{V3}$ | $2.4\times {10}^{-10}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [40] |

V(IV) diffusion coefficient | ${D}_{V4}$ | $3.9\times {10}^{-10}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [40] |

V(V) diffusion coefficient | ${D}_{V5}$ | $3.9\times {10}^{-10}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [40] |

HSO_{4}^{−} diffusion coefficient | ${D}_{{\mathrm{HSO}}_{4}^{-}}$ | $1.33\times {10}^{-9}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [41] |

SO_{4}^{2−} diffusion coefficient | ${D}_{{\mathrm{SO}}_{4}^{2-}}$ | $1.065\times {10}^{-9}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [41] |

H^{+} diffusion coefficient | ${D}_{{\mathrm{H}}^{+}}$ | $9.312\times {10}^{-9}\text{}{\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ [41] |

Dynamic viscosity | $\mu $ | $4.9238\times {10}^{-3}\text{}\mathrm{Pa}\text{}\mathrm{s}$ [42] |

Term | Symbol | Value |
---|---|---|

Electronic conductivity | ${\sigma}_{s}$ | $1\times {10}^{3}\text{}\mathrm{S}\text{}{\mathrm{m}}^{-1}$ [42] |

Porosity | $\epsilon $ | $0.929$ [43] |

Specific surface area | $a$ | $1.62\times {10}^{4}\text{}{\mathrm{m}}^{2}$ [43] |

Kozeny–Carman constant | ${\text{}k}_{ck}$ | $4.28$ [42] |

Electrode fibre diameter | ${d}_{f}$ | $1.76\times {10}^{-5}\text{}\mathrm{m}$ [43] |

Term | Symbol | Value |
---|---|---|

Cathodic transfer coefficient | ${\alpha}_{+}$ | $0.5$ [42] |

Anodic transfer coefficient | ${\alpha}_{-}$ | $0.5$ [42] |

Standard rate constant for positive reaction | ${k}_{+}$ | $6.8\times {10}^{-7}\text{}\mathrm{m}\text{}{\mathrm{s}}^{-1}$ [40] |

Standard rate constant for negative reaction | ${k}_{-}$ | $1.7\times {10}^{-7}\text{}\mathrm{m}\text{}{\mathrm{s}}^{-1}$ [44] |

Standard equilibrium potential for positive side | ${E}_{+}^{\prime}$ | $1.004\text{}\mathrm{V}$ [38] |

Standard equilibrium potential for negative side | ${E}_{-}^{\prime}$ | $-0.255\text{}\mathrm{V}$ [38] |

Term | Symbol | Value |
---|---|---|

Temperature | T | 298 K |

State of Charge | SOC | 50% |

Volumetric flow rate | Q | $60\text{}\mathrm{mL}\text{}{\mathrm{min}}^{-1}$ |

Outlet pressure | ${p}_{out}$ | $0\text{}\mathrm{Pa}$ |

Electrode thickness | ${L}_{e}$ | $0.003\text{}\mathrm{m}$ [42] |

Electrode width | ${w}_{e}$ | $0.025\text{}\mathrm{m}$ [42] |

Electrode length | ${h}_{e}$ | $0.02\text{}\mathrm{m}$ [42] |

Membrane thickness | ${L}_{m}$ | $125\text{}\mathsf{\mu}\mathrm{m}$ [42] |

Method | Computational Time | Speed Up |
---|---|---|

CFD | 50 s | - |

ANN1 (Voltage) | 0.0032 s | 15,625 |

ANN2 (Overpotential) | 0.1875 s | 266.7 |

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## Share and Cite

**MDPI and ACS Style**

Martínez-López, J.; Portal-Porras, K.; Fernández-Gamiz, U.; Sánchez-Díez, E.; Olarte, J.; Jonsson, I.
Voltage and Overpotential Prediction of Vanadium Redox Flow Batteries with Artificial Neural Networks. *Batteries* **2024**, *10*, 23.
https://doi.org/10.3390/batteries10010023

**AMA Style**

Martínez-López J, Portal-Porras K, Fernández-Gamiz U, Sánchez-Díez E, Olarte J, Jonsson I.
Voltage and Overpotential Prediction of Vanadium Redox Flow Batteries with Artificial Neural Networks. *Batteries*. 2024; 10(1):23.
https://doi.org/10.3390/batteries10010023

**Chicago/Turabian Style**

Martínez-López, Joseba, Koldo Portal-Porras, Unai Fernández-Gamiz, Eduardo Sánchez-Díez, Javier Olarte, and Isak Jonsson.
2024. "Voltage and Overpotential Prediction of Vanadium Redox Flow Batteries with Artificial Neural Networks" *Batteries* 10, no. 1: 23.
https://doi.org/10.3390/batteries10010023