Introduction

Over the last decades, renewable energy sources (RES) have been actively developing worldwide aimed at reducing the carbon footprint. Due to the variable nature of generation from these sources, the electric grid stability becomes an acute problem. Large-scale energy storage systems (ESS) are the most recommended solution to address this issue, allowing for the minimization of frequency and power fluctuations from RES, ensuring reliable electricity production and consumption.

There are various types of ESS [1], including electrochemical systems, pumped hydro storage plants, storage systems based on flywheels and compressed air. Among them, electrochemical storage systems are the most common and rapidly growing. Specifically, systems which utilizes external storage for reactants, are among the most suitable for use in large grids due to their long lifespan and highly flexible modularity. One of the most promising types of such systems are redox flow batteries (RFBs), which offer significant advantages in large-scale, long-duration energy storage and grid applications [2]. There are several types of RFB, however, vanadium redox flow batteries (VRFB), which utilize vanadium ions in different oxidation states, are the most developed and widely adopted technology. These features allow VRFB to be used in grids with high power and capacity requirements.

Large-scale systems typically occupy significant space and are often installed outdoors, exposing them to seasonal climate variations. These conditions result in rapid temperature fluctuations of the main components of VRFB, which results in changes in the kinetics and thermodynamics of storage systems. The kinetics of temperature-dependent reactions in VRFB are thoroughly examined in Ref. [3]. These studies indicate that peak oxidation and reduction currents increase with temperature, indicating an overall enhancement in reaction kinetics. However, the kinetics of side reactions, such as hydrogen evolution, also improve, leading to a reduction in coulombic efficiency. Additionally, critical temperatures significantly impact the electrolyte, enhancing the stability of V2+, V3+, and V4+ ions while decreasing the stability of the V5+ ions, which precipitates when the temperature exceeds 40 °C or falls below −5 °C [[4], [5], [6]]. Furthermore, experimental evidence from Refs. [[7], [8], [9]] demonstrates that lower temperatures increase the viscosity of the electrolyte, substantially affecting pressure loss in the hydraulic system and reducing battery efficiency. Therefore, proper temperature control and monitoring of the main components of VRFB is necessary to maintain their high efficiency.

Mathematical modeling serves as a valuable tool for the prediction of temperature changes under various operational conditions. Using these models, advanced control strategies can be developed and optimal operating conditions can be selected, in order to extend the lifetime of the battery. Most non-isothermal VRFB models are based on the conservation of energy and mass describing the dynamics of heat processes occurring inside the battery. The main sources of generated heat include electrochemical reactions, electrical current (Joule heating), overpotentials, hydraulic friction, crossover reactions, and shunt currents [10]. For large-scale systems (with a power of more than 5 kW), the dominant sources are electrochemical heating and Joule heating.

There are two main groups of non-isothermal models for VRFB: models with distributed parameters [[11], [12], [13], [14]] and models with lumped parameters [[15], [16], [17], [18], [19], [20], [21], [22], [23], [24]]. The first group considers the spatial distribution of temperature throughout the volume of the main system’s components. Such models can identify points of maximum heating, which can be useful in the designing stage. In Ref. [13], a two-dimensional model is presented, based on the principles of mass, energy and charge conservation, which was on the first reported non-isothermal models for VRFB. This model considers solely for the heating of the VRFB due to chemical reactions and the flow of electric current through the stack (Joule heating). Despite its simplicity, this model served as the foundation for more advanced non-isothermal models. Other studies [11,12] introduce three-dimensional models that consider the temperature distribution throughout the entire VRFB volume. The authors observed similar results, indicating a uniform temperature distribution across the electrodes at a certain state of charge (SOC). In the study [14], the authors used a similar three-dimensional non-isothermal model to analyze time constants of heat and mass transfer processes, and find the temperature stabilization time across the system. Thus, multi-dimensional models offer detailed insights into the spatial temperature distribution but demand substantial computational resources, limiting their practicality for real-time monitoring and control applications. The second group only considers the change in time of the temperature averaged over the volume. Such models do not require significant computational resources and are more suitable for fast temperature prediction in control and monitoring tasks. In particular, the zero-dimensional models were used in various real-time control problems. For instance, the study in Ref. [15] introduces a basic zero-dimensional model with three state variables, considering only heating due to electric current passage. Despite not considering all types of heat loss, this model successfully predicted battery temperature behavior under various ambient temperature profiles. The authors later expanded this model to include heating from crossover and shunt currents [18]. Another research group [19,22,25,26] employed a similar model to analyze the performance of a large VRFB system (9 kW/24 kWh). Given the specific setup, they examined heat transfer between cells in a stack, enabling them to obtain temperature distribution results across cells under different operating conditions. Additionally, the model in Ref. [16] includes both thermal effects and hydraulic processes within the system. The authors found significant temperature gradients at low electrolyte flow rates, while high flow rates resulted in nearly uniform temperature distribution throughout the system. They also suggested using a serpentine-parallel channel structure in the stack to achieve a more uniform distribution of electrolyte flow and system temperature. It is worth noting that the mentioned models cannot be used to analyze the VRFB at low temperatures, since they do not take into account the effect of increasing the viscosity of the electrolyte. However, they made it possible to accurately predict the behavior of the battery at room temperatures.

To achieve more efficient and long-scale battery operation, it is necessary to control the temperature of the VRFB, keeping it within specified limits. Temperature control approaches are divided into passive and active strategies. In passive strategies [[21], [22], [23],27] temperature is controlled without additional equipment. Such approaches include, for example, the selection of the optimal electrolyte flow rate, the choice of reservoir geometry and the hydraulic system. Passive strategies are simple for implementation, but they may be ineffective under critical operating conditions. Active approaches [18,24,28] use additional equipment for thermal management that can speed up heat transfer in VRFB and improve temperature control. These approaches include, for example, installing a liquid heat exchanger in the hydraulic system and cooling through air conditioning. With active strategies, VRFB temperature can be maintained within the specified limits under almost any operating conditions, but requires additional costs for the installation of cooling systems.

Analysis and control of VRFB operation at low temperatures (about 0 °C) is another important issue. Under such conditions, the electrolyte becomes more viscous, which leads to a pressure loss in the hydraulic system resulting in a decrease in battery capacity. One possible way to increase capacity is increasing pump power, but this can lead to an overall loss in efficiency. At the moment, only a few works [[29], [30], [31]] have been devoted to the analysis of the operation of the VRFB at low temperatures. In Refs. [29,30], the properties of the electrolyte and electrode were experimentally measured at a temperature of −10 °C, and a significant drop in efficiency was shown when under these conditions. Further, these results were used to develop a stationary model, which predicts the output voltage depending on temperature [31]. However, in the above-mentioned works, no approach was proposed to describe the dynamics of temperatures and concentrations under low temperature conditions. Also, no approach has yet been proposed for increasing the VRFB temperature. Another comprehensive experimental results combined with 3D modeling of VRFB operating at different temperatures was presented in the study [9]. Authors measured viscosity of the electrolyte under different temperatures and SOC, fitted this data and used in for multi-filed modeling of the VRFB performance. The results include several important investigations, including the fact, that the Coulombic efficiency increase and overpotentials decrease at high temperatures. However, the model used in this study was validated only at lab-scale experimental setup and computation resources required for the simulation was high enough due to the using of multi-dimensional approach. To the best authors knowledge, no approach has yet been proposed to analyze the thermal behavior of a battery at low temperatures, which would take into account the change in electrolyte viscosity with temperature and could be used to predict the behavior of a large-scale battery in real time.

In this paper, we present a new zero-dimensional non-isothermal model based on energy and mass balance laws. The model takes into account the temperature-dependent viscosity of the electrolyte and allows for simulation of various hydraulic properties of a VRFB under different operating temperatures. Additionally, it predicts changes in a wide range of VRFB parameters during cycling, including voltage, vanadium ion concentrations, stack and tank temperatures, pressure drop, electrolyte flow rate, capacity, and power. The model has been validated using experimental data from two large-scale installations over an extended period. A detailed analysis of VRFB performance at low ambient temperatures is conducted using the developed model. A comparison with previously reported models is provided in Table 1, highlighting the advantages of the proposed approach. A preliminary version of this non-isothermal VRFB model, which considers low-temperature conditions, was previously presented by the authors at the IEEE 7th International Electrical and Energy Conference (CIEEC) [32]. In this paper, we significantly extend that work by expanding the modeling parameter range and enhancing the modeling methodology. Specifically, the impact of heat sources on overall VRFB dynamics is analyzed across a wider time and temperature range. The behavior of battery power and capacity is analyzed for different temperatures and load currents. Two modes of battery operation are considered: constant flow rate and constant pump power. The first mode is commonly used in theoretical modeling and advantageous for battery state control. However, in real systems, the variable viscosity of the electrolyte requires continuous adjustment of the pump power to maintain this mode. The second mode is simpler to implement in practice, as most commercial pumps operate at constant power. However, this mode alters the electrolyte flow through the system, which necessitates monitoring the battery’s condition. Finally, a new capacity restoration approach is proposed, which involves controlling the power of the pumps. The next section provides the equations of the non-isothermal VRFB model. The results of model validation and simulations of the temperature and capacity of the VRFB under various operating conditions are presented in Section 3. Finally, Section 4 presents the conclusions.

Section snippets

General laws

The energy conservation law is used to describe thermal processes occurring in the VRFB. In this work, the following heating sources are taken into account: conductivity, convection, electrochemical heating and Joule heating:∂∂t(ρC‾pT)+∇(v→ρCp‾T)−λ‾∇2T=qch+qel,where ρ is the volume-averaged electrolyte density, Cp is the volume-averaged thermal capacity, v is the electrolyte velocity, λ is the volume-averaged thermal conductivity, qel is the source of electrical heat losses and qch is the

Model parameters

The simulations were carried out for a 5 kW VRFB with tanks filled with 45 L of electrolyte each, having a total vanadium concentration of 1.45 mol/L, which was analyzed in the previous study of authors [40]. The stack consisted of 38 cells with porous electrodes, with a surface area of 848 cm2. The thermal parameters of the battery were assumed to be similar to those presented in the previous paper [33]. The values of all VRFB parameters are listed in Table 2.

All simulation tests were carried

Conclusions

This paper presents a new non-isothermal model of a vanadium redox flow battery (VRFB) based on the evolution of ion concentrations and temperature inside the battery resulting from the physical processes occurring within it. This model accurately predicts the behavior of the battery over a wide range of temperatures (from 5 °C to 40 °C) and operating parameters (current ranging from 24 mA/cm2 to 142 mA/cm2, electrolyte flow rate from 4 to 16 L/min). Using this model, a detailed analysis of the

CRediT authorship contribution statement

Stanislav Bogdanov: Writing – original draft, Software, Methodology, Formal analysis, Conceptualization. Federico Martin Ibanez: Writing – review & editing, Validation, Supervision. Chuanyu Sun: Writing – review & editing. Sergei Parsegov: Writing – review & editing. Mikhail Pugach: Writing – review & editing, Supervision, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research was supported by RSF (project No. 23-79-01239).

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