## 1. Introduction

The shortage of primary energy and environmental problems have led to increased development of renewable energy in all countries of the world. However, renewable energy has the characteristics of discontinuity, instability, and uncontrollability. Large-scale integration of renewable energy into power grids will bring serious impact on the safe and stable operation of power grids, resulting in a large number of abandoned light and wind [

1]. Large-scale energy storage technology is one of the effective methods to solve this problem [

2,

3,

4]. Among them, the liquid flow battery has attracted wide attention in the home and abroad because of its independent capacity, flexible location, safety, and reliability. In view of the problems of ion cross-contamination and high cost of ion exchange membrane in traditional dual-flow batteries, Professor Pletcher of Cape Town University had proposed single-flow lead–acid batteries [

5,

6,

7,

8] in 2004. Due to the obvious advantages of single-flow batteries over dual-flow batteries, different series of single-flow batteries have been developed at home and abroad, such as zinc–nickel single-flow batteries [

9], lead dioxide/copper single-flow batteries [

10], and quinone/cadmium [

11] single-flow batteries. Among them, zinc–nickel single-flow batteries have attracted wide attention due to their long life, high energy efficiency, safety, and environmental protection [

9]. In recent years, the research and development of zinc–nickel single-flow batteries have been mainly based on experiments, including the selection and testing of key materials [

12,

13,

14], electrolyte composition addition [

15,

16,

17,

18], and flow structure design [

19,

20,

21,

22] to improve the performance of zinc–nickel single-flow batteries and promote large-scale zinc–nickel single-flow battery systems (ZNBs) to form an energy storage system for engineering applications [

23].

Establishing a general electrical model that can accurately reflect the external characteristics of the stack is the premise of predicting and analyzing the parameters of ZNBs energy storage system and optimizing its operation, and then building an efficient battery stack management system. At present, there are few studies on the electrical model construction of zinc–nickel single-flow battery stacks, and the development of more complete vanadium redox flow batteries can be referred to. Barote et al. [

24,

25] and Chahwan et al. [

26] proposed the basic equivalent circuit model of the vanadium redox flow battery. The model used a controlled current source and a fixed resistance to represent parasitic loss, reaction resistance, and electrode capacitance, and a voltage source to represent stack voltage. However, their models do not take into account the dynamic characteristics of batteries and lack of experimental verification. Recently, Ankur et al. [

27] aimed to make vanadium redox flow batteries further oriented to renewable energy sources, and built an equivalent circuit model of vanadium redox flow batteries considering electrolyte flow rate, pump loss, and self-discharge. Accurate estimation of battery stack terminal voltage and dynamic SOC was achieved, and the optimal range of variable electrolyte flow under dynamic SOC was investigated, which provided support for the design of flow controller. On the basis of the above, reference [

28] further estimated the parameters of the internal electrical components of the equivalent circuit of the vanadium redox flow battery under different electrolyte flow rates, charge–discharge current densities, and charge states, and coupled the obtained parameters with the simulation model. The comparison with the experimental results showed that the accuracy of the model has been significantly improved. For the zinc–nickel single-flow battery stack studied in this paper, Yao Shou-guang et al. [

29,

30], based on the working principle of zinc–nickel single-flow batteries, built the PNGV (the Partnership for a New Generation Vehicles) equivalent circuit model, and further obtained the PNGV model parameters by parameter identification based on the experimental data of the pulse discharge of the battery at 100 A. Then, the high-order polynomial and exponential function fitting method was used to obtain the analytical formula of each model parameter. Xiao M. et al. [

31] proposed an improved Thevenin equivalent circuit model of the zinc–nickel single-flow battery, based on the principle of parameter identification and the least-squares curve-fitting method to obtain the parameters of the improved model, and then the discrete mathematical model of each parameter in the improved model was obtained by discretization. However, the above equivalent circuit model established for the zinc–nickel single-flow battery does not consider the effects of self-discharge, electrolyte flow, and pump loss.

Based on the preliminary work, a general electrical model considering the factor of flow rate, self-discharge, and pump loss which can accurately reflect the external characteristics of the stack is proposed in the paper. In addition to this, another significant contribution of this paper is to use flow factor multiplied by the theoretical minimum flow and genetic algorithm to determine an optimal flow rate for minimum loss in the ZNBs system, considering both the internal power loss and pump power loss. Such a comprehensive modeling of zinc–nickel single-flow batteries has not been reported in the literature available at home and abroad. The general electrical model is simulated in MATLAB/Simulink and is verified by a zinc–nickel single-flow battery stack composed of 23 single batteries in parallel. The simulation model can support the design of efficient battery management systems for large-scale ZNBs energy storage system.

## 2. Equivalent Circuit Model

The positive electrode of the zinc–nickel single-flow battery adopts a nickel oxide electrode used in a secondary battery; the negative electrode is an inert metal current collector (nickel-plated steel strip), and 10 mol/L KOH + 5 g/L LiOH + 0.5 mol/L ZnO solution is used as the base electrolyte. The positive electrode reaction is completed in the porous nickel positive electrode, and the negative electrode reaction is a surface deposition/dissolution reaction.

Figure 1 is a schematic diagram of the basic structure of a zinc–nickel single-flow battery stack (300 Ah), which comprises 23 parallel cells, and the electrolyte is driven by a pump to flow through the stack from the bottom during the charge and discharge cycle.

Figure 2 is a schematic structural view of a partially parallel single cell, and d

_{1} is an interval between the positive and negative electrodes. The specific structural parameters of the model are shown in

Table 1.

The active substance in the nickel oxide electrode undergoes a chemical reaction during charge and discharge. The charge–discharge reaction process is as shown in Equation (1). The zinc negative electrode is accompanied by deposition and dissolution during charge and discharge. The charge–discharge reaction process is as shown in Equation (2). The total reaction in the zinc–nickel single-flow battery is shown in Equation (3).

Taking the above-mentioned zinc–nickel single-flow battery stack (300 Ah) as the research object, the equivalent circuit model considering the flow rate, pump power loss, and self-discharge is built. The final general electrical model of the zinc–nickel single-flow battery stack is shown in

Figure 3. The following

Section 2.1,

Section 2.2,

Section 2.3,

Section 2.4 and

Section 2.5 elaborate on each module of the general electrical simulation model of the zinc–nickel single-flow battery.

#### 2.1. Internal Loss

Experimental tests show that the system efficiency of the zinc–nickel single-flow battery stack (300 Ah) is about 69% when the charge–discharge current is 100 A, and the remaining 31% is internal loss. The actual power inside the stack can be calculated by Equation (4). The internal loss of the stack can be divided into ohmic loss and polarization loss. The effect on the stack can be reflected in the equivalent circuit model as ohmic loss resistance (

${\mathrm{R}}_{\mathrm{resistive}}$) and polarization loss resistance (

${\mathrm{R}}_{\mathrm{reaction}}$), which can be calculated by Equation (4) [

32].

In Equation (4),

${\mathrm{P}}_{\mathrm{rate}}$ is rated power and

${\mathsf{\eta}}_{\mathrm{system}}$ is system efficiency. In Equation (5),

$\mathrm{K}$ is power loss coefficient,

${\mathrm{I}}_{\mathrm{max}}$ is the maximum charge–discharge current of the battery stack, and R is the internal loss resistance (ohmic loss resistance or polarization loss resistance). Equivalent circuit model parameters are calculated under very bad conditions [

32], that is, when the charge–discharge current is the maximum current and SOC is 0.2. This paper is based on the function expression of the ohmic loss resistance (

${\mathrm{R}}_{\mathrm{resistive}}$) and the polarization loss resistance (

${\mathrm{R}}_{\mathrm{reaction}}$) of the zinc–nickel single-flow battery stack (300 Ah) proposed in reference [

29]. When the SOC is 0.2, the values of

${\mathrm{R}}_{\mathrm{resistive}}$ and

${\mathrm{R}}_{\mathrm{reaction}}$ are respectively 0.623 mΩ and 0.2504 mΩ, and then the ohmic loss coefficient (

${\mathrm{K}}_{\mathrm{resistive}}$) and polarization loss coefficient (

${\mathrm{K}}_{\mathrm{reaction}}$) are calculated by Equation (5) to be 10.8% and 4.35%, respectively, and the parasitic loss is about 15.85% of the total loss.

#### 2.2. Pump Loss Model

The pump loss model of the zinc–nickel single-flow battery is shown in

Figure 4. The pump loss is characterized by fixed loss (

${\mathrm{R}}_{\mathrm{fix}}$) and pump current loss (

${\mathrm{I}}_{\mathrm{pump}}$). Fixed loss resistance (

${\mathrm{R}}_{\mathrm{fix}}$) is calculated by Equation (6), in which

${\mathrm{U}}_{\mathrm{min}}$ is the minimum voltage of the stack and

${\mathrm{P}}_{\mathrm{fix}}$ is the fixed loss power, which is experimentally measured to account for about 2% of

${\mathrm{P}}_{\mathrm{stack}}$.

The function relationship between pump loss current (

${\mathrm{I}}_{\mathrm{pump}}$) and pump power (

${\mathrm{P}}_{\mathrm{mech}}$) in the electrical model is shown in Equation (7). The pump loss coefficient (

$\mathrm{M}$) is related to pump loss power. Definition of M see Equation (8).

The mechanical loss (

${\mathrm{P}}_{\mathrm{mech}\_\mathrm{loss}}$) includes two parts: the mechanical loss (

${\mathrm{P}}_{\mathrm{pipe}\_\mathrm{loss}}$) caused by the electrolyte flowing through the pipeline connecting the stack and the external storage tank, and the mechanical loss (

${\mathrm{P}}_{\mathrm{stack}\_\mathrm{loss}}$) caused by the electrolyte flowing through the stack. The total loss (

${\mathrm{P}}_{\mathrm{mech}\_\mathrm{loss}}$) is shown in Equation (9).

When the electrolyte of the zinc–nickel single-flow battery flows through pipes, valves, and liquid storage tanks, it will cause a certain pressure drop, which is collectively called pipeline pressure drop. The pressure drop equation of the pipeline can be obtained by the Bernoulli equation, which is related to electrolyte flow rate, loss along the pipeline, local loss, and height difference between inlet and outlet of the pipeline. Pipeline pressure drop and mechanical loss can be expressed as Equations (10) and (11). The pressure drop of the tube outside the stack is estimated to be about 65.5 kPa.

The pressure drop in the stack is determined by the flow rate of the electrolyte and the resistance of the electrolyte, so the expressions of pressure drop and mechanical loss in the stack are as follows:

In Equation (12),

$\tilde{\mathrm{R}}$ is the hydraulic resistance of the stack, and its value can be seen in the previous research work of our group [

33]. The formula for calculating

${\mathrm{P}}_{\mathrm{stack}}$ is shown in Equation (13).Considering the pump efficiency, the total mechanical loss of the battery system can be defined as Equation (14).

#### 2.3. Self-Discharge Loss

The self-discharge of the zinc–nickel single-flow battery is mainly caused by the negative reaction of the negative electrode, which forms a microprimary battery on the surface of the negative electrode, which has a significant influence on the attenuation of the battery capacity. In this paper, the self-discharge effect is equivalent to the loss resistance (

${\mathrm{R}}_{\mathrm{self}}$) in the equivalent circuit model. The calculation formula is shown in Equation (15), where

${\mathrm{P}}_{\mathrm{self}}$ is the power loss caused by self-discharge, and its expression is given by Equation (16). For the self-discharge power loss coefficient (

$\mathrm{f}$), the calculation formula is shown in Equation (17), where

${\mathrm{U}}_{1}$ and

${\mathrm{U}}_{2}$ are the changes of battery voltage with time in the charge–discharge process without considering self-discharge effect and considering self-discharge effect, respectively.

#### 2.4. Voltage Estimation Model

The voltage estimation module of the zinc–nickel single-flow battery stack is shown in

Figure 5. The ion activity should be used when calculating the battery electromotive force using the Nernst equation. When the ionic strength is not large, and the valence state of the oxides and the reductants is not high, the battery electromotive force can be directly calculated by using the ion concentration. In the zinc–nickel single-liquid battery, the valence states of the hydroxide ion and zincate ion are −1 and −2, respectively. The active material nickel oxide of the positive electrode is not present in the battery in the form of ions, and its ion activity cannot be further measured. Only the proton concentration of hydrogen can be used to indicate the content of nickel hydroxide. Whether it is theoretical analysis or comparison with experimental results, it is shown that the error caused by the calculation of the voltage of the stack using the ion concentration is small and within an acceptable range. The potentials of the positive and negative electrodes are as follows:

${\mathrm{E}}^{+}$ is the positive equilibrium potential,

${\mathrm{E}}^{-}$ is the negative equilibrium potential, T is the ambient temperature, and

$\mathrm{n}$ is the electron transfer number in the electrode reaction. The concentration of positive active substance can be replaced by H proton concentration. Equation (18) can be rewritten as follows:

The battery stack potential is as follows:

Based on the above-mentioned calculations in Equations (18)–(21) for the potential of the zinc–nickel single-flow battery stack, combined with the range of concentration of each substance in

Table 2, the battery potential can be further expressed by SOC as Equation (22), where

${\mathrm{E}}^{0}$ is 1.705 V. Under different operating conditions, the terminal voltage is affected by internal loss and self-discharge. The terminal voltage is estimated by Equation (23), where “±” indicates the charging process and the discharging process.

${\mathrm{E}}_{\mathrm{self}-\mathrm{discharge}}$ is the average voltage drop caused by the self-discharge during charge and discharge, which is 3.65 mV and 6.9 mV, respectively [

33].

#### 2.5. SOC Estimation Model

SOC is used to characterize the state of charge of batteries. Its estimation module is shown in

Figure 6. Based on the change of concentration of

$\mathrm{Zn}{\left(\mathrm{OH}\right)}_{4}^{2-}$, the dynamic SOC value of the zinc–nickel single-flow battery is reflected in Equation (24). “±” indicates the charging and discharging process. The value of

${\mathrm{C}}_{\mathrm{max}}^{\mathrm{Zn}{\left(\mathrm{OH}\right)}_{4}^{2-}}$ can be obtained as 1 mol/L from

Table 2.

The SOC of the zinc–nickel single-flow battery stack storage system is divided into

${\mathrm{SOC}}_{\mathrm{tank}}$ in the tank and

${\mathrm{SOC}}_{\mathrm{stack}}$ in the stack. The SOC in the stack is given by Equation (25). To simplify the estimation of the SOC, the formula for calculating the dynamic SOC of the stack is shown in Equation (26). Equation (27) is a formula for calculating the

${\mathrm{SOC}}_{\mathrm{stack}}$. When charging, b takes a value of 1, and when discharged, it is −1. The simulation parameters involved in the model are shown in

Table 3.

## 4. Conclusions

In this paper, the zinc–nickel single-flow battery stack is taken as the research object, and a general electrical model considering self-discharge, pump loss, and flow is built by using MATLAB/Simulink software. The self-discharge module, pump loss module, SOC, and voltage estimation module in this model are described in detail in

Section 2. In order to evaluate the accuracy of the electrical model, the charging and discharging experiments of the zinc–nickel single-flow battery stack (300 Ah) were carried out under different charging and discharging currents (50 A, 100 A, 150 A). The results are compared with the simulation values (considering self-discharging and without considering self-discharging). The results show that the simulation values obtained by the simulation model considering self-discharging are closer to the experimental results. The minimum error of voltage in charging is 0–0.02%, the maximum error is 1.1–2.61%, the minimum error of voltage in discharging is 0.002–0.02%, and the maximum error is 1.8–3.85%. In addition, the Coulombic efficiency of the complete charge and discharge cycle of the simulation model is estimated. Under the operating conditions of rated electrolyte flow rate (0.09 L/S), charging current 100 A, and discharge current 50 A, 100 A, and 150 A, the comparison with experimental data shows that the simulation model has high accuracy in estimating Coulomb efficiency. The flow rate of electrolytes is one of the most influential parameters in the operation of battery stacks. Excessive flow rate of electrolytes will cause high pump loss, and too low a flow rate of electrolyte will increase the internal loss of the battery stack. Therefore, there exists a time-varying optimal electrolyte flow rate to maximize the system efficiency of the zinc–nickel single-flow battery stack corresponding to the dynamic SOC. In this paper, the overall power loss (pump loss, internal loss) of the system is taken as the objective function, and two methods, genetic algorithm and theoretical minimum flow multiplied by different flow factors, are used to optimize the flow rate. The results show that, compared with the rated flow rate (0.09 L/s), the optimized flow rate of electrolytes improves the system efficiency significantly. The results show that under the constant charge and discharge power, the above two optimization methods have significantly improved the system performance, and the flow factor optimization method is more convenient. However, in the face of the “peak-valley” phenomenon of charge and discharge power in actual engineering, the optimization method of fixed flow factor may not achieve the expected effect, and the genetic algorithm can optimize the electrolyte flow in real time to provide better flow control strategy.