Energy Storage and Management for Electric Vehicles Printed Edition of the Special Issue Published in Energies www.mdpi.com/journal/energies James Marco, Quang Truong Dinh and Stefano Longo Edited by Energy Storage and Management for Electric Vehicles Energy Storage and Management for Electric Vehicles Special Issue Editors James Marco Quang Truong Dinh Stefano Longo MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Quang Truong Dinh University of Warwick UK Special Issue Editors James Marco University of Warwick UK Stefano Longo Cranfield University UK Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Energies (ISSN 1996-1073) in 2019 (available at: https://www.mdpi.com/journal/energies/special issues/ Energy for EV) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. 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Contents About the Special Issue Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface to ”Energy Storage and Management for Electric Vehicles” . . . . . . . . . . . . . . . . ix Shouguang Yao, Xiaofei Sun, Min Xiao, Jie Cheng and Yaju Shen Equivalent Circuit Model Construction and Dynamic Flow Optimization Based on Zinc–Nickel Single-Flow Battery Reprinted from: Energies 2019 , 12 , 582, doi:10.3390/en12040582 . . . . . . . . . . . . . . . . . . . . 1 Chaofeng Pan, Yanyan Liang, Long Chen and Liao Chen Optimal Control for Hybrid Energy Storage Electric Vehicle to Achieve Energy Saving Using Dynamic Programming Approach Reprinted from: Energies 2019 , 12 , 588, doi:10.3390/en12040588 . . . . . . . . . . . . . . . . . . . . 18 Seyed Saeed Madani, Erik Schaltz and Søren Knudsen Kær Simulation of Thermal Behaviour of a Lithium Titanate Oxide Battery Reprinted from: Energies 2019 , 12 , 679, doi:10.3390/en12040679 . . . . . . . . . . . . . . . . . . . . 37 Jos ́ e Luis Sampietro, Vicen ̧ c Puig, and Ramon Costa-Castell ́ o Optimal Sizing of Storage Elements for a Vehicle Based on Fuel Cells, Supercapacitors, and Batteries Reprinted from: Energies 2019 , 12 , 925, doi:10.3390/en12050925 . . . . . . . . . . . . . . . . . . . . 52 Stefan Englberger, Holger Hesse, Daniel Kucevic and Andreas Jossen A Techno-Economic Analysis of Vehicle-to-Building: Battery Degradation and Efficiency Analysis in the Context of Coordinated Electric Vehicle Charging Reprinted from: Energies 2019 , 12 , 955, doi:10.3390/en12050955 . . . . . . . . . . . . . . . . . . . . 79 Henry Miniguano, Andr ́ es Barrado, Cristina Fern ́ andez, Pablo Zumel and Antonio L ́ azaro A General Parameter Identification Procedure Used for the Comparative Study of Supercapacitors Models Reprinted from: Energies 2019 , 12 , 1776, doi:10.3390/en12091776 . . . . . . . . . . . . . . . . . . . 96 Hongjie Liu, Tao Tang, Jidong Lv and Ming Chai A Dual-Objective Substation Energy Consumption Optimization Problem in Subway Systems Reprinted from: Energies 2019 , 12 , 1876, doi:10.3390/en12101876 . . . . . . . . . . . . . . . . . . . 116 Xiangdong Sun, Jingrun Ji, Biying Ren, Chenxue Xie and Dan Yan Adaptive Forgetting Factor Recursive Least Square Algorithm for Online Identification of Equivalent Circuit Model Parameters of a Lithium-Ion Battery Reprinted from: Energies 2019 , 12 , 2242, doi:10.3390/en12122242 . . . . . . . . . . . . . . . . . . . 144 Deidre Wolff, Lluc Canals Casals, Gabriela Benveniste, Cristina Corchero and Llu ́ ıs Trilla The Effects of Lithium Sulfur Battery Ageing on Second-Life Possibilities and Environmental Life Cycle Assessment Studies Reprinted from: Energies 2019 , 12 , 2440, doi:10.3390/en12122440 . . . . . . . . . . . . . . . . . . . 159 Bizhong Xia, Yadi Yang, Jie Zhou, Guanghao Chen, Yifan Liu, Huawen Wang, Mingwang Wang and Yongzhi Lai Using Self Organizing Maps to Achieve Lithium-Ion Battery Cells Multi-Parameter Sorting Based on Principle Components Analysis Reprinted from: Energies 2019 , 12 , 2980, doi:10.3390/en12152980 . . . . . . . . . . . . . . . . . . . 178 v In-Ho Cho, Pyeong-Yeon Lee and Jong-Hoon Kim Analysis of the Effect of the Variable Charging Current Control Method on Cycle Life of Li-ion Batteries Reprinted from: Energies 2019 , 12 , 3023, doi:10.3390/en12153023 . . . . . . . . . . . . . . . . . . . 195 Woo-Yong Kim, Pyeong-Yeon Lee, Jonghoon Kim and Kyung-Soo Kim A Nonlinear-Model-Based Observer for a State-of-Charge Estimation of a Lithium-ion Battery in Electric Vehicles Reprinted from: Energies 2019 , 12 , 3383, doi:10.3390/en12173383 . . . . . . . . . . . . . . . . . . . 206 vi About the Special Issue Editors James Marco is a Chartered Engineer and a Fellow of the Institution of Engineering and Technology (FIET). After graduating with an Engineering Doctorate from Warwick in 2000, James worked for several years within the automotive industry, leading engineering research teams for Ford (North America and Europe), Jaguar Cars, Land Rover, and DaimlerChrysler (Germany). In 2006, James returned to academia, taking a research position at Cranfield University and later joined WMG, University of Warwick, in 2013. Since joining WMG, his research has focused on (1) the design of novel thermal management solutions for energy storage systems, (2) the integration of electric vehicles into a future charging infrastructure—vehicle-to-grid (V2G) operation and optimising fast charging to mitigate battery degradation, (3) the design of new control functions to support second-life energy storage applications, and (4) the design of accelerated energy storage characterisation techniques to enable the broader adoption of circular economy strategies for used vehicle batteries. Quang Truong Dinh is Assistant Professor in Energy Management and System Control. He was awarded his B.E. mechatronics degree, first-class, from the Mechanical Engineering Department at Ho Chi Minh City University of Technology, Vietnam, in March 2006. He completed his first-class Ph.D. mechatronics degree in the School of Mechanical Engineering at University of Ulsan, South Korea, in early 2010. After obtaining his Ph.D. degree, he worked in the School of Mechanical Engineering (University of Ulsan) as a Postdoctoral Researcher for two years and as a Research Professor for three-and-a-half years. From August 2015, he joined WMG, the University of Warwick. Dr Dinh’s research interests are mainly concentrated on the following fields: Mechatronics—advanced mechatronic systems, integrated systems with both sensing, communication and automation technologies, system engineering and power management and control; Control theories and applications—nonlinear control (robust, adaptive, predictive, hybrid), networked control, remote control, fault tolerant control; Renewable energy—wind/wave energy converter, self-performance optimization control to maximize energy harvesting efficiency. vii Stefano Longo received his MSc in Control Systems from the Department of Automatic Control and System Engineering at the University of Sheffield in 2007 and completed his PhD in Control Systems in the Department of Mechanical Engineering at the University of Bristol in 2011. His PhD thesis was awarded the prestigious Institution of Engineering and Technology (IET) Control and Automation Prize for significant achievements in the area of control engineering. In November 2010, he was appointed for the position of Research Associate in the Department of Electrical and Electronic Engineering at Imperial College London, where he worked in the intersection of the Control & Power, and Circuit & Systems Research Groups. He retained the position of Honorary Research Associate until 2016. He was appointed Lecturer (Assistant Professor) at Cranfield University in 2012 and promoted to Senior Lecturer (Associate Professor) in 2017. His work and research interests gravitate around the problem of designing and implementing advanced control and estimation algorithms in hardware, where algorithm design and hardware implementation are not seen as two separate and decoupled problems but, rather, as a single interconnected problem. These ideas have been applied to networked control systems (whilst at the University of Bristol), to parallelisable hardware for constrained optimal control and real-time optimisation (whilst at Imperial College London) and, more recently, to alternative powertrain and chassis control. viii Preface to ”Energy Storage and Management for Electric Vehicles” One of the main drivers for technological development and innovation within the automotive and road transport sectors in recent years is the need to reduce the fuel consumption and exhaust emissions of vehicles while concurrently exceeding consumer expectations of quality, driveability, refinement, and vehicle range. To meet this challenge, engineers and researchers have worked together to design, integrate, and validate future powertrain technologies for the next generation of hybridised and fully electric vehicles. Within the context of many electrified vehicle applications, the design and management of high-voltage battery systems represents the greatest element of research novelty. The aim of this Special Issue of Energies is to explore research innovation within the battery systems engineering domain that incorporates optimization, mathematical modelling, control engineering, thermal management, mechanical design, and component sizing and packaging. James Marco, Quang Truong Dinh, Stefano Longo Special Issue Editors ix energies Article Equivalent Circuit Model Construction and Dynamic Flow Optimization Based on Zinc–Nickel Single-Flow Battery Shouguang Yao 1, *, Xiaofei Sun 1 , Min Xiao 1 , Jie Cheng 2 and Yaju Shen 2 1 School of Energy and Power Engineering, Jiangsu University of Science and Technology, Zhengjiang 212000, China; ntsunxf@126.com (X.S.); xiaomin_just@126.com (M.X.) 2 Zhangjiagang Zhidian Fanghua Storage Research Institute, Zhangjiagang 215600, China; chengjie_chj@126.com (J.C.); syjee7766@163.com (Y.S.) * Correspondence: zjyaosg@126.com; Tel.: +86-15051110000 Received: 15 January 2019; Accepted: 11 February 2019; Published: 13 February 2019 Abstract: Based on the zinc–nickel single-flow battery, a generalized electrical simulation model considering the effects of flow rate, self-discharge, and pump power loss is proposed. The results compared with the experiment show that the simulation results considering the effect of self-discharge are closer to the experimental values, and the error range of voltage estimation during charging and discharging is between 0% and 3.85%. In addition, under the rated electrolyte flow rate and different charge–discharge currents, the estimation of Coulomb efficiency by the simulation model is in good agreement with the experimental values. Electrolyte flow rate is one of the parameters that have a great influence on system performance. Designing a suitable flow controller is an effective means to improve system performance. In this paper, the genetic algorithm and the theoretical minimum flow multiplied by different flow factors are used to optimize the variable electrolyte flow rate under dynamic SOC (state of charge). The comparative analysis results show that the flow factor optimization method is a simple means under constant charge–discharge power, while genetic algorithm has better performance in optimizing flow rate under varying (dis-)charge power and state of charge condition in practical engineering. Keywords: zinc–nickel single-flow battery; equivalent circuit model; self-discharge; dynamic flow rate optimization; genetic algorithm 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 – 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 – 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 Energies 2019 , 12 , 582; doi:10.3390/en12040582 www.mdpi.com/journal/energies 1 Energies 2019 , 12 , 582 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 – 14 ], electrolyte composition addition [ 15 – 18 ], and flow structure design [ 19 – 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. 2 Energies 2019 , 12 , 582 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. Figure 1. Basic structure of zinc–nickel single-flow battery. Figure 2. Basic structure of partially parallel single cells. Table 1. Size parameters of the initial model. Main Components Size Parameters Height of porous nickel electrode (mm) 240 Width of porous nickel electrode (mm) 186 Thickness of porous nickel electrode (mm) 0.64 Height of negative pole (mm) 240 Width of negative pole (mm) 186 Thickness of negative pole (mm) 0.08 Distance between anode and cathode(d 1 /mm) 160 Electrolyte density (kg · m − 3 ) 1456.1 Electrolyte viscosity (kg · m − 1 · s − 1 ) 0.003139 No. of parallel cells in stack 23 Inner diameter of the pipeline (mm) 15 Length of pipeline (cm) 40 Pipeline import and export height difference (cm) 5 Number of bends 3 3 Energies 2019 , 12 , 582 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). 2NiOOH + 2H 2 O + 2e − 2Ni ( OH ) 2 + 2OH − E 0 = 0.49 V (1) Zn + 4OH − Zn ( OH ) 2 − 4 + 2e − E 0 = − 1.215 V (2) 2NiOOH + 2H 2 O + Zn 2Ni ( OH ) 2 + Zn ( OH ) 2 E 0 = 1.705 V (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 Sections 2.1–2.5 elaborate on each module of the general electrical simulation model of the zinc–nickel single-flow battery. Figure 3. Generalized electrical model of zinc–nickel single-flow battery stack. 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 (R resistive ) and polarization loss resistance (R reaction ), which can be calculated by Equation (4) [32]. P stack = P rate η system (4) R = K · P stack I 2 max (5) In Equation (4), P rate is rated power and η system is system efficiency. In Equation (5), K is power loss coefficient, I 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 4 Energies 2019 , 12 , 582 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 ( R resistive ) and the polarization loss resistance ( R reaction ) of the zinc–nickel single-flow battery stack (300 Ah) proposed in reference [ 29 ]. When the SOC is 0.2, the values of R resistive and R reaction are respectively 0.623 m Ω and 0.2504 m Ω , and then the ohmic loss coefficient ( K resistive ) and polarization loss coefficient ( K 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 ( R fix ) and pump current loss ( I pump ). Fixed loss resistance ( R fix ) is calculated by Equation (6), in which U min is the minimum voltage of the stack and P fix is the fixed loss power, which is experimentally measured to account for about 2% of P stack R fix = U 2 min P fix (6) The function relationship between pump loss current ( I pump ) and pump power ( P mech ) in the electrical model is shown in Equation (7). The pump loss coefficient (M) is related to pump loss power. Definition of M see Equation (8). Figure 4. Pump loss model of zinc–nickel single-flow battery stack. I pump = P mech_loss U stack = M · ( I stack SOC ) U stack (7) M = P mech · SOC worse I max (8) The mechanical loss ( P mech_ loss ) includes two parts: the mechanical loss ( P pipe_ loss ) caused by the electrolyte flowing through the pipeline connecting the stack and the external storage tank, and the mechanical loss ( P stack_ loss ) caused by the electrolyte flowing through the stack. The total loss (P mech_ loss ) is shown in Equation (9). P mech_loss = P stack_loss + P pipe_loss (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 5 Energies 2019 , 12 , 582 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. Δ P pipe = − γ ( Δ V 2 s 2g + Δ Z + h f + h m ) (10) P pipe = Δ P pipe × Q (11) 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: Δ P stack = Q × ̃ R (12) P stack = Δ P stack × Q (13) In Equation (12), ̃ 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 P stack is shown in Equation (13).Considering the pump efficiency, the total mechanical loss of the battery system can be defined as Equation (14). P mech_loss = P pipe_loss + P stack _ loss η pump (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 ( R self ) in the equivalent circuit model. The calculation formula is shown in Equation (15), where P self is the power loss caused by self-discharge, and its expression is given by Equation (16). For the self-discharge power loss coefficient ( f ), the calculation formula is shown in Equation (17), where U 1 and 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. R self = U 2 min P self (15) P self = f · P stack (16) f = ∫ t 2 t 1 U 1 I 1 dt − ∫ t 2 t 1 U 2 I 2 dt ∫ t 2 t 1 U 1 I 1 dt (17) 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 6 Energies 2019 , 12 , 582 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: Positive electrode potential : E + = E 0 + + RT nF ln ( C NiOOH C Ni ( OH ) 2 · C OH − ) 2 (18) Negative electrode potential : E − = E 0 − + RT nF ln ( C Zn ( OH ) 2 − 4 C OH − 4 ) (19) E + is the positive equilibrium potential, E − is the negative equilibrium potential, T is the ambient temperature, and 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: Positive electrode potential : E + = E 0 + + RT nF ln ( C H max − C H C H · C OH − ) 2 (20) The battery stack potential is as follows: E stack = E 0 + RT F ln ( C H max − C H C H × C OH − C Zn ( OH ) 2 − 4 1/2 ) (21) Figure 5. Open-circuit voltage estimation model of zinc–nickel single-flow battery stack. 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 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. E self - 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]. E stack = E 0 + RT nF ln ( ( SOC 1 − SOC ) 2 × ( 1.4SOC + 9.6 ) 2 1 − 0.7SOC ) (22) E terminal = E stack ( OCV ) ± I stack ( R reaction + R resistive ) − E self − discharge (23) 7 Energies 2019 , 12 , 582 Table 2. Range [28]. Parameters Unit Range C H mol · m − 3 0–35,300 C OH − mol · m − 3 9600–11,000 C Zn ( OH ) 2 − 4 mol · m − 3 300–1000 C H max mol · m − 3 35,300 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 Zn ( OH ) 2 − 4 , 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 C Zn ( OH ) 2 − 4 max can be obtained as 1 mol/L from Table 2. SOC = 1 − C Zn ( OH ) 2 − 4 initial ± C Zn ( OH ) 2 − 4 variable C Zn ( OH ) 2 − 4 max (24) The SOC of the zinc–nickel single-flow battery stack storage system is divided into SOC tank in the tank and SOC 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 SOC 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. SOC stack = SOC stack_in + SOC stack_out 2 = SOC tank + SOC tank + I stack F × Q × C 2 (25) SOC stack_t = SOC tank_t + I stack 2 × F × Q × C (26) SOC tank_t = SOC tank_initial + b × ∫ t 2 t 1 I stack_t dt F × V × C (27) Figure 6. SOC estimation model of zinc–nickel single-flow battery stack. 8 Energies 2019 , 12 , 582 Table 3. Parameters [29–34]. Parameters Unit Value Rated voltage V 1.6 I max A 200 U min V 1.2 P rate W 160 Stack capacity Ah 300 Number of parallel cells - 23 Operating temperature range ◦ C − 40~40 Volume of electrolyte L 8.5 R resistive Ω 0.00064 R reaction Ω 0.00036 K resistive - 10.8% K reaction - 4.35% R f ix Ω 0.313 C electrode F 138 R sel f Ω 0.16 f - 0.039 η pump - 0.8 F C/mol − 1 96485 SOC worse - 0.2 ̃ R Pa/m 3 14186843 n - 2 T K 298 C mol/L 1 3. Results and Discussions 3.1. Terminal Voltage Estimation and Error Analysis of the Charging This section compares the voltage values of the zinc–nickel single-flow battery stacks obtained from experimental and simulation models at different charging currents (50 A, 100 A, 150 A). Figure 7a shows the comparison between the terminal voltage value of the stack obtained by the experiment and the voltage of the stack of the equivalent circuit model (considering self-discharge and without considering self-discharge) when the charging current is 100 A. The results show that the simulation results without considering the self-discharge effect have a large error with the experimental values. When the model considers the capacity loss and voltage drop caused by self-discharge, the charging time and voltage value obtained by the simulation are more consistent with the experimental values, so as to avoid the undercharge phenomenon caused by the large voltage estimation error. Figure 7b is a relative error analysis of the model simulation voltage value considering the self-discharge effect and the experimental value, and the error range is between 0.001% and 2.61%. Figure 7. ( a ) Simulation results and experimental verification of ZNBs voltage at 100 A charging current; ( b ) relative error between simulation results considering self-discharge and experimental results. 9