# Overview of Simplified Mathematical Models of Batteries

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1 Overview of Simplified Mathematical Models of Batteries Sergei Melentjev, Deniss Lebedev Tallinn University of Technology (Estonia) bstract This paper describes the composition of the mathematical model of battery behaviour with subsequent implementation of this model in virtual simulation environment Three battery types are modelled and each one is described through the appropriate model Models are based on the realworld counterparts, so after the simulation of different models with a number of common parameters and same load, it is possible to evaluate the effectiveness of each battery type for supplying specific load in theory I INTODUCTION One of the most significant barriers in the development, commercialization and popularization of electric vehicles is the battery capacity Even though modern technological advances offer numerous ways to preserve electrical energy in different devices, the mass and environmental restrictions of electric vehicles allow primarily accumulator batteries as the source of current for the electric drive in a car Batteries are also of numerous varieties, but all of them share common flaws, such as low energy capacity versus size and poor perfomance in cold conditions lso, there is a problem of battery capacity derating, as effective capacity value is getting lower and lower in time with such factors as temperature and battery improper handling speeding up the rate of the derating process Proper storage and work conditions can help with slowing down the battery ageing process, as well as careful charge and discharge control However, these measures can become quite expensive, so there is a need to properly examine their effect on the battery cycle life This makes the task of choosing the right battery type of utmost importance, as even the slightest differences in parameters may cause drastic changes in the behaviour of electric vehicle There are also economic considerations, such as price and lead-times for mass production, life expectancy and maintenance requirements However, the choice for the latter part can be considerably simplified by choosing from the most popular battery types currently on the market, as they tend to match the required economic criteria more, than those developed for specific purposes For this research, three of the common battery types were chosen for model development and software simulation with approximated load of an electric vehicle, so that it may be possible to propose most suitable technological solution for real-life experiments Overview II BTTEY MODELS Most existing models for the simulation of battery behaviour can be divided into three sub-groups: Experimental Electrochemical Electrical First two models are not suitable to correctly simulate battery dynamics [1] However, specially developed electriccircuit based models can be used for accurate prediction of charge and discharge of batteries, taking state of charge into account The model used in this article allows simulation of battery dynamics using only the data from battery manufacturers datasheet, and also there are only minor differences in models representing different battery types Because all the different parameters for comparison of different kinds of battery types are many, at the moment of writing this paper only mathematics of charging and discharging are considered This model allows easy modification and further development, for example reaction to temperature, charging algorithms according to price and battery ageing phenomenon can be added at a later stage of work B Model Description The general equation for the used battery model is the following equation: it i C, (1) it it V actual battery voltage (V) E battery constant voltage (V) polarization resistance (Ω) battery capacity (h) it actual battery charge (h) exponential zone amplitude (V) B exponential zone time constant inverse (h -1 ) battery internal resistance (Ω) i actual battery current () C exponential voltage (V) ll of the parameters, mentioned above, should be openly available from the manufacturer s datasheet However, polarization resistance, exponential zone amplitude and exponential zone time constant inverse should be calculated from the discharge curve of the battery The necessary parameters for calculations are shown on Fig 1 231

2 ( B it ) C = e (1) Charge it i i + C, (11) it it,1 C is defined by (9) ( B it ) C = e (12) Fig 1 Example discharge curve The equations are as follows: = V full (2) V exp 3 B = (3) exp To calculate E, the following equation must be used [2]: E = V full + + i (4) Due to the different nature of the chemical materials, used in different types of batteries, equations for the simulation of these batteries also differ: Lead cid Discharge it i C, (5) it it Charge C = B i ( C + ) (6) it i i + C, (7) it it,1 Li-Ion Discharge C = B i ( C ) (8) it i C, (9) it it Ni-Mh Discharge mode is defined by (5) and (6) Charging is given in equations (8) and (13) it i i + C (13) it it,1 C Load s For the simulated load of the battery, electric car drive was chosen, as it is a well-demanded application and also a good testing ground for the batteries due to its dynamic nature, so it satisfies both demands for practical usage of the test result and suitable simulation environment The dynamic behavior of the electric car drive in motion is the same as of the inner combustion engine [3] Typically, there are periods of acceleration and deceleration as well as prolonged periods of driving with constant speed or waiting Typical load cycle of the electric vehicle in natural conditions is presented in Table I TBLE I BTTEY LOD CYCLES Multiple of discharge current C Time (s) It is assumed that the mentioned cycle occurs again and again until battery charge level drops below some defined state of charge Then, we imagine that the vehicle is brought to the charging station and charged with appropriate charge current until it is sufficiently charged Then the cycle is repeated once again until the simulation time expires llowable levels of charge and discharge, as well as appropriate current values are given in manufacturer s datasheet For this research, it is stated that batteries operate at nominal temperature, and that electrochemical processes inside the batteries cause no heating lso, no aging effects are taken into account for the current version of the model 232

3 ll of the simulated batteries share the same input signals, but are configured and limited by their nominal and maximum or minimum required parameters Comparison is also based on the common outputs for all three batteries Overview III BTTEY MODELS The model is built using the MTLB 212a Simulink software by The MathWorks, Inc The model itself represents the equations described in previous section but is built from the Simulink function blocks The equation is subdivided into several sub-sections for the ease of presentation and modification B Current Block Current block -6 Charge current, it = idt (14) The result of (14) is in mpere-seconds, so we need to additionally divide this value by 36 to get standard mpere-hours ctual state of charge is calculated in relation to nominal charge and varies between 1 (fully charged) and (fully discharged) D Polarization Voltage Block Fig 4 shows the polarization voltage according to (1) Exact formula for polarization voltage, which the according Simulink block is based upon, is given in the (15) V pol Polarization voltage block = it (15) it [SOC] Charge limits > [] Load cycle Charge/Discharge [] 5 Discharge current, [] [Vpol] Polarization voltage Fig 2 Block for current calculation Fig 2 shows Simulink diagram for the calculation of charge or discharge current Charging is made with constant current, while discharge current follows the load profile Switching between charging and discharging is made using the state of charge signal When the battery reaches certain level of discharge, then battery begins to charge up to a certain level; afterwards the next load cycle begins s of the state of charge signal, at which the switching occurs, are defined by the battery type and manufacturer Fig 4 Polarization Voltage Block E Polarization esistor Block Fig 5 shows the polarization resistance calculation for different types of batteries and according to charge and discharge regimes Polarization resistance block C State of Charge Block Fig 3 shows the calculation of the state of charge through calculating the actual charge first SOC block [] [] [pol] Polarization resistance 1 s -1/36 s s to h [] > Charge/Discharge2 1 [] [SOC] u Li-Ion or Lead/Ni-Mh Fig 5 Polarization esistance Block Fig 3 State of charge block The Simulink model represents the following equations Charge of the battery is calculated with the following formula pol = (16) it 233

4 pol pol = (17) it,1 = (18) it,1 Equation (16) is for discharging of all types of batteries, (17) is for charging of Li-Ion or Lead-cid batteries and (18) is for discharging of Ni-Mh type of batteries F Exponential Block Fig 6 demonstrates the Simulink modeling of the exponential part of battery voltage This part is described through equations (6), (8) and (1) Li-Ion [] [B] e u Li-Ion HETE 12V/3H TBLE III LI-ION BTTEY PMETES 3 h 5 V B 3 h -1 E 12 V 3 Ω 1 Ω Ni-Mh EVB TECHNOLOGY 1/GP3EVH TBLE IV NI-MH BTTEY PMETES 3 h 15 V B 4 h -1 E 1277 V 11 Ω 6 Ω -1 Lead-cid and Ni-Mh [B] u Li-Ion/Lead or Ni-Mh [C] For the battery parameters it is assumed, that the internal resistance is constant, the nominal capacity is constant, there is no self-discharge, there is no memory effect, maximum SOC is 1%, temperature has no effect on parameters, there is an unlimited cycle life with no derating and there are no environmental considerations V SIMULTION ESULTS [] > Charge/Discharge1 [C] -1 Fig 6 Exponential Block Overview Simulation starts with zero charge, and then continues to rise until 75% of SOC is achieved Then, battery is put to work until it reaches 3% of discharge, and it starts charging up to 75% again Charging is done with nominal 1 hour charge current corresponding to each battery, and discharge current C is 2 The simulation results are presented in graphs of 2 cycle and number of full cycles for 24 hours B Lead-cid Fig 7 shows two charge/discharge cycles IV BTTEY DT The target of this research is to compare and analyze the behavior of three types of batteries that can be used for energy source in electric vehicle mong each battery type, the most typical variants were chosen among The corresponding batteries are as follows: Lead cid Panasonic VL LC-XC1228 TBLE II LED-CID BTTEY PMETES 28 h 11 V B 652 h -1 E 1167 V 8 Ω 12 Ω Fig 7 Lead-cid Cycle Simulation of 24 hours resulted in 3 full cycles It also should be noted that for VL batteries a discharge down to 3% of nominal capacity results in cycle life of about 6 to 8 charge/discharge cycles 234

5 C Li-Ion Fig 8 shows two charge/discharge cycles Fig 8 Li-Ion Cycle Simulation of 24 hours resulted in 29 full cycles For Li-Ion it is also common to have 6 to 8 full charge/discharge cycles down to 3% of original capacity D Ni-Mh Fig 9 shows two charge/discharge cycles VI CONCLUSION Simulation esult Evaluation ccording to the test results, there is only a minor difference in the performance of different battery types However, since real-life experiments with the same samples are not done at the moment, no evaluation of model value can be done lso, since there are numerous simplifications in calculations, simulation results cannot serve as a basis for economically right choice for the power supply of the electric vehicle There are numerous things that need further attention before model effectiveness can be verified However, since the target of this work was to create working model and perform three simulations of different batteries using it, the mission can be considered successfully accomplished B Issues, Which equire Solving Firstly, comparison with real-life test data is required to verify how much model deviates from reality in its current state Secondly, battery selection criteria should be redefined, since even batteries of the same type and nominal capacity can vary greatly in parameters, so comparing batteries of different quality has no point lso, battery parameters should be calculated based on the real measurement, not on the manufacturer data Some manufacturers provide inaccurate data, and some unconventional parameters are just roughly estimated nd lastly, in order to be able to make full use of the model, cycle life deterioration should be introduced, as well as temperature effects CNOWLEDGMENT Fig 9 Ni-Mh Cycle Simulation of 24 hours resulted in 29 full cycles For Ni- Mh batteries it is expected to discharge to 3% of nominal charge capacity in about 6 to 8 cycles, same as Lead- cid and Li-Ion batteries Considering more or less the same theoretical life expectancy of different batteries, it should be noted that environmental conditions and charging/discharging behavior affect life expectancy of different batteries in a different way uthors thank the Estonian rchimedes Foundation (Project Doctoral School of Energy and Geotechnology II ) for financial support EFEENCES [1] O Tremblay, and L Dessaint, Experimental validation of a battery dynamic model for EV applications World Electric Vehicle Journal, vol 3, May 29 [2] D Istardi Modeling and energy consumption determination of an electric Go-kart Chalmers University of Technology, 29 [3] E Martinez-osas, Vasquez-Medrano and Flores-Tlacuahuac, Modeling and simulation of lithium-ion batteries Computers and Chemical Engineering, vol 35, pp , May

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