electric vehicles (PHEVs)/electrical vehicle (EVs), interest of researchers is increased significantly in Li-ion batteries as these are leading energy storage ...
ICATE 2013 Paper Identification Number-85
1
Temperature Dependent Circuit-based Modeling of High Power Li-Ion Battery for Plug-in Hybrid Electrical Vehicles A. Panday and H.O. Bansal
Abstract— To increase the electrical range of plug-in hybrid electric vehicles (PHEVs)/electrical vehicle (EVs), interest of researchers is increased significantly in Li-ion batteries as these are leading energy storage component. An accurate modeling of battery with knowledge of all its parameters will optimize the vehicle performance. For achieving better performance of vehicle, correct knowledge of state of charge (SOC) of battery is required. Battery characteristics get affected due to several parameters such as temperature, aging and cycle life etc. In this paper, a new expression of temperature dependent battery SOC is developed and effect of temperature variation on this is investigated. Change in temperature influences the current, open circuit voltage (OCV), internal resistance and capacity of battery which contribute in SOC estimation. To demonstrate the real time characteristics and behavioral change of battery with respect to temperature; two dynamic battery models using MATLAB/SIMULINK environment are implemented. Influence of temperature on OCV, resistance, capacity and SOC is observed and simulation results are compared with recent studies. The simulation results show the effectiveness of models proposed. Index Terms— Discharge current, OCV, SOC, PHEV, self discharge current, temperature effect
I. INTRODUCTION
A
utomobiles have made great contribution to the growth of modern society by satisfying many needs for mobility in everyday life. The development of internal combustion (IC) engine has a major role in automobile sector. But due to the toxic emissions in form of carbon dioxide (CO2), carbon monoxide (CO), nitrogen oxides (NOx), unburned hydrocarbons (HCs) etc. in a large amount by IC engine has caused pollution problems, global warming and badly affecting the ozone layer which led to cause serious problems to environment and human life. As petroleum resources are limited, consumption of petroleum should be reduced. One solution to reduce the petroleum consumption is by adopting an alternate transportation technology which uses current IC engine as primary power source and batteries/electric motor as peaking power source. Environmental and economical advantages can also be gained Manuscript sent for review on September 27, 2012.
by applying this alternative transportation technology as it is reducing toxic emissions. This concept brought new transportation medium as electric vehicles (EVs), hybrid electric vehicles (HEVs) and PHEVs in picture, which are clean, efficient and environment friendly [1]. PHEV is combination of HEV and EV consisting of an IC engine and a large on-board rechargeable battery which is having the advantages of large driving range with easy refueling and at the same time reducing the fuel consumption and toxic emissions. Thus PHEV is a means to reduce the energy demand and replacing the liquid fuel consumption by storing electrical energy in large on-board rechargeable batteries with high fuel economy [2]. The main component behind the PHEV’s eminent performance is large on-board rechargeable battery which basically works according to the charge available in the battery. Battery performance and cost are essential factors for development of PHEVs/EVs. Li-ion batteries are getting very much attention from researchers and replacing Nickel-metal hydride and lead-acid batteries because of its much higher specific energy, specific density, durability and lower self-discharge rate in comparison to the other types of batteries with excellent life cycle and with no memory effect [3]. These characteristics make Li-ion batteries promising for next generation vehicles. To model the battery performance, it is very much required to design the battery mathematically to extract the real amount of charge without affecting the health of battery. A number of parameters like aging, temperature, load variation, shaking, cycle life and resistance etc. dominate the battery performance. Proper modeling of battery will lead to better estimation of its SOC and ultimately the vehicle performance and fuel economy will get affected. II. PROPOSED SOC CALCULATION The vehicle performance is characterized by SOC or depth of discharge (DOD) of battery. Battery SOC is defined as the ratio of remaining capacity to fully charged capacity. SOC = SOC 0 −
1 idt SOC 0 ∫
(1)
where SOC0 is the initial SOC level of battery. Exact estimation of SOC is very much desired for both driver notification and energy management system knowledge to optimize the performance of vehicle and to reduce the fuel consumption.
ICATE 2013 Paper Identification Number-85
2
A large number of researchers investigated about how to calculate exact SOC of battery. Pang et al. used open circuit voltage method to calculate the SOC (voltage based SOC i.e. SOCv) of battery [4]. The open-circuit voltage based SOC estimation technique is advantageous in various aspects like i) OCV versus SOC characteristic is independent of the age of the Li-ion battery [5], ii) this is very accurate requires some rest time [6]. Piller et al. concluded from the study that ampere-hour counting method (current based SOC i.e. SOCi) is a suitable method to estimate SOC of battery as it is easy, direct and easily implementable [7]. If the current measurement is accurate, this method is reliable also. But it may have some initial value or accumulated error problems. To overcome the shortcomings of both, these two can be combined together. Tang et al. identified contribution of both SOCv and SOCi together to estimate accurate SOC of battery [8, 9] but does not include effect of temperature. Li-ion battery has a very eminent effect of temperature on its performance and on its various parameters. Under optimal temperature range batteries behaves as prescribed but outside this range the battery cell experiences severe loss of capacity. Therefore it is of utmost importance to analyze the effect of temperature on various other parameters of the battery, so that the same can be utilized without deteriorating the battery health. To characterize the battery performance under influence of temperature, this paper includes thermal effect during modeling of battery which is giving the correct information about the status of different parameters affected due to temperature change. The derived temperature dependent SOC expression is as follows:
SOC = wSOC v + (1 − w)(SOC i − η ) Where η = ⎛⎜1 −
⎝
(2)
SOC 0 ⎞ ⎟ is a correction factor, SOCv and 100 ⎠
So the consideration of this OCV with temperature effect will lead to modify
SOCv and will contribute in final SOC
calculation. B. SOCi Calculation Coulomb counting method involves the current integration flowing through the battery to get the SOC i
(
)
1 Cp − ∫ idt Cp where Cp is battery capacity in Ah. SOCi =
(5)
Due to change in temperature the cell reaction rate gets changed which has been depicted here using following (6), (7) and (8). Arrhenius equation gives the reaction rate as (6) K = K 0 * e ( − Ea / RT ) where K=reaction constant, R=gas constant and Ea=activation energy. During the electron transfer reaction, electrons require the additional amount of energy to surmount the energy barrier is called activation energy (Ea=J-mol-1) which depends on temperature. As for every 10°C temperature increase current gets double so for ∆T temperature change, reaction rate ratio becomes
K (T + ΔT ) (7) = 2 ( ΔT /10) K (T ) K is reaction rate (mole/s) and can be expressed as current. Suppose K1 is reaction rate at temperature (T+∆T) and K2 at temperature T K1= K (T+∆T) and K2 = K (T) and using (6) and (7), we get ln
K (T + Δ T ) E a ⎛ 1 1 ⎜ = − K (T ) R ⎜⎝ TT TT + ΔT
⎞ ⎛ ΔT ⎞ ⎟⎟ = ⎜ ⎟ ln 2 ⎠ ⎝ 10 ⎠
SOCi are temperature dependent as derived below. This single SOC expression suits to every proposed model here.
leads to get an expression for
A. SOCV Calculation Cell voltage under reversible conditions i.e. all the reactions are balanced is called equilibrium voltage which is occasionally referred as OCV or rest voltage. Using the OCV and SOC relationship, battery SOC can be estimated. With this OCV, voltages based SOC (SOCv) can be estimated easily using following equation
⎛ R ⎞⎛ T ⎞ E a = ln 2⎜ ⎟⎜ ⎟ ⎝ 10 ⎠⎝ T + 10 ⎠ (8) shows the dependence of Ea on temperature [10].
1 SOCv = (OCV − a0 ) a1
(3)
where a0 = battery terminal voltage when SOC = 0% and a1 = battery terminal voltage when SOC = 100%. But due to change in temperature, equilibrium voltage of battery at any temperature T gets changed like (4)
⎛ dV ⎞ (4) V(T ) = V( 298) + ⎜ ⎟(T − 298) ⇒ OCV ⎝ dT ⎠ dV is temperature coefficient and is constant for the dT considered temperature range.
Ea to relate with temperature (8)
Effect of temperature on self-discharge current can be ⎛
demonstrated as
i( s − d )
⎛ R ⎞⎛ T
⎞⎞
−⎜⎜ ln 2 ⎜ ⎟ ⎜ ⎟ ⎟⎟ 1 = K 0 (T )e ⎝ ⎝ 10 ⎠⎝ T +10 ⎠ ⎠ Cp
III. MODELING OF BATTERY INVOLVING THERMAL EFFECT Battery involves different areas of science like chemistry and physics and electrical engineering. Here circuit based modeling using different electrical components to analyze the behavior of battery is performed. To analyze the effect of temperature on SOC, authors have chosen two battery models. Proposed models are capable of simulating the noticeable influence of temperature for short and long term behavior of battery with minimal complexity and are suitable to be utilized in PHEV/EV applications. Proposed models are constructed considering real-life operating conditions of power assist
ICATE 2013 Paper Identification Number-85
3
vehicles. The one is with internal resistance only showing the voltage drop and another is with a capacitor-resistance combination (double layer voltage) with internal resistance. OCV and current for both models are calculated to get SOCv and SOCi and finally an accurate SOC is calculated. A. Model 1 Model 1 consists of power source and an internal resistance. When power source delivers the current, output voltage is lower than the no load voltage because there is a voltage drop due to the internal resistance offered by the battery illustrated in Fig. 1. The internal resistance consists of both, the ohmic resistance and the polarization resistance [11]. To determine the SOCv of model 1 , (9) V (t ) = V (t ) − i(t ) * Rint V(t) is OCV and from (4) we get OCV at temperature T and from (3), SOCv(T) can be calculated as follows SOCv (T) =
{
}
⎤ 1 ⎡ ' ⎛ dV ⎞ V (t) + (i *2(ΔT / 10) + i(s−d ) ) * Rint + ⎜ ⎟(T − 298) − a0 ⎥ (10) a1 0 ⎢⎣ dT ⎝ ⎠ ⎦
SOCi at temperature T, we get SOCi (T ) =
[
{
} ]
1 Cp − ∫ i * 2 ( ΔT / 10) + i( s −d ) dt Cp
(11)
Now SOC of battery at any temperature T can be calculated using (2), (10) and (11). B. Model 2 To predict the run-time behavior of the battery, transient R and C (in parallel) are connected in series with the internal resistance as illustrated in Fig. 2. The time constant (RC) characterize here the time varying response of battery. Due to the double-layer formation at electrode/solution interface capacitive effects arise [12]. This capacitance consists of purely electrical polarization capacitance and diffusion capacitance [13]. Transient response of the battery is influenced by double layer and diffusion capacitance when the rates of reactions are high. This effect is modeled using a single lumped capacitance in parallel with the resistance [14]. To determine the SOCv of model 2 ⎡ ⎧⎛ R + 2 Rint ⎞ − (t / RC ) ⎫⎤ ' V (t ) = V (t ) − i (t ) ⎢ Rint − ⎨⎜ ⎟e ⎬⎥ ⎢⎣ ⎭⎥⎦ ⎩⎝ RC ⎠
(12)
Like (10) and (11) get the SOCv and SOCi for model 2 and then using (2) get the new SOC value. IV. MODEL EXTRACTION AND SIMULATION Knowledge of exact SOC during vehicle driving is important to optimally utilize the battery capability. Li-ion battery performs very well at ambient environment (room temperature = 25°C). A change in temperature due to any reason (may be environmental effect or may be due to battery pack utilization) alters the characteristic of battery so it is advisable to keep the battery temperature in a limited range. The variation may be as much significant sometimes that it may alter the complete performance characteristic of the battery. To validate the temperature dependent characteristics, the proposed battery models are used with a load set to draw a constant current of 25.6 A, but with the environmental temperature set to various values. The battery was then discharged repeatedly from full SOC to zero SOC at each temperature. The simulation data are analyzed to obtain the
relation between the open circuit voltage & SOC and effect on internal resistance and capacity of battery as given in the following sections. A. Open Circuit Voltage The OCV is a function of chemical composition, pressure and temperature as presented in [15]. OCV represents batteries’ chemical reaction phenomenon. In Li-ion batteries concentration of lithium ions of the solid phase with respect to intercalating materials (Li-ions) determines the open circuit potential [16]. Battery SOC consists of OCV as the major component which defines the thermodynamic properties of the battery. To get the effect of temperature on OCV, proposed models have been simulated from -20°C to 60°C to see the response for a wide temperature range and corresponding curves are shown in fig 3 (a) and (b). At different SOC values ranging from 10% to 90%, a variation in OCV is seen due to temperature change. OCV increases as temperature gets higher than room temperature and decreases with decreasing temperature. From -20°C to 40°C, OCV curves are very close to each other. OCV values changes a lot in comparison to the average OCV after 40°C. Results obtained here are compared with the practical results given [17]-[18] and are found to be optimistic. As temperature increases, OCV values gets differ for both models at various SOC ranges. Fig. 4 (a) and (b) shows OCV at 0°C and 25°C respectively with not much difference in OCV values for both proposed models. But beyond 45°C results differ for both models and there is a large difference found at 60°C shown in fig 4 (c). B. Capacity Amount of charge contained by battery is useable capacity. Capacity of the battery extensively varies with operating conditions and strongly depends on the internal impedance of the battery. It gets much influenced by the temperature change. At high temperatures chemical reaction requires less activation energy so with a very less amount of energy more intercalation and de-intercalation of Li-ions occurs resulting in a higher cell voltage. Due to the higher temperatures Li-ions can diffuse faster, which means a higher current can flow which increases the power and discharge capacity of Li-ion cells. At low temperatures, intercalation and de-intercalation requires higher activation energy for the chemical reactions. So less lithium ions can participate in the active cell mechanism which results in a temporary loss of capacity [19]. From the simulation results, at 0°C the capacity diminishes by 11.9 % from the ambient for both models and at higher temperatures capacity increases which is shown in table 1. Fig. 5 (a) and (b) depicts the actual remaining capacity of battery as temperature gets lower down. Here in model implementations, self-discharge current is considered so the battery capacity at room temperature also is not 100% in the graph. From negative to room temperature, there is no change in capacity offered by both models as in Fig. 5(c). C. Resistance There is a decrease in terminal voltage of cell during
ICATE 2013 Paper Identification Number-85 discharge because of a voltage drop across the internal resistance of the cell. It also requires the larger voltage to charge the cell. Hence reduces its effective capacity as well as decreases its charging/discharging efficiency. Higher discharge rates give rise to higher internal voltage drops which explains the lower voltage discharge curves at high C-rates [20]. Internal resistance of the battery varies with change in various parameters and can never be considered as constant even though the manufactures label it as constant. It majorly depends on C-rate, SOC and temperature. At low temperatures cell may be very inefficient due to larger impedance and reduces the capacity. But at higher temperatures efficiency improves due to the lesser internal impedance rate of chemical reactions increases. For lower internal resistance, however there is an increase in the self discharge rate and cycle life also deteriorates at high temperatures. Fig. 6 (a) and (b) shows the effect of temperature on resistance for both models. The results for various temperatures are demonstrating the similar results as in [21]. The effect of temperature on the resistance of both models below and at room temperature is similar i.e. no difference in characteristics. As room temperature gets crossed, resistance offered by model 1 is larger than offered by model 2 which is replicated in Fig. 6 (c). Thus it is seen that RC combination is imposing an effect at higher temperatures but doesn’t alter the performance at room temperature and below this. Varying from 100% SOC to 10% SOC, the resistance offered by both models is observed. Fig. 7 (a) and (b) very clearly depicts that a minute change occurs in resistances for the whole range of SOC at any particular temperature. For utilizing the larger capacity of the battery, resistance offered should be less and as resistance increases capacity gets decreased. Fig. 8 (a) and (b) demonstrate relation of resistance and battery capacity. The battery models are designed in matlab/simulink environment which suits for dynamic simulations consisting of the controlled voltage source and considers the SOC only as a state variable.
4 model 2 at higher temperature. The higher temperature effect on the battery performance may seem to have a positive effect, but in the long term this will deteriorate the life of battery. Based on these results, it is also concluded that increasing the number of RC combinations in the model will increase the complexity of circuit but may not lead to significant improvement in results. REFERENCES [1] [2]
[3] [4]
[5]
[6]
[7]
[8]
[9]
[10] [11]
[12] [13]
V. CONCLUSION Appropriate battery model in PHEVs/EVs improves the overall performance of vehicle and reduces the liquid fuel consumption. Both the proposed models are successfully implemented in the matlab/simulink environment. As selfdischarge current concept is also included hence proposed battery models are capable of providing the information about different parameters more accurately. The effect of temperature on various parameters like capacity, resistance, current and OCV are shown through different curves and are compared with the results published earlier. It is also concluded that the proposed SOC expression suits to both of the models hence no need to derive it separately for every model. By comparing results of both the models, it is observed that below and at room temperature characteristic of both the proposed models are same. But beyond room temperature, transient effect becomes vital and hence influencing OCV, capacity and resistance, therefore it is suggested to work with
[14]
[15] [16]
[17] [18]
[19]
[20] [21]
U.S. Environmental Protection Agency, Automobile emissions: An overview.1993 Jan .EPA 400-F-92-007, Fact sheet OMS-5. K. Parks, P. Denholm, and Markel T, “Costs and emissions associated with plug-in hybrid electric vehicle charging in the xcel energy Colorado service territory,” National Renewable Energy Laboratory, May 2007. Report No.: NREL/TP-640-41410. Contract No. DE-AC36-99GO10337. X. Zhang, and C. Mi, Vehicle power management, London: Springer, , 2011, ,ch. 3. S.Pang, J. Farrell, and J. Du, M. Barth, “Battery State-of-Charge Estimation,” in Proc. American Control Conference, vol. 2,. IEEE Press, Arlington, VA, 2001, pp. 1644-1649. S. Abu-Sharkh, and D. Doerffel, “Rapid test and non-linear model characterisation of solid-state lithium-ion batteries,” J. Power Sources, vol.130, pp. 266–274, December 2004. S. J. Lee, J. H. Kim, J. M. Lee, and B. H. Cho, “The state and parameter estimation of an li-ion battery using a new ocv-soc concept,” Power Electro. Specialists Conf., 2007, pp 2799–2803. S. Piller, M. Perrin, and A. Jossen, “Methods of state of charge determination and their applications,” J. Power Sources, vol. 96, no. 1, pp. 113-120, 2001. X.Tang, X.Mao, J. Lin, and B. Koch, “Li-ion battery parameter estimation for state of charge,” in Proc. American control conf., USA, 2011, pp. 941-946. M. Verbrugge, and E. Tate, “adaptive state of charge algorithm for nickel metal hydride batteries including hysteresis phenomena,” J. Power Sources, vol. 126, pp. 236-249, 2004. D. Berndt, Maintenance-free batteries based on aqueous electrolyte, India: Overseas Press, 2006, ch. 2. H. Culcua, B. Verbruggeb, N. Omara, P. Van Den Bosscheb, and J. Van Mierloa, “internal resistance of cells of li-ion battery modules with freedon CAR model”, Worlf Electric Vehicle J., vol. 3, pp. 1-9, 2009 [EVS24 International Battery, Hybrid and Fuel Cell Electric Vehicle Symposium] E. Cliffs, “Electrochemical Systems”, 2nd ed., Prentice-Hall, 1991. B. E. Conway, “Transition from ‘Supercapacitor’ to ‘Battery’ behaviour in electrochemical energy storage,” J. Electrochem. Soc., vol. 138, no. 6, pp. 1539-1548, 1991. M. Chen, and G.A. Rincon-Mora, “Accurate electrical battery model capable of predicting runtime and I-V performance,” IEEE Trans. Energy Conv. Vol. 21, no. 2, pp. 504-511,2006. K. Ozawa, “Lithium Ion Rechargeable batteries”, 1st ed., Germany: Wiley-VCH, 2009, pp. 67-102. R. Elger, “On the behaviour of the li-ion battery in hev application”, Ph. D thesis, Dept. Chem. Eng. and Tech., Applied Electrochemistry,Kungliga Tekniska Högskolan, Stockholm, 2004. V.A. Johnson, “Battery performance model in advisor,” J. Power Sources, vol. 110, pp. 321-329, 2002. V.A. Johnson, and T. Sack, “Temperature-dependent battery models for high power lithium-ion batteries,” 17th Annual Electric Vehicle Symposium, NREL/CP-540-28716, Canada, 2000, pp. 1-14. L. Lam, “A practical circuitbased model for state of health estimation of li-ion,” M. Sc.Thesis, Dept. Electrical. Sustainable Energy, University of Technology Delft, Netherlands, 2011. http://www.mpoweruk.com/performance.htm J. Kim, S. Lee, and B. Cho, “The state of charge estimation employing empirical parameters measurements for various temperature,” in Proc. IEEE Int. Power Electr. Motion Control Conf., China, 2009, pp 939-944.
ICATE 2013 Paper Identification Number-85
5
385 MODEL 2 380
375
OCV (volt)
Aishwarya Panday received her B.Tech degree from Uttar Pradesh Technical University, U. P. and M. Tech degree from Birsa Institute of Technology, Jharkhand, India in 2008 and 2010 respectively. She is currently pursuing Ph.D. degree in Electrical and Electronics Engineering at Birla Institute of Technology & Sciences (B.I.T. S), Pilani, India. Her research area is modelling of Li-ion battery and power optimisation techniques used in hybrid electrical vehicle and plug-in hybrid electric vehicle.
253K 263K 273K 283K 298K 313K 323K 333K AVERAGE
370
365
360
355 10
30
40
50 SOC (%)
60
70
80
90
(b) Figure 3. Effect of temperature on OCV 374 MODEL 2 MODEL 1
372 370 368
O C V (v olt)
Hari Om Bansal received his B.E. degree in Electrical Engineering from University of Rajasthan, jaipur, M.E. degree in Electrical Power Systems from Malviya National Institute of Technology (M.N.I.T), Jaipur and Ph.D. Degree from Birla Institute of Technology & Sciences (B.I.T.S), Pilani, India in 1998, 2000, 2005 respectively. He worked as research assistant in M.N.I.T, Jaipur From 2000 to 2001,then as a Lecturer at the Mody College of Engineering & Technology, Sikar, India. He joined as an Assistant Lecturer in 2001 and worked as Lecturer since 2002-2005 and currently Assistant Professor at B.I.T.S, Pilani. His major interest is in research areas as applications of artificial intelligence techniques in Power Systems, Control Systems, distributed generation and solar energy and hybrid vehicle technology. He has written a book chapter on rural electrification and published number of papers in solar thermal system, power distribution generation using artificial techniques.
20
366 364 362 360 358 356
AT 273 K
354 90
80
70
60
50 SOC (%)
40
30
20
10
(a) 374 MODEL 2 MODEL 1
372 370 368 366 364
R
362 360 358 AT 298 K 356 90
V’
V
80
70
60
50
40
30
20
10
(b) 386 MODEL 2 MODEL 1
384
Figure1. Battery equivalent circuit model 1
382
O C V (v o lt)
380
C
378 376 374 372
R
370 368
R
V’
V
AT 333 K
366 90
80
70
60
50 SOC (%)
40
30
20
10
(c) Figure 4. A comparison results of OCV at different temperatures
Figure 2. Battery equivalent circuit model 2
70 MODEL 1 60
A V A ILA B LE C A P A C I TY (A h)
385 MODEL 1 380
O C V (v o lt)
375
370 253 K 263 K 273 K 283 K 298 K 313 K 313 K 323 K AVERAGE
365
360
355 10
20
30
40
50 SOC (%)
(a)
60
70
80
50
40
30
20
10
0 298
90
293
288
283
278
273 268 TEMPERATURE (K)
(a)
263
258
253
248
243
ICATE 2013 Paper Identification Number-85
6
70
100 MODEL 2
90
MODEL 1
80
50
273K 288K 298K 308K
70 R E S IS T A N C E (O h m )
AVA ILA BLE CA PACITY (Ah)
60
40
30
20
60 50 40 30
10
20 0 293
288
283
278
273
268 263 TEMPERATURE (K)
258
253
248
243
238
10
(b)
0 100
90
80
70
60
40
30
20
10
(a)
MODEL 2 MODEL 1 DIFFERENCE IN CAPACITY OF BOTH MODELS
90 80
100
70
90 60
80 50
MODEL 2 70
40
R E S IS T A N C E (o h m )
A V A ILA B LE C A P A CI TY (% )
50 SOC (%)
100
30 20 10 0 298
288
278
268
258
248 238 TEMPERATURE (K)
298
288
278
268
60
AT 278 K AT 288 K AT 298 K AT 308 K
50 40 30
258
20
(c)
10 0 100
Figure 5. Variation in capacity for change in temperature
90
80
70
60
50
40
30
20
10
SOC (%)
(b) 400
Fig. 7: Variation in resistances for different temperatures at different SOC ranges
MODEL 1
350
R E S IS T A N C E (O h m )
300 250 400
200 RESISTANCE AND AVAILABLE CAPACITY AT DIFFERENT TEMPERATURES
350
MODEL 1
150 300 R E S IS T A N C E (oh m )
100 50 0 243
268
293
318 TEMPERATURE (k)
343
268
(a)
250 200 150 100 50
400 MODEL 2
350
0 4.06 at 243K
4.38 at 248K
5.23 at 253K
6.69 at 258K
8.95 at 263K
12.29 at 268K 17.12 at 273K 24.02 at 278K 33.84 at 283K 67.46 at 293K 95.36 at 298K AVAILABLE CAPACITY (%)
(a)
250 400
200 150
350
100
300 R E S IS T A N C E (ohm )
R E S IS T A N C E (O h m )
300
50 0 243
268
293
318 TEMPERATURE (K)
343
368
(b)
RESISTANCE AND AVAILABLE CAPACITY AT DIFFERENT TEMPERATURE
MODEL 2
250 200 150 100
-3
3
x 10
50
D IF F E R E N C E IN R E S S T A N C E (O h m )
FOR MODEL 1 AND MODEL 2 0 4.63 at 243K
2.5
52.23 at 253
66.15 at 258
75.97 at 263
82.87 at 268 87.70 at 273 AVAILAVLE CAPACITY (%)
91.04 at 278
93.30 at 283
94.76 at 288
95.61 at 293
95.93 at 298
(b) Fig. 8: Relation between resistance and capacity at different temperature
2
1.5
TABLE I.
1
0.5
32.53 at 248
303
328
353
378 TEMPERATURE (K)
303
328
353
(c) Variation of resistance at higher temperature for both models Figure 6. Temperature Effect on resistance
TEMPERATURE EFFECT ON CAPACITY
Tempe rature (°C)
Increase in Capacity (%) Model 1
Model 2
35
4.3
4.73
40
4.88
5.41
45
6.2
6.27
