Modeling of Lithium-Ion Battery for Energy Storage System Simulation

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then explained in terms of the battery SOD in a n th order polynomial. Also, from Fig. 1, the equilibrium potential E(t,T,i,l) is also seen as a function of V(i,T,t,l) and current i(t). These relationships are expressed as (1) and (2). 2) Secondly, the discharge rate and temperature corresponding to the reference curve are treated as the reference discharge rate and temperature. Therefore in view of the above, expressions for E, V(i,T,t,l), SOD and Rint are Eit [ ( ), T( t), tl , ] = vit [ ( ), T( t), tl , ] + R i( t) (1) n � int r v[ i( t), T( t), t, l] = c SOD [ i( t), T( t), t, l] (2) k k = 0 k t 1 SOD[(), i t T (), t t, l] = � i() t dt (3) Q 0 R = R + R (4) int 1 2 Where ck is the coefficient of the k th -order term in the polynomial representation of the reference curve and Qr is the battery capacity referred to the cutoff voltage for the reference curve. For k = 0, E = c0 is the open-circuit voltage at the beginning of discharge at the reference temperature of the reference curve. B. Description of the Internal Resistance Normally, the internal resistance Rint will increase with the state of discharge. In this model, Rint has two components R1 and R2. R1 is defined as the internal resistance of the lithiumion battery at SOD=0. It depends on the discharge condition, i.e. temperature, current level and lifecycle. R2 is the increase in Rint as SOD increases. R2 can also be affected by the temperature. However, it is proposed that an n th -order polynomial is used instead to describe the relationship between R2 and SOD. A correction factor �(T) will then be used later to compensate for variation of R2 with T. Based on the above, R can be defined as a function of 1 discharge current, temperature and lifecycle, as follows, R = f( i( t), T( t), l) (5) 1 Take derivative on both sides of (5), δ f δ f δ f ∂ R = ∂ i+ ∂ T + ∂l (6) 1 δi δT δl R1 can be calculated by dividing the initial voltage drop (shown in Fig. 2) by the discharge current i(t) at SOD=0. From T 1 n the experimental data, [ ∂R ∂R ] 1 1 1 1 1 �∂i ∂T ∂l � � � � � n �∂i � n ∂T � � can be easily calculated. � n ∂l � � Expressed in matrix form, 1 1 �∂R � �∂i � � � � = � � � � n n �∂R � � � i 1 � �∂ 1 ∂T � n ∂T 1 ∂l � ��δ f � �� δi n ∂l � � δ f δT δ f � δl �� 1 T � and (7) 17 V 16.5 16 15.5 15 0 0.1 0.2 SOD T=25� I=2 A Fig. 2 Determination of the voltage drop at SOD=0 for different discharge condition Then using least-square method to obtain the values δ f δ f δ f of , , δiδT δ l . Then R1 can be obtained as δ f δ f δ f R = ( i− i ) + ( T − T ) + ( l − l ) + R (8) 1 ref ref ref 1_ref δi δT δl where, iref, Tref, lref are the discharge rate, temperature and lifecycle of the reference curve. R1_ref is the internal resistance of the reference curve at SOD=0. Returning to R2, in general, R2 can be defined as a function of SOD and temperature, as follows: R = g( T( t), SOD) (9) 2 Firstly choose another temperature discharge curve which is of the same discharge rate (the reference discharge rate). Then define i*R2_ref as the voltage drop (the difference between this curve and reference curve) at the same SOD, as for example shown in Fig. 3. The selection of the SOD point can be arbitrarily because as shown subsequently, it does not cause significant difference to the final simulation result. Normally, selection of the SOD point at the middle of these curves yields higher overall accuracy. An n th -order polynomial can be used to fit to that relationship between R2_ref and SOD. The same order polynomial as that with the potential E is recommended, again to yield higher accuracy. A correction term �(T) is used to compensate for the variation of R2 at different discharge condition. This is illustrated as follows. The method to determine R2 is illustrated in Fig. 3, where experimental data from the ULTRALIFE UBBL10 lithium-ion battery is used. The reference curve and another curve at -20� are chosen to calculate R2_ref, which is shown in Fig. 4. Then, n k R = � r * SOD [ i( t), t] (10) 2_ref k k = 0 Fig. 3. Determination of the R2 for discharge condition at different temperatures. Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.
In (10), rk is the coefficient of the kth order term in the polynomial representation of R2_ref. SOD selected is less than the SOD level at the termination of discharge. The terminal SOD chosen in this paper is 0.7. Then the correction term is i* R ( T) 2 α ( T ) = (11) i* R2_ref From (9)-(11), R2 of the internal resistance can be expressed as n k R = α ( T) * r * SOD [ i( t), t] (12) 2 � k k = 0 Fig. 4. R2_ref for the lithium-ion battery based on the 25�and - 20� curves. C. Description of the Thermal Characteristics Since E is temperature dependent, temperature must be calculated dynamically so that it is available for computation of E during each time step [6]. The temperature change of the battery is governed by the thermal energy balance [7] described by� dT () t 2 m* c * = i() t *( R + R ) −h A[ T() t −T ] (13) p 1 2 c a dt In (13), m is the battery mass (in kg), cp is the specific heat (J/kg/K), hc is the heat transfer coefficient (W/m 2 ), A is the battery external surface area (m 2 ), Tc is the ambient temperature. The heat power terms include resistive heating and heat exchange to the surroundings. Heat generation due to entropy change or phase change, changes in the heat capacity and mixing have all been ignored here because from [5], it has been concluded that such omission will not cause significant loss of model accuracy. D. Implementation based on the VHDL model in SIMPLORER Equations (1)-(4), (8), (12) and (13) provide complete description of the battery. From the derived mathematical model, the model can be simulated using VHDL-AMS method in SIMPLORER [8]. Unfortunately, lifecycle has not been considered in the battery model implementation since there is insufficient experimental data at the time of the writing of this paper. III. COMPARISON OF SIMULATION AND TEST RESULTS A. Discharge Characteristics A dynamic model of the ULTRALIFE UBBL10 lithiumion battery on the methodology given above was constructed for use in the SIMPLORER software. The parameters are given in TABLE 1. The rate dependence of potential was validated by testing the Ultralife UBBL10 lithium-ion VHDL model. The initial SOD is set to 0. The battery is maintained at room temperature (25° C) by setting a large cooling coefficient (hc = 100 W/m 2 K). This simulates the idealized constant-temperature case. Comparison of the simulation and test results of discharge currents at 1 A and 4 A are shown in Fig. 5. As can be seen from the figure, a most satisfactory match between the model and test data has been obtained for the reference curve (2A) and while good agreement has also been obtained for all other discharge rates. TABLE I. SPECIFICATIONS OF LITHIUM BATTERY TESTED IN THIS PAPER Model Tested UltraLife UBBL10 Type of Battery Cylindrical 18650 Li-ion cells assembly Operating Temperature -32°C to 60°C Storage Temperature Range -32°C to 60°C Voltage 16.33 V Capacity 6.2 Ah Heat capacity 925(J/kg/K) Fig. 5. Simulation and test data at different current levels for the Ultralife UBBL10 lithium-ion batter The simulation data were processed to obtain the relation between the voltage and the SOD. The results are compared with the test data, as shown in Fig. 6. Again excellent match was achieved for the operating temperatures of -30� and 60 �. �� Simulation and test data at different temperature levels for the Ultralife UBBL10 lithium-ion battery Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:54 from IEEE Xplore. Restrictions apply.
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