Depending on the required detail, the circuit model of batteries can represented by simple equivalent circuit consisting of a voltage source in series with an internal resistor or other more complete models based on impedance spectroscopy. Accurate modelling of the dynamical behaviour of electrochemical power sources is an important issue in the simulation of EV power systems. In addition, in practical applications the battery monitoring and battery management systems require some form of dynamic battery models, which are continuously adapted to the battery state. In all applications several issues need to be considered,
• Batteries are not static devices as energy is chemically converted during the discharge and charge process.
• Electrochemical systems are highly non-linear, and this nonlinearity is significant for most power sources under normal operating conditions.
• Their dynamical behaviour depends on many parameters such as temperature, state-of-charge, history of operation, and operating frequency.
Thevenin models
The basic Thevenin-based model shown in Figure 3.12 (a) uses two series resistors, each with a corresponding blocking diode to represent the variation in internal resistance during discharging and charging modes. In Figure 3.12 (b), replacing the voltage source with a bulk capacitor emulates the non-linear open circuit voltage as well as the charge depletion of the battery. Adding an additional series resistor as shown in Figure 3.12(c) to capture the losses during over charging and over discharging increases the accuracy of the battery model but may be excluded if such a condition is protected elsewhere in the system. The effect of self- discharge may be included into the model by adding a parallel resistance branch to the circuit. This is shown in Figure 3.12(d). The Thevenin model of batteries may be refined even further to include much more parameters including temperature effects to produce an even closer representation of the actual intended battery system.
Voc Ri
Vbatt Rdis
Rchg
Voc Ri
Vbatt Rdis
Rchg
Voc Ri
Vbatt Rdis
Rchg
Rs
Voc Ri
Vbatt Rdis
Rchg
Rs
Rsd Cbatt
(a) (b)
(c) (d)
Figure 3.12 Variation of the Thevinin battery circuit model
Descriptions of the model parameters for Figure 3.12 are as follows;
Voc and Vbatt is the battery open circuit voltage and terminal voltage respectively Ri is the lumped internal resistance
Rdis is the discharge resistance to account for the battery charge release rate Rchg is the discharge resistance to account for the battery charge acceptance rate
Rs is the series resistance that accounts for over discharging and over charging (gassing) Rsd is the parallel resistance that models the self-discharge of the battery
Cbatt is the capacitor to model the bulk charge instead of a SoC dependent voltage source
Impedance spectroscopy model
Impedance-based models employ the method of electrochemical impedance spectroscopy to obtain an ac-equivalent impedance model in the frequency domain. The model uses complex R-L-C networks to fit the impedance spectra. By impedance spectroscopy measurements,
Surewaard et al.[99] produced an electric circuit representation of a battery model. In the circuit of Figure 3.13, Ri is the battery internal resistance, Lbatt represents the battery inductance, VOC is the open circuit voltage, Rgas and Vogas represent the gassing reactions of the battery. The complex inductance of the model is represented by ZArc1 and ZArc2 and the impedance depression between the semicircles is approximated by the number of RC circuits. Depending on the number of depressions, additional RC circuits can be taken to approximate the battery impedance measurements. Impedance-based models however only work for a fixed SOC and temperature settings [100].
ZARC1 ZARC2
ZArc2
ZArc1 Im(Z)
Re(Z) LBatt
0
Impedance measurement
ZArc1 ZArc2
VBatt
LBatt Ri
Rgas Vo,gas
+ +
V0CV
Figure 3.13 Approximation of measured impedance spectroscopy line by electrical elements (Reproduced from Surewaard et al [101])
VHDL –AMS Model
Battery models can also be defined as a Very high-speed integrated circuit Hardware Description Language (VHDL) entity. The VHDL model permits both long and short term effects of the battery to be included during simulations of extended drive profiles and cycle life. The VHDL battery model used in this work is based on the standard VHDL- AMS (Analog Mixed Signal) entity in the SIMPLORER® [102] simulation package. Table 3-2 shows the VHDL input parameters and the corresponding values that was used to model four VRLA batteries in a series configuration.
Description [Unit] Parameter ID Value
Initial Acid Density [g/cm²] ad0 1.27
Battery Temperature [°C] temperature 30
Rated Capacity [Ah] cap 500
Rated Discharge Current [A] r_curr 250
Rated Discharge Time [h] r_time 2
Internal Resistance at full charge and nominal temperature [Ω] nom_res 40m
Number of Cells [/] num_cells 24
Acid Density at Full Charge [g/cm³] ad_full 1.28
Acid Density at Complete Discharge [g/cm³] ad_disc 1.01
Nominal Temperature [°C] nom_temp 25
Fraction of Capacity at Low Temperature [/] f_low_cap 0.6
Gain Limit of Capacity at High Temperature [/] f_hi_cap 1.02
Low Temperature where f_low_cap is Specified [°C] low_temp -20
Fraction of Capacity near plate [/] f_plate_cap 0.3
Capacity Gain in the Slow Discharging Limit [/] f_slow_cap 1.03
Self Discharge Rate per Day [%/day] sdpd 0.25
Temperature Dependency of Self-Discharge [°C] sd_t 16.37
Float Current [A/Ah] flt_curr 2m
Cell Voltage where Float Current is Reached [V/cell] flt_volt 2.3
Gassing Threshold Voltage at 25°C [V/cell] gass_th 2.39
Scaling of Gassing Current with Terminal Voltage [V/cell] gass_sl 0.2
Temperature Coefficient of OCV [V/cell/°C] t_coeff_ocv 0.15m
Temperature Coefficient of Full Charge Internal Resistance [1/°C] t_coeff_res 7.5m Coefficient for Internal Resistance Variation with SOC soc_coeff_res 0.5 Table 3-2 VHDL-AMS input parameters of a lead acid battery model
Simulation results of the battery VHDL model and an extended Thevenin model ( Figure 3.11 -b) were compared with empirical measurements for model validation. Figure 3.14 presents the comparison of simulated and measured battery terminal voltages. The plots on the left show the comparison with the VHDL model while the plots on the right are the comparisons with the Thevinin model. Both these models produced very similar results and shows good agreement with the measured values. However, the VHDL model proves more accurate as the battery state of charge reduces. As the simulations and experiments were carried out sequentially, the VHDL model shows that it accounts for parameters not considered in the Thevenin model. This can be seen in the fourth comparison set of Figure 3.14. The highlighted region of the graph shows that the VHDL model produces a smaller no-load battery voltage error compared to the Thevenin model. Therefore for extended battery run times, the VHDL model is more accurate. Since the execution time for both
models are very similar (20 seconds in this case), adopting the VHDL model for the rest of this work proved viable.
Simulation Measured
Simulation Measured
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Measured
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Measured
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Measured Simulation
Measured Simulation
Measured VHDL
VHDL
VHDL
VHDL
Thevenin
Thevenin
Thevenin
Thevenin Time (s)
Time (s)
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Time (s) Voltage
(V)
Voltage (V)
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error error
Figure 3.14 Comparison of terminal voltages between VHDL-AMS and Thevinin models against measured values