A New Dynamic Model for Lead-Acid Batteries N. Jantharamin*, L. Zhangt School of Electronic and Electrical Engineering, University of Leeds, United Kingdom *
[email protected],
[email protected]
effect, which consists of ohmic voltage drop and overvoltage effects. The ohmic voltage drop is caused by the resistances of the active materials, the supportive grids within the electrodes and the porous separators. On the other hand, the overvoltage effects represent the extra energy needed to force the electrochemical reaction to proceed at a required rate [2]. Consequently, the terminal voltage of the battery is always less than the EMF during discharge, while higher during charging. The polarisation effect depends on the state-ofcharge, current, temperature and the operating mode of the battery.
Keywords: Modelling, battery, lead-acid, state of charge, polarisation.
Abstract A new equivalent circuit model for lead-acid batteries is presented, taking into account internal losses due to selfdischarge and polarisation effect within a battery. This model is compact in describing both the ohmic voltage drop and overvoltage effects in a combined form of polarisation effect, using a single equivalent resistance for each operating mode. The battery electromotive force is represented by a dependent voltage source whose value changes with the state of charge. The circuit parameters expressed in non-linear function forms are used to describe the charging and discharge characteristics. The model has been derived from the manufacturer's data and validated by experiment.
1 Introduction Lead-acid batteries have been the most widely used energy storage units in stand-alone photovoltaic (PV) applications. They are cheap and most readily available. Besides, they have high nominal voltage (2 V/cell) and correspondingly high energy efficiency [1]. During charge/discharge cycles, the dynamic characteristics of the batteries depend on the state of charge, charge/discharge rate and temperature. The state of charge, SoC, in per cent is defined as
SoC = SoC o
+(_1 JIb .dTJX 100 ' en
(1)
0
where SoCo is the initial state of charge in per cent, Cn is the nominal capacity in amp-hours, and I b is the battery current in amperes, which is defined positive during charging. and negative during discharge. The battery may suffer from premature failure due to overcharge and deep discharge. In order to devise a promising control and management scheme for a battery, not only the state-of-charge estimation but also terminal voltage prediction of the battery is required. This can be achieved by deriving a mathematical model for the battery, which can be used to simulate the battery behaviour and hence design promising control strategies.
Various attempts have been made to develop a detailed mathematical model of the battery. The simplest battery model uses a series connection of an ideal voltage source and a simple resistor. Since the model parameters are constant, this model is not realistic. In addition to the series connection of the voltage source E b and the ohmic resistance R b , the Thevenin battery model includes a parallel resistor-capacitor network (RoJ/Cov ) in order to describe the overvoltage effects [3], as shown in Figure 1. However, all parameters are assumed constant, thus this model is inaccurate. An improvement upon the Thevenin model is that proposed by [4] as shown in Figure 2. The battery EMF is denoted by the voltage across the capacitor Cb • The shunt resistor Rsd represents self-discharge losses. During charging, the current flows through the ohmic resistance R be and the overvoltage resistance Rove, while during discharge the current flows through R bd and R ovd• Ideal diodes are connected in their respective paths to allow the current flowing through the desired resistances. All values of the model parameters vary with the battery EMF. Using curve-fitting technique, a very complicated exponential function is chosen to describe each parameter. Despite increased accuracy, tuning of fitting coefficients is not easy. Besides, experiments must be carried out to obtain the data for curve fitting. However, clear information about the data acquisition was not given, and the effect of charge/discharge rates on the battery performance was not presented.
2 Battery Modelling The modelling and model parameter determination is complicated by the battery's dynamic behaviour. The difference between the electromotive force (EMF) and the terminal voltage of the battery is caused by the polarisation
Figure 1: Thevenin battery model
86
4.1 Battery EMF, E b According to the data sheet, the relationship between the open-circuit voltage and the remaInIng capacity is approximately linear as shown with the dashed-line boundaries in Figure 4. Using a linear approximation technique, a linear function between E b and SoC is established: E b = O.OI375·(SoC) + 11.5 .
(2)
Figure 2: Dynamic battery model proposed by [4] 13
In [5], a modification of the model in Figure 2 is proposed. The EMF changes with the current, remaining energy and temperature of the battery. However, exhausting tests must be carried out in order to find an expression for the EMF.
12.5
~
~
w
u:
~ ~
................................
.0
.
12
:2 w
In practice, the ohmic voltage drop and overvoltage effects are identified together [2,4,5]. Therefore, the polarisation resistance can be described by a single equivalent resistance for each operating mode, and thus the model becomes more compact.
11.5
•••••••••. Data boundaries - - Approxi mation
40
20
60
80
100
State of charge, SoC [%]
Figure 4: Variation in the EMF with SoC
3 A New Dynamic Battery Model
4.2 Self-Discharge Resistance, R sd
As shown in Figure 3, a new dynamic computer model for lead-acid batteries is proposed by the authors. The battery EMF, E b , is denoted by a dependent voltage source whose value varies directly and linearly with the battery state-ofcharge. The self-discharge is represented by a shunt resistance R sd• The polarisation effect is described by a resistor-capacitor network. Ideal diodes are connected to dictate the direction of current flowing through the desired polarisation resistances, i.e. R dch during discharge and R ch during charging. Non-linear expressions for the circuit parameters can be established by means of curve-fitting technique directly using the manufacturer's data, except for R ch ' whose data must be obtained from a charging test.
The information on the remaining capacity against the storage time in the data sheet is used to model the self-discharge resistance R sd• U sing a piecewise linear function approximation, the self-discharge current I sd at 20°C is assumed to be constant within each interval of the storage time: I = t:SoC . C n , (3) ~t
100
sd
where ~t is the storage time interval and equals one month in this estimation, and ~SoC is a decrease in SoC in per cent. Subsequently, R sd is estimated as R sd
=~.
(4)
I
sd
Accordingly, R sd is plotted against SoC as shown in Figure 5. To use curve fitting technique, a quadratic polynomial function is chosen to express R sd in kn as
Rsd = -O.039·(SOC)2 + 4.27·(SoC) - 19.23 .
(5)
Equation (5) is used to simulate the changes in SoC with storage time, and the results are in agreement with those given in the data sheet, as shown in Figure 6. Figure 3: The dynamic battery model proposed by the authors
4 Model Validation using Real Battery Data In this work, a Yuasa NP4-12 battery is modelled using the proposed battery dynamic model and the battery's data sheet. The battery is the valve-regulated lead-acid type with the nominal capacity of 4 Ah and the nominal voltage of 12 V. Since the temperature effect on the discharge curves is not given by the manufacture, the battery modelling is carried out for the characteristics at 20°C or room temperature.
50
60
70
80
90
100
State of charge, SoC [0/0]
Figure 5: Variation in the self-discharge resistance with SoC
87
10 A Data sheet - - Simulation
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0.8
5
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10
15
0.2
10
468
2
20
12
Discharge current [A]
Storage time [months]
Figure 6: Capacity loss due to the self-discharge at 20°C
Figure 8: Variation in Rbdi with the discharge current
4.3 Discharge Polarisation Resistance, R tkh
From Equations (6) and (7) after the transients die away, the resistance R bd for each discharge rate can be derived as R =Eb-Vbt _ R . (10)
From the proposed model in Figure 3, the terminal voltage during discharge is expressed as Vbt = E b - IbR dch
•
[1-
exp(
Hence, the values of R bd for each discharge rate is calculated and plotted in Figure 9. It can be observed that most curves lie close to each other, except the resistance curve for 0.2 A. To simplify the modelling process, the mathematical expression for Rbd is established without consideration of the resistance curve for 0.2 A, and therefore assumed to depend only on the state of charge. Using a common exponential function, R bd can be described as
(6)
In this modelling process, the resistance R dch is divided into two components:
(7) where R bdi explains the change in the terminal voltage from the EMF during transient interval and therefore depends on the discharge current, and R bd indicates variation of R dch with the state of charge as the discharge proceeds. The discharge characteristic curves of the Yuasa NP4-12 are shown in Figure 7. As can be observed, the first sample of the terminal voltage for each curve is taken at one minute after the discharge has started. For each discharge rate, R bdi can be estimated as R
- Eb bdi -
Rbd = 2.926·exp(-0.042·SoC) .
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where R bci describes the change in the terminal voltage from the EMF during transient interval and therefore depends on the charging rate, and R bc represents variation of the internal resistance with the state of charge. Observation of the charging characteristic shows that the transient dies away after about 5 minutes. The resistance Rbci can be estimated as
= Vb! I-Eb
R
(16)
bei·
b
-Simulation
10
The varIatIon of R bc with state of charge Figure 12.
2
Time [min]
IS
shown In
2
Figure 10: Comparison of discharge curves
E .l::.
.-
1.5
.Q,
The advantage of the proposed model is that it can reasonably predict discharge curves for discharge rates other than those given by the manufacture. Figure 11 shows the curves predicted for discharge currents of 0.1, 0.3, 0.6 and 1.2 A, which are not given in the battery's data sheet.
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--.-_. Calculation -Curve fitting
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100
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State of charge, SoC [0/0]
-
I I I 1 II II I I I 1 II I III -1- -4 -l-l +.-1-+1+ I I I I1II11
I -
~
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Figure 12: Variation in R bc with SoC
I I I I I I -I-I-I-l +-1 -I I I1III1
To use curve fitting technique, a quadratic expression is chosen to describe the variation of R bc with SoC, and given as R bc = 9.32 x l0- 5·(SoC)2 +O.OI·(SoC) +0.028 . (17)
2
10
Time [min]
Figure 11: Predicted discharge curves
4.5 Charging Polarisation Resistance, R ch Since no data of charging characteristics is available in the data sheet, a charging test must be undertaken to acquire information for modelling R ch • The charging test is carried out at constant charging voltage of 14.4 V with the maximum charging current limited to 0.5 A. However, the effect of charging rate on the charging characteristics is not investigated in this work.
Based on the settling time of the voltage response, Cov for charging can be estimated. In this case the value of 40 F is obtained, which is found identical to that estimated for discharge process. Thus, one capacitance is used in the proposed model. Then, the charging characteristic at the charging rate of 0.5 A is simulated, and the results are in agreement with the testing results, as shown in Figure 13. 14
~ 13.5 ;;0 cD
C)
oS
13
"0
> "iii
12.5
c:
.~
Regarding the proposed model shown in Figure 3, the terminal voltage during charging process is expressed as
Vbt=Eb+lbReh-[l-eXP(-
~ J]. R eh COy
CD
....•....,Experiment
12
- - Simulation
t-
11.5
(13)
0
50
100
150
200
250
300
350
400
Time [min]
Figure 13: Charging curve section with 0.5A charging rate
The internal equivalent resistance Rch during charging process is divided into two components:
All parameters in the proposed model have been determined. Now, the model can be used to predict the terminal voltage
(14)
89
adjusted to match other batteries of the NP series with different nominal capacity. According to the manufacturer's manual [6], all batteries in the NP series have the same discharge characteristics for discharge currents scaled in proportion to the nominal capacity. Therefore, the fitting coefficients of the mathematical expressions of the presently proposed model can also be scaled in the same way:
and the state of charge for a continuous operation with varying currents as shown in Figure 14. 14 r--------.-----r---------r--r-------~
~
13
~
12
(1) C)
>
100
200
400
300
500
C n1 .[RatCnl land [RatCn2 l= C
Time [min]
(a) ~ 100
[Cat' at CnJ = Cn2
. [Cov at Cnl ], (19) Cn1 where R refers to R sd' Rch , and Rdch in the proposed model, Cn } is the nominal capacity of the battery modelled in this work (4 Ah), and Cn2 is the nominal capacity of any battery in the NP series required to be modelled. Moreover, the relationship between the EMF and state of charge is identical for all batteries in the series, and hence the coefficients of the EMF equation remain unchanged.
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s 5
en
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(18)
n2
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400
Time [min]
(b) Figure 14: Simulation of charge/discharge operation: (a) EMF and terminal voltage and (b) State of charge
7 Conclusions 5 Discharge Test with Varying Currents
A new compact equivalent circuit model for lead-acid batteries has been proposed. The model can be used to predict the state of charge and the terminal voltage. Most model parameters can be directly extracted from the manufacturer's data, and thus the modelling process can be quickly carried out without several exhausting experiments. The proposed model is validated using real battery data and predicts the terminal voltage with reasonable accuracies. With its reduced complexity, the model is expected to be useful both for designing battery management systems and for studying the battery ageing process.
The NP4-12 battery is now tested at room temperature. The battery is discharged at different rates during continuous operation. The experimental and simulation results are compared, as shown in Figure 15(a). The model can predict the terminal voltage with reasonable accuracies. The state of charge is also predicted under these operating conditions as shown in Figure 15(b).
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References
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800
20
1
40
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80
[4]
1
100
12-8.
8
Time [min]
(b) [5]
Figure 15: Discharge test with varying currents: (a) Terminal voltage and (b) State of charge
6 Scaling Procedures
[6]
Although the parameters of the proposed model are tuned to match the data of a given battery (Yuasa NP4-12), they can be
90
D. U. Sauer. "Electrochemical storage for photovoltaics", Handbook ofphotovoltaic science and engineering, A. Luque and S. Hegedus, Eds. Chichester, UK: John Wiley & Sons Ltd., pp. 799-862, (2003). D. Berndt. "Maintenance-free batteries", 2nd ed. Taunton, UK: Research Studies Press Ltd., 1997. H. L. Chan and D. Sutanto. "A new battery model for use with battery energy storage systems and electric vehicles power systems", IEEE Power Engineering Society Winter Meeting, volume 1, pp.470-475, (2000). Z. M. Salameh, M. A. Casacca, and W. A. Lynch. "A mathematical model for lead-acid batteries", IEEE Transactions on Energy Conversion, volume 7, no. 1, pp. 93-98, (1992). M. Diirr, A. Cmden, S. Gair, and J. R. McDonald. "Dynamic model of a lead acid battery for use in a domestic fuel cell system", Journal ofPower Sources, volume 161, no. 2, pp. 1400-1411, (2006). ---. "NP valve regulated lead acid battery manual", Yuasa Battery Sale (UK) Ltd., (1991).
