Technical Report from www.RERinfo.ca
A Lead-Acid Battery Model for Hybrid System Modelling
Prepared by: Michael Ross 2180 rue Valois Montreal, Quebec H1W 3M5
[email protected]
Presented to: Dave Turcotte CANMET Energy Diversification Research Laboratory Natural Resources Canada
Under contract #0-1238DT 2 March, 2001
Summary A simple battery model for hybrid system simulation has been developed. Given battery temperature and charge or discharge current, the model calculates the battery state-ofcharge, voltage, gassing current, heat evolution, and ageing. The model has been implemented in Simulink.
CONTENTS Introduction......................................................................................................................... 1 Inputs, Outputs and Capabilities ......................................................................................... 1 The Model........................................................................................................................... 2 Modelling State-of-Charge .............................................................................................. 2 Modelling Battery Voltage and Gassing Current ............................................................ 3 Modelling Ageing............................................................................................................ 5 Modelling Self-Discharge................................................................................................ 6 Modelling Heat Evolution ............................................................................................... 7 Modelling Gassing and Loss of Water ............................................................................ 7 Model Calibration ............................................................................................................... 8 Model Implementation........................................................................................................ 8 Recommendations............................................................................................................... 8 References........................................................................................................................... 9
Introduction The battery is one of the principal components of a hybrid power system. It is a complex device, with non-linear behaviour and memory over the short-term, medium-term and long-term. Any attempt to simulate hybrid power systems will require a model of the battery. The simulation requirements will dictate the nature of the model used. For optimising energy flows within a hybrid system, a relatively simple battery model should suffice. A lead-acid battery model has been developed for the hybrid system modelling being done at CEDRL. It is a semi-empirical model, partly taken from the literature and partly founded on basic electrochemistry. The model has been fit to measured data from two lead-acid batteries, the Global Yuasa 45T15 tubular flooded-vented 2V cell and the GNB Absolyte IIP 50A-05 12V battery. The model was originally written and fit to the measured data in Matlab; subsequently it has been implemented in Simulink. This report describes the inputs and outputs of the model, discusses its capabilities, documents the mathematical relations in the model, overviews the fitting and validation procedure, and recommends further developments to the model. Inputs, Outputs and Capabilities The inputs to the model are: 1) The current (positive for charge, negative for discharge, zero for open circuit) 2) The temperature of the battery (in ºC) The outputs of the model are: 1) The battery voltage 2) The battery State-of-Charge (SOC), given in terms of the equivalent state-of-charge for a discharge rate of C/20 at 25ºC. 3) A vector of maintenance requirements: a) A battery replacement counter, which indicates how many battery lifetimes (or the fraction thereof) have been used. b) A watering counter, which indicates how many times the battery has required watering (or the fraction thereof). 4) The heat evolved from the battery (in W, with positive indicate net heat production). 5) The battery gassing rate (in L/hr of hydrogen gas evolved). In order for the model to generate these outputs, the user must specify a set of parameters describing the battery bank to be modelled. This set of parameters consists of: 1) The nominal battery voltage (e.g., 12V, 24V, 48V, etc.) 2) The C/20 battery capacity at 25ºC (in Ah) 3) The battery technology (Global Yuasa Tubular or Absolyte IIP) 4) The self-discharge of the battery in one month (in terms of % of capacity at the outset of the month), at a temperature specified by the user. 5) The float life of the battery (in years) at a temperature specified by the user.
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6) The method by which the cycle life is to be estimated, linear or log-linear. The user specifies two points on the cycles versus SOC curve (with SOC being given for a C/20 rate at 25ºC). With the linear estimation method, these two points define a linear relationship between cycle life and SOC. With the log-linear estimation, the two points define a straight line when the natural logarithm of cyclelife is graphed against SOC. 7) The cell water loss that necessitates watering (in kg). 8) The starting SOC for the simulation, given in terms of a C/20 rate at 25ºC. The model accounts for certain elements of battery behaviour and ignores others. Note that the model accounts battery self-discharge and for the effect of temperature (over a range of approximately –5ºC to 35ºC) on battery capacity, gassing, self-discharge, and resistance. Note, however, that the model does not account for stratification of the battery nor changes in the battery operation due to ageing or sulfation. The Model Modelling State-of-Charge State-of-charge is an ill-defined metric and using it within a battery model causes several problems. Rather than deal with such problems as negative state-of-charge, state-ofcharge over 100%, etc., this battery model uses another metric of the energy stored within the battery—the charge (i.e., number of Ah) that would have to be discharged from a fully charged battery to bring it to its current state. This is an unambiguous metric that has definite physical meaning. Every time step, the “Ah discharged” variable is updated according to AhD = − ∫ (I − max (I gas , I SD ))dt + AhD0 where AhD is the number of Ah that would have to be discharged from a fully charged battery to bring it to its current state; I is the normalized charge current (in amps, negative for discharge); Igas is the gassing current (in amps, calculated elsewhere in the model, zero or positive on charge and zero on discharge); ISD is self-discharge current (in amps, always positive); t is time; and AhD0 is AhD at the beginning of the simulation. This last variable is calculated from AhD0 = C ref − C ref SOC 0 where Cref is the reference cell capacity (i.e., the capacity of the cell to which the model was fit) in Ah; and SOC0 is the initial state-of-charge specified by the user (in terms of C/20 at 25ºC). The normalized charge current, I, is calculated by
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I = I sim
C ref C sim
where Isim is the charge current input to the model (i.e., the charge current outside of the battery model) in amps; and Csim is the capacity of the battery specified for the simulation (in Ah).
Modelling Battery Voltage and Gassing Current When the charge current, I, is negative, the battery is discharging and the cell voltage, V, is calculated from a relation loosely based on [Copetti et al., 1993]: ∂V V = DVOC + OC (T − 25) − DVslope (1 − AhR ) + ∂T DP3 DP1 I (1 − DaTr (T − 25)) + DP DP DK T 1 + (− I ) 2 ( AhR + 0.0001) 4 where DVoc is a fit parameter roughly corresponding to the open-circuit voltage of the battery when it is fully charged; T is the battery temperature; DVslope is a fit parameter roughly corresponding to the partial derivative of the open circuit voltage with respect to state-of-charge; AhR is the useable amount of charge remaining in the battery; DKT is the discharge capacity of the reference cell at the temperature T, in Ah; DP1,2,3,4 are fit parameters; and DaTr is a parameter describing the change in internal battery resistance with temperature. The variable DKT is calculated by: DK T = DK (1 + DaTc(T − 25)) where DK is a fit parameter corresponding to the discharge capacity of the reference cell at an infinitesimal discharge current (i.e., the capacity of active material in the cell) in Ah; and DaTc is a parameter describing the change in the useable battery capacity with temperature.
From this, AhR is calculated by: AhR = 1 −
AhD DK T
The partial derivative of the open circuit voltage with respect to temperature (in V/ºC) is calculated based on a curve fit to data in [Vinal, 1955, p. 194]:
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∂VOC − 44.500655 + 120.18547 SG 2 − 120.79444SG 4 + 53.966821SG 6 − 9.0628981SG 8 = ∂T 1000 where SG is the specific gravity of the battery (assumed fully mixed, i.e., no stratification) measured at 25ºC and is calculated from AhD(SG full − SGempty ) SG = SG full − DK where the subscripts “full” and “empty” refer to the battery being fully charged and fully discharged.
When the charge current, I, is zero, the battery is open circuited and the cell voltage, V, equals VOC and is calculated from a curve fit to data in [Vinal, 1955, p. 193]: 0.00077765039 VOC = -168.22968 + 174.13606SG-1.4836919e SG -169.0027 SG ⋅ ln (SG)ln (SG)
When the charge current, I, is positive, the battery is charging and the cell voltage, V, is calculated based on a relation adapted from [Copetti et al., 1993]: Vmax , ∂VOC (T − 25) + CVslope AhC + V = min CVOC + ∂T CP3 I CP1 + (1 − CaTr (T − 25)) CK 1 + I CP2 1 − ( AhC + 0.0001)CP4 where Vmax is the maximum attainable voltage, i.e., the voltage that would occur with all charge current being used in the gassing reaction; CVoc is a fit parameter roughly corresponding to the open-circuit voltage of the battery when it is fully discharged; CVslope is a fit parameter roughly corresponding to the partial derivative of the open circuit voltage with respect to state-of-charge; AhC is the useable amount of charge remaining in the battery; CK is a fit parameter corresponding to the capacity of the reference cell, in Ah; CP1,2,3,4 are fit parameters; and CaTr is a parameter describing the change in internal battery resistance with temperature. The useable amount of charge remaining in the battery is calculated from: AhD AhC = 1 − CK The gassing current, Igas, is based solely upon the voltage of the battery and the temperature (the charge current and state-of-charge are only implicitly involved). The
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rate of the gassing reaction is assumed to be described by the Tafel equation, which is linearized at voltages just above the onset of gassing. Between the voltage of the onset of gassing, Vgas, and the voltage at which the linearization no longer applies, Vgas+VTafel, there is a smooth transition between the linear and the exponential curves. This is achieved by linearly weighting the two equations between these two points: Vdiff CK ⋅ τ 1 ⋅ e τ 3 Vdiff ≥ VTafel VTafelf I gas = Vdiff ⋅ CK ⋅ τ 1 ⋅ e τ 3 Vdiff V ⋅ 1 − diff + CK ⋅ τ ⋅ e τ 3 ⋅ Vdiff 0 ≤ Vdiff < VTafel 1 V V V Tafel Tafel Tafel where τ1 and τ3 are fit parameters. The gassing current is actually based on Vdiff, the difference between the battery voltage and the voltage at the onset of gassing, Vgas: Vdiff = max[0, V − (τ 2 (1 − CaTg (T − 25)))] where τ2 is a fit parameter roughly corresponding to the voltage of the onset of gassing, and CaTg is a parameter describing the change in gassing voltage with temperature. The maximum attainable voltage given the charge current, Vmax, is found from solving for Vdiff in the equation for Igas, with Igas=I, the charge current. This is an iterative process in some cases.
Modelling Ageing Three counters are used to estimate the number of battery lives used. The float life counter holds the estimate of the fraction of the “current” battery’s float life that has been used. The cycle life counter holds the estimate of the fraction of the “current” battery’s cycle life that has been used. The overall life counter holds the number of battery lives used (a counting number) plus the fraction of the “current” battery that has been used. This fraction is estimated to be equal to whichever is greater of the float life and the cycle life counters. Both the float life and cycle life counters are reset to zero, and one is added to the overall life counter, whenever either the float life or the cycle life counter reaches one. Note that this is similar to, but not exactly equivalent to, the approach used in the WATSUN PV model [WATSUN Simulation Laboratory, 1997], which at each time step decreases the remaining battery life by the greater of the cycle life ageing and float life ageing in that time step. It is generally considered that the float life of a lead-acid battery halves for every 8ºC (for non- or low-antimonial grids) to 10ºC (for antimonial grids) rise in temperature [Spiers et al., 1995]. Below 20ºC, float ageing is considered invariant with temperature [Spiers et al., 1996]. Thus the time rate of float life ageing (with t in hours) can be
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estimated from the following equation. This is integrated with respect to time to generate the float life counter. dAge float 1 1 = max , mT 20 m 8760 ⋅ β dt ⋅ 10 8760 ⋅ β ⋅ 10 float float -2 -2 where m is –3.01x10 for antimonial grids and –3.76x10 for non- or low-antimonial grids; and βfloat is calculated based on a float life (in years) and associated temperature specified by the model user. It is not clear how best to estimate the cycle life ageing. Some of the literature suggests log-linear ageing curves [Linden, 1984] while manufacturers sometimes indicate linear ageing curves (e.g., Global Yuasa). The model used here calculates that from time t to time t+dt, the cycle life ageing will be equal to: 1 1 1 dAgecycle = − 2 Cycles (SOC t + dt ) Cycles (SOCt ) This expression can be summed over time steps to determine the fraction of the battery’s cycle life that has been used. The state-of-charge is calculated from: AhD SOC = 1 − C ref With a linear cycle life ageing estimation method, the number of cycles to failure as a function of SOC is given by: Cycles(SOC ) = mcyc SOC + β cyc whereas with a log-linear cycle life ageing estimation method, the number of cycles to failure as a function of SOC is given by: m SOC Cycles(SOC ) = β cyc 10 cyc
where mcyc and βcyc are calculated based on the pair of points on the cycles versus SOC curve specified by the user. Note that if the user specifies a very flat cycle life curve, this approach will underestimate the rate of cycle life ageing. That is, if the SOC has little effect on the number of cycles before failure, this approach will not be appropriate. For most real batteries, and certainly most lead-acid batteries, the above approach appears reasonable.
Modelling Self-Discharge The rate of self-discharge rises exponentially with temperature; low- or no-antimony grids have a lower rate of self-discharge than high antimony grids. The self-discharge current is calculated by: − ln 1 − β SD ⋅ 10 mSDT I SD = (C ref − AhD ) ⋅ 720
( (
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))
where mSD is 0.027, and based on the slope of a graph given in [Linden, 1984, p.3-18] and information given in [Spiers et al., 1998]; and βSD is calculated based on the monthly self-discharge rate furnished by the user: SD β SD = mmonth 10 SDT where SDmonth is the fraction of the battery’s capacity that is discharged in one month at a temperature of T.
Modelling Heat Evolution Three sources of heat are modelled here: 1) “Joule” Heat: The heat due to the losses causing the overpotential, i.e., the current multiplied by the entire polarisation overpotential. 2) “Reversible” Heat: The heat, liberated by the battery on charge and absorbed on discharge, corresponding to the difference between the heat of reaction and the useful work done. 3) “Gassing” Heat: The heat absorbed when water is electrolyzed in the gassing reaction and the hydrogen and oxygen gasses thus produced escape the battery; it corresponds to the difference between the heat of reaction and the electric work done to perform the electrolysis. On discharge, current is negative and the rate of heat generation (in Watts, with positive indicating heat liberated from the battery) is calculated by: ∂V Pheat = − I ⋅ VOC − V − I ⋅ (T + 273.15) ⋅ OC ∂T The former term represents the joule heat and the latter term the reversible heat. On charge, the rate of heat generation is calculated by: ∂VOC 48.44 × 10 3 Pheat = I ⋅ VOC − V + I ⋅ (T + 273.15) ⋅ + I gas* 1.23 − V − I gas* ⋅ (1 − η recomb ) ∂T 2 ⋅ 96500 where Igas* is the greater of ISD and Igas; and ηrecomb is the fraction of oxygen gas evolved that is recombined within the battery. In the above equation, the first term represents the joule heat of the charge reaction, the second the reversible heat of the charge reaction, the third the joule heat of the gassing reaction, and the fourth the gassing heat.
Modelling Gassing and Loss of Water The accumulated water loss from the battery, MH2O, expressed as a fraction of the maximum permissible water loss, MH2O (in kg), is calculated from:
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M H 2O = ∫
3.357 × 10 −4 ⋅ I gas* Μ H 2O
dt
The rate of hydrogen gas evolution, in L/hr, is equal to the 0.4178Igas*. Model Calibration
The model was calibrated, i.e., the fit parameters were determined, by a nonlinear leastsquares method. The current-voltage-SOC relations of the model were implemented as MATLAB functions (one for charge and the other for discharge). These functions were then repeatedly called by a MATLAB nonlinear least-squares method until the best fit was achieved for a set of charge and discharge curves for Global Yuasa and Absolyte IIP batteries. These curves were taken from battery testing done a CEDRL in 1997 and 1998. Four curves were used, corresponding to roughly C/10, C/20, C/50, and C/200 rates. In order to weight each curve equally, all curves were adjusted so that they had the same number of data points. One problem that arose in this fitting procedure was the lack of data at high voltages, i.e., in the gassing region. In order to fit the Global Yuasa battery in this region, the C/10 rate curve was extended above 2.5 V based on curves for tubular batteries in [Copetti et al., 199X]. For the Absolyte IIP 50A-05 battery, the gassing parameters were calculated directly in order to achieve a gassing plateau of 15.5V (2.58V per cell) at a current of 11A (roughly C/10) and a temperature of 25ºC, with gassing starting at 2.28V per cell. That is, under the above conditions, when all current is causing gassing, the battery would be at 15.5V. Another problem that arose was the lack of data at temperatures below 25ºC. Because only a few curves were available, and these only at temperatures of –5ºC and –15ºC, the temperature adjustment factors found in [Copetti et al., 1993] were used. Model Implementation
The model was implemented in Simulink. It was tested and debugged using a series of basic current and temperature cycling regimes. The only important consideration in the operation of the model is that within the Simulink solver zero crossing checking must be disabled. Recommendations
1) The model works quite well and has a number of somewhat novel elements that are arguably improvements on the models found in the literature. As it is presently implemented, it could form the basis of a conference article. With further work it might form the basis of a journal article. At minimum, a conference article should be published. 2) Further battery testing should be done at temperatures below 25ºC, at high voltages, and at high currents. Furthermore, the cycling behaviour of the battery should be
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documented and compared with that predicted by the model in order to determine how seriously stratification will affect the model accuracy. 3) A number of possible improvements to the model should be investigated: • Rather than fit the parameters DVOC, DVslope, CVOC, and CVslope to the curves, the open circuit voltage should be calculated using the curve for VOC. This would simplify the model, put it on a sound electrochemical foundation, and possibly improve the low temperature charge performance of the model, particularly for the Absolyte IIP battery. • Rather than apply a temperature correction term to the whole of the “overpotential” part of the charge and discharge voltage relations, the temperature correction should be applied to only the term with current in the denominator. It may be possible to infer this relation from the change in resistivity of sulfuric acid with temperature. • Change the form of the gassing equation, such that gassing commences at the theoretical onset of gassing, rises linearly to a fit parameter describing the voltage at which the exponential nature of the Tafel equation becomes evident. At this voltage, the numerator of the exponent would be 0, not VTafel, as it is now. This would improve the model’s ability to predict battery efficiency. • Directly incorporate the gassing current into the voltage relation for charging. That is, Igas should be subtracted from I for the term of the charge relation that includes AhC in the denominator. This would improve the model’s performance when the battery in nearly fully charged. • Attempt to account for stratification, perhaps by modelling the cell as three or more parallel grids, in different concentration solutions of sulfuric acid. • After the above improvements, see if fit exponents can be replaced with more physically justified exponents. 4) The model should be made available, in its Simulink implementation, on the Internet. This would permit outside parties to comment on and suggest improvements to the model. References
Copetti, J.B., E. Lorenzo, and F. Chenlo. “A General Battery Model for PV System Simulation”. Progress in Photovoltaics. Vol. 1, pp. 283-292 (1993). Copetti, J.B., F. Chenlo, and E. Lorenzo. “Comparison between Charge and Discharge Batteries Models and Real Data for PV Applications”. Source unknown. 199X. Linden, David. Handbook of Batteries and Fuel Cells. New York: McGraw-Hill Publishing Company, 1984. Spiers, David J. and Asko D. Rasinkoski. “Predicting the service lifetime of lead/acid batteries in photovoltaic systems”. Journal of Power Sources. Vol. 53, pp. 245-253 (1995).
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Spiers, David J. and Asko A. Rasinkoski. “Limits to Battery Lifetime in Photovoltaic Applications”. Solar Energy. Vol. 58, pp. 147-154 (1996). Spiers, David and Jimmy Royer. Guidelines for the Use of Batteries in Photovoltaic Systems. Neste Advanced Power Systems/Natural Resources Canada, 1998. Vinal, George Wood. Storage Batteries (4th edition). New York: John Wiley & Sons, 1955. Watsun Simulation Laboratory. Watsun-PV 6.0 User’s Manual and Program Documentation. Waterloo, Ontario: University of Waterloo, 1997.
Appendix: Changes to Battery Model
(2002/01/08) When implemented in Simulink, the combination of charge controller and battery was not stable when on float charge at low voltages (e.g., 2.2 to 2.3 V per cell). The battery voltage was overly sensitive to charge current; this caused the variable time step solver to reduce the time step, resulting in a slow simulation. Perturbations to the available charge current (i.e., the output of the array) would often cause the simulation to halt with an error message indicating an instability. The way that the model deals with gassing was changed in order to make the model more stable. At the low voltages typical of float charge, gassing now occurs at a low rate. This means that the currents at low voltages are higher than before, making the battery voltage less sensitive to small perturbations in the charge current. The new expression for Igas on charge is:
I gas
Vonset −τ 2 V −τ 2 τ3 − CKτ 1e τ 3 min (I − I SD ), CKτ 1e = 0
V ≥ Vonset V < Vonset
where τ1, τ2, and τ3 are fit parameters and Vonset is a constant describing the lowest voltage at which any gassing occurs. This is set to 2.17V, which should be lower than any realistic float setpoint. In effect, the new gassing curve is an exponential curve, with the curve shifted downwards such that Igas intersects the I=0 axis at the voltage Vonset. Since this ignores the linearity of the Tafel curve at low voltages, this is somewhat less realistic than the previous relation, in theory. In practice, it gives a good fit to the measured battery curves.
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The low gassing current suggested by the model when the battery is on float is quite realistic. Even though the gassing voltage for a lead acid cell is commonly given as 2.39 V at 25ºC, this is the voltage at which the exponential nature of the tafel curve becomes evident. Electrolysis can begin at voltages as low as 1.2 V, and gassing can be observed visually in batteries well below 2.39 V. The equation for the battery voltage on charge has been changed slightly as well. Whereas before the voltage was limited to Vmax, the voltage attained when all current was used in gassing, now there is no limit on voltage but the gassing current is limited to the total charge current, I-ISD. The current used to calculate the battery voltage is now I-ISD, i.e., it is reduced by the self-discharge current. These changes result in more realistic battery output, simplify the model, and make for more stable operation when combined with the charge controller model in Simulink. A change has also been made to how the self-discharge current is treated when calculating the state of charge. Now this is calculated as: AhD = − ∫ (I − (I gas + I SD ))dt + AhD0
That is, self-discharge is considered to occur at all times, even during gassing. This is physically justified, but leads to some peculiar behaviour when all the current is going into gassing: state-of-charge reaches a plateau below a full state-of-charge. This minor problem arises out of inadequacies in the way the gassing current is calculated, not in the assumption that self-discharge occurs at all times.
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