[american institute of aeronautics and astronautics 35th intersociety energy conversion engineering...

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AOO-37830 AIAA-2000-2976

STOCHASTIC MODELING OF RECHARGEABLE BATTERY LIFEIN A PHOTOVOLTAIC POWER SYSTEM

Angel Urbina, Thomas L. PaezSandia National Laboratories

Validation and Uncertainty Quantification Processes, 9133Mail Stop 0557

Albuquerque, NM 87185-0557

Rudolph G. JungstSandia National Laboratories

Lithium Battery R&D Department, 2521Mail Stop 0613

Albuquerque, NM 87185-0613

ABSTRACT

We have developed a stochastic model for the powergenerated by a photovoltaic (PV) power supply systemthat includes a rechargeable energy storage device. Theultimate objective of this work is to integrate thisphotovoltaic generator along with other generationsources to perform power flow calculations to estimatethe reliability of different electricity gridconfigurations. For this reason, the photovoltaic powersupply model must provide robust, efficient realizationsof the photovoltaic electricity output under a variety ofconditions and at different geographical locations. Thishas been achieved by use of a Karhunen-Loeveframework to model the solar insolation data. Thecapacity of the energy storage device, in this case alead-acid battery, is represented by a deterministicmodel that uses an artificial neural network to estimatethe reduction in capacity that occurs over time. Whencombined with an appropriate stochastic load model, allthree elements yield a stochastic model for thephotovoltaic power system. This model has beenoperated on the Monte Carlo principle in stand-alonemode to infer the probabilistic behavior of the system.In particular, numerical examples are shown toillustrate the use of the model to estimate battery life.By the end of one year of operation, there is a 50%probability for the test case shown that the battery willbe at or below 95% of initial capacity.

1 Copyright © 2000 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved.

INTRODUCTION

Renewable energy generators (PV, wind, etc.) arefinding increased use for providing power to remoteareas that do not have access to the electricity grid.Another useful characteristic of these power sources isSow operating cost compared to traditional generationsources such as combustion turbines. Because they canoften be sited close to the power demand in distributedgeneration systems, renewable generators also areclaimed to have a positive impact on the reliability ofthe electricity grid in general. Sandia NationalLaboratories has been developing power grid models toquantify the reliability of the National power grid andstudy how it can be improved. In order to includerenewable generation in this study, Sandia hasdeveloped the stochastic model for a photovoltaicpower system.

An energy storage system is necessary to allow the PVgenerator power to be dispatchable and to continue toservice the load during periods of low PV output. Thephotovoltaic-based power supply must be sized tosatisfactorily serve the system load and also tosufficiently charge the energy storage sub-system(typically a lead-acid battery) that will continue tosupply the system load while the photovoltaic system isnot generating power. Optimal sizing of the systemmust therefore account for the facts that the powergenerated by the system and the power used by the loadare stochastic processes and that continued deepdischarge of the energy storage device can causeaccelerated capacity loss in the case of a lead-acid

995


battery. Another use for the stochastic renewablegenerator model is therefore to provide a predictivecapability for the performance of the battery undervarious design and operating options. This can be usedto optimize the system for the best trade off amongcost, load requirements, and battery life.

The stochastic photovoltaic power system model willbe described for each of its three elements. Then,numerical example will be presented showing theperformance and capabilities of the software.Ultimately the model may be used to design andoptimize renewable energy power systemsincorporating energy storage.

PV/RECHARGEABLE ENERGY STORAGEMODEL

The present investigation creates a framework for theanalysis of the performance of a photovoltaic powersupply system. The power output of the PV array ismodeled as a non-Gaussian stochastic process. Thestochastic process characteristics are known primarilythrough statistics of solar insolation taken from datameasured at various geographic locations. Thisinvestigation models insolation at particular locationsusing measured statistical data in a Karhunen-Loeveframework. This robust expansion permits therepresentation of insolation in terms of a small numberof deterministic eigenfunctions and non-Gaussianrandom variable coefficients. Our approach uses acanonical variate analysis (CVA) technique to simulatetwo main components of the solar radiation, directnormal and diffuse horizontal radiation. In addition, theload profile is modeled as deterministic and/orstochastic, depending on its operating characteristics.The stochastic photovoltaic system and load modelsallow us to simulate the charge/discharge cyclesexperienced by the energy storage device during use inthe photovoltaic supply environment.

The behavior of the rechargeable energy storage device,in particular the damage caused in a lead-acid batteryby prolonged periods of deep discharge, is modeled bymeans of an artificial neural network (ANN). Thisdeterministic battery model reflects the damage tocapacity caused by battery use at low state-of-chargelevels. Damage has been observed experimentally, butbecause the functional form of the relationship betweencapacity and deep discharge is unknown, and ispresumed to vary with battery type, we model it with anANN whose parameters are to be trained withexperimental data. A generalization of the radial basisfunction ANN is used to represent battery damage as a

function of depth of discharge and duration ofdischarge. Data obtained experimentally serve asexemplars for training the ANN. At this time, onlylimited battery data that characterize damage associatedwith discharge periods are available. Therefore, thesedata were augmented with plausible, synthetic data fortraining.

All these elements are combined into a singleframework to yield a stochastic model for the powersupply/energy storage/load system. Figure 1 shows aschematic representation of this system. The model isoperated on the Monte Carlo principle to yieldrealizations of the stochastic processes characteristic ofthe operational phenomena, and these can be analyzedusing the tools of statistics to infer the probabilisticbehavior of the system. We shall now discuss the solarresource and energy storage device simulators in moredetail.

Solar Resource Photovoltaic Array

Batteries

Loads

Figure 1: Power supply/energy storage/load system

Soiar Resource Simulator

In previous work, we used a bivariate Markov processto simulate the direct normal and diffuse horizontalsolar radiation data. This technique yields accuraterealizations of the random process but it is very CPUintensive. In light of this, we seek a faster method thatwill be as accurate as, or more accurate than our currenttechnique.

The main focus of this investigation is to implement aCVA approach to modeling the daily direct normal anddiffuse horizontal radiation components. CVA hasbeen successfully used to model random processes,such as the dynamic response of a nonlinear mechanicalsystem (see Paez and Hunter1). In the following

996


section we describe the CVA approach, and the resultof its implementation is presented later in the paper.

Canonical Variate Analysis

Canonical variate analysis is an extension of the auto-regressive moving average (ARMA) modelingtechnique. Originally developed by Hotelling2, CVAwas improved by Larimore3. Larimore'simplementation of CVA permits accurate and efficientsimulation of random processes. CVA is described indetail in several references (see Larimore3, Hunter4).

CVA involves three critical transformations, namely,the measurement to state transformation, the evolutionof past states to future states, and the future state toestimated measurement transformation. Equations 1through 4, below, implement these steps.

Let *\(jl i =...,-1,0,1,..., denote a (possiblymultivariate) measured excitation, at times*j = JrJ = .-.,-1,0,1,... , and let y ( t j } j = ...-1,0,1,... ,denote the corresponding (possibly multivariate)measured response. The first operation of CVAtransforms measurements of the system "past", p, into astate space s.

where the system past, relative to time t\ , is defined:

y(t}-lr) x(t})(la)

The transformation matrix, J, will be defined below.The ensemble of past signals, P, and the ensemble offuture signals, F, of the system are defined as follows,relative to time ?; , in terms of measured excitation andresponse.

P =

(t\ - 7) - • • y(h - IT)-jr)

y(tk -r}-y(tk -lr}x(tk}-x(tk - j

(2)

F = (2a)

Past states are evolved into future states, and futurestates are inverse transformed into the measuredresponse space using the standard modal analysisframework. That framework is characterized by theequations:

s(t + T) = As(t)+ Bx(t)+ e(t]y(t) = Cs(t)+ Dx(t)+ Ee(t)+ w(t) (3)

where A, B, C, and D, are system matrices, e(t] andw(t) are noise vectors, and E accounts for state modelnoise in the state to measurement transformation. Themeasurement to state transform defined by the matrix Jin Equation 1 is developed as follows:

SVD1/2

= UWVT(4)

where SVD[.] indicates the singular valuedecomposition. The gist of the operations carried out inthese equations is to, first, establish the autocorrelationmatrices of the system past, P, and future, F, usingensembles of segments of the measured excitation andresponse. These autocorrelations are decomposed usingthe Karhunen-Loeve expansion (see Ghanem andSpanos5). The principal components are retained, andtheir cross-correlations are orthogonalized. This yieldsthe transformation matrix, /, a measurement to statespace transformation that yields an optimal relationbetween past and future principal components. Next allmeasured data segments are transformed into the statespace using /. The state transition and other state spacematrices, A, B, C, and D, are identified using linearleast squares. The transformation and state spaceparameters are identified using P and F, ensembles ofsignals. Response predictions operate onp to predict/.

A global model can be created using all the data, or alocal model can be created using data in theneighborhood of a particular past, p. In this way, wecreate global and local linear models. In thisinvestigation, local linear models are used tocharacterize the joint behavior of direct normal anddiffuse horizontal insolation. Further, we take theexcitation, j, to be zero in this application.

997


Equation 4 results from minimizing the mean squareerror in the prediction of the future / from the past p.The process of transforming the measurements p tocritical waveforms s, transforming past states s to futurestates, and finally returning to the measurement domainfor / may seem awkward, but in fact is considerablymore stable than directly predicting / from p.Intuitively, the gain comes from minimizing the numberof parameters necessary in the past to future predictionprocess.

With the approach outlined above, we generaterealizations of daily direct normal and diffusehorizontal radiation and subsequently use them as thepower source for our model. In the next section weexamine the energy storage component of our model.

RECHARGEABLE ENERGY STORAGE MODEL

A lead-acid battery will serve as the energy storagedevice for this model, allowing the power from the PVsource to be dispatchable. Due to the random nature ofthe PV resource, it is expected that the battery willexperience periods of deficit charging, which areknown to cause a degradation of the battery capacity. Ifthese periods are frequent, they can eventually lead tobattery failure. One goal of this research is to modelthis damage mechanism via a non-phenomenologicaltechnique, in this case using an ANN. The followingsections describe the particular ANN used in this work.

Artificial Neural Networks

A focus of this investigation is to use an ANN to modelthe damage caused in a rechargeable lead-acid batterydue to extended discharge periods. Due to the complexbehavior of a rechargeable battery, an explicit formrelating the capacity loss to the depth and duration of adischarge cannot be easily derived. This is partly due tothe expected variability in the results obtained from thedifferent batteries that will be tested and partly due tothe complexity of the electrochemical reactionsinvolved in the process. It is thus convenient tosimulate the experimental data using a non-phenomenological modeling technique to characterizethe relationship between the inputs and desired outputs.In this case, the output is the damage introduced to thebattery due to discharge cycles and the inputs are theduration and depth of the discharge. Previous work,such as O'Gorman et al6, Urbina et al1, Chan et a/8,Yamazaki et af and Peng et al10, have successfullydemonstrated the modeling of primary and secondary

battery systems using ANN. In the following section, abrief presentation of the specific ANN algorithm usedin this project is given.

The Multivariate Polynomial Spline ANN

The multivariate polynomial spline (MVPS) network isan artificial neural network of the radial basis functiontype. The radial basis function ANN, which wasdeveloped by Moody and Darken", simulates mappingsvia the superposition of radial basis functions. Theradial basis function ANN is an accurate localapproximator, and although it trains rapidly, it has thepotential for size difficulties as the dimension of theinput space grows. A generalization of the radial basisfunction ANN is the connectionist normalized linearspline (CNLS) network. This was developed by Joneset al.12 and seeks to simulate a mapping by using radialbasis functions in a higher order approximation than theradial basis function network. The MVPS networkgeneralizes the CNLS network to multiple outputdimensions and higher degree local approximations.The development of the mathematical frameworkbehind the MVPS network is presented in thefollowing.

First, let X be an n-dimensional input vector to thesystem being modeled, and let Z be its correspondingm-dimensional output vector. The mapping from X to Zis denoted by:

(5)

Assume that the function g(X) is deterministic but thatits form and parameters are unknown. Write an identitythat equates g(X) to itself, then multiply both sides by aweighting function (radial basis function) that iscentered at Cf.

g(X) W(X, = g(X) w(X,(6)

Expand the function g(X) on the right hand side ofEquation 6 using the first three terms in the Taylorseries:

g(X) w(X,(X -

(7)

where A0, A\ and A2 are the coefficients of the localmodels, C, is the "center" of the/h local approximation(also referred to as center vector or center), £ contains

998

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

the quadratic and cross terms obtained from theelements in X, /3 is a parameter related to the width ofthe radial basis function and w(X,Cj,/7) are the weightsattached to the local models (in this case a multivariateGaussian probability density function is chosen as theweighting expression).

This approximation can be optimized in theneighborhood of Cj using least squares or weightedleast squares and is accurate in the vicinity of the dataused to develop it as long as the behavior of themapping in the neighborhood is closely represented bythe Taylor series. Approximations can be developed inall neighborhoods of the input vector space. Thesubscript/ (where j = 0,1..JV, and N is the number ofregions of approximation) accounts for this inEquations 6 and 7.

The local approximations of the X to Z mapping cannow be combined to create an approximate, global map.To accomplish this, several of the approximations aresuperimposed in a series. Each component in the seriesis weighted according to its distance from the inputvector X. Local models that are near X are weightedheavily, whereas those that are further away areweighted less. The series is obtained by summing theterms in Equation 3 as follows:

(8)

Since g(X) on the left-hand side of Equation 8 isindependent of the index j, it can be removed from thesummation. The resulting expression can benormalized to obtain:

OJ (X -

(9)

Equation 9 represents the parametric form of the MVPSnetwork. Figure 2 shows a single input/single outputexample of local polynomial models fitted to arbitraryinput/output data. It can be seen from the figure that fora highly nonlinear response, the local polynomialmodeling is quite appropriate. Ultimately, and asdescribed in the above paragraphs, the local models aresplined together to form a global model.

The MVPS network is used in feed forward operationby specifying the input vector X, evaluating the weightsw, calculating <£ substituting the weights and £ intoEquation 9, and evaluating the output g(X). This outputis an interpolation among the training outputs. Notethat the range of the summation index/ is not specifiedin Equation 9. It is clear that the summation should becarried out over those local models nearest the inputvector X. The ANN computer code used in the presentinvestigation accepts a user-defined number of localmodels and the ones nearest, in Cartesian space, to theinput vector are chosen to make each prediction. Usingthe MVPS ANN, we developed a model to represent thedamage that accumulates in a lead-acid battery whenexposed to periods of deficit charging.

Figure 2: Local polynomial models

ANN-Based Lead-Acid Battery Damage Model

The objective of the rechargeable battery portion of thepower supply model is to simulate the loss of capacityof the battery with time as it is subjected to a cyclingenvironment typical of photovoltaic applications. Inaddition to the normal wear out of the battery thatoccurs as a function of number of cycles, depth ofdischarge, discharge rate, and temperature, batteries arefrequently exposed to periods of deficit charging in PVapplications. These accelerate degradation of capacityin lead-acid systems due to sulfation of the batteryplates. It is assumed in this study that deficit chargingis the major factor causing accelerated loss of capacityand therefore shortened life of batteries in PV powersupplies. Experimental measurements will be aimed atdetermining what particular deficit charge conditionshave the greatest effect on capacity loss. Twoparameters were selected to characterize the deficitcharge, namely maximum depth of discharge, D, andthe duration of the discharge, T. Figure 3 shows thesevariables.

999


The maximum depth of discharge, D is defined as thedeepest discharge, (as measured from a referencethreshold, Cd) encountered during a deficit chargeperiod. The duration of the discharge, T is defined asthe time it takes to return the capacity to the referencelevel, Cd. (This is a function of the amount of solarinsolation available.) A moving average was used torepresent the average daily capacity at any time.

the ANN description of the damage surface can beeasily updated and refined. It is important to note thatalthough reasonable, this initial model is only based onlimited data and must be revised.

0

„ -0.05c.<£

5 -0.1%I -0,5•sI -0.2

-0,250

Depth of discharge (percent)100 600

Duration of discharge (hrs)

800 !000 1200Time (hre)

1400 1600 1800 2000 Figure 4: Proposed battery damage model

Figure 3: Definition of depth and duration of discharge

For initial calibration of the battery damage, the modeldraws on data collected earlier at the Florida SolarEnergy Center (FSEC). Different charge to load ratiosand discharge rates were studied to determine the effecton cycle life. For details on these tests see Reference13. Although these data give some indication of thereduction in cycle life that occurs due to extended timesat low state-of-charge (SOC), there are too few datapoints to completely define the battery damage surface.Consequently, a new set of deficit charge cycle tests hasbeen planned that will provide a large enough data setto complete training of the ANN.

The focus in the new tests is to be on the damage due todeficit charging, so only a single discharge rate, C/20,will be used. A few sustaining cycles will be run beforeand after the deficit charge region before measuring thecapacity loss; however, batteries will not be cycled toend-of-life. This will shorten the experiments to a fewweeks maximum and will allow a greater variety ofdeficit charge characteristics (depth and duration ofdischarge) to be examined. Complex impedance datawill also be collected to investigate whether smallamounts of battery degradation can be detected earlierin the tests by this method.

Figure 4 shows the battery damage model which wesimulated using an MVPS ANN. As experimentalmeasurements of battery capacity loss are completed,

TEST CASE AND RESULTS

A test case was set up to exercise the PV/rechargeablebattery/load model. The model was operated on theMonte Carlo principle to yield realizations of the hourlysolar insolation data, using the CVA techniquedescribed earlier. The solar insolation along with theload profile was input to the lead-acid battery modeland hourly capacity cycling data were produced.

The specific test case consisted of a PV array located inAlbuquerque, NM (latitude 35° 03'). The tilt angle wasset at 50° (which enhances power generation in thewinter months) and the azimuth angle was set to 0°(which means that the array is facing due South). ThePV array is rated at 2.1 amps at standard test conditions(i.e. 1000 w/m2 and 25°C temperature) and has anominal voltage of 12 volts. Three PV modules wereconnected in parallel to give a current of 6.3 amps. A105 Ah lead-acid battery serves as the energy storagedevice for this example. It is assumed that the capacitythreshold, Cd, below which a discharge starts to causedamage, is 52 Ah (50% of initial capacity). The batterydamage surface shown in Figure 4 is simulated via theMVPS ANN. The applied load consists of a constantload during nighttime and a randomly applied loadduring the daytime. This load profile requires anaverage of 30 Ah per day/night.

1000


The following results were obtained using simulatedPV data based on 10 years of actual insolationmeasurements for the Albuquerque, NM area.

Figure 5a shows measured hourly insolation data for themonth of January 1978. Figure 5b shows a realizationof hourly insolation data for a simulated January asobtained with our PV simulator software. It isimportant to note that these realizations will not beidentical because they are random process realizations,but their characters should be similar because therandom process source of Figure 5b is meant tosimulate the random process source of Figure 5 a. Thatis, there are days in which the insolation is low andothers in which it is high.

1.2

f 1

§ 0.8a0

I"£2 0.4°

0.2

0

i

i

ft

ii1

1 i

I Ml

_

-

-

"1

0 100 200 300 400 SOO 600 700 800Hours

Figure 5a: Measured total hourly insolation for January1978

300 400 500 600 700Hours

Figure 5b: Simulated total hourly insolation for 31January days

Figure 6 shows a time history for one simulated monthof estimated available capacity data (shown as a solidline) and the maximum potential capacity, M (shown as

a heavy solid line). M is the initial capacity minus theirreversible capacity loss that accumulates due to deepdischarge events and also due to regular daily cycling.As expected, there is a drop in the maximum potentialcapacity where the deep discharges occur. This isconsistent with our ANN model shown in Figure 4.That is, more damage is caused to the battery duringdeep discharges with long duration.

-r 105.2

!03.894 187 280 373 466 559 652

Hours

Figure 6: Capacity simulation using a random load

A first passage probability analysis was performed forthe maximum potential capacity, M, and it is shown inFigure 7. This shows the probability that a batterypasses a specific capacity threshold prior to any giventime. The results show that by approximately 4500hours of operation (approximately 6 months), thebattery's maximum potential capacity will be below99% of the initial capacity with probability one. By theend of one year of operation, there is about an 80%probability that the battery will be at or below 97% ofthe initial capacity and about 50% probability that itwill be at 95% or less of the initial capacity.

Figure 7 also shows a comparison of first passageprobability distributions for maximum potentialcapacity, M, at several different barrier levels. The solidlines are the first passage probability distributions thatresult from Monte Carlo analysis of 100 insolationrealizations. The dashed lines are the first passageprobability distributions that result from Monte Carloanalysis of ten years of measured insolation data. (Thelatter curves possess a stepped character because thenumber of years of data measurement is relativelysmall.) Some possible reasons for disagreement in theshort term are (1) the limited amount of measured data,and (2) the detailed behavior of the simulation.Importantly, as the probability of first passage failureincreases, the results of the simulations converge to theresults based on measured data.

1001


This type of analysis allows us to make decisions suchas when the optimal time to replace a battery has beenreached and how to size the PV/battery system toachieve the desired reliability for a particularapplication.

99% of

97% of

0 1000 2000 3000 4000 5000 6000 7000 8000 9(iOO

HoursFigure 7: First passage probability of M (solid line =using simulated data; dashed line = using actual data)

One of our objectives in implementing the CVAtechnique was to reduce the computational timerequired to generate yearly realizations of the solarinsolation. Using the Markov approach of reference 7,the CPU time to generate a year-long realization was2.4 minutes. Using this new CVA approach the time isreduced to 1.2 minutes without a loss in accuracy. Thistime reduction is critical since our PV generator willeventually be integrated into a larger power flowanalysis code.

SUMMARY

The objective of this investigation was to develop aframework for simulating the power output from aphotovoltaic system, including an energy storagedevice. The model has also been demonstrated to becapable of assessing the reliability and life ofrechargeable batteries in the PV use environment. Thisinformation can be applied to optimize PV systemdesigns and establish more cost-effective batteryreplacement schedules. A means for simulating thesolar insolation random process was required, and wesought an efficient technique for the generation ofrandom process realizations, where the random processis only characterized by historical records of thephenomenon. We chose to use the canonical variateanalysis approach, a technique that has been widelyused to simulate dynamical systems. The CVA

approach was used to generate realizations of the twomain components of solar insolation.

The current system uses lead-acid batteries for energystorage. It is acknowledged that damage can accumulatein rechargeable batteries when they are used in a deficitcharge environment, such as PV. Damage modelingwas performed using a multivariate polynomial splineartificial neural network. The system load was modeledas a combined deterministic/stationary random process.

The results of the investigation and numerical examplesindicate that plausible simulations of available batterycapacity and maximum potential capacity, M, can begenerated. Indeed, the important measure of modelperformance embodied in the first passage probabilitydistributions for M shows that the system performancesimulations are accurate. Calculations for a test casepredict good retention of battery capacity for at leastone year. It remains to investigate more metrics ofsystem performance to assure that system simulationsare accurate in all details.

ACKNOWLEDGEMENTS

Sandia National Laboratories is a multi-programlaboratory operated by Sandia Corporation, a LockheedMartin Company, for the United States Department ofEnergy under Contract DE-AC04-94AL8500Q.

REFERENCES

1. Paez, T., Hunter, N., (1999), "Dynamical SystemModeling Via Signal Reduction and NeuralNetwork Modeling," Proceedings of the 70th Shockand Vibration Symposium.

2. Retelling, H., (1936), "Relations Between TwoSets of Variates," Biometrica, V. 28, pp. 321-377.

3. Larimore, W., (1983), "System Identification,reduced Order Filtering, and Modeling ViaCanonical Variate Analysis," Proceedings of theAmerican Control Conference, H. Rao, P. Dorato,eds.

4. Hunter, N., (1997), "State Analysis of NonlinearSystems Using Local Canonical Variate Analysis,"Proceedings of the 30?h Hawaii InternationalConference on System Science.

5. Ghanem, R., Spanos, P., (1991), Stochastic FiniteElements: A Spectral Approach, Springer-Verlag,New York.

1002


6. O'Gorman, C. C., R. G. Jungst, D. Ingersoll, and T.L. Paez, (1998), "Artificial Neural NetworkSimulation of Battery Performance", Proceedingsof the 194th Electrochemical Society Meeting,Abstract #66.

7. Urbina, A., T. L. Paez, C. C. O'Gorman P. Barney,R. G. Jungst and D. Ingersoll, (1999), "Reliabilityof Rechargeable Batteries in a Photovoltaic PowerSupply System", Journal of Power Sources 80, pp.30-38.

8. Chan, C. C., E. W. C. Lo, and S. Weixiang, (2000),"The available capacity computation model basedon artificial neural network for lead-acid batteriesin electric vehicles," Journal of Power Sources 87,pp. 201-204.

9. Yamazaki, T., K. Sakurai, and K.-I. Muramoto,(1998), "Estimation of Residual Capacity of sealedLead-Acid Batteries by Neural Network,"Proceedings of the Twentieth InternationalTelecommunications Energy Conference, pp. 210-214.

10. Peng, 1, Y. Chen, and R. Eberhart, (2000),"Battery Pack State of Charge Estimator DesignUsing Computational Intelligence Approaches,"Proceedings of the Fifteenth Annual BatteryConference on Applications and Advances, pp.173-177.

11. Moody, J. and C. Darken, (1989), NeuralComputation, (1)2, pp. 281-294.

12. Jones, R. D., et al., (1990), Cognitive Modeling inSystem Control, The Santa Fe Institute.

13. Hund, T, (1996), Battery Testing for PhotovoltaicApplications, Sandia National Laboratories, reportnumber: SAND96-2794C. (This report can also befound at the following Internet address:http://www.sandia.gov/pv/bos/batteries.htm).

1003

[american institute of aeronautics and astronautics 35th intersociety energy conversion engineering...

Documents