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State-of-Charge/Health W Waag and DU Sauer, RWTH Aachen University, Aachen, Germany & 2009 Elsevier B.V. All rights reserved.
Introduction State-of-charge (SoC) and state-of-health (SoH) are two important figures of merit that describe the state of a battery. In particular, in this article these figures of merit are discussed in the context of lead–acid batteries (LABs). A wide range of definitions exist, but many of them are often not clear and unequivocal. The following sections present the complexity of these definitions as well as methods for determination of SoC and SoH in laboratory and field applications under real-world conditions.
•
State-of-Charge State-of-charge is generally defined as an actually available amount of charge in a given battery (Q ) related to the maximum available amount of charge, which can be taken from this battery after a 100% full charging (C ) and is usually expressed as a percentage: SoC ¼
actually available amount of charge ðQ Þ 100% ½1 maximum available amount of charge ðCÞ
This definition for LABs is not clear and unequivocal. The reason for this is that both used values, the reference value ‘maximum available amount of charge’, the so-called ‘battery capacity’, and ‘actually available amount of charge’ can be defined and accordingly measured in different ways. The reference test for Q is a discharge with a certain defined current until a predefined cutoff voltage at a certain predefined battery temperature. The reference test for the battery capacity C is a full charge followed by a discharge under similar conditions as described before. Depending on discharge current rate, battery temperature, cutoff voltage, and the definition of ‘full charge’ different values for Q , C, and hence for SoC can be obtained. To understand the definition of SoC ‘full charge’ should be defined first. Generally, it is defined by a charge procedure resulting in a fully charged battery. However, ‘full’ is not ‘full’ and depends strongly on the defined charge procedure. Some often-used definitions of ‘fully charged battery’ are as follows: Physical full means that all available active masses are in • charged state. In new batteries, all active masses are available for charge. In aged batteries, parts of the active masses can become loose because of erosion, might not be accessible for charge current because of corrosion
•
layers on electrodes, or might be transferred into irreversible sulfates, and are therefore not available for charge anymore. Physical full is achieved at the point of time where further charging current is utilized 100% into side reactions such as gassing or corrosion. Nominal full is achieved when a charge procedure prescribed by a battery manufacturer or by a given standard is applied. For new batteries, this is usually almost the same state as a physical full. In aged batteries, for example, coarse-grained lead-sulfate crystals form during operation or because of recrystallization processes. These crystals often cannot be dissolved by standard charging procedures. Therefore, parts of the active masses remain in discharged state after the nominal full charge. To achieve the physical full state modified charging strategies must be applied such as charging at elevated temperatures or for longer periods of time. For example, an international standard (EN 50342–1:2006) for six-cell flooded starter–light–ignition (SLI) batteries define as a nominal charge CCCVcharge by 2535 1C and (16.0070.01) V with current limitation of 5Inominal for 24 h. In aged batteries there may still be some lead sulfates left after this charge procedure. They can be widely dissolved if an additional charge by at least 40 1C is applied. Operational full is defined as the highest possible SoC of a battery, which can be achieved under field conditions in a given application. Nominal charge conditions often cannot be applied for batteries that are used in realworld applications due to the system design, restrictions concerning the maximum charge voltage, the battery temperature, and the available charging time. As a result, the battery, whether new or aged, cannot even reach the nominal full charge state. For example, in conventional vehicles, the system voltage usually cannot exceed about 15 V (which is lower than 16 V defined for nominal charge) and charge periods are limited to the driving times (usually much lower than 24 h at once), so that even fresh SLI battery cannot be fully charged in terms of nominal charge.
As it follows from reference tests for C and Q , the battery is defined as empty when by discharging it with defined nominal current at a defined temperature the predefined cutoff voltage is reached. The discharge procedure with the mentioned parameters is known as a standard capacity test. This definition is more practicable than physically fully discharged battery, where all active masses are in discharged state, because of several reasons.
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First, LAB cannot be physically discharged completely without causing irreversible damage to it. Second, in most applications the battery should provide a certain voltage level even if it is ‘empty’. Third, a total physical discharge would last almost infinite long time. The manufacturer or the user of the battery can define nominal discharge rate, end-of-discharge voltage, and temperature. Therefore, it is necessary to mention the parameters for the determination of the capacity by a capacity test. Otherwise, results are not comparable. Once meanings of ‘full’ and ‘empty’ battery are clearly defined, different unambiguous definitions of battery capacity can be introduced: Nominal capacity or rated capacity C . The nominal or • rated capacity is the value for the capacity given by N
the manufacturer at nominal operating conditions
SoC ¼
•
•
•
The general definition of SoC according to eqn [1] is useful when SoC should be measured by reference tests, because for both values, Q and C, the amount of charge can be calculated during discharge as a discharge current multiplied with the discharge time. If a certain SoC should be set up (so that battery has defined amount of charge Q), it is not possible to discharge the LAB until it is empty and then charge it again and calculate the stored amount of charge by integration of charge current. The reason is that because of higher battery voltage during charging, a significant part of charge current flows into gassing reaction and thus the actually stored amount of charge is lower than calculated by the integration of the charge current. Therefore, in order to set up a certain SoC of the battery, it should be fully charged (to 100% SoC) and then a certain amount of charge Qd should be removed from the battery by discharging, so that
maximum available amount of charge ðCÞ removed amount of charge ðQd Þ 100% maximum available amount of charge ðCÞ
(defined by temperature, discharge current, and endof-discharge voltage similar to the standard capacity test). Initial capacity C0. Initial capacity is the measured capacity of a new battery. The reference measurement consists of a nominal full charge followed by a standard capacity test as defined above. For a given LAB this value might be slightly higher or lower than nominal capacity CN because of production tolerance, systematic oversizing by the manufacturer, or missing initialization cycles which can increase the capacity at the beginning of the lifetime. Actual capacity Ca. Actual capacity is the measured capacity of the battery in its present condition. The reference measurement is the same as one for the initial capacity. Consequently, for a fresh battery Ca ¼ C0. In the case of aged batteries CaoC0 because of aging processes that lead to capacity losses. However, this is not necessarily correct in all cases. Some LABs show an increase in the actual capacity Ca over several months or even years. This has been observed especially for valve-regulated lead–acid (VRLA) batteries. Available capacity Cav. Available capacity is the capacity of a given new or aged battery accessible in a given application. The reference measurement often is an operational full charge followed by a discharge, with nominal current, until an application-defined end-of-discharge voltage is reached at actual battery temperature.
Now it is possible to define SoC, but before this an important point is to be noted.
½2
This is actually a slightly different definition of SoC, but if C, Q , and Qd are measured at similar discharge conditions (temperature, discharge current, end-off-discharge voltage, and the same age of the battery), then C ¼Q þQ
d
½3
and this definition of SoC is equivalent to the one given in eqn [1]. If ‘SoC’ is mentioned, usually the actual available capacity related to the nominal capacity CN is meant. Since CN is often not a measured value for a given battery, condition [3] is not fulfilled. In that case, using eqns [1] or [2] different values for SoC can be obtained. From this point of view for a new battery, SoC related to the initial capacity (C0) is more preferable because condition [3] is fulfilled. For example, a fresh SLI battery with a nominal capacity of CN ¼ 100 Ah is given. The battery may have an initial capacity of C0 ¼ 105 Ah. In this case if the battery should be set up to 50% SoC (related to CN), then Qd ¼ 50 Ah should be discharged from the battery according to eqn [2]. However, by discharging the battery under nominal conditions, capacity of 55 Ah can be removed from the battery until it is empty. It would mean that the SoC (related to CN) according to definition [1] was 55%. For aged batteries, SoC related to the initial capacity and using definitions [1] or [2] would not be consistent. In this case, SoC related to the actual capacity (SoCa) should be used. For the same reason in application only SoC related to the available capacity (SoCav) using definitions [1] and [2] is correct.
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The relation between different SoCs can be explained using an example shown in Figure 1. In this example an aged LAB with initial capacity C0 ¼ 100 Ah is given. Because of large lead-sulfate crystals physical full charge cannot be obtained within the limited time of a nominal charge procedure. Therefore, capacity of 5 Ah remains uncharged. At the given end-of-discharge voltage criteria the battery delivers less capacity due to aging compared with the new battery. In this example, this amounts to an additional capacity loss of 20 Ah. This results in an actual capacity of Ca ¼ 75 Ah. The SoCa window between 0% and 100% can be mapped to the SoC0 window between 20% and 95%. In certain applications the available capacity of the battery might be only Cav ¼ 65 Ah because operational full charging leaves a significant amount of active masses in the discharged state. SoCav can be mapped to the SoC0 window between 20% and 85% or in other words in the given application the battery can only operate between 20% and 85% of SoC related to its initial capacity. All above definitions for capacity and SoC take nominal temperature or at least similar temperature at all times for granted. Since temperature has a significant influence on the battery capacity, considerable other values of these figures of merit would be obtained by other temperatures. It is worse to mention that another problem for the precise definition of the SoC can occur. Owing to different rates of the side reactions in the positive and the negative electrode, it can happen that the SoC of the two electrodes deviates. Generally, the SoC is defined for the battery in total, but for some purposes, the individual performance of the electrodes is of relevance. Similar to this problem is the inhomogeneous SoC of cells in a series connection. Typically, cells are not exactly at the same temperature and therefore side reactions occur at different rates; hence, the SoC of the cells deviates.
SoCa
100% 95%
5 Ah
100%
SoCav 15 Ah
SoC0 Physical full
Cav = 65 Ah
100% C a = 75 Ah
C 0 = 100 Ah
85%
0%
0%
20 Ah
20%
0%
Figure 1 Schematic visualization of the relations between different state-of-charge (SoC) definitions.
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Depth-of-Discharge Depth of discharge (DoD) is a figure of merit that is often used instead of SoC. It is defined as an amount of charge removed from the battery at the given state (Qd) related to the total amount of charge, which can be stored in this battery (C) and usually expressed as a percentage: DoD ¼
removed amount of charge ðQd Þ 100% ½4 maximum available amount of charge ðCÞ
For this definition the same complexity as described above for SoC is valid. If SoC is calculated according to eqn [2], it is easy to see that DoD ¼ 100% SoC
½5
In some cases DoD is more handy than SoC, because it starts from a fully charged battery state, which is more independent from external conditions (in particular, for new batteries) than an empty battery.
State-of-Health State-of-health (SoH) is a figure of merit that reflects the general condition of the battery and its ability to deliver the specified performance compared to a fresh battery. It is often described using fuzzy statements like ‘new’, ‘fresh’, ‘old’, ‘aged’, or ‘worn out’. State-of-health relates to the different battery parameters, for example, internal resistance, but is normally used as a figure of merit that describes relation between actual battery capacity (Ca) and nominal battery capacity (CN) expressed as a percentage: SoHa ¼
actual battery capacity ðCa Þ 100% nominal battery capacity ðCN Þ
½6
Since new batteries usually have a slightly higher capacity than declared by the manufacture, according to this definition SoH >100% can occur. It is necessary to mention that a battery user might use also the definition SoHav ¼
actual available battery capacity ðCav Þ 100% nominal battery capacity ðCN Þ
½7
According to these definitions, the battery from Figure 1 has an SoHa of 75% and an SoHav of 65%. This reflects on the different views of the battery SoH from the point of battery manufacturer and a battery user. The battery manufacturer can insist on the SoHa because after sufficient charging it delivers 75% of the nominal capacity. The battery user on the contrary can use in a certain application only 65% of the capacity.
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The battery reaches its end-of-life, when SoH falls below a defined value. The end-of-life criterion is typically defined for certain applications depending on the ‘use profile’ of the battery. For uninterruptible power supplies (UPSs), typically 80% SoH is used, in other applications batteries are still in operation at 50% SoH.
State-of-Function In many cases, neither the SoC nor the SoH in any of the above definitions can answer the question, if the battery still meets the requirements in the given application under actual operating conditions. The state-of-function (SoF) is an additional figure of merit or a set of figures of merit that provide information about the capability of the battery to perform a certain task, which is relevant for the functionality or reliability of the system. Thus, SoF brings together the battery state parameters and the requirements of a given application. Therefore, no general definition of SoF can be made. It must be defined for certain applications. Some typical examples of SoF definitions are as follows: Charge storage capability. Charge storage capability is an • amount of charge, which can be removed from the bat-
• •
•
tery in the given application by discharging with a current typical for this application at its present condition (SoH and temperature) after operational full charge. Available charge. Available charge is an amount of charge, which can be removed from the battery in the given application by discharging with a current typical for this application in its present condition (SoC, SoH, and temperature) without any previous charges or discharges. Cranking capability (CC). The CC is an important SoF for SLI batteries. It describes the battery voltage response on a given current pulse after a defined time interval. The load case is defined by a time-dependent discharge current profile, which generally reflects the worst-case scenario. The essential information of the voltage response Vbat,load is the lowest battery voltage during the load case. If the voltage drops below a predefined voltage threshold Vthreshold, the SOF with regard to this function is zero, that is, the operation – cranking, braking, or whatever else demands power from the battery – will fail. The reference test for the CC is a discharge with a predefined current profile. The charge acceptance (CA). The CA is the charging current that the battery accepts at its present SoC, SoH, and temperature, if it is charged with a given maximum charging voltage Vlim. Charge acceptance is especially interesting for applications with limited available charging time. This is the case, for example, for SLI batteries used in vehicles with regenerative braking, where charging events take only few seconds and high amount of
energy should be fed into the battery, or for batteries in photovoltaic (PV) systems, where charging times are limited to the sunshine hours. A useful definition of CA is the average current during a defined time interval while the battery voltage is at the desired charging voltage. The reference test for CA is an immediate charge of the battery with the maximum allowed charging voltage to determine the current to which the battery can be charged within the defined interval.
Methods for Determination of State-ofCharge and State-of-Health While SoC and SoH of LABs can be determined in the laboratory in a relatively easy way, in real-world application it is always a challenge because of a wide range of conditions like temperature, load profile, current range, and their unpredictable changes during the lifetime of the product. A precise determination of these figures of merit, and therefore also for all kinds of SoFs, is often not possible, and most of the methods, which can be used in real-world applications, can only estimate values of SoC and SoH rather than determining them. Often, only a combination of different approaches allows an accurate calculation of SoC and SoH. A technical system including battery data measurement, determination of SoC, SoH, and eventually other battery state variables based on these measured data is called battery diagnosis. A battery management uses information from the battery diagnosis to operate the battery and the system at optimal conditions. An overview of different techniques for battery diagnosis for LABs is given below. Discharge Test For each definition of SoC and SoH a reference test is defined, which is a discharge of the battery (with previous charge if needed) under defined conditions (temperature, discharge current, and cutoff voltage). Usually such a test is only practicable in a laboratory. In realworld application, the operation of the system needs to be interrupted for a long period. Furthermore, this test would result in energy losses and would accelerate aging due to the full cycling of the battery. Therefore, it cannot be considered for many applications, but it is used, for example, quite often in UPS, for example, on a regular annual basis. However, at the end of the day, the discharge test is the one and only precise and unquestionable method to determine SoC and SoH. Ah Balance Ah balance is the most common technique for the calculation of SoC in real-world applications. It is mainly an
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integration of battery current and therefore a balancing of charged and discharged ampere-hours. The SoC can be determined as a relation between Ah balance and a battery capacity (C), provided that the starting point (SoC(t0)) is known: SoCðt Þ ¼ SoCðt0 Þ þ
1 C
tZ 0 þt
Ibat ðtÞ dt
½8
t0
There are mainly two difficulties that make this method not ideal: First, there is always an inaccuracy in current mea• surement. Even small offsets in measured current
•
values accumulate because of integration and can lead to significant errors in calculated SoC after a while. For example, for a 100 Ah battery, an offset of 10 mA leads to an error of about 7% after only 1 month. Second, a part of the battery terminal current, which is the only value that can be measured, flows into side reaction (Iside). Gassing is the most relevant side reaction in LABs, but others such as corrosion or microshort circuits due to dendrites can also occur. This current which goes into side reactions should be actually subtracted from the measured battery current before integration, otherwise it influences the Ah balance similar to measurement errors: 1 SoCðt Þ ¼ SoCðt0 Þ þ C
tZ 0 þt
ðIbat ðtÞ Iside ðtÞÞ dt
½9
t0
However, side reactions typically behave highly nonlinear and therefore the share of the terminal current going into the side reaction can vary from approximately zero to nearly 100% if the battery is fully charged. There is no way to measure the current going into the side reactions online and in situ. Nevertheless, the simplest solution for minimizing the influence of a gassing current on the Ah balance is based on an assumption that only a certain percent of charge current flows into the main reaction and thus contributes to the battery charge. Therefore, only 80–98% of the charge current depending on the battery type and the charge conditions is considered by the integration. The improved solution is to estimate the current share going into the gassing reaction as the main side reaction by a gassing model. It is assumed that under normal operation conditions (no internal short circuits) more than 90% of the side reactions are given by the gassing reaction. During gassing, water from the electrolyte is split into hydrogen and oxygen. In VRLA batteries (gel or AGM) the majority of the gas is recombined with water, but this is of no relevance for the Ah balance. Since the gassing reaction is mostly influenced by the battery voltage and temperature, and both
797
parameters are usually measured in a system, a simple implementation is possible. However, the parameters of such models are strongly dependent on battery type and they change during the lifetime of the battery. Therefore, self-learning and self-adapting algorithms are necessary to estimate the gassing current correctly throughout the battery lifetime. Generally, both difficulties are only critical if the battery is not charged completely on a regular basis. If the batteries receive a full charge, for example, everyday as it happens in forklift truck applications, none of the above effects is of relevance. The Ah balance is reset everyday and therefore the error that can amount to measurement errors or side reactions is very small and negligible. Since errors in Ah balancing cannot be completely eliminated and because the initial value of SoC has to be known, additional technique should be used in any case to obtain recalibration points. Anyway, due to its simplicity, this method is normally used as a basic technique for SoC calculation in almost all battery diagnostic systems. For the determination of the dynamic change in the SoC it is the best solution anyway and therefore is very accurate for all periods between other recalibration points. Measurement of Acid Concentration or Density In LABs the electrolyte, which is diluted sulfuric acid, is not only used as an ion conductor but also takes part in the main chemical reaction. By discharging the battery, sulfuric acid reacts with lead and lead dioxide to give lead sulfate and therefore is used up; during charging, the reaction is reversed and sulfuric acid is therefore released back to the electrolyte. This means that the concentration of the sulfuric acid is linear proportional to the SoC of the battery. Thus, if the relationship between acid concentration and SoC is known, then by measuring the former the latter can be determined. Unfortunately, the acid concentration itself is not easy to measure. However, since the concentration is almost linearly proportional to the density assuming constant temperature and pressure, the density is often used for determining the SoC. Often, the term specific gravity is used. Different sensors exist (e.g., hydrometers) but in any case a direct intervention into a battery for a measurement is required, so that this technique can only be used in laboratory or offline. This principle, for example, finds its use in SoC indicators that are built in flooded SLI batteries and which are based on colored balls floating in the electrolyte. Some complications that reduce the accuracy of this method or its usability are as follows: This method is only feasible with flooded LABs. No • sensors are available to measure the electrolyte density in VRLA batteries.
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Batteries from different manufacturers as well as from • different product lines of one manufacturer may have
•
significantly different amounts of electrolyte related to the active masses of the cell. The acid concentration in a fully charged battery cell may also be different. Therefore, for each product line, the dependence between SoC and electrolyte density must be known or measured before using it for the SoC determination. Moreover, even within a certain product line, there are variations in electrolyte density, and its amount in a specific cell compared to the designed value. It leads to additional inaccuracies. During the lifetime of a battery, the amount of water in the battery cell changes similarly to losses due to gassing. As sulfuric acid molecules remain in the battery, the overall concentration increases with increasing water loss. The acid density is usually measured at the top of the cell, but the acid concentration distribution after the battery was in operation may not be constant at all. Acid stratification is a well-known effect in flooded batteries and consists of a significant acid concentration gradient in the vertical direction in a cell. Especially after partial cycling, the acid concentration at the bottom of the cell is significantly higher compared with the top of the battery. In particular, above the electrode plates, where acid concentration or density is usually measured, it can be much lower. It leads to the fact that the SoC determined from the acid density above the electrodes may be lower than the actual SoC of the battery. Owing to diffusion, the acid stratification disappears, but a considerable time period of several weeks is necessary. Other more efficient measures involve air bubbling systems, shaking of the batteries, or extensive gassing of the batteries for stirring the electrolyte.
Open-Circuit Voltage and Electromotive Force Open-circuit voltage (OCV) is the terminal voltage of the battery under open-circuit condition. Physically OCV of a LAB is mostly defined by its electromotive force (EMF), which is the internal driving force of a battery for providing energy to a load. Physically, the EMF is the difference between the reduction potentials of the half-cell reactions and depends on the electrolyte concentration. Therefore, EMF and OCV are related to the SoC due to their dependency on the acid concentration. This relation allows one to use OCV and EMF for SoC determination. Open-circuit voltage method for SoC determination refers to a measurement of the battery terminal voltage under open-circuit conditions, whereas EMF method refers to the estimation of EMF (which is very close to OCV) from the battery terminal voltage while moderated charge or discharge current is applied.
Open-Circuit Voltage method
Since acid concentration is the link between SoC and OCV, all complications that reduce the accuracy of SoC determination using measuring of acid concentration (variations in battery production, acid stratification, and gassing reaction) are also valid for this method. Admittedly, acid stratification influences the determined SoC value differently and gassing reaction has additional influence: While acid density measurement usually considers the • electrolyte at a top part of the battery, the OCV of the
•
cell with acid stratification is dominated by the higher potential caused by the higher concentration of sulfuric acid at the bottom of the battery. In fact, the SoC calculated from OCV in a battery with stratification is typically overestimated as long as no correction algorithms for the compensation of the acid stratification are used. The additional role of the gassing reaction is that the measured OCV is actually a mixed potential resulting from the discharge reaction and the gassing reaction. Since gassing reaction is highly temperature dependent, OCV also shows a non-negligible dependency on the temperature.
Additional sources of inaccuracy are the overvoltages after charge or discharge. A significant period is required to reach steady-state conditions. Moreover, in many applications even in rest periods a small current required for monitoring devices, clocks, and so forth, flows through the battery, so that there are actually no opencircuit conditions at all. Nevertheless, the OCV method is widely used in realworld applications. It is easy to implement because OCV is almost linear to SoC, at least in a range of about 30–100% SoC, and this method can be used for all types of LABs. An individual calibration of the relation between OCV and SoC is necessary in the same way as it was discussed for direct acid concentration measurements. For many typical LABs an empirical equation OCVðVÞ ¼ 0:86 þ acid density ðg cm3 Þ
½10
can be used as a rule of thumb. Electromotive Force Method
There are different techniques allowing the estimation of the EMF of a battery by measuring the battery terminal voltage: Model-based approaches, which are described below, • can be used to calculate overvoltages that should be subtracted from battery terminal voltage to obtain the EMF value. The accuracy of this method is defined by
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•
All complications described for the OCV method are also valid for the EMF method, but they are only of secondary order since the inaccuracy of the first step – estimation of EMF from battery terminal voltage – is significantly higher than the inaccuracy of the next step – SoC determination from EMF. The inaccuracy of the EMF method for LABs is quite high. It can only give a wide range of about 20–50% depending on where the SoC could be. Therefore, it is only used in combination with other methods. Impedance Measurements The electrochemical impedance of a battery is a frequency-dependent complex number Z( f ) and represents a real and imaginary part or its modulus and phase. A set of impedances measured at different frequencies in a wide range from few millihertz to some hundred hertz is called impedance spectrum and can be represented as a Nyquist diagram (imaginary part vs real part). Since batteries show a capacitive behavior, at an interesting frequency range, electrochemists generally plot the negative of the imaginary part on the ordinate axis, so that the capacitive part of the impedance spectrum appears in the upper quadrants. Alternatively, a representation as a Bode diagram (absolute value and argument vs frequency) can be used. As an example an impedance spectra of an SLI battery at different SoC levels is presented in Figure 2. Measuring an impedance spectrum is called impedance spectroscopy. In the laboratory, impedance spectroscopy is normally performed using an impedance spectroscope, which generates a small sinusoidal current or voltage signal and feeds it in the battery. The voltage response versus the current response is measured and the complex impedance can be determined. The hardware is very costly and therefore cannot be used in the majority of the real-world applications. Therefore, another technique called passive impedance spectroscopy can be used instead under certain circumstances. In this case, current fluctuations, which are already presented in the system, are analyzed. Since no active network is required and impedance can be calculated using digital filters, which
90% SoC 70% SoC 40% SoC 20% SoC
−25 −20 −15 Im(Z) (mΩ)
•
the properties of the used battery model and its parametrization. Linear interpolation method. In this case, the average battery voltage is calculated, when the battery is charged and discharged with same moderated currents within a short time, so that SoC does not significantly change. Voltage relaxation method. After charge or discharge, the battery voltage will relax to the EMF value (or actually OCV value, but it is quite the same as explained above). This relaxation process can be monitored and the EMF can be predicted.
−10 −5 0 5 10 15 20 5
10
15
20
25 30 Re(Z) (mΩ)
35
40
45
Figure 2 Typical spectrum of a lead–acid battery depending on the SoC (60 Ah flooded starter light iginition (SLI) battery (OEM product); the battery was discharged with 1.5 A during the measurement).
are easy to implement, this method is suitable for use in applications at low costs. Nevertheless, a precise measurement of current and voltage is necessary. However, depending on the excitations available in the system, only impedance at certain frequencies can be determined with sufficient accuracy. Therefore, this method cannot offer the same amount of information as it can be obtained with an active impedance spectroscope in the laboratory. In addition, the accuracy of passive impedance spectroscopy is lower than by using impedance spectroscope. Pure impedance data can be hardly used for SoC and SoH determination, but the results from impedance measurements are valuable information, which can be used in an overall algorithm. Most useful applications of impedance measurements for battery diagnostic are as follows: Measurement of the internal resistance (R ) of the bat• tery. Internal resistance of LAB can be measured by i
impedance spectroscopy at relatively high frequencies (some 100 Hz). Since internal resistance is affected by the acid concentration and the conductivity of the active masses, a dependence between Ri and SoC exists and can be used for estimation of the latter. However, this dependence is only significant at SoCs lower than about 30–50% (Figure 3). Moreover, the internal resistance is influenced by the temperature, it has some variations for a given product line and it increases significantly during the lifetime because of aging. In total it makes accurate determination of SoC from Ri extremely difficult and this method can only be used in combination with other techniques. The determination of SoH defined as a
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10
For accurate measurement, a significant discharge current is needed, because only noticeable voltage drops can be measured with high accuracy. Direct current resistance has the same correlation with SoC and SoH as the internal resistance measured using impedance spectroscopy and described above. However, the requirements for the measurement of current and voltage for determining DCR are lower compared with passive or active impedance spectroscopy.
9 8 7
Ri (mΩ)
6 5 4 3
Model-Based Approach
2 1 0
0
20
40
60
80
100
SoC (%)
Figure 3 Internal resistance of a lead–acid battery depending on state of charge (70 Ah flooded new starter–light–ignition (SLI) battery (OEM product)). SoC, state-of-charge.
•
relation between an actual battery capacity and a nominal battery capacity is almost impossible. Even though internal resistance increases during aging, it is mostly defined by corrosion – an aging process that does not influence the capacity loss directly. Owing to the increase of internal resistance, the end-of-discharge voltage is reached earlier by the discharge, but this indirect influence of the internal resistance to the battery capacity is not significant. However, there is some correlation between the increase of the internal resistance and a capacity loss. This fact is used in some systems for SoH determination. The correlation is quite good in UPS systems and it is quite poor for batteries with partial SoC cycling such as in SLI or PV system applications. The measurement of Ri is mostly useful if resistive overvoltages should be calculated. It is also a basic parameter for the determination of some SoFs (e.g., CC in vehicles), and owing to the easy and cost-effective measurements with modern devices it is widely used in different applications. Parametrization of battery models. By using modelbased approaches as described below, the model parameters must be continuously adapted. Passive impedance spectroscopy can be used as a supporting method for online adaptation.
Measurement of Direct Current Resistance The term direct current resistance (DCR) is used to describe the internal resistance of a battery measured as a relation between the terminal voltage drop (DV) caused by a change in the terminal current (DI): DCR ¼
DV DI
½11
The main idea of the model-based approach is to have a relatively simple dynamic battery model with only few parameters in a diagnostic system. The measured battery terminal current can be fed into this model and a voltage response can be compared with the measured voltage to track changes in the battery parameters. These changes can be used to calculate changes in SoC and SoH of the battery. Thereby, the Kalman filter is applied. It is a recursive filter, which estimates the state of a dynamic system from a series of noisy measurements and is well known from system theory. Three main steps are required to design such a system: 1. The battery model must be chosen. The used model affects significantly the quality of the diagnostic system. Linear models are easy to implement, but because of the strong nonlinear behavior of LABs they lead to high inaccuracies of the diagnostic system. The use of nonlinear models can yield better results, but their implementation is difficult and special extensions of Kalman filter should be used. The main criterion for choosing the battery model is the possibility to present it as state variable equations: x_ðt Þ ¼ A xðt Þ þ B uðt Þ yðt Þ ¼ C xðt Þ þ D uðt Þ
½12
where u(t) is the vector of applied inputs, x(t) is the state vector, y(t) is the vector of measured outputs, A is the state matrix, B is the input matrix, C is the output matrix, and D is the feed through matrix. In addition, the observability, which can be verified by investigation of state variable equations, should be guaranteed. Two examples of models, which can be used for LABs, are shown in Figure 4. The deviation in the behavior of the battery model from the real battery and inaccuracies in measurement should be additionally considered in a state variable model: x˙ ðt Þ ¼ A xðt Þ þ B uðt Þ þ G sðt Þ zðt Þ ¼ yðt Þ þ mðt Þ ¼ C xðt Þ þ D uðt Þ þ mðt Þ
½13
where s(t) is the additive vector that represents system disturbances and model inaccuracies, m(t) is the additive
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health
801
Cs Ri
Ri Rs
U
Cb
Rd
Rs
Re
Cs
Cb
U
Figure 4 Battery models used in a model-based approach: Rd – self-discharge, Cb – main battery capacity, Ri – internal resistance, Rs and Cs – transient effects, Re – end resistance, U – terminal voltage.
vector that represents effects of measurement noise, z(t) is the vector of measured outputs corrupted by noise, and G is the coupling matrix that regulates the influence of the disturbances on each state. Often additional state variables are employed to obtain better connection between state vector and SoC or SoH of the battery. 2. The Kalman filter should be designed. It is generally an algorithm to track changes of the state variables of the model using comparison of the model outputs to the measured values under consideration of measurement noises and disturbances caused by the model inaccuracy. Changes of state variables are used to calculate changes in SoC and SoH of the battery. In addition, a new parameter set for the battery model can be calculated, that is, the model is adapted to the new conditions. For a simple linear battery model, the standard Kalman filter can be used. For use with nonlinear models, modern extension of Kalman filter, such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) can be applied. 3. The parametrization should be performed to find initial values of the model parameters. The model and the system should be also analyzed to determine the coupling matrix G as well as the noise vectors s(t) and m(t). The quality of this analysis is a crucial factor for a successful design. This method can be used in combination with supporting algorithms that improve the parametrization of the battery model and consequently adapt it to the battery, since aging and temperature significantly influence the characteristic of the battery. Hence, an impedance measurement can be applied. Model-based approach can be effectively used in systems with high dynamics, for example, in vehicle applications. Methods of Artificial Intelligence Two methods of artificial intelligence are used in battery diagnostic quite often to determine SoC: expert systems
and artificial neural networks. While expert systems attempt to simulate the knowledge and analytical skills of one or more human experts that are able to express their knowledge as a set of rules, which can be transferred into mathematical expressions, artificial neural networks attempt to use the concepts of human brain to find an output based on several nonlinear correlated input state variables. Expert systems
Expert systems for SoC determination can be implemented as a set of simple rules. Rules can generally use all observable or measurable state variables such as current, voltage, and temperature to make a statement about SoC. If more information is available, such as that on impedance at different frequencies, pressure, or acid concentration, they can be used as additional state variables as well. First or second time derivatives of the state variables can be used as input in the same way as output from models which can do a pre-analysis of the data. Thereby, often fuzzy statements like ‘current high’ or ‘temperature low’ are used. As an example, the following rules can be applied: If battery cell voltage is higher or equal to 2.5 V per • cell, the temperature is above 20 1C and the time
•
derivative of the charge current is small, then the SoC is above 70%. If discharge current is low, temperature is high, and voltage is low, then SoC is low.
After applying the rules, the so-called de-fuzzyfication takes place to transfer the statements into numbers, which can be used by other algorithms or control systems. One of the combination methods described in the next section can be used to generate one statement about SoC at each time using these rules. When applying an expert system to batteries, typically only a certain range of SoC can be determined. Typically a rule can be applied only from time to time (most rules, e.g., require quite stable conditions rather than highly dynamic battery operation) and therefore other
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algorithms are used in combination to achieve a continuous SoC output. The Ah balance as described in eqn [9] is perfectly suited to follow the changes of the SoC even during periods of high accuracy until the expert system delivers a new SoC estimation. Artificial neural network
A principle of using neural networks is shown in Figure 5. The measured battery parameters (charge/discharge current, voltage, and temperature) or derived parameters (amp-hours, di/dt, dv/dt, etc.) as well as the previous SoC can be used as input values for the neural network. An interconnected group of nodes is applied to calculate SoC. One or more layers of nodes are used between the input and the output layer. The information of the neural network is based on weight factors for all input variables with regard to a certain node. The value zi j of a node i in layer j is given by the sum of the k weighted inputs from the previous layer j 1. Nodes are used as new input variables for the next layer of nodes: zji ¼
X
wzj 1 zj zjk1
½15
i
k
k
Network topology and used node functions should be carefully selected depending on the application, desired network complexity, and available input parameters. The advantage of this method is that no mathematical model for the correlation between the input and the output variables is required. The disadvantage is that a huge amount of training data are required to train the neural network. The training data consist of a complete set of input variables and an exact known output vector. The data are used to determine the weight factors
Input
Rules
Approaches for Combination of Different Methods As mentioned before, often only a combination of different methods can deliver reliable information about SoC of the battery. Different approaches implemented in one system estimate SoC either continuously or at certain points of time, whenever valid input data for a given algorithm is available. The usage of different algorithms requires concepts for the combination of their respective outputs. Some common techniques are listed below: Fuzzy logic. This technique is used when different • fuzzy statements about SoC should be combined and
•
½16
wzj 1 zj k
from eqn[15]. If such a neural network should be adapted according to the changing performance of the battery during the lifetime, this method requires a significant data memory and must be able to deliver valid sets of input and output variables. One option to overcome the problem with large data set storage is to use special neural gas models, which can be adapted instantaneously by any new available learning point. In any case, for selflearning algorithms it is necessary to implement an additional independent algorithm for delivering reference points.
i
Output
SoCprev
Temp SoC i
• dv/dt
Figure 5 Artificial neural network used for state-of-charge (SoC) estimation.
qualitative information about SoC based thereon should be delivered as a result. This method is often used, for example, in expert systems or as a part of a neural network. Boundary consideration. Many methods can only make statements whether SoC is higher than a certain value or lower than a certain value or both. Furthermore, even for methods, which are able to deliver more precise information about SoC, upper and lower boundaries can be defined by considering inaccuracies of this method. Usually these methods are used in combination with the Ah balance as the recalibration points or in other words, as limits for Ah counter. If the Ah balance bounces against one of the boundaries, the Ah balance will be limited to the boundary value accordingly. It is advantageous for this technique if three Ah balance counters are used: Ah balance according to the measured value, Ah balance taking into account the maximum positive error occurring from the measurement system and from model errors with regard to side reactions (see eqn [9]), and an Ah balance considering the maximum negative error. This method is illustrated in Figure 6. Kalman filter. In model-based approaches, the Kalman filter is used in combination with a battery model, but Kalman filters can be used as a method for combination of SoC values delivered by different techniques as well. In this case a model for SoC of the battery is employed based on an equation representing the SoC
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health Ah balance
Ri
C
u(t ) V wi x(t ) y(t ) z(t )
Ahmax
Ahmiddle Ahmin
0
Time Boundaries from supporting algorithms
Figure 6 Example for using boundaries from different state-ofcharge (SoC) algorithms in combination with a triple Ah balance approach.
as a weighted sum of SoCs estimated by different methods: SoCðt Þ ¼ aðt Þ þ DSoCðDt Þ aðt Þ ¼ w1 SoC1 ðt Dt Þ þ ? þ wn SoCn ðt Dt Þ X þ ð1 wi Þ SoCðt Dt Þ
Z(f) DI Dt DV l(t ) r(t )
803
internal resistance of the battery (ohm) vector of the applied inputs voltage weighting factors state vector vector of measured outputs vector of measured outputs corrupted by noise frequency-dependent complex number change in current time step voltage drop additive vector representing effect of measurement noise additive vector representing system disturbances and model inaccuracies
Abbreviations and Acronyms
½17
where Dt is the time step, DSoC(Dt) is the SoC change calculated by current integration during the duration of the time step Dt, a(t) is the weighted sum of SoCs, SoC1, y, SoCn are the SoCs delivered from different methods, and w1, y, wn are the weighting factors considering inaccuracies of the respective method. The additional advantage of using different methods in one system is that self-learning algorithms are possible in this case. Whenever a method delivers a reliable SoC value, all other algorithms can adapt themselves accordingly.
Nomenclature
AGM CA CC CCCV DCR DoD EMF LAB OCV PV SLI SoC SoF SoH UPS VRLA
absorbent glass mat charge acceptance cranking capability constant voltage/constant current (charging strategy) direct current resistance depth-of-discharge electromotive force lead–acid battery open-circuit voltage photovoltaic starter light ignition state-of-charge state-of-function state-of-health uninterruptible power supply valve-regulated lead–acid
Symbols and Units a(t ) A B C C C0 Ca Cav CN D G Ibat Iside Q Qd
weighted sum of SoCs state matrix input matrix output matrix capacity of a battery (Ah) initial capacity actual capacity available capacity nominal capacity feed through matrix coupling matrix battery terminal current a part of the battery terminal current that flows into side reaction amount of charge (Ah) discharged amount of charge (Ah)
See also: Batteries: Capacity; Secondary Batteries – Lead–Acid Systems: Overview; Lifetime Determining Processes; Batteries: Partial-State-of-Charge.
Further Reading Bhangu BS, Bentley P, Stone DA, and Bingham CM (2005) Nonlinear observers for predicting state-of-charge and state-of-health of lead– acid batteries for hybrid-electric vehicles. IEEE Transactions on Vehicular Technology 54(3): 783--794. Blanke H, Bohlen O, Buller S, et al. (2005) Impedance measurements on lead–acid batteries for state-of-charge, state-of-health and cranking capability prognosis in electric and hybrid electric vehicles. Journal of Power Sources 144: 418--425. Bohlen O (2008) Impedance-Based Battery Monitoring. ISBN 978-38322-7606-5. Aachen: Shaker Verlag.
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Cai CH, Du D, and Liu ZY (2003) Battery state-of-charge (SOC) estimation using adaptive neuro-fuzzy inference system (ANFIS). Proceedings of the 12th IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1068–1073. St. Louis, USA, 25–28 May. Chiasson J and Vairamohan B (2005) Estimating the state of charge of a battery. IEEE Transactions on Control Systems Technology 13(3): 465--470. Coleman M, Lee CK, Zhu C, and Hurley WG (2007) State-of-charge determination from EMF voltage estimation: Using impedance, terminal voltage, and current for lead–acid and lithium-ion batteries. IEEE Transactions on Industrial Electronics 54(5): 2550--2557. Doerffel D and Sharkh SA (2006) A critical review of using the Peukert equation for determining the remaining capacity of lead–acid and lithium-ion batteries. Journal of Power Sources 155: 395--400. Feder DO, Croda TG, Champlin KS, and Hlavac MJ (1992) Field and laboratory studies to assess the state-of-health of valve-regulated lead–acid batteries. Conductance/capacity correlation studies. Proceedings of the 14th International Telecommunications Energy Conference, pp. 218–233. Gould CR, Bingham CM, Stone DA, and Bentley P (2008) Novel battery model of an all-electric personal rapid transit vehicle to determine state-of-health through subspace parameter estimation and a Kalman estimator. Proceedings of the International Symposium on Power Electronics, Electrical Drives, Automation and Motion, pp. 1217–1222. Ischia (Italy), 11–13 June. Huet F (1998) A review of impedance measurements for determination of the state-of-charge or state-of-health of secondary batteries. Journal of Power Sources 70: 59--69. Kaiser R (2007) Optimized battery-management system to improve storage lifetime in renewable energy systems. Journal of Power Sources 168: 58--65. Kray D and Sauer DU (1999) A modified neural gas network learns online unevenly distributed data from accumulator without training sets. Proceedings of the 7th European Congress on Intelligent Techniques and Soft Computing (EUFIT’99). Aachen.
Lee DS, Shiah SJ, Lee CM, and Wang YC (2007) State-of-charge estimation for electric scooters by using learning mechanisms. IEEE Transactions on Vehicular Technology 56(2): 544--556. Meissner E and Richter G (2003) Battery monitoring and electrical energy management precondition for future vehicle electric power systems. Journal of Power Sources 116: 79--98. Okoshi T, Yamada K, Hirasawa T, and Emori A (2006) Battery condition monitoring (BCM) technologies about lead–acid batteries. Journal of Power Sources 158: 874--878. Piller S, Perrin M, and Jossen A (2001) Methods for state-of-charge determination and their applications. Journal of Power Sources 96: 113--120. Pop V, Bergveld HJ, Notten PHL, and Regtien PPL (2005) State-of-theart of battery state-of-charge determination. Measurement Science and Technology 16(12): 93--110. Sauer DU, Bohlen O, Sanders T, Waag W, Schmidt R, and Gerschler J (2007) Batteriezustanderkennung: Mo¨gliche Verfahrens- und Algorithmenansa¨tze, Grenzen der Batteriezustandserkennung. In: Scho¨llmann M (ed.) Energiemanagement und Bordnetze II, ISBN 13 978-3-8169-2649-8, pp. 1--30. Renningen: Expert-Verlag. Sauer DU, Karden E, Fricke B, et al. (2007) Charging performance of automotive batteries – an underestimated factor influencing lifetime and reliable battery operation. Journal of Power Sources 168(1): 22--30. Streuer P (2003) Ladezustandsanzeiger fu¨r eine Batterie. European Patent EP 1,369,952 A2. Paris: Jouve. Valdez MAC, Valera JAO, Jojutla Ma, and Arteaga OP (2006) Estimating SoC in lead–acid batteries using neural networks in a microcontroller-based charge-controller. Proceedings of the International Joint Conference on Neural Networks, pp. 2713–2719. Vasebi A, Bathaee SMT, and Partovibakhsh M (2008) Predicting state of charge of lead–acid batteries for hybrid electric vehicles by extended Kalman filter. Energy Conversion and Management 49: 75--82.
Introduction State-of-charge (SoC) and state-of-health (SoH) are two important figures of merit that describe the state of a battery. In particular, in this article these figures of merit are discussed in the context of lead–acid batteries (LABs). A wide range of definitions exist, but many of them are often not clear and unequivocal. The following sections present the complexity of these definitions as well as methods for determination of SoC and SoH in laboratory and field applications under real-world conditions.
•
State-of-Charge State-of-charge is generally defined as an actually available amount of charge in a given battery (Q ) related to the maximum available amount of charge, which can be taken from this battery after a 100% full charging (C ) and is usually expressed as a percentage: SoC ¼
actually available amount of charge ðQ Þ 100% ½1 maximum available amount of charge ðCÞ
This definition for LABs is not clear and unequivocal. The reason for this is that both used values, the reference value ‘maximum available amount of charge’, the so-called ‘battery capacity’, and ‘actually available amount of charge’ can be defined and accordingly measured in different ways. The reference test for Q is a discharge with a certain defined current until a predefined cutoff voltage at a certain predefined battery temperature. The reference test for the battery capacity C is a full charge followed by a discharge under similar conditions as described before. Depending on discharge current rate, battery temperature, cutoff voltage, and the definition of ‘full charge’ different values for Q , C, and hence for SoC can be obtained. To understand the definition of SoC ‘full charge’ should be defined first. Generally, it is defined by a charge procedure resulting in a fully charged battery. However, ‘full’ is not ‘full’ and depends strongly on the defined charge procedure. Some often-used definitions of ‘fully charged battery’ are as follows: Physical full means that all available active masses are in • charged state. In new batteries, all active masses are available for charge. In aged batteries, parts of the active masses can become loose because of erosion, might not be accessible for charge current because of corrosion
•
layers on electrodes, or might be transferred into irreversible sulfates, and are therefore not available for charge anymore. Physical full is achieved at the point of time where further charging current is utilized 100% into side reactions such as gassing or corrosion. Nominal full is achieved when a charge procedure prescribed by a battery manufacturer or by a given standard is applied. For new batteries, this is usually almost the same state as a physical full. In aged batteries, for example, coarse-grained lead-sulfate crystals form during operation or because of recrystallization processes. These crystals often cannot be dissolved by standard charging procedures. Therefore, parts of the active masses remain in discharged state after the nominal full charge. To achieve the physical full state modified charging strategies must be applied such as charging at elevated temperatures or for longer periods of time. For example, an international standard (EN 50342–1:2006) for six-cell flooded starter–light–ignition (SLI) batteries define as a nominal charge CCCVcharge by 2535 1C and (16.0070.01) V with current limitation of 5Inominal for 24 h. In aged batteries there may still be some lead sulfates left after this charge procedure. They can be widely dissolved if an additional charge by at least 40 1C is applied. Operational full is defined as the highest possible SoC of a battery, which can be achieved under field conditions in a given application. Nominal charge conditions often cannot be applied for batteries that are used in realworld applications due to the system design, restrictions concerning the maximum charge voltage, the battery temperature, and the available charging time. As a result, the battery, whether new or aged, cannot even reach the nominal full charge state. For example, in conventional vehicles, the system voltage usually cannot exceed about 15 V (which is lower than 16 V defined for nominal charge) and charge periods are limited to the driving times (usually much lower than 24 h at once), so that even fresh SLI battery cannot be fully charged in terms of nominal charge.
As it follows from reference tests for C and Q , the battery is defined as empty when by discharging it with defined nominal current at a defined temperature the predefined cutoff voltage is reached. The discharge procedure with the mentioned parameters is known as a standard capacity test. This definition is more practicable than physically fully discharged battery, where all active masses are in discharged state, because of several reasons.
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First, LAB cannot be physically discharged completely without causing irreversible damage to it. Second, in most applications the battery should provide a certain voltage level even if it is ‘empty’. Third, a total physical discharge would last almost infinite long time. The manufacturer or the user of the battery can define nominal discharge rate, end-of-discharge voltage, and temperature. Therefore, it is necessary to mention the parameters for the determination of the capacity by a capacity test. Otherwise, results are not comparable. Once meanings of ‘full’ and ‘empty’ battery are clearly defined, different unambiguous definitions of battery capacity can be introduced: Nominal capacity or rated capacity C . The nominal or • rated capacity is the value for the capacity given by N
the manufacturer at nominal operating conditions
SoC ¼
•
•
•
The general definition of SoC according to eqn [1] is useful when SoC should be measured by reference tests, because for both values, Q and C, the amount of charge can be calculated during discharge as a discharge current multiplied with the discharge time. If a certain SoC should be set up (so that battery has defined amount of charge Q), it is not possible to discharge the LAB until it is empty and then charge it again and calculate the stored amount of charge by integration of charge current. The reason is that because of higher battery voltage during charging, a significant part of charge current flows into gassing reaction and thus the actually stored amount of charge is lower than calculated by the integration of the charge current. Therefore, in order to set up a certain SoC of the battery, it should be fully charged (to 100% SoC) and then a certain amount of charge Qd should be removed from the battery by discharging, so that
maximum available amount of charge ðCÞ removed amount of charge ðQd Þ 100% maximum available amount of charge ðCÞ
(defined by temperature, discharge current, and endof-discharge voltage similar to the standard capacity test). Initial capacity C0. Initial capacity is the measured capacity of a new battery. The reference measurement consists of a nominal full charge followed by a standard capacity test as defined above. For a given LAB this value might be slightly higher or lower than nominal capacity CN because of production tolerance, systematic oversizing by the manufacturer, or missing initialization cycles which can increase the capacity at the beginning of the lifetime. Actual capacity Ca. Actual capacity is the measured capacity of the battery in its present condition. The reference measurement is the same as one for the initial capacity. Consequently, for a fresh battery Ca ¼ C0. In the case of aged batteries CaoC0 because of aging processes that lead to capacity losses. However, this is not necessarily correct in all cases. Some LABs show an increase in the actual capacity Ca over several months or even years. This has been observed especially for valve-regulated lead–acid (VRLA) batteries. Available capacity Cav. Available capacity is the capacity of a given new or aged battery accessible in a given application. The reference measurement often is an operational full charge followed by a discharge, with nominal current, until an application-defined end-of-discharge voltage is reached at actual battery temperature.
Now it is possible to define SoC, but before this an important point is to be noted.
½2
This is actually a slightly different definition of SoC, but if C, Q , and Qd are measured at similar discharge conditions (temperature, discharge current, end-off-discharge voltage, and the same age of the battery), then C ¼Q þQ
d
½3
and this definition of SoC is equivalent to the one given in eqn [1]. If ‘SoC’ is mentioned, usually the actual available capacity related to the nominal capacity CN is meant. Since CN is often not a measured value for a given battery, condition [3] is not fulfilled. In that case, using eqns [1] or [2] different values for SoC can be obtained. From this point of view for a new battery, SoC related to the initial capacity (C0) is more preferable because condition [3] is fulfilled. For example, a fresh SLI battery with a nominal capacity of CN ¼ 100 Ah is given. The battery may have an initial capacity of C0 ¼ 105 Ah. In this case if the battery should be set up to 50% SoC (related to CN), then Qd ¼ 50 Ah should be discharged from the battery according to eqn [2]. However, by discharging the battery under nominal conditions, capacity of 55 Ah can be removed from the battery until it is empty. It would mean that the SoC (related to CN) according to definition [1] was 55%. For aged batteries, SoC related to the initial capacity and using definitions [1] or [2] would not be consistent. In this case, SoC related to the actual capacity (SoCa) should be used. For the same reason in application only SoC related to the available capacity (SoCav) using definitions [1] and [2] is correct.
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health
The relation between different SoCs can be explained using an example shown in Figure 1. In this example an aged LAB with initial capacity C0 ¼ 100 Ah is given. Because of large lead-sulfate crystals physical full charge cannot be obtained within the limited time of a nominal charge procedure. Therefore, capacity of 5 Ah remains uncharged. At the given end-of-discharge voltage criteria the battery delivers less capacity due to aging compared with the new battery. In this example, this amounts to an additional capacity loss of 20 Ah. This results in an actual capacity of Ca ¼ 75 Ah. The SoCa window between 0% and 100% can be mapped to the SoC0 window between 20% and 95%. In certain applications the available capacity of the battery might be only Cav ¼ 65 Ah because operational full charging leaves a significant amount of active masses in the discharged state. SoCav can be mapped to the SoC0 window between 20% and 85% or in other words in the given application the battery can only operate between 20% and 85% of SoC related to its initial capacity. All above definitions for capacity and SoC take nominal temperature or at least similar temperature at all times for granted. Since temperature has a significant influence on the battery capacity, considerable other values of these figures of merit would be obtained by other temperatures. It is worse to mention that another problem for the precise definition of the SoC can occur. Owing to different rates of the side reactions in the positive and the negative electrode, it can happen that the SoC of the two electrodes deviates. Generally, the SoC is defined for the battery in total, but for some purposes, the individual performance of the electrodes is of relevance. Similar to this problem is the inhomogeneous SoC of cells in a series connection. Typically, cells are not exactly at the same temperature and therefore side reactions occur at different rates; hence, the SoC of the cells deviates.
SoCa
100% 95%
5 Ah
100%
SoCav 15 Ah
SoC0 Physical full
Cav = 65 Ah
100% C a = 75 Ah
C 0 = 100 Ah
85%
0%
0%
20 Ah
20%
0%
Figure 1 Schematic visualization of the relations between different state-of-charge (SoC) definitions.
795
Depth-of-Discharge Depth of discharge (DoD) is a figure of merit that is often used instead of SoC. It is defined as an amount of charge removed from the battery at the given state (Qd) related to the total amount of charge, which can be stored in this battery (C) and usually expressed as a percentage: DoD ¼
removed amount of charge ðQd Þ 100% ½4 maximum available amount of charge ðCÞ
For this definition the same complexity as described above for SoC is valid. If SoC is calculated according to eqn [2], it is easy to see that DoD ¼ 100% SoC
½5
In some cases DoD is more handy than SoC, because it starts from a fully charged battery state, which is more independent from external conditions (in particular, for new batteries) than an empty battery.
State-of-Health State-of-health (SoH) is a figure of merit that reflects the general condition of the battery and its ability to deliver the specified performance compared to a fresh battery. It is often described using fuzzy statements like ‘new’, ‘fresh’, ‘old’, ‘aged’, or ‘worn out’. State-of-health relates to the different battery parameters, for example, internal resistance, but is normally used as a figure of merit that describes relation between actual battery capacity (Ca) and nominal battery capacity (CN) expressed as a percentage: SoHa ¼
actual battery capacity ðCa Þ 100% nominal battery capacity ðCN Þ
½6
Since new batteries usually have a slightly higher capacity than declared by the manufacture, according to this definition SoH >100% can occur. It is necessary to mention that a battery user might use also the definition SoHav ¼
actual available battery capacity ðCav Þ 100% nominal battery capacity ðCN Þ
½7
According to these definitions, the battery from Figure 1 has an SoHa of 75% and an SoHav of 65%. This reflects on the different views of the battery SoH from the point of battery manufacturer and a battery user. The battery manufacturer can insist on the SoHa because after sufficient charging it delivers 75% of the nominal capacity. The battery user on the contrary can use in a certain application only 65% of the capacity.
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The battery reaches its end-of-life, when SoH falls below a defined value. The end-of-life criterion is typically defined for certain applications depending on the ‘use profile’ of the battery. For uninterruptible power supplies (UPSs), typically 80% SoH is used, in other applications batteries are still in operation at 50% SoH.
State-of-Function In many cases, neither the SoC nor the SoH in any of the above definitions can answer the question, if the battery still meets the requirements in the given application under actual operating conditions. The state-of-function (SoF) is an additional figure of merit or a set of figures of merit that provide information about the capability of the battery to perform a certain task, which is relevant for the functionality or reliability of the system. Thus, SoF brings together the battery state parameters and the requirements of a given application. Therefore, no general definition of SoF can be made. It must be defined for certain applications. Some typical examples of SoF definitions are as follows: Charge storage capability. Charge storage capability is an • amount of charge, which can be removed from the bat-
• •
•
tery in the given application by discharging with a current typical for this application at its present condition (SoH and temperature) after operational full charge. Available charge. Available charge is an amount of charge, which can be removed from the battery in the given application by discharging with a current typical for this application in its present condition (SoC, SoH, and temperature) without any previous charges or discharges. Cranking capability (CC). The CC is an important SoF for SLI batteries. It describes the battery voltage response on a given current pulse after a defined time interval. The load case is defined by a time-dependent discharge current profile, which generally reflects the worst-case scenario. The essential information of the voltage response Vbat,load is the lowest battery voltage during the load case. If the voltage drops below a predefined voltage threshold Vthreshold, the SOF with regard to this function is zero, that is, the operation – cranking, braking, or whatever else demands power from the battery – will fail. The reference test for the CC is a discharge with a predefined current profile. The charge acceptance (CA). The CA is the charging current that the battery accepts at its present SoC, SoH, and temperature, if it is charged with a given maximum charging voltage Vlim. Charge acceptance is especially interesting for applications with limited available charging time. This is the case, for example, for SLI batteries used in vehicles with regenerative braking, where charging events take only few seconds and high amount of
energy should be fed into the battery, or for batteries in photovoltaic (PV) systems, where charging times are limited to the sunshine hours. A useful definition of CA is the average current during a defined time interval while the battery voltage is at the desired charging voltage. The reference test for CA is an immediate charge of the battery with the maximum allowed charging voltage to determine the current to which the battery can be charged within the defined interval.
Methods for Determination of State-ofCharge and State-of-Health While SoC and SoH of LABs can be determined in the laboratory in a relatively easy way, in real-world application it is always a challenge because of a wide range of conditions like temperature, load profile, current range, and their unpredictable changes during the lifetime of the product. A precise determination of these figures of merit, and therefore also for all kinds of SoFs, is often not possible, and most of the methods, which can be used in real-world applications, can only estimate values of SoC and SoH rather than determining them. Often, only a combination of different approaches allows an accurate calculation of SoC and SoH. A technical system including battery data measurement, determination of SoC, SoH, and eventually other battery state variables based on these measured data is called battery diagnosis. A battery management uses information from the battery diagnosis to operate the battery and the system at optimal conditions. An overview of different techniques for battery diagnosis for LABs is given below. Discharge Test For each definition of SoC and SoH a reference test is defined, which is a discharge of the battery (with previous charge if needed) under defined conditions (temperature, discharge current, and cutoff voltage). Usually such a test is only practicable in a laboratory. In realworld application, the operation of the system needs to be interrupted for a long period. Furthermore, this test would result in energy losses and would accelerate aging due to the full cycling of the battery. Therefore, it cannot be considered for many applications, but it is used, for example, quite often in UPS, for example, on a regular annual basis. However, at the end of the day, the discharge test is the one and only precise and unquestionable method to determine SoC and SoH. Ah Balance Ah balance is the most common technique for the calculation of SoC in real-world applications. It is mainly an
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health
integration of battery current and therefore a balancing of charged and discharged ampere-hours. The SoC can be determined as a relation between Ah balance and a battery capacity (C), provided that the starting point (SoC(t0)) is known: SoCðt Þ ¼ SoCðt0 Þ þ
1 C
tZ 0 þt
Ibat ðtÞ dt
½8
t0
There are mainly two difficulties that make this method not ideal: First, there is always an inaccuracy in current mea• surement. Even small offsets in measured current
•
values accumulate because of integration and can lead to significant errors in calculated SoC after a while. For example, for a 100 Ah battery, an offset of 10 mA leads to an error of about 7% after only 1 month. Second, a part of the battery terminal current, which is the only value that can be measured, flows into side reaction (Iside). Gassing is the most relevant side reaction in LABs, but others such as corrosion or microshort circuits due to dendrites can also occur. This current which goes into side reactions should be actually subtracted from the measured battery current before integration, otherwise it influences the Ah balance similar to measurement errors: 1 SoCðt Þ ¼ SoCðt0 Þ þ C
tZ 0 þt
ðIbat ðtÞ Iside ðtÞÞ dt
½9
t0
However, side reactions typically behave highly nonlinear and therefore the share of the terminal current going into the side reaction can vary from approximately zero to nearly 100% if the battery is fully charged. There is no way to measure the current going into the side reactions online and in situ. Nevertheless, the simplest solution for minimizing the influence of a gassing current on the Ah balance is based on an assumption that only a certain percent of charge current flows into the main reaction and thus contributes to the battery charge. Therefore, only 80–98% of the charge current depending on the battery type and the charge conditions is considered by the integration. The improved solution is to estimate the current share going into the gassing reaction as the main side reaction by a gassing model. It is assumed that under normal operation conditions (no internal short circuits) more than 90% of the side reactions are given by the gassing reaction. During gassing, water from the electrolyte is split into hydrogen and oxygen. In VRLA batteries (gel or AGM) the majority of the gas is recombined with water, but this is of no relevance for the Ah balance. Since the gassing reaction is mostly influenced by the battery voltage and temperature, and both
797
parameters are usually measured in a system, a simple implementation is possible. However, the parameters of such models are strongly dependent on battery type and they change during the lifetime of the battery. Therefore, self-learning and self-adapting algorithms are necessary to estimate the gassing current correctly throughout the battery lifetime. Generally, both difficulties are only critical if the battery is not charged completely on a regular basis. If the batteries receive a full charge, for example, everyday as it happens in forklift truck applications, none of the above effects is of relevance. The Ah balance is reset everyday and therefore the error that can amount to measurement errors or side reactions is very small and negligible. Since errors in Ah balancing cannot be completely eliminated and because the initial value of SoC has to be known, additional technique should be used in any case to obtain recalibration points. Anyway, due to its simplicity, this method is normally used as a basic technique for SoC calculation in almost all battery diagnostic systems. For the determination of the dynamic change in the SoC it is the best solution anyway and therefore is very accurate for all periods between other recalibration points. Measurement of Acid Concentration or Density In LABs the electrolyte, which is diluted sulfuric acid, is not only used as an ion conductor but also takes part in the main chemical reaction. By discharging the battery, sulfuric acid reacts with lead and lead dioxide to give lead sulfate and therefore is used up; during charging, the reaction is reversed and sulfuric acid is therefore released back to the electrolyte. This means that the concentration of the sulfuric acid is linear proportional to the SoC of the battery. Thus, if the relationship between acid concentration and SoC is known, then by measuring the former the latter can be determined. Unfortunately, the acid concentration itself is not easy to measure. However, since the concentration is almost linearly proportional to the density assuming constant temperature and pressure, the density is often used for determining the SoC. Often, the term specific gravity is used. Different sensors exist (e.g., hydrometers) but in any case a direct intervention into a battery for a measurement is required, so that this technique can only be used in laboratory or offline. This principle, for example, finds its use in SoC indicators that are built in flooded SLI batteries and which are based on colored balls floating in the electrolyte. Some complications that reduce the accuracy of this method or its usability are as follows: This method is only feasible with flooded LABs. No • sensors are available to measure the electrolyte density in VRLA batteries.
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Batteries from different manufacturers as well as from • different product lines of one manufacturer may have
•
significantly different amounts of electrolyte related to the active masses of the cell. The acid concentration in a fully charged battery cell may also be different. Therefore, for each product line, the dependence between SoC and electrolyte density must be known or measured before using it for the SoC determination. Moreover, even within a certain product line, there are variations in electrolyte density, and its amount in a specific cell compared to the designed value. It leads to additional inaccuracies. During the lifetime of a battery, the amount of water in the battery cell changes similarly to losses due to gassing. As sulfuric acid molecules remain in the battery, the overall concentration increases with increasing water loss. The acid density is usually measured at the top of the cell, but the acid concentration distribution after the battery was in operation may not be constant at all. Acid stratification is a well-known effect in flooded batteries and consists of a significant acid concentration gradient in the vertical direction in a cell. Especially after partial cycling, the acid concentration at the bottom of the cell is significantly higher compared with the top of the battery. In particular, above the electrode plates, where acid concentration or density is usually measured, it can be much lower. It leads to the fact that the SoC determined from the acid density above the electrodes may be lower than the actual SoC of the battery. Owing to diffusion, the acid stratification disappears, but a considerable time period of several weeks is necessary. Other more efficient measures involve air bubbling systems, shaking of the batteries, or extensive gassing of the batteries for stirring the electrolyte.
Open-Circuit Voltage and Electromotive Force Open-circuit voltage (OCV) is the terminal voltage of the battery under open-circuit condition. Physically OCV of a LAB is mostly defined by its electromotive force (EMF), which is the internal driving force of a battery for providing energy to a load. Physically, the EMF is the difference between the reduction potentials of the half-cell reactions and depends on the electrolyte concentration. Therefore, EMF and OCV are related to the SoC due to their dependency on the acid concentration. This relation allows one to use OCV and EMF for SoC determination. Open-circuit voltage method for SoC determination refers to a measurement of the battery terminal voltage under open-circuit conditions, whereas EMF method refers to the estimation of EMF (which is very close to OCV) from the battery terminal voltage while moderated charge or discharge current is applied.
Open-Circuit Voltage method
Since acid concentration is the link between SoC and OCV, all complications that reduce the accuracy of SoC determination using measuring of acid concentration (variations in battery production, acid stratification, and gassing reaction) are also valid for this method. Admittedly, acid stratification influences the determined SoC value differently and gassing reaction has additional influence: While acid density measurement usually considers the • electrolyte at a top part of the battery, the OCV of the
•
cell with acid stratification is dominated by the higher potential caused by the higher concentration of sulfuric acid at the bottom of the battery. In fact, the SoC calculated from OCV in a battery with stratification is typically overestimated as long as no correction algorithms for the compensation of the acid stratification are used. The additional role of the gassing reaction is that the measured OCV is actually a mixed potential resulting from the discharge reaction and the gassing reaction. Since gassing reaction is highly temperature dependent, OCV also shows a non-negligible dependency on the temperature.
Additional sources of inaccuracy are the overvoltages after charge or discharge. A significant period is required to reach steady-state conditions. Moreover, in many applications even in rest periods a small current required for monitoring devices, clocks, and so forth, flows through the battery, so that there are actually no opencircuit conditions at all. Nevertheless, the OCV method is widely used in realworld applications. It is easy to implement because OCV is almost linear to SoC, at least in a range of about 30–100% SoC, and this method can be used for all types of LABs. An individual calibration of the relation between OCV and SoC is necessary in the same way as it was discussed for direct acid concentration measurements. For many typical LABs an empirical equation OCVðVÞ ¼ 0:86 þ acid density ðg cm3 Þ
½10
can be used as a rule of thumb. Electromotive Force Method
There are different techniques allowing the estimation of the EMF of a battery by measuring the battery terminal voltage: Model-based approaches, which are described below, • can be used to calculate overvoltages that should be subtracted from battery terminal voltage to obtain the EMF value. The accuracy of this method is defined by
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•
All complications described for the OCV method are also valid for the EMF method, but they are only of secondary order since the inaccuracy of the first step – estimation of EMF from battery terminal voltage – is significantly higher than the inaccuracy of the next step – SoC determination from EMF. The inaccuracy of the EMF method for LABs is quite high. It can only give a wide range of about 20–50% depending on where the SoC could be. Therefore, it is only used in combination with other methods. Impedance Measurements The electrochemical impedance of a battery is a frequency-dependent complex number Z( f ) and represents a real and imaginary part or its modulus and phase. A set of impedances measured at different frequencies in a wide range from few millihertz to some hundred hertz is called impedance spectrum and can be represented as a Nyquist diagram (imaginary part vs real part). Since batteries show a capacitive behavior, at an interesting frequency range, electrochemists generally plot the negative of the imaginary part on the ordinate axis, so that the capacitive part of the impedance spectrum appears in the upper quadrants. Alternatively, a representation as a Bode diagram (absolute value and argument vs frequency) can be used. As an example an impedance spectra of an SLI battery at different SoC levels is presented in Figure 2. Measuring an impedance spectrum is called impedance spectroscopy. In the laboratory, impedance spectroscopy is normally performed using an impedance spectroscope, which generates a small sinusoidal current or voltage signal and feeds it in the battery. The voltage response versus the current response is measured and the complex impedance can be determined. The hardware is very costly and therefore cannot be used in the majority of the real-world applications. Therefore, another technique called passive impedance spectroscopy can be used instead under certain circumstances. In this case, current fluctuations, which are already presented in the system, are analyzed. Since no active network is required and impedance can be calculated using digital filters, which
90% SoC 70% SoC 40% SoC 20% SoC
−25 −20 −15 Im(Z) (mΩ)
•
the properties of the used battery model and its parametrization. Linear interpolation method. In this case, the average battery voltage is calculated, when the battery is charged and discharged with same moderated currents within a short time, so that SoC does not significantly change. Voltage relaxation method. After charge or discharge, the battery voltage will relax to the EMF value (or actually OCV value, but it is quite the same as explained above). This relaxation process can be monitored and the EMF can be predicted.
−10 −5 0 5 10 15 20 5
10
15
20
25 30 Re(Z) (mΩ)
35
40
45
Figure 2 Typical spectrum of a lead–acid battery depending on the SoC (60 Ah flooded starter light iginition (SLI) battery (OEM product); the battery was discharged with 1.5 A during the measurement).
are easy to implement, this method is suitable for use in applications at low costs. Nevertheless, a precise measurement of current and voltage is necessary. However, depending on the excitations available in the system, only impedance at certain frequencies can be determined with sufficient accuracy. Therefore, this method cannot offer the same amount of information as it can be obtained with an active impedance spectroscope in the laboratory. In addition, the accuracy of passive impedance spectroscopy is lower than by using impedance spectroscope. Pure impedance data can be hardly used for SoC and SoH determination, but the results from impedance measurements are valuable information, which can be used in an overall algorithm. Most useful applications of impedance measurements for battery diagnostic are as follows: Measurement of the internal resistance (R ) of the bat• tery. Internal resistance of LAB can be measured by i
impedance spectroscopy at relatively high frequencies (some 100 Hz). Since internal resistance is affected by the acid concentration and the conductivity of the active masses, a dependence between Ri and SoC exists and can be used for estimation of the latter. However, this dependence is only significant at SoCs lower than about 30–50% (Figure 3). Moreover, the internal resistance is influenced by the temperature, it has some variations for a given product line and it increases significantly during the lifetime because of aging. In total it makes accurate determination of SoC from Ri extremely difficult and this method can only be used in combination with other techniques. The determination of SoH defined as a
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10
For accurate measurement, a significant discharge current is needed, because only noticeable voltage drops can be measured with high accuracy. Direct current resistance has the same correlation with SoC and SoH as the internal resistance measured using impedance spectroscopy and described above. However, the requirements for the measurement of current and voltage for determining DCR are lower compared with passive or active impedance spectroscopy.
9 8 7
Ri (mΩ)
6 5 4 3
Model-Based Approach
2 1 0
0
20
40
60
80
100
SoC (%)
Figure 3 Internal resistance of a lead–acid battery depending on state of charge (70 Ah flooded new starter–light–ignition (SLI) battery (OEM product)). SoC, state-of-charge.
•
relation between an actual battery capacity and a nominal battery capacity is almost impossible. Even though internal resistance increases during aging, it is mostly defined by corrosion – an aging process that does not influence the capacity loss directly. Owing to the increase of internal resistance, the end-of-discharge voltage is reached earlier by the discharge, but this indirect influence of the internal resistance to the battery capacity is not significant. However, there is some correlation between the increase of the internal resistance and a capacity loss. This fact is used in some systems for SoH determination. The correlation is quite good in UPS systems and it is quite poor for batteries with partial SoC cycling such as in SLI or PV system applications. The measurement of Ri is mostly useful if resistive overvoltages should be calculated. It is also a basic parameter for the determination of some SoFs (e.g., CC in vehicles), and owing to the easy and cost-effective measurements with modern devices it is widely used in different applications. Parametrization of battery models. By using modelbased approaches as described below, the model parameters must be continuously adapted. Passive impedance spectroscopy can be used as a supporting method for online adaptation.
Measurement of Direct Current Resistance The term direct current resistance (DCR) is used to describe the internal resistance of a battery measured as a relation between the terminal voltage drop (DV) caused by a change in the terminal current (DI): DCR ¼
DV DI
½11
The main idea of the model-based approach is to have a relatively simple dynamic battery model with only few parameters in a diagnostic system. The measured battery terminal current can be fed into this model and a voltage response can be compared with the measured voltage to track changes in the battery parameters. These changes can be used to calculate changes in SoC and SoH of the battery. Thereby, the Kalman filter is applied. It is a recursive filter, which estimates the state of a dynamic system from a series of noisy measurements and is well known from system theory. Three main steps are required to design such a system: 1. The battery model must be chosen. The used model affects significantly the quality of the diagnostic system. Linear models are easy to implement, but because of the strong nonlinear behavior of LABs they lead to high inaccuracies of the diagnostic system. The use of nonlinear models can yield better results, but their implementation is difficult and special extensions of Kalman filter should be used. The main criterion for choosing the battery model is the possibility to present it as state variable equations: x_ðt Þ ¼ A xðt Þ þ B uðt Þ yðt Þ ¼ C xðt Þ þ D uðt Þ
½12
where u(t) is the vector of applied inputs, x(t) is the state vector, y(t) is the vector of measured outputs, A is the state matrix, B is the input matrix, C is the output matrix, and D is the feed through matrix. In addition, the observability, which can be verified by investigation of state variable equations, should be guaranteed. Two examples of models, which can be used for LABs, are shown in Figure 4. The deviation in the behavior of the battery model from the real battery and inaccuracies in measurement should be additionally considered in a state variable model: x˙ ðt Þ ¼ A xðt Þ þ B uðt Þ þ G sðt Þ zðt Þ ¼ yðt Þ þ mðt Þ ¼ C xðt Þ þ D uðt Þ þ mðt Þ
½13
where s(t) is the additive vector that represents system disturbances and model inaccuracies, m(t) is the additive
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health
801
Cs Ri
Ri Rs
U
Cb
Rd
Rs
Re
Cs
Cb
U
Figure 4 Battery models used in a model-based approach: Rd – self-discharge, Cb – main battery capacity, Ri – internal resistance, Rs and Cs – transient effects, Re – end resistance, U – terminal voltage.
vector that represents effects of measurement noise, z(t) is the vector of measured outputs corrupted by noise, and G is the coupling matrix that regulates the influence of the disturbances on each state. Often additional state variables are employed to obtain better connection between state vector and SoC or SoH of the battery. 2. The Kalman filter should be designed. It is generally an algorithm to track changes of the state variables of the model using comparison of the model outputs to the measured values under consideration of measurement noises and disturbances caused by the model inaccuracy. Changes of state variables are used to calculate changes in SoC and SoH of the battery. In addition, a new parameter set for the battery model can be calculated, that is, the model is adapted to the new conditions. For a simple linear battery model, the standard Kalman filter can be used. For use with nonlinear models, modern extension of Kalman filter, such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) can be applied. 3. The parametrization should be performed to find initial values of the model parameters. The model and the system should be also analyzed to determine the coupling matrix G as well as the noise vectors s(t) and m(t). The quality of this analysis is a crucial factor for a successful design. This method can be used in combination with supporting algorithms that improve the parametrization of the battery model and consequently adapt it to the battery, since aging and temperature significantly influence the characteristic of the battery. Hence, an impedance measurement can be applied. Model-based approach can be effectively used in systems with high dynamics, for example, in vehicle applications. Methods of Artificial Intelligence Two methods of artificial intelligence are used in battery diagnostic quite often to determine SoC: expert systems
and artificial neural networks. While expert systems attempt to simulate the knowledge and analytical skills of one or more human experts that are able to express their knowledge as a set of rules, which can be transferred into mathematical expressions, artificial neural networks attempt to use the concepts of human brain to find an output based on several nonlinear correlated input state variables. Expert systems
Expert systems for SoC determination can be implemented as a set of simple rules. Rules can generally use all observable or measurable state variables such as current, voltage, and temperature to make a statement about SoC. If more information is available, such as that on impedance at different frequencies, pressure, or acid concentration, they can be used as additional state variables as well. First or second time derivatives of the state variables can be used as input in the same way as output from models which can do a pre-analysis of the data. Thereby, often fuzzy statements like ‘current high’ or ‘temperature low’ are used. As an example, the following rules can be applied: If battery cell voltage is higher or equal to 2.5 V per • cell, the temperature is above 20 1C and the time
•
derivative of the charge current is small, then the SoC is above 70%. If discharge current is low, temperature is high, and voltage is low, then SoC is low.
After applying the rules, the so-called de-fuzzyfication takes place to transfer the statements into numbers, which can be used by other algorithms or control systems. One of the combination methods described in the next section can be used to generate one statement about SoC at each time using these rules. When applying an expert system to batteries, typically only a certain range of SoC can be determined. Typically a rule can be applied only from time to time (most rules, e.g., require quite stable conditions rather than highly dynamic battery operation) and therefore other
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algorithms are used in combination to achieve a continuous SoC output. The Ah balance as described in eqn [9] is perfectly suited to follow the changes of the SoC even during periods of high accuracy until the expert system delivers a new SoC estimation. Artificial neural network
A principle of using neural networks is shown in Figure 5. The measured battery parameters (charge/discharge current, voltage, and temperature) or derived parameters (amp-hours, di/dt, dv/dt, etc.) as well as the previous SoC can be used as input values for the neural network. An interconnected group of nodes is applied to calculate SoC. One or more layers of nodes are used between the input and the output layer. The information of the neural network is based on weight factors for all input variables with regard to a certain node. The value zi j of a node i in layer j is given by the sum of the k weighted inputs from the previous layer j 1. Nodes are used as new input variables for the next layer of nodes: zji ¼
X
wzj 1 zj zjk1
½15
i
k
k
Network topology and used node functions should be carefully selected depending on the application, desired network complexity, and available input parameters. The advantage of this method is that no mathematical model for the correlation between the input and the output variables is required. The disadvantage is that a huge amount of training data are required to train the neural network. The training data consist of a complete set of input variables and an exact known output vector. The data are used to determine the weight factors
Input
Rules
Approaches for Combination of Different Methods As mentioned before, often only a combination of different methods can deliver reliable information about SoC of the battery. Different approaches implemented in one system estimate SoC either continuously or at certain points of time, whenever valid input data for a given algorithm is available. The usage of different algorithms requires concepts for the combination of their respective outputs. Some common techniques are listed below: Fuzzy logic. This technique is used when different • fuzzy statements about SoC should be combined and
•
½16
wzj 1 zj k
from eqn[15]. If such a neural network should be adapted according to the changing performance of the battery during the lifetime, this method requires a significant data memory and must be able to deliver valid sets of input and output variables. One option to overcome the problem with large data set storage is to use special neural gas models, which can be adapted instantaneously by any new available learning point. In any case, for selflearning algorithms it is necessary to implement an additional independent algorithm for delivering reference points.
i
Output
SoCprev
Temp SoC i
• dv/dt
Figure 5 Artificial neural network used for state-of-charge (SoC) estimation.
qualitative information about SoC based thereon should be delivered as a result. This method is often used, for example, in expert systems or as a part of a neural network. Boundary consideration. Many methods can only make statements whether SoC is higher than a certain value or lower than a certain value or both. Furthermore, even for methods, which are able to deliver more precise information about SoC, upper and lower boundaries can be defined by considering inaccuracies of this method. Usually these methods are used in combination with the Ah balance as the recalibration points or in other words, as limits for Ah counter. If the Ah balance bounces against one of the boundaries, the Ah balance will be limited to the boundary value accordingly. It is advantageous for this technique if three Ah balance counters are used: Ah balance according to the measured value, Ah balance taking into account the maximum positive error occurring from the measurement system and from model errors with regard to side reactions (see eqn [9]), and an Ah balance considering the maximum negative error. This method is illustrated in Figure 6. Kalman filter. In model-based approaches, the Kalman filter is used in combination with a battery model, but Kalman filters can be used as a method for combination of SoC values delivered by different techniques as well. In this case a model for SoC of the battery is employed based on an equation representing the SoC
Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health Ah balance
Ri
C
u(t ) V wi x(t ) y(t ) z(t )
Ahmax
Ahmiddle Ahmin
0
Time Boundaries from supporting algorithms
Figure 6 Example for using boundaries from different state-ofcharge (SoC) algorithms in combination with a triple Ah balance approach.
as a weighted sum of SoCs estimated by different methods: SoCðt Þ ¼ aðt Þ þ DSoCðDt Þ aðt Þ ¼ w1 SoC1 ðt Dt Þ þ ? þ wn SoCn ðt Dt Þ X þ ð1 wi Þ SoCðt Dt Þ
Z(f) DI Dt DV l(t ) r(t )
803
internal resistance of the battery (ohm) vector of the applied inputs voltage weighting factors state vector vector of measured outputs vector of measured outputs corrupted by noise frequency-dependent complex number change in current time step voltage drop additive vector representing effect of measurement noise additive vector representing system disturbances and model inaccuracies
Abbreviations and Acronyms
½17
where Dt is the time step, DSoC(Dt) is the SoC change calculated by current integration during the duration of the time step Dt, a(t) is the weighted sum of SoCs, SoC1, y, SoCn are the SoCs delivered from different methods, and w1, y, wn are the weighting factors considering inaccuracies of the respective method. The additional advantage of using different methods in one system is that self-learning algorithms are possible in this case. Whenever a method delivers a reliable SoC value, all other algorithms can adapt themselves accordingly.
Nomenclature
AGM CA CC CCCV DCR DoD EMF LAB OCV PV SLI SoC SoF SoH UPS VRLA
absorbent glass mat charge acceptance cranking capability constant voltage/constant current (charging strategy) direct current resistance depth-of-discharge electromotive force lead–acid battery open-circuit voltage photovoltaic starter light ignition state-of-charge state-of-function state-of-health uninterruptible power supply valve-regulated lead–acid
Symbols and Units a(t ) A B C C C0 Ca Cav CN D G Ibat Iside Q Qd
weighted sum of SoCs state matrix input matrix output matrix capacity of a battery (Ah) initial capacity actual capacity available capacity nominal capacity feed through matrix coupling matrix battery terminal current a part of the battery terminal current that flows into side reaction amount of charge (Ah) discharged amount of charge (Ah)
See also: Batteries: Capacity; Secondary Batteries – Lead–Acid Systems: Overview; Lifetime Determining Processes; Batteries: Partial-State-of-Charge.
Further Reading Bhangu BS, Bentley P, Stone DA, and Bingham CM (2005) Nonlinear observers for predicting state-of-charge and state-of-health of lead– acid batteries for hybrid-electric vehicles. IEEE Transactions on Vehicular Technology 54(3): 783--794. Blanke H, Bohlen O, Buller S, et al. (2005) Impedance measurements on lead–acid batteries for state-of-charge, state-of-health and cranking capability prognosis in electric and hybrid electric vehicles. Journal of Power Sources 144: 418--425. Bohlen O (2008) Impedance-Based Battery Monitoring. ISBN 978-38322-7606-5. Aachen: Shaker Verlag.
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Secondary Batteries – Lead–Acid Systems | State-of-Charge/Health
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