Device and method for determining characteristic variables for batteries

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First Claim
1. Method for determination of characteristic variables for electrical states of an energy storage battery, having the following steps:
 a) Subdividing the electrolyte volume (v) of the energy storage battery into at least two electrolyte volume components (v_{i}) with associated electrolyte balancing areas (S_{i});
b) Defining of at least two electrode plate balancing areas (P_{k}) by subdividing the total resistance (R^{M}) of the electrode plates in the energy storage battery into resistance components (R_{k}^{M}) for the defined electrode plate balancing areas (P_{k}), and the total energy storage capacity (K^{M}) of the electrode plates in the energy storage battery into energy storage capacity components (K_{k}^{M}) for the defined electrode plate balancing areas (P_{k});
c) Determining of the electrolyte concentration (r_{i}) of the electrolyte volume components (v_{i}) for the defined electrolyte balancing areas (S_{i});
d) Determining of the amounts of charge (KE_{k}^{M}) which are in each case held in the electrode plates in the electrode plate balancing areas (P_{k}); and
e) Determining of at least one characteristic variable for associated electrical states of the energy storage battery by means of a mathematical model for describing an electrical equivalent circuit at least by means of the variables of the resistance components (R_{k}^{M}), of the energy storage capacity components (K_{k}^{M}), of the electrolyte concentration (r_{i}) and of the amounts of charge (KE_{k}^{M}) held.
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Abstract
A method for determination of characteristic variables for electrical states of an energy storage battery has the following steps:a) the electrolyte volume (v) of the energy storage battery is subdivided into at least two electrolyte volume components (vi) with associated electrolyte balancing areas (Si); b) at least two electrode plate balancing areas (Pk) are defined by subdividing the total resistance (RM) of the electrode plates in the energy storage battery into resistance components (RkM) for the defined electrode plate balancing areas (Pk), and the total energy storage capacity (KM) of the electrode plates in the energy storage battery into energy storage capacity components (KkM) for the defined electrode plate balancing areas (Pk); c) the electrolyte concentration (ri) of the electrolyte volume components (vi) for the defined electrolyte balancing areas (Si) is determined; d) the amounts of charge (KEkM) which are in each case held in the electrode plates in the electrode plate balancing areas (Pk) are determined; and e) at least one characteristic variable is determined for associated electrical states of the energy storage battery by means of a mathematical model for describing an electrical equivalent circuit at least by means of the variables of the resistance components (RkM), of the energy storage capacity components (KkM), of the electrolyte concentration (ri) and of the amounts of charge (KEkM) held.
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Battery voltage warning device  
Patent #
US 4,943,777 A
Filed 12/02/1989

Current Assignee
Sanshin Kogyo Kabushiki Kaisha

Sponsoring Entity
Sanshin Kogyo Kabushiki Kaisha

Process for charging maintenancefree lead batteries with a fixed electrolyte  
Patent #
US 4,952,861 A
Filed 09/06/1988

Current Assignee
VB AUTOBATTERIE GMBH

Sponsoring Entity
Varta Batterie AG

Polyphase alternator and dual voltage battery charging system for multiple voltage loads  
Patent #
US 4,816,736 A
Filed 03/12/1987

Current Assignee
Johnson Controls Technology Company

Sponsoring Entity
GlobeUnion Inc.

Dynamic stateofcharge indicator for a battery and method thereof  
Patent #
US 4,876,513 A
Filed 12/05/1988

Current Assignee
Johnson Controls Technology Company

Sponsoring Entity
GlobeUnion Inc.

Vehicle battery diagnostic device  
Patent #
US 4,719,427 A
Filed 08/07/1986

Current Assignee
Mitsubishi Electric Corporation

Sponsoring Entity
Mitsubishi Electric Corporation

Device for indicating the fully charged state of a battery  
Patent #
US 4,642,600 A
Filed 04/02/1985

Current Assignee
VB AUTOBATTERIE GMBH

Sponsoring Entity
Varta Batterie AG

Microcomputer controlled electronic alternator for vehicles  
Patent #
US 4,659,977 A
Filed 10/01/1984

Current Assignee
Continental Automotive GmbH

Sponsoring Entity
CHRYSLER CORPORATION HIGHLAND PARK MI A DE CORP

Method and apparatus for monitoring and indicating the condition of a battery and the related circuitry  
Patent #
US 4,665,370 A
Filed 09/15/1980

Current Assignee
AUTOMOTIVE DESIGNS INC. A CORP. OF MICHIGAN

Sponsoring Entity
AUTOMOTIVE DESIGNS INC. A CORP. OF MICHIGAN

Battery state of charge gauge  
Patent #
US 4,595,880 A
Filed 08/08/1983

Current Assignee
Ford Global Technologies LLC

Sponsoring Entity
Ford Motor Company

Automatic battery analyzer including apparatus for determining presence of single bad cell  
Patent #
US 4,322,685 A
Filed 02/29/1980

Current Assignee
Johnson Controls Technology Company

Sponsoring Entity
Johnson Controls International plc

Automatic battery analyzer  
Patent #
US 4,193,025 A
Filed 12/23/1977

Current Assignee
Johnson Controls International plc

Sponsoring Entity
GlobeUnion Inc.

Apparatus and method for calibrated testing of a vehicle electrical system  
Patent #
US 4,207,611 A
Filed 12/18/1978

Current Assignee
Ford Motor Company

Sponsoring Entity
Ford Motor Company

Device for determining the charge condition for a secondary electric storage battery  
Patent #
US 4,153,867 A
Filed 11/14/1977

Current Assignee
Akkumulatorenfabrik Dr. Leopold Jungfer

Sponsoring Entity
Akkumulatorenfabrik Dr. Leopold Jungfer

Method of measuring the charge condition of galvanic energy sources and apparatus for carrying out this method  
Patent #
US 3,906,329 A
Filed 08/27/1973

Current Assignee
Firma Deutsche Automobilgesellschaft mbH DT

Sponsoring Entity
Firma Deutsche Automobilgesellschaft mbH DT

19 Claims
 1. Method for determination of characteristic variables for electrical states of an energy storage battery, having the following steps:
 a) Subdividing the electrolyte volume (v) of the energy storage battery into at least two electrolyte volume components (v_{i}) with associated electrolyte balancing areas (S_{i});
b) Defining of at least two electrode plate balancing areas (P_{k}) by subdividing the total resistance (R^{M}) of the electrode plates in the energy storage battery into resistance components (R_{k}^{M}) for the defined electrode plate balancing areas (P_{k}), and the total energy storage capacity (K^{M}) of the electrode plates in the energy storage battery into energy storage capacity components (K_{k}^{M}) for the defined electrode plate balancing areas (P_{k});
c) Determining of the electrolyte concentration (r_{i}) of the electrolyte volume components (v_{i}) for the defined electrolyte balancing areas (S_{i});
d) Determining of the amounts of charge (KE_{k}^{M}) which are in each case held in the electrode plates in the electrode plate balancing areas (P_{k}); and
e) Determining of at least one characteristic variable for associated electrical states of the energy storage battery by means of a mathematical model for describing an electrical equivalent circuit at least by means of the variables of the resistance components (R_{k}^{M}), of the energy storage capacity components (K_{k}^{M}), of the electrolyte concentration (r_{i}) and of the amounts of charge (KE_{k}^{M}) held.
 a) Subdividing the electrolyte volume (v) of the energy storage battery into at least two electrolyte volume components (v_{i}) with associated electrolyte balancing areas (S_{i});
 2. Method according to claim 1, characterized in that the acid density of the electrolyte volume components (vi) is a measure of the electrolyte concentration.
 3. Method according to claim 1 or 2, characterized by the rest voltages (U00k) being determined in the defined electrode plate balancing areas (Pk) as a function of the electrolyte concentration (ri) in an associated electrolyte balancing area (Si).
 4. Method according to claim 3, characterized by the rest voltages (U00k) also being determined in the defined electrode plate balancing areas (Pk) as a function of the battery temperature (T) and of the amounts of charge (KEkM) held.
 5. Method according to one of the preceding claims, characterized by the currents (Ii) for the electrolyte balancing areas (Si) being calculated from the total current (I) flowing to the connecting terminals of the energy storage battery and from a division ratio which is dependent on the electrolyte concentration (ri) in the respective electrolyte balancing area (Si).
 6. Method according to one of the preceding claims, characterized by the currents (Ik) for the electrode plate balancing areas (Pk) being calculated from the total current (I) flowing to the connecting terminals of the energy storage battery and from a division ratio which is dependent on the rest voltage (U00k) in the respective electrode plate balancing area (Pk) and the resistance components (RkM).
 7. Method according to claim 6, characterized in that, for the discharge currents, the currents (Ii) for the electrolyte balancing areas (Si) are assumed to be equal to the currents (Ik) calculated for the associated electrode plate balancing areas (Pk).
 8. Method according to one of the preceding claims, characterized by the amount of charge (KEkM) held in an electrode plate balancing area (Pk) after a time period (dt) being calculated from the energy storage capacity (KEkM) held in this electrode plate balancing area (Pk) until immediately before this time period and the charge (IkM×dt) which has flowed through the electrode plate balancing area (Pk) in the time period (dt):
 9. Method according to one of the preceding claims, characterized by the amount of charge (RKkM) which can still be drawn from the electrode plates in the individual electrode plate balancing areas (Pk) being calculated from the energy storage capacity component (KkM) of the respective electrode plate balancing area (Pk) minus the amount of charge (KEkM) which was drawn from the active mass in the electrode plate balancing area (Pk) prior to the time period (dt), and minus the charge (IkM×dt) drawn from the electrode plates in the electrode plate balancing area (Pk) in the time period (dt).
 10. Method according to one of the preceding claims, characterized by the amounts of charge (RKis) which can still be drawn from the electrolyte volume components (Vi) in the individual electrolyte balancing areas (Si) being calculated as a function of the electrolyte concentration (ri) at that time and the electrolyte volume (vi) in the respective electrolyte balancing area (Si) and the battery temperature (T).
 11. Method according to one of the preceding claims, characterized by the remaining capacity (RKg) which can still be drawn from the energy storage battery being calculated as a characteristic variable as a function of the amounts of charge (RKkM) which have been determined for the electrode plate balancing areas (Pk) and can still be drawn from the active masses of the respective electrode plate balancing areas (Pk), and as a function of the amounts of charge (RKiS) which have been determined for the electrolyte balancing areas (Si) and can still be drawn from the electrolyte volume components (vi) in the respective electrolyte balancing areas (Si).
 12. Method according to one of the preceding claims, characterized by the characteristic variable being determined as a function of the charge reversal current (IU) on the electrode plates between points where the rest voltage (U00,x) is relatively high and points where the rest voltage (U00,y) is relatively low, with the charge reversal current (IU) being calculated from the polarization resistance (Rp) on the electrode plate surfaces and the sum of the grid resistances (Rn) between the points (x, y), using the formula:
 13. Method according to one of the preceding claims, characterized by the rest voltage (U00,g) of the energy storage battery being calculated as a characteristic variable as a function of the state of charge of the individual electrode plate balancing areas (Pk), of the rest voltages (U00,k) in the electrode plate balancing areas (Pk), of the resistance components (RkM) of the polarization resistance (Rp) and of the charge reversal current (IU) by reversing the charge on in each case one individual electrode plate.
 14. Method according to one of the preceding claims, characterized by the critical temperature (Tcrit) for the start of ice crystal formation in the balancing areas being determined as a function of the electrolyte concentration (ri) in the electrolyte balancing areas (Si), the temperature (T) and the electrolyte volume components (vi).
 15. Method according to one of the preceding claims, characterized by the ice crystal volume (vice) in the electrolyte balancing areas (Si) being determined from a defined relationship between the equilibrium concentration (CGGi) of the acid in the electrolyte and the temperature (T) as well as the acid concentration (Ci) in the electrolyte using the formula:
 16. Method according to claim 14 or 15, characterized by a characteristic variable for the performance of the energy storage battery being determined as a function of the proportion of the ice crystal volume (vice) to the electrolyte volume (vi) in the electrolyte balancing areas (Si) and the location (Si) at which the ice crystal formation occurs.
 17. Method according to one of the preceding claims, characterized by the state values for the electrolyte concentration (ri) in the electrolyte balancing areas (Si) being adapted as a function of the difference between the actual rest voltage (U00) and the calculated rest voltage (U00,g).
 18. Method according to one of the preceding claims, characterized in that separate electrolyte balancing areas (Si) are assigned to outer areas of the energy storage battery and are assessed separately from the other electrolyte balancing areas (Si).
 19. Monitoring device for an electrochemical energy storage battery having a measurement unit for measurement of the battery terminal voltage (U), of the battery terminal current (I) and of the battery temperature (T), and having an evaluation unit, characterized in that the evaluation unit is designed to carry out the method according to one of the preceding claims, preferably by programming of a microprocessor unit.
1 Specification
The invention relates to a method for determination of characteristic variables for electrical states of an energy storage battery.
The invention also relates to a monitoring device for an electrochemical energy storage battery having a measurement unit for measurement of the battery terminal voltage, of the battery terminal current and of the battery temperature, and having an evaluation unit.
There is a requirement to determine or to predict the instantaneous state of an electrochemical energy storage battery, such as the state of charge or the capability to be loaded with a high current. By way of example, the capability of a starter battery to start a motor vehicle with an internal combustion engine is governed by the state of charge and the state of ageing and the apparent capacity loss of the battery, since the current level which can be drawn from the starter battery and its power output are limited. It is of major importance to determine the state of charge and the starting capability of a starter battery in situations in which, for example, the engine is being operated intermittently, since the vehicle power supply system together with its loads is still operated during the periods when the engine is stopped, even though the generator is not producing any current. The monitoring of the state of charge and of the starting capability of the energy storage battery must in situations such as these ensure that the energy content of the energy storage battery always remains sufficient to still start the engine.
Widely differing methods are known for measurement of the state of charge and for determination of the load behavior of energy storage batteries. For example, integrating instruments are used (Ah counters), with the charging current possibly being taken into account weighted with a fixed charging factor. Since the useable capacity of an energy storage battery is highly dependent on the magnitude of the discharge current and on the temperature, methods such as these also do not allow any satisfactory statement to be made about the useable capacity which can still be drawn from the battery.
By way of example, it is known from DE 22 42 510 C1 for the charging current to be weighted with a factor which is itself dependent on the temperature and on the state of charge of the battery, for a method for measurement of the state of charge.
DE 40 07 883 A1 describes a method in which the starting capability of an energy storage battery is determined by measurement of the battery terminal voltage and of the battery temperature, and by comparison with a state of charge family of characteristics which is applicable to the battery type to be tested.
DE 195 43 874 A1 discloses a calculation method for the discharge characteristic and remaining capacity measurement of an energy storage battery, in which the current, voltage and temperature are likewise measured, and in which the discharge characteristic is approximated by a mathematical function with a curved surface.
DE 39 01 680 C1 describes a method for monitoring the cold starting capability of a starter battery, in which the starter battery is loaded with a resistance at times. The voltage dropped across the resistance is measured and is compared with empirical values to determine whether the cold starting capability of the starter battery is still sufficient. In this case, the starter battery is loaded by the starting process.
Furthermore, DE 43 39 568 A1 discloses a method for determination of the state of charge of a motor vehicle starter battery, in which the battery current and rest voltage are measured, and the state of charge is deduced from them. In this case, the battery temperature is also taken into account. The charging currents measured during different time periods are compared with one another, and a remaining capacity is determined from them.
DE 198 47 648 A1 describes a method for learning a relationship between the rest voltage and the state of charge of an energy storage battery for the purpose of estimation of the storage capability. A measure for the electrolyte capacity of the electrolyte in the energy storage battery is determined from the relationship between the rest voltage difference and the amount of current drawn during the load phase. This makes use of the fact that the rest voltage rises approximately linearly with the state of charge in the higher state of charge ranges which are relevant in practice.
The problem of determining the state of an electrochemical energy storage battery using the already known methods is that wear occurs not only when rechargeable energy storage batteries are being discharged and charged but also when they are stored without any load being applied, and the relevant wear factors are not all considered.
In the case of a leadacid rechargeable battery, the electrolyte is composed of dilute sulfuric acid, that is to say a solution of sulfuric acid in water. In the completely charged state, this is typically an approximately 4 to 5 molar solution. During the discharge reaction, sulfuric acid in the electrolyte is consumed at both electrodes on the basis of the reaction equation:<FORM>Positive electrode: PbO<sub>2</sub>+H<sub>2</sub>SO<sub>4</sub>+2H<sup>+</sup>+2e<sup>−</sup>→PbSO<sub>4</sub>+2H<sub>2</sub>O </FORM><FORM>Negative electrode: Pb+H<sub>2</sub>SO<sub>4</sub>→Pb+2H<sup>+</sup>+2e<sup>−</sup></FORM>and, furthermore, H<sub>2</sub>O is formed at a positive electrode. In consequence, the concentration and the relative density of the electrolyte fall during discharging, while they rise again during the charging reaction, which takes place in the opposite sense.
If the sulfuric acid which is formed during the charging reaction has the capability for convection in the earth's field of gravity, then it has the tendency to fall in layers to the bottom of the cell vessel for the leadacid rechargeable battery cells. The electrolyte in the lower area of the respective cell vessel thus has a higher concentration than that in the upper area of the cell vessel. In the case of a leadacid rechargeable battery, this state is referred to as acid stratification.
Since both the charging and discharge reaction as well as the parasitic reactions, such as gas development, corrosion etc., are in general influenced by the electrolyte concentration, acid stratification leads to the cell state not being uniform.
One object of the invention is thus to provide an improved method for determination of characteristic variables for electrical states of an energy storage battery.
According to the invention, the object is achieved by the following steps:
 a) the electrolyte volume of the energy storage battery is subdivided into at least two electrolyte volume components with associated electrolyte balancing areas;
 b) at least two electrode plate balancing areas are defined by subdividing the total resistance of the electrode plates in the energy storage battery into resistance components for the defined electrode plate balancing areas, and the total energy storage capacity of the electrode plates in the energy storage battery into energy storage capacity components for the defined electrode plate balancing areas;
 c) the electrolyte concentration of the electrolyte volume components for the defined electrolyte balancing areas is determined;
 d) the amounts of charge which are in each case held in the electrode plates in the electrode plate balancing areas are determined; and
 e) at least one characteristic variable is determined for associated electrical states of the energy storage battery by means of a mathematical model for describing an electrical equivalent circuit at least by means of the variables of the resistance components, of the energy storage capacity components, of the electrolyte concentration and of the amounts of charge held.
The subdivision of the energy storage battery into electrolyte balancing areas and electrode plate balancing areas makes it possible to record the complex physical and chemical processes in an energy storage battery, including acid stratification, by means of a mathematical model for describing an electrical equivalent circuit of the energy storage battery, in order to determine characteristic variables for electrical states of energy storage batteries.
In this case, the acid density may be a measure of the electrolyte concentration.
It is advantageous for rest voltages to be determined in the defined electrode plate balancing areas as a function of the electrolyte concentration or acid density in an associated electrolyte balancing area. In this case, one electrode plate balancing area is in each case connected to an associated electrolyte balancing area, but not to the other electrolyte balancing areas. The calculation of the rest voltage in one electrode plate balancing area takes account of the fact that this is significantly influenced by the electrolyte concentration in the associated electrolyte balancing area, the amount of charge stored and, possibly, the battery temperature.
It is particularly advantageous for currents for the electrolyte balancing areas to be calculated from the total current flowing to the connecting terminals of the energy storage battery and from a division ratio which is dependent on the electrolyte concentration in the respective electrolyte balancing area. In the same way, currents for the electrode plate balancing areas can be calculated from the total current flowing to the connecting terminals of the energy storage battery and from a division ratio which is dependent on the rest voltage in the respective electrode plate balancing area and the resistance components.
The distribution of the current between the electrolyte and electrode plate balancing areas is preferably weighted in the charging direction as a function of the distribution of the current in the discharge direction, with the currents for the electrolyte balancing areas corresponding during discharging to the currents which have been calculated for the electrode plate balancing areas.
For charging processes, on the other hand, there is a difference between the division ratios for calculation of the currents for the electrolyte balancing area and the division ratios for calculation of the currents for the electrode plate balancing areas.
It is also advantageous to calculate the amount of charge drawn from one electrode plate balancing area in a time period by subtracting the charge which has flowed through the electrode plate balancing area in the time period under consideration from the amount of charge stored in this electrode plate balancing area until immediately before this time period.
The amount of charge which can still be drawn from the electrode plates in the individual electrode plate balancing areas can thus be calculated from the energy storage capacity component of the respective electrode plate balancing area minus the amount of charge which was drawn from the electrode plates in the electrode plate balancing area prior to the time period under consideration, and minus the charge drawn from the electrode plates in the electrode plate balancing area in that time period.
Furthermore, the amounts of charge which can still be drawn from the electrolyte volume components in the individual electrolyte balancing areas can be calculated as a function of the instantaneous electrolyte concentration and the electrolyte volume in the respective electrolyte balancing area, and the battery temperature.
The remaining capacity which can be still be drawn from the energy storage battery is preferably used as a characteristic variable as a function of the amounts of charge determined for the electrode plate balancing areas and can still be drawn for the electrode plates in the respective electrode plate balancing areas, and as a function of the amounts of charge which have been determined for the electrolyte balancing areas and can still be drawn from the electrolyte volume components in the respective electrolyte balancing areas.
The rest voltage of the energy storage battery can also be calculated as a characteristic variable, as a function of the amount of charge stored, possibly with respect to the energy storage capacity in the individual electrode plate balancing areas, of the rest voltages in the electrode plate balancing areas, of the resistance components, of the polarization resistance and the charge reversal current by charge reversal on in each case one individual electrode plate. The charge reversal current to be taken into account occurs as a result of the active material being discharged at the point where the rest voltage is high, and by charging taking place at the point where the rest voltage is relatively low. This is associated with a voltage drop across the polarization resistance, which represents the polarization resistance at the plate surface for small currents, and the individual grid resistances of the electrode plate grids. This results in a rest voltage whose value is between the rest voltages of the individual balancing areas (mixed potential).
It is thus advantageous to determine the characteristic variable as a function of the charge reversal current on the electrode plates between points where the rest voltage is relatively high and points where the rest voltage is low, with the charge reversal current being calculated from the polarization resistance on the electrode plate surfaces and the sum of the grid resistances between those points, using the formula:<maths id="MATHUS00001" num="1"><math overflow="scroll"><mrow><msub><mi>I</mi><mi>U</mi></msub><mo>=</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>y</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mrow><msub><mi>R</mi><mrow><mi>P</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mi>P</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>+</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mi>x</mi></mrow><mi>y</mi></munderover><mo></mo><msub><mi>R</mi><mi>n</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></maths>
It is also advantageous to determine the critical temperature for the start of ice crystal formation in the balancing areas as a function of the electrolyte concentration in the electrolyte balancing areas, the temperature and the electrolyte volume components. In this case, the ice crystal volume in the electrolyte balancing areas can be calculated from a defined relationship between the equilibrium concentration of the acid in the electrolyte and the temperature as well as the acid concentration in the electrolyte using the formula:<FORM>v<sub>ice,i′</sub>=(1−C<sub>i</sub>/CGG<sub>i</sub>)·v<sub>i </sub></FORM>
The ice formation in different zones can thus be assessed by consideration of individual balancing areas and calculation of the acid density in these balancing areas. The information about the volume of ice in each electrolyte balancing area makes it possible to state whether the performance of the energy storage battery is being adversely affected by the ice. This can be done in such a way, for example, that, when ice crystals occur only in the upper electrolyte balancing area, it is assumed that the battery will have its full performance while, in contrast, it is assumed that the battery will no longer be as powerful when ice crystals occur in the lowermost electrolyte balancing area.
It is also advantageous to adapt the state values for the electrolyte concentration in the electrolyte balancing areas as a function of the difference between the actual rest voltage and the calculated rest voltage. This takes account, for example, of any mixing of the acid caused by movement of the battery, which cannot be taken into account in the model since only the battery terminal current and the battery terminal voltage are measured.
Owing to the fact that outer areas of the energy storage battery may have a different behavior, it is also advantageous for separate electrolyte balancing areas, which are weighted separately to the other electrolyte balancing areas, to be assigned to outer areas.
A further object of the invention is to provide an improved monitoring device.
The object is achieved by the monitoring device of this generic type by the evaluation unit for carrying out the method as described above being formed, for example, by programming of a microprocessor unit.
The invention will be explained in more detail in the following text with reference to the attached drawings, in which:
FIG. 1 shows an electrical equivalent circuit of an energy storage battery, which is subdivided into acid balancing areas and electrode plate balancing areas, during discharging;
FIG. 2 shows an electrical equivalent circuit of the energy storage battery shown in FIG. 1, with the currents associated with the acid balancing areas and the electrode plate balancing areas, during charging;
FIG. 3 shows a graph of a function f<sub>L </sub>as a function of the amount of charge drawn, related to an energy storage capacity component, in an electrode plate balancing area in order to calculate the charging current distribution in the electrode plate balancing areas;
FIG. 4 shows a graph of a function f<sub>E </sub>as a function of the amount of charge drawn, related to an energy storage capacity component, in order to calculate the discharge current distribution;
FIG. 5 shows a graph of the state of charge of an energy storage battery, calculated from the charge balance, against time;
FIG. 6 shows a graph of the rest voltage, calculated from the model of the energy storage battery using the electrical equivalent circuit, and the actually measured unloaded voltage of an energy storage battery, against time;
FIG. 7 shows a graph of the acid density in the acid balancing areas of an energy storage battery against time;
FIG. 8 shows a graph of the loss of energy storage capability of an energy storage battery against time;
FIG. 9 shows a graph of the remaining capacity in the acid and associated electrode plate balancing areas, against time; and
FIG. 10 shows a graph of the energy storage capacity which can be drawn, against time.
FIGS. 1 and 2 show an electrical equivalent circuit of an energy storage battery during charging and discharging, which is subdivided into three electrolyte balancing areas S<sub>1</sub>, S<sub>2 </sub>and S<sub>3</sub>, which are located horizontally one above the other, with the respective electrolyte volume components v<sub>1</sub>, v<sub>2 </sub>and v<sub>3</sub>. Furthermore, the energy storage battery is subdivided horizontally in the same way into three electrode plate balancing areas P<sub>1</sub>, P<sub>2 </sub>and P<sub>3</sub>, which are located one above the other, for the electrode plates. For this purpose, the total energy storage capacity K<sup>M </sup>of the electrode plates in the energy storage battery is subdivided into three energy storage capacity components K<sub>1</sub><sup>M</sup>, K<sub>2</sub><sup>M </sup>and K<sub>3</sub><sup>M</sup>. In addition, the grid resistances R<sub>1</sub>, R<sub>2 </sub>and R<sub>3 </sub>(which occur in these areas) of the electrode plate grids are associated with the individual electrode plate balancing areas P<sub>1</sub>, P<sub>2 </sub>and P<sub>3</sub>. The resistance R represents the output conductor resistance.
The subdivision of the electrolyte balancing areas S<sub>i </sub>and electrode plate balancing areas P<sub>k</sub>, where i=1 to 3 and k=1 to 3, results in only the electrode plate balancing area P<sub>1 </sub>being connected to the electrolyte balancing area S<sub>1</sub>. In a corresponding manner, only the electrode plate balancing area P<sub>2 </sub>is connected to the electrolyte balancing area S<sub>2</sub>, and only the electrode plate balancing area P<sub>3 </sub>is connected to the electrolyte balancing area S<sub>3</sub>.
As can also be seen, a charging and discharge current I<sub>1</sub>, I<sub>2 </sub>and I<sub>3 </sub>can be associated with the respectively associated electrolyte and electrode plate balancing areas S<sub>i</sub>, P<sub>k</sub>, the sum of which currents corresponds to the total current I flowing to the connecting terminals of the energy storage battery. This total current I as well as the terminal voltage U of the energy storage battery can be measured very easily throughout the life of the energy storage battery, and can be used as an input variable for determination of the characteristic variables for electrical states of the energy storage battery.
FIGS. 1 and 2 also show that amounts of charge KE<sub>k</sub><sup>M </sup>which, at most, can assume the value of the respective energy storage capacity component K<sub>k</sub><sup>M </sup>can be drawn from the respective electrode plate balancing area P<sub>k</sub>.
When the energy storage battery is being charged, as is sketched in FIG. 1, the distribution of the total current I is separate in the charging direction for the electrolyte balancing areas S<sub>i </sub>and the electrode plate balancing areas P<sub>K</sub>. This is because the reduction in the more concentrated sulfuric acid in the energy storage battery in the downward direction results in additional charging of the electrolyte volume components v<sub>1</sub>, v<sub>2 </sub>in the electrolyte balancing areas S<sub>1</sub>, S<sub>2 </sub>at the bottom. Mathematically, this can be regarded as an additional current flow. For the sake of simplicity, this effect is dealt with as if the electrolyte balancing areas S<sub>i </sub>had a different charging current I<sub>i</sub><sup>s </sup>applied to them.
The charging current components I<sub>i</sub><sup>s </sup>for the electrolyte balancing areas S<sub>i </sub>are calculated as follows:<maths id="MATHUS00002" num="2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mi>i</mi><mi>S</mi></msubsup><mo>=</mo><mrow><mfrac><msub><mi>q</mi><mi>i</mi></msub><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo></mo><msub><mi>q</mi><mi>n</mi></msub></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000022" num="2.2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mi>N</mi><mi>S</mi></msubsup><mo>=</mo><mrow><mi>I</mi><mo></mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo></mo><mn>1</mn></mrow></munderover><mo></mo><msubsup><mi>I</mi><mi>n</mi><mi>S</mi></msubsup></mrow></mrow></mrow></math></maths>where N represents the number of electrolyte balancing areas, and q represents the division factors.
The equation system for the three electrolyte balancing areas S<sub>1</sub>, S<sub>2 </sub>and S<sub>3 </sub>illustrated by way of example is as follows:<maths id="MATHUS00003" num="3"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>1</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mfrac><msub><mi>q</mi><mn>1</mn></msub><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><mo>+</mo><msub><mi>q</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000032" num="3.2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>2</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mfrac><msub><mi>q</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><mo>+</mo><msub><mi>q</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000033" num="3.3"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>3</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mi>I</mi><mo></mo><msubsup><mi>I</mi><mn>1</mn><mi>S</mi></msubsup><mo></mo><msubsup><mi>I</mi><mn>2</mn><mi>S</mi></msubsup></mrow></mrow></math></maths>
The following equations apply to the division factors q<sub>i</sub>:<maths id="MATHUS00004" num="4"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>=</mo><mi/><mo></mo><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>·</mo><mrow><mo>(</mo><mrow><msub><mi>r</mi><mi>L</mi></msub><mo></mo><msub><mi>r</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>·</mo><mrow><mi>max</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><msubsup><mi>KE</mi><mn>3</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>3</mn><mi>M</mi></msubsup></mfrac><mo>,</mo><mfrac><msubsup><mi>KE</mi><mn>2</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>2</mn><mi>M</mi></msubsup></mfrac><mo>,</mo><mfrac><msubsup><mi>KE</mi><mn>1</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>1</mn><mi>M</mi></msubsup></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>q</mi><mn>2</mn></msub><mo>=</mo><mi/><mo></mo><mrow><msub><mi>c</mi><mn>2</mn></msub><mo>·</mo><mrow><mo>(</mo><mrow><msub><mi>r</mi><mi>L</mi></msub><mo></mo><msub><mi>r</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>·</mo><mrow><mi>max</mi><mo></mo><mrow><mo>(</mo><mrow><mfrac><msubsup><mi>KE</mi><mn>3</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>3</mn><mi>M</mi></msubsup></mfrac><mo>,</mo><mfrac><msubsup><mi>KE</mi><mn>2</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>2</mn><mi>M</mi></msubsup></mfrac></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>q</mi><mn>3</mn></msub><mo>=</mo><mi/><mo></mo><mrow><msub><mi>c</mi><mn>3</mn></msub><mo>·</mo><mrow><mo>(</mo><mrow><msub><mi>r</mi><mi>L</mi></msub><mo></mo><msub><mi>r</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mfrac><msubsup><mi>KE</mi><mn>3</mn><mi>M</mi></msubsup><msubsup><mi>K</mi><mn>3</mn><mi>M</mi></msubsup></mfrac><mo>)</mo></mrow></mrow></mrow></mtd></mtr></mtable></math></maths>
The variables c<sub>1 </sub>to c<sub>3 </sub>are empirical constants which must be determined experimentally for each energy storage battery type. The variable r<sub>L </sub>is the acid density which is formed on the electrode plates during charging. For simplicity, this is assumed to be constant and has a typical value of 1.46 g/cm<sup>3</sup>.
The values r<sub>1 </sub>to r<sub>3 </sub>are the acid densities in the electrolyte balancing areas S<sub>1</sub>, S<sub>2 </sub>and S<sub>3</sub>. The values (KE<sub>k</sub><sup>M</sup>) are the amounts of charge which are drawn from the respective electrode plate balancing area P<sub>k </sub>and which, at most, can assume the value of the energy storage capacity components K<sub>k</sub><sup>M </sup>defined by the subdivision of the energy storage battery.
The amounts of charge KE<sub>k</sub><sup>M </sup>which are drawn are reduced, for a time period dt under consideration, by the current component I<sub>k</sub><sup>M</sup>, integrated over the time period dt, of the respective electrode plate balancing area P<sub>k</sub>. If the current component I<sub>k</sub><sup>M </sup>is assumed to be constant for the time period dt, the value of the amount of charge drawn KE<sub>k</sub><sup>M </sup>is given by:<FORM>KE<sub>k</sub><sup>M</sup>(t<sub>1</sub>+dt)=KE<sub>k</sub><sup>M</sup>(t<sub>1</sub>)+I<sub>k</sub><sup>M</sup>·dt </FORM>where t<sub>1 </sub>is the start of the time period dt. In the initial state after the energy storage battery has been brought into use for the first time, the amount of charge drawn is calculated to be:<FORM>KE<sub>k</sub><sup>M</sup>(t<sub>0</sub>)=K<sub>k</sub><sup>M</sup>+I<sub>k</sub><sup>M</sup>·dt. </FORM>
These current components I<sub>k</sub><sup>M </sup>for the electrode plate balancing areas P<sub>k</sub>, that is to say the currents flowing in the electrode plates and solid masses, are described as follows:<maths id="MATHUS00005" num="5"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mi>k</mi><mi>M</mi></msubsup><mo>=</mo><mrow><mrow><mfrac><msubsup><mo>ⅆ</mo><mi>k</mi><mi>L</mi></msubsup><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>1</mn></mrow><mrow><mi>N</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>1</mn></mrow></munderover><mo></mo><msubsup><mo>ⅆ</mo><mi>n</mi><mi>L</mi></msubsup></mrow></mfrac><mo>·</mo><mi>I</mi></mrow><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mi>and</mi></mrow></mrow></math></maths><maths id="MATHUS000052" num="5.2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mi>N</mi><mi>M</mi></msubsup><mo>=</mo><mrow><mi>I</mi><mo></mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo></mo><mn>1</mn></mrow></munderover><mo></mo><msubsup><mi>I</mi><mi>n</mi><mi>M</mi></msubsup></mrow></mrow></mrow></math></maths>where N is the number of electrode plate acid balancing areas P<sub>k</sub>, and d<sub>n</sub><sup>L </sup>are division factors.
The equation system for three electrode plate balancing areas P<sub>1</sub>, P<sub>2</sub>, P<sub>3 </sub>is as follows:<maths id="MATHUS00006" num="6"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>1</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mfrac><msubsup><mo>ⅆ</mo><mn>1</mn><mi>L</mi></msubsup><mrow><msubsup><mo>ⅆ</mo><mn>1</mn><mi>L</mi></msubsup><mo></mo><mrow><mo>+</mo><mrow><msubsup><mo>ⅆ</mo><mn>2</mn><mi>L</mi></msubsup><mo></mo><mrow><mo>+</mo><msubsup><mo>ⅆ</mo><mn>3</mn><mi>L</mi></msubsup></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000062" num="6.2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>2</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mfrac><msubsup><mo>ⅆ</mo><mn>2</mn><mi>L</mi></msubsup><mrow><msubsup><mo>ⅆ</mo><mn>1</mn><mi>L</mi></msubsup><mo></mo><mrow><mo>+</mo><mrow><msubsup><mo>ⅆ</mo><mn>2</mn><mi>L</mi></msubsup><mo></mo><mrow><mo>+</mo><msubsup><mo>ⅆ</mo><mn>3</mn><mi>L</mi></msubsup></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000063" num="6.3"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>3</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mi>I</mi><mo></mo><msubsup><mi>I</mi><mn>1</mn><mi>M</mi></msubsup><mo></mo><mrow><msubsup><mi>I</mi><mn>2</mn><mi>M</mi></msubsup><mo>.</mo></mrow></mrow></mrow></math></maths>
In this case, the division factors d<sub>k</sub><sup>L </sup>are as follows:<maths id="MATHUS00007" num="7"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>1</mn><mi>L</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>1</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>/</mo><mn>2</mn></mrow><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>R</mi><mn>3</mn></msub><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><msub><mi>f</mi><mi>L</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>1</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>1</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths><maths id="MATHUS000072" num="7.2"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>2</mn><mi>L</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>2</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>2</mn></msub><mo>/</mo><mn>2</mn></mrow><mo>+</mo><msub><mi>R</mi><mn>3</mn></msub><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><msub><mi>f</mi><mi>L</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>2</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>2</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths><maths id="MATHUS000073" num="7.3"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>3</mn><mi>L</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>3</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>3</mn></msub><mo>/</mo><mn>3</mn></mrow><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><msub><mi>f</mi><mi>L</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>3</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>3</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths>where U is the total battery voltage related to one cell, measured at the connecting terminals of the energy storage battery.
The rest voltages U<sub>00,k </sub>on the respective electrode plate balancing areas P<sub>k </sub>are a function of the acid density r<sub>i </sub>in the associated electrolyte balancing areas S<sub>i </sub>and the battery temperature T.
FIG. 3 sketches an example of a function f<sub>L </sub>as a function of the amount of charge KE<sub>k</sub><sup>M </sup>drawn, related to the energy storage capacity component K<sub>k</sub><sup>M </sup>in an electrode plate balancing area P<sub>k</sub>. Up to a ratio of KE<sub>k</sub><sup>M </sup>of 0.1, the function f<sub>L </sub>rises linearly and steeply to the value 0.9. After this, the value of the function f<sub>L </sub>rises linearly and slowly to the value 1 until a ratio<maths id="MATHUS00008" num="8"><math overflow="scroll"><mrow><mfrac><msubsup><mi>KE</mi><mi>k</mi><mi>M</mi></msubsup><msubsup><mi>K</mi><mi>k</mi><mi>M</mi></msubsup></mfrac><mo>=</mo><mn>1</mn></mrow></math></maths>is reached.
During discharging, the total current I in the electrode plate balancing areas P<sub>k </sub>is calculated in a similar manner to that described above, but using a different function f<sub>E </sub>for the relationship between the amount of charge KE<sub>k</sub><sup>M </sup>drawn and the energy storage capacity components K<sub>k</sub><sup>M</sup>:<maths id="MATHUS00009" num="9"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>1</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mfrac><msubsup><mo>ⅆ</mo><mn>1</mn><mi>E</mi></msubsup><mrow><msubsup><mo>ⅆ</mo><mn>1</mn><mi>E</mi></msubsup><mo></mo><mrow><mo>+</mo><mrow><msubsup><mo>ⅆ</mo><mn>2</mn><mi>E</mi></msubsup><mo></mo><mrow><mo>+</mo><msubsup><mo>ⅆ</mo><mn>3</mn><mi>E</mi></msubsup></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000092" num="9.2"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>2</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mfrac><msubsup><mo>ⅆ</mo><mn>2</mn><mi>E</mi></msubsup><mrow><msubsup><mo>ⅆ</mo><mn>1</mn><mi>E</mi></msubsup><mo></mo><mrow><mo>+</mo><mrow><msubsup><mo>ⅆ</mo><mn>2</mn><mi>E</mi></msubsup><mo></mo><mrow><mo>+</mo><msubsup><mo>ⅆ</mo><mn>3</mn><mi>E</mi></msubsup></mrow></mrow></mrow></mrow></mfrac><mo>·</mo><mi>I</mi></mrow></mrow></math></maths><maths id="MATHUS000093" num="9.3"><math overflow="scroll"><mrow><msubsup><mi>I</mi><mn>3</mn><mi>M</mi></msubsup><mo>=</mo><mrow><mi>I</mi><mo></mo><msubsup><mi>I</mi><mn>1</mn><mi>M</mi></msubsup><mo></mo><mrow><msubsup><mi>I</mi><mn>2</mn><mi>M</mi></msubsup><mo>.</mo></mrow></mrow></mrow></math></maths>
In this case, the division factors d<sub>k</sub><sup>E </sup>are as follows:<maths id="MATHUS00010" num="10"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>1</mn><mi>E</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>1</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>/</mo><mn>2</mn></mrow><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub><mo>+</mo><msub><mi>R</mi><mn>3</mn></msub><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><msub><mi>f</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>1</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>1</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths><maths id="MATHUS000102" num="10.2"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>2</mn><mi>E</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>2</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>2</mn></msub><mo>/</mo><mn>2</mn></mrow><mo>+</mo><msub><mi>R</mi><mn>3</mn></msub><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><msub><mi>f</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>2</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>2</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths><maths id="MATHUS000103" num="10.3"><math overflow="scroll"><mrow><msubsup><mi>d</mi><mn>3</mn><mi>E</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mn>3</mn></mrow></msub><mo></mo><mi>U</mi></mrow><mrow><mrow><msub><mi>R</mi><mn>3</mn></msub><mo>/</mo><mn>3</mn></mrow><mo>+</mo><mi>R</mi></mrow></mfrac><mo>·</mo><mrow><mrow><msub><mi>f</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>KE</mi><mn>3</mn><mi>M</mi></msubsup><mo>,</mo><msubsup><mi>K</mi><mn>3</mn><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></math></maths>
The values for the amounts of charge KE<sub>k</sub><sup>M </sup>drawn are calculated as described above by reduction by the current component I<sub>k</sub><sup>M</sup>, integrated over the time period dt, for the electrode plate balancing areas P<sub>k</sub>.
By way of example, FIG. 4 shows the function f<sub>E </sub>as a function of the amount of charge KE<sub>k</sub><sup>M </sup>drawn related to the energy storage capacity component K<sub>k</sub><sup>M</sup>. The value for f<sub>E </sub>is equal to unity up to a ratio<maths id="MATHUS00011" num="11"><math overflow="scroll"><mrow><mfrac><msubsup><mi>KE</mi><mi>k</mi><mi>M</mi></msubsup><msubsup><mi>K</mi><mi>k</mi><mi>M</mi></msubsup></mfrac><mo>=</mo><mrow><mn>0.9</mn><mo>.</mo></mrow></mrow></math></maths>The value of f<sub>E </sub>falls linearly from unity to a value of 0 in the range from<maths id="MATHUS00012" num="12"><math overflow="scroll"><mrow><mfrac><mrow><mi>KE</mi><mo></mo><mfrac><mi>M</mi><mi>k</mi></mfrac></mrow><mrow><mi>K</mi><mo></mo><mfrac><mi>M</mi><mi>k</mi></mfrac></mrow></mfrac><mo>=</mo><mrow><mn>0.9</mn><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mi>to</mi><mo></mo><mstyle><mtext> </mtext></mstyle><mo></mo><mn>1.</mn></mrow></mrow></math></maths>
During discharging, the current component I<sub>i</sub><sup>s </sup>for the electrolyte balancing areas S<sub>i </sub>corresponds to the value of the current component I<sub>k</sub><sup>M </sup>for the associated electrode plate balancing area P<sub>k</sub>.
Once current has been drawn or added, the acid density r<sub>i </sub>for an electrolyte balancing area S<sub>i </sub>is calculated as a function:<FORM>r<sub>i</sub>(t<sub>1</sub>+dt)=f(r<sub>i</sub>(t<sub>1</sub>), v<sub>i</sub>, I<sub>i</sub><sup>s</sup>·dt, T). </FORM>
In this case, r<sub>i </sub>is the acid density in the electrolyte balancing area S<sub>i </sub>before the time period dt, v<sub>i </sub>is the electrolyte volume component in the electrolyte balancing area S<sub>i</sub>, T is the temperature of the energy storage battery, and I<sub>i</sub><sup>s</sup>·dt is the charge added to or drawn from the electrolyte balancing area S<sub>i </sub>in the time period dt.
By way of example, the amount of charge which can still be drawn from the energy storage battery can be calculated as a characteristic variable for states of the energy storage battery, in order to estimate the remaining capacity of the energy storage battery.
For this purpose, the amount of charge RK<sub>k</sub><sup>M </sup>(remaining capacity) which can still be drawn from the active mass is calculated for each electrode plate balancing area P<sub>k </sub>using:<FORM>RK<sub>k</sub><sup>M</sup>=K<sub>k</sub><sup>M</sup>−(KE<sub>k</sub><sup>M</sup>+I<sub>k</sub><sup>M</sup>·dt). </FORM>
The amount of charge RK<sub>i</sub><sup>s </sup>which can be drawn from the corresponding electrolyte balancing area S<sub>i </sub>is then calculated as a function of the electrolyte volume v<sub>i </sub>and the instantaneous acid density r<sub>i </sub>in the electrolyte balancing area S<sub>i</sub>, as well as the temperature T of the energy storage battery, using the function:<FORM>RK<sub>i</sub><sup>s</sup>=f(v<sub>i</sub>, r<sub>i</sub>, T). </FORM>
The amount of charge RK<sub>g </sub>which can still be drawn from the energy storage battery is calculated on the basis of the amount of charge RK<sub>k</sub><sup>M </sup>which can be drawn in the electrode plate balancing areas P<sub>k </sub>and the amount of charge RK<sub>i</sub><sup>s </sup>which can be drawn in the electrolyte balancing areas S<sub>i</sub>, with the following situation distinction which is necessary to take account of the “dropoff” of higher density acid from an upper balancing area to a lower balancing area. This is based on the assumption that complete discharging takes place first of all in the lower area of the energy storage battery, then in the central area and finally at the top.
 a) If RK<sub>1</sub><sup>s</sup><RK<sub>1</sub><sup>M</sup>:
Missing acid capacity is taken at most from the electrolyte volume v<sub>2 </sub>until the acid in the electrolyte volume v<sub>2 </sub>has been consumed. The acid capacity RK<sub>2</sub>s in the electrolyte volume v<sub>2 </sub>is reduced corresponding to the acid consumed in the electrolyte volume v<sub>1</sub>. RK<sub>1</sub><sup>s </sup>is thus increased.
 b) If RK<sub>2</sub><sup>s</sup><RK<sub>2</sub><sup>M</sup>.
Missing acid capacity is taken at most from the electrolyte volume v<sub>3 </sub>until the acid in the electrolyte volume v<sub>3 </sub>has been consumed. The acid capacity RK<sub>3</sub><sup>s </sup>in the electrolyte volume v<sub>3 </sub>is reduced corresponding to the acid consumed in the electrolyte volume v<sub>2</sub>. The new value for RK<sub>2</sub><sup>s </sup>can then be calculated using step a).
This then results in:<maths id="MATHUS00013" num="13"><math overflow="scroll"><mrow><msub><mi>RK</mi><mn>8</mn></msub><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo></mo><mrow><mi>min</mi><mo></mo><mrow><mo>(</mo><mrow><msubsup><mi>RK</mi><mi>k</mi><mi>s</mi></msubsup><mo>,</mo><msubsup><mi>RK</mi><mi>k</mi><mi>M</mi></msubsup></mrow><mo>)</mo></mrow></mrow></mrow></mrow></math></maths>
The rest voltage U<sub>00,g </sub>of the energy storage battery can be calculated from the individual rest voltage levels U<sub>00,k </sub>of the balancing areas S<sub>i</sub>, P<sub>k </sub>as a further characteristic variable for the state of the energy storage battery. In this case, however, it is also necessary to take account of the charge reversal current I<sub>U </sub>from points where the rest voltage U<sub>00 </sub>is relatively high to points where the rest voltage U<sub>00 </sub>is low on the same electrode plate. This charge reversal current I<sub>U </sub>is a result of the active material at the point where the rest voltage U<sub>00 </sub>is high being discharged, and charging taking place at the point where the rest voltage U<sub>00 </sub>is relatively low. This is associated with a voltage drop across the polarization resistance R<sub>p</sub>, which represents the polarization resistance on the plate surface when the currents are low, and the individual grid resistances R<sub>k</sub>. This results in a rest voltage U<sub>00 </sub>whose value is between the rest voltages U<sub>00,k </sub>of the individual balancing areas S<sub>i</sub>, P<sub>k </sub>(mixed potential).
The charge reversal current I<sub>U </sub>is calculated as follows:<maths id="MATHUS00014" num="14"><math overflow="scroll"><mrow><msub><mi>I</mi><mi>U</mi></msub><mo>=</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>y</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>·</mo><mrow><mo>(</mo><mrow><msub><mi>R</mi><mrow><mi>P</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>+</mo><msub><mi>R</mi><mrow><mi>P</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>+</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mi>x</mi></mrow><mi>y</mi></munderover><mo></mo><msub><mi>R</mi><mi>n</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></maths>
The total rest voltage U<sub>00,g </sub>of the energy storage battery is calculated as follows:<maths id="MATHUS00015" num="15"><math overflow="scroll"><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>g</mi></mrow></msub><mo>=</mo><mrow><msub><mi>U</mi><mrow><mn>00</mn><mo>,</mo><mi>x</mi></mrow></msub><mo></mo><mrow><msub><mi>I</mi><mi>U</mi></msub><mo>·</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>R</mi><mrow><mi>P</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>+</mo><mrow><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mi>x</mi></mrow><mi>y</mi></munderover><mo></mo><msub><mi>R</mi><mi>n</mi></msub></mrow></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mrow></mrow></mrow></math></maths>
If appropriate, the charge reversal current I<sub>U </sub>and the total rest voltage U<sub>00,g </sub>must be related to the number of cells.
The indices x, y for the rest voltage U<sub>00,x</sub>, U<sub>00,y </sub>relate to a respective electrode plate balancing area P<sub>k</sub>, with a case distinction being made since a charge reversal current I<sub>U </sub>can flow only when the point at which the rest voltage U<sub>00 </sub>is relatively high still has mass which can be discharged, and the point where the rest voltage U<sub>00 </sub>is relatively low still has mass which can be charged. The case distinction is as follows:
 a) K<sub>1</sub><sup>M </sup>is partially charged or fully charged, K<sub>3</sub><sup>M </sup>is partially charged or discharged, and K<sub>2</sub><sup>M </sup>is irrelevant:<FORM>X=1, Y=3 </FORM><FORM>R<sub>i</sub>=R<sub>1</sub>/2+R<sub>2</sub>+R<sub>3</sub>/2 </FORM>
 b) K<sub>1</sub><sup>M </sup>is partially charged or fully charged, K<sub>2</sub><sup>M </sup>is partially charged or discharged, and K<sub>3</sub><sup>M </sup>is fully charged:<FORM>X=1, Y=2 </FORM><FORM>R<sub>i</sub>=R<sub>1</sub>/2+R<sub>2</sub>/2 </FORM>
 c) K<sub>2</sub><sup>M </sup>is partially charged or fully charged, K<sub>3</sub><sup>M </sup>is partially charged or discharged, and K<sub>1</sub><sup>M </sup>is discharged:<FORM>X=2, Y=3 </FORM><FORM>R<sub>i</sub>=R<sub>2</sub>/2+R<sub>3</sub>/2 </FORM>
 d) K<sub>1 </sub>is fully charged or discharged, and K<sup>m</sup><sub>2 </sub>and KM are fully charged:<FORM>U<sub>00,g</sub>=U<sub>00,1 </sub></FORM>
 e) K<sub>2</sub><sup>M </sup>is fully charged or partially charged, K<sub>1</sub><sup>M </sup>is discharged, and K<sub>3</sub><sup>M </sup>is fully charged:<FORM>U<sub>00,g</sub>=U<sub>00,2 </sub></FORM>
 f) all other cases:<FORM>U<sub>00,g</sub>=U<sub>00,3</sub>. </FORM>
Furthermore, the ice formation can be calculated and assessed as a characteristic variable. This can be done by analysis of individual electrolyte balancing areas S<sub>i </sub>and calculation of the acid density r<sub>i </sub>in these electrolyte balancing areas S<sub>i </sub>for different zones. The temperature T and the acid density r<sub>i </sub>can then be used to calculate the volume of ice crystals that has formed in the individual electrolyte balancing areas S<sub>i</sub>.
Each temperature T has an associated equilibrium concentration CGG of the sulfuric acid in the electrolyte. If the concentration falls below this level, then pure water freezes out and ice crystals are formed until this equilibrium concentration CGG is reached in the rest of the liquid phase of the electrolyte. The volume of ice crystals is then calculated ignoring the volume extent of the water as follows:<FORM>v<sub>ice,I</sub>=(1−C<sub>i</sub>/CGG<sub>i</sub>)·v<sub>i</sub>, </FORM>where the equilibrium concentration CGG is a defined function of the temperature in the respective electrolyte balancing area S<sub>i</sub>:<FORM>CGG<sub>I</sub>=f(T<sub>i</sub>) </FORM>C<sub>i </sub>is the concentration of sulfuric acid in the electrolyte volume v<sub>i</sub>, and can be calculated from the acid density r<sub>i </sub>and the temperature T<sub>i </sub>in the respective electrolyte balancing area S<sub>i</sub>.
The information about the ice volume vice in each electrolyte balancing area S<sub>i </sub>can then be used to make a statement as to whether the ice crystals are adversely affecting the performance of the energy storage battery. This may be done, for example, in such a way that the battery is assumed to have its full performance when ice crystals occur only in the uppermost electrolyte balancing area S<sub>i</sub>. In contrast, the energy storage battery is no longer regarded as being powerful when ice crystals occur in the lowermost electrolyte balancing area S<sub>i</sub>.
In this case, one or more parameters and/or state variables for the battery model can be determined by adaptation of the model results by means of a parameter and/or state estimation method.
For example, the acid density r<sub>i </sub>can be matched to measured values. By calculation of the rest voltage U<sub>00,g </sub>in the model and by comparison with the actual rest voltage U<sub>00,g </sub>which can be obtained by other methods or can be determined by waiting for a long rest phase, it is possible to make a statement on how well the model is mapping the battery state at that time. For example, movement of the battery may result in the acid being mixed in a way which the model cannot take account of since only the current and voltage are measured. In situations such as this, adaptation must then be carried out. This is done on the basis of the difference between the rest voltage U<sub>00,g </sub>calculated in the model and the actual rest voltage U<sub>00,g </sub>by adaptation of the acid densities r<sub>i </sub>in the electrolyte balancing areas S<sub>i </sub>as follows:
If, for example, the difference between the measured rest voltage and the rest voltage calculated from the model is negative and the rest voltage U<sub>00,g </sub>is determined solely by the acid density r<sub>i </sub>in the central balancing area S<sub>2 </sub>in the case of a model with three electrolyte balancing areas S<sub>i </sub>then the acid density r<sub>i </sub>in the second balancing area S<sub>i </sub>is reduced. The acid densities r<sub>i </sub>in the other balancing areas S<sub>1</sub>, S<sub>3 </sub>will have to be adapted on the basis of the boundary conditions:
 a) amount of acid remains constant
 b) acid density r<sub>i </sub>rises monotonally from top to bottom, and
 c) specific limit values are not undershot or exceeded.
The method has been verified on the basis of an actual experiment with measurements on a 110 Ah battery. In the process, acid stratification was caused to occur to an ever greater extent in the battery. The experiment was carried out by charging and discharging in steps, with the experiment being started in the discharged state. During the process, the energy storage battery was discharged until the final discharge voltage of 10.5 volts was reached.
The experimental and comparative data are plotted against the observation time t in FIGS. 5 to 10.
FIG. 5 shows the state of charge SOC determined by calculation of the charge throughput. As can be seen, the energy storage battery was discharged and charged in cycles. The discharge cycles can be identified by the falling state of charge SOC, and the charging cycles by the rising state of charge SOC.
FIG. 6 shows a comparison of the measured unloaded terminal voltage U of the energy storage battery at zero current (I=0), and of the rest voltage U<sub>00,g </sub>calculated from the model described above. The measured unloaded voltage U is symbolized by the dots.
Since the unloaded voltage U measured at zero current does not in itself represent the rest voltage U<sub>00</sub>, which does not occur until after a rest phase of several hours, the first measured values in the rest phase are different to the other values. Nevertheless, a good match can be seen between the measurement and the rest voltage calculated from the model.
FIG. 7 shows the acid densities r<sub>1</sub>, r<sub>2 </sub>and r<sub>3 </sub>calculated for the three electrolyte balancing areas S<sub>1</sub>, S<sub>2 </sub>and S<sub>3 </sub>plotted against time. This clearly shows that the acid density increases more sharply in the charging phase in the lower area, that is to say in the electrolyte balancing area S<sub>1</sub>, than in the electrolyte balancing areas S<sub>2 </sub>and S<sub>3 </sub>above it.
FIG. 8 shows the loss of energy storage capacity ΔK for the theoretical case of full charging. The remaining energy storage capacity at any given time t can in this way be determined from the energy storage capacity K of the energy storage battery when new.
FIG. 9 shows the remaining capacity RK<sub>k </sub>which is still available in the individual electrode plate balancing areas P<sub>k</sub>, plotted against time t. This clearly shows that, in particular, the electrolyte balancing area S<sub>3 </sub>and the electrode plate balancing area P<sub>3 </sub>substantially govern the loss of energy storage capacity, since there is virtually no acid here once the balancing areas S<sub>1</sub>, S<sub>2 </sub>and P<sub>1</sub>, P<sub>2 </sub>have been discharged.
FIG. 10 shows the energy storage capacity RK<sub>g </sub>which can be drawn as a characteristic variable for the state of the energy storage battery which was calculated from the model described above, against time. In the ideal situation, when the described method is working perfectly, all of the discharge processes end on the zero line. As can be seen, there was an error of about 10%*RK<sub>g </sub>only towards the end of the experiment. There is therefore a good match between the model and the measurement.
Furthermore, the measured and calculated variables can be used to predict the highcurrent behavior and other characteristic variables.