WO2015183250A1

WO2015183250A1 - Estimation of the remaining lifetime of a supercapacitor

Info

Publication number: WO2015183250A1
Application number: PCT/US2014/039680
Authority: WO
Inventors: Fredrik GÖTH; Viktor NORDGREN; Samir ALAGIC
Original assignee: Schneider Electric Buildings Llc
Priority date: 2014-05-28
Filing date: 2014-05-28
Publication date: 2015-12-03

Abstract

There is provided an iterative method for estimating the remaining lifetime of a supercapacitor being charged by a charge voltage which is based on a capacitance value of the supercapacitor.

Description

ESTIMATION OF THE REMAINING LIFETIME OF A SUPERCAPACITOR

Technical field

The present invention relates to the field of supercapacitors. In particular, it relates to a method, an arrangement, and a system for estimating the remaining lifetime of a supercapacitor.

Background

Supercapacitors, also known as ultra capacitors or electrochemical double layer capacitors (EDLC), are used as high power storage devices. In many applications, such as in embedded systems, the supercacitors replace conventional batteries with the goal of having a maintenance free backup supply.

Supercapacitors surpass most battery technologies in terms of lifetime. However, aging of supercapacitors is still an issue. It is known that several factors, of which charge voltage and temperature are the most prominent, influence the aging of supercapacitors. Increasing the voltage and

temperature exponentially accelerates electrochemical reactions in the supercapacitors, thereby leading to a decrease in capacitance as well as in conductance.

Optimizing the amount of energy that can be stored in the backup supply, and hence the lifetime, is one of the most important design

considerations for an embedded system since it directly impacts the duration of the system, see e.g. WO 2013/107497 A1 . It is therefore critical to develop accurate life estimation techniques for embedded systems. Most systems of today that use supercapacitors as a backup source are over dimensioned in capacity. However, as more products with stricter requirements with respect to cost, size, user friendliness, and reliability start using supercapacitors as a power source, the need for proper monitoring and lifetime estimation will be necessary.

Rajan and Rahman, "Lifetime Analysis of Super Capacitor for Many

Power Electronics Applications", IOSR Journal of Electrical and Electronics Engineering, Vol. 9, Issue 1 , 2014, propose to estimate the lifetime of a supercapacitor by multiplying a rated lifetime by a number of lifetime factors which depend on temperature, voltage and moisture. The proposed analysis is disadvantageous in that it assumes that the temperature, voltage, and moisture are constant throughout the life of the capacitor.

An approach which tries to take the time dependency of the voltage and temperature into account is disclosed in US 2006/0012378 A1 . This document discloses determining instantaneous ultracapacitor life based on measurements of instantaneous voltage and temperature at different time points. An estimated lifetime is then determined by combining the determined instantaneous ultracapcitor life corresponding to the different time points. A drawback with this method is that it when combining the instantaneous ultracapacitor life typically will average or smooth out the dependency with respect to temperature and voltage. There is thus room for improvements.

Summary of the invention

In view of the above, it is thus an object of the present invention to provide an improved estimation of the remaining lifetime of a supercapacitor.

According to a first aspect of the invention, the above object is achieved by a method for estimating the remaining lifetime of a

supercapacitor, SC, being charged by a charge voltage which is based on a capacitance value of the SC, comprising:

at a first point in time,

receiving a measured capacitance value of the SC, a measured time derivative of the capacitance of the SC, and an initial charge voltage of the SC,

initializing a time indicator,

initializing an estimated capacitance value of the SC to the measured capacitance value of the SC,

as long as the estimated capacitance value is larger than a threshold value which indicates that the life of the SC has come to an end, iterating the steps of: calculating a charge voltage of the SC based on the estimated capacitance value,

calculating an estimated decrease in capacitance during a time interval corresponding to the current iteration based on the time derivative of the capacitance of the SC, a length of the time interval, and a factor which is indicative of an aging behavior of the SC, wherein the factor depends on a difference between the calculated charge voltage and the initial charge voltage,

updating the estimated capacitance value by subtracting the estimated decrease in capacitance from the estimated capacitance value,

increasing the time indicator by an amount corresponding to the length of the time interval corresponding to the current iteration, and estimating the remaining lifetime of the SC as the value of the time indicator from the last iteration.

The above method is thus an iterative method which predicts, i.e. estimates, the capacitance value of the supercapacitor as a function of time. When predicting the capacitance value, the accelerated aging of the supercapacitor caused by the charge voltage is taken into account. The remaining lifetime is estimated as the value when the predicted capacitance value reaches below a threshold which defines the lifetime of the

supercapacitor.

The method assumes that the supercapacitor is controlled according to a strategy which at each point in time sets the charge voltage on basis of the capacitance value, e.g. as explained in WO 2013/107497 A1 . In other words, there is a known relation between the capacitance value and the charge voltage. With knowledge of that strategy and relation, the evolution of the charge voltage over time may be predicted from the capacitance value.

Accordingly, the above method proposes to start up from a measured capacitance value and then iteratively updating the charge voltage based on the capacitance value, and then updating the capacitance value by taking the aging effects of the charge voltage into account. The lifetime of the supercapacitor may be defined as the time until the capacitance decreases below a threshold. The threshold may e.g. correspond to 50% of the original capacitance value of the supercapacitor.

By initial charge voltage is meant the charge voltage applied to the supercapacitor at the first point in time.

The relation between the charge voltage and the capacitance value may be given by a (pre-defined) function. This function reflects the assumed control strategy of the charge voltage of the supercapacitor. For example, the charge voltage of the supercapacitor may be calculated based on the estimated capacitance value according to a function which increases as the estimated capacitance value decreases.

The energy stored in the capacitor is given by E_stored = -^■ C ^■ V² , where

C is the capacitance value and V is the charge voltage. Thus, a strategy which increases the charge voltage as the capacitance value decreases may compensate for a decrease in stored energy due to a decreasing capacitance value caused by aging.

For example, the charge voltage of the supercapacitor may be calculated based on the estimated capacitance value such that the energy Estored stored in the supercapacitor is kept essentially constant. Such strategy is advantageous in that it optimizes the charge voltage such that, at each time point, a low as possible charge voltage which is chosen at the same time as the energy in the supercapacitor matches the energy demand.

According to another example, the charge voltage may be calculated based on the estimated capacitance value such that the energy which may be discharged from the supercapacitor is kept essentially constant. In more detail, due to surrounding circuit components, the supercapacitor may only be capable of discharging a portion of its stored energy E_stored . Specifically, there may be a lower energy limit E_Umit of the stored energy which cannot be discharged. In other words, the supercapacitor may only be capable of discharging a portion of its stored energy that exceeds a lower limit, i.e.

Estored - Eumit ■ In such a case, the charge voltage of the supercapacitor is calculated based on the estimated capacitance value such that the portion of the stored energy exceeding the lower limit is kept essentially constant. The supercapacitor has an equivalent representation which comprises a capacitive part and a resistive part, the conductance being the inverse of an equivalent series resistance (ESR) of the resistive part of the supercapacitor. Due to aging of the capacitor, the capacitance as well as the conductance decreases over time. In addition to the capacitance, the method may take the conductance value of the supercapacitor into account when calculating the charge voltage. More precisely, the SC may be charged by a charge voltage which is based on the capacitance value and a conductance value of the SC, and the SC charge voltage of the supercapacitor may further be calculated based on an estimated conductance value of the supercapacitor according to a function which increases as the estimated capacitance value and

conductance value decrease.

As a current flows from the capacitor during discharge, there will according to Ohm's law be a voltage V_ESR across the supercapacitor being equal to the product of the equivalent series resistance and the current flowing from the supercapacitor. As a result of the ohmic voltage drop V_ESR of the capacitor, all of the energy stored in the capacitor E_stored may not be discharged since there will be power losses in the resistive component in the supercapacitor. More precisely, the energy E_backup which e.g. may be used to power a load is given by E_backup = E_stored - E_ESR . In other words, the conductance value of the supercapacitor is associated with a resistive part of the supercapacitor which give rise to an ohmic voltage drop, and the charge voltage of the may be adjusted based on the estimated conductance value to compensate for the ohmic voltage drop. For example, the charge voltage may be calculated based on the capacitance and conductance value such that Ebackup is kept essentially constant.

According to another example, the charge voltage may be calculated based on the capacitance and conductance value such that E_stored - E limit - EESR is kept essentially constant. In other words, in case the supercapacitor is only capable of discharging a portion of its stored energy that exceeds a lower limit, and the charge voltage of the SC is calculated based on the estimated capacitance value and the estimated conductance value such that the portion of the stored energy exceeding the lower limit is kept essentially constant. The conductance is typically not constant but decreases in a similar fashion as the capacitance. Therefore, the method may comprise iteratively updating the estimated conductance value in a similar manner as the capacitance value.

The estimated decrease in capacitance during a time interval corresponding to the current iteration may be calculated by multiplying the time derivative of the capacitance of the supercapacitor by the length of the time interval, and the factor which is indicative of an aging behavior of the supercapacitor. In this way, the time derivative of the capacitance, which was measured at the first time point, is multiplied by the aging factor such that the product reflects the aging speed of the supercapacitor in view of the charge voltage.

The factor which is indicative of the aging behavior of the

supercapacitor may depend on the difference between the charge voltage and the initial charge voltage according to a non-linear function. In particular, the non-linear function may be an exponential function.

As discussed above, the inventive method takes the time-varying charge voltage into account when estimating the remaining lifetime of the supercapacitor. Alternatively, or additionally, the method may also take the temperature into account in order to further improve the lifetime estimate. For that purpose, the method may further comprise:

at the first point in time, measuring an initial temperature, and during each iteration, estimating a temperature, wherein the factor which is indicative of the aging behavior of the SC further depends on a difference between the estimated temperature and the initial temperature.

For example, the temperature may be estimated according to a model which takes the temporal variability of the temperature into account and/or may be predicted from previously measured temperature values.

According to embodiments, the factor which is indicative of the aging behavior of the supercapacitor may depend on the difference between the estimated temperature and the initial temperature according to an exponential function. The method may further comprise generating an output which indicates the remaining lifetime. For example, the indication of the remaining lifetime may be used to update a time indicator or time gauge which shows how long the remaining lifetime of the supercapacitor is. Further, the output may include an alarm event which indicates the remaining lifetime of the

supercapacitor.

The time indicator is typically set to zero at initialization. Further, the received time derivative of the capacitance of the supercapacitor is measured with respect to a time interval prior to the first point in time.

According to a second aspect of the invention, the above object is achieved by computer program product comprising computer code

instructions stored on computer-readable medium for performing the method of the first aspect when executed on a device having processing capability.

According to a third aspect of the invention, the above object is achieved by a device for estimating the remaining lifetime of a supercapacitor, SC, being charged by a charge voltage which is based on a capacitance value of the SC, comprising:

a receiving component configured to receive a measured capacitance value of the SC, an initial charge voltage of the SC, and a measured time derivative of the capacitance of the SC at a first point in time,

a processing component configured to:

at the first point in time,

initialize a time indicator,

initialize an estimated capacitance value of the SC to the measured capacitance value of the SC,

as long as the estimated capacitance value is larger than a threshold value which indicates that the life of the SC has come to an end, iterate the steps of:

calculating a charge voltage of the SC based on the estimated capacitance value,

increasing the time indicator by an amount corresponding to the length of the time interval corresponding to the current iteration, and estimate the remaining lifetime of the SC as the value of the time indicator from the last iteration.

The device for estimating the remaining lifetime of a supercapacitor may typically be included in a product, such as in an embedded system.

According to a fourth aspect of the invention, the above object is achieved by a system for estimating the remaining lifetime of a

a SC, an arrangement for measuring a capacitance value and a time derivative of the capacitance of the SC, and

a device for estimating the remaining lifetime of the SC according to the third aspect.

The second, third, and fourth aspect may generally have the same features and advantages as the first aspect. It is further noted that the invention relates to all possible combinations of features unless explicitly stated otherwise.

Generally, all terms used in the claims are to be interpreted according to their ordinary meaning in the technical field, unless explicitly defined otherwise herein. All references to "a/an/the [device, event, message, alarm, parameter, step etc.]" are to be interpreted openly as referring to at least one instance of said device, event, message, alarm, parameter, step etc., unless explicitly stated otherwise. The steps of any method disclosed herein do not have to be performed in the exact order disclosed, unless explicitly stated. Brief Description of the Drawings

The above, as well as additional objects, features and advantages of the present invention, will be better understood through the following illustrative and non-limiting detailed description of preferred embodiments of the present invention, with reference to the appended drawings, where the same reference numerals will be used for similar elements, wherein:

Fig. 1 schematically illustrates a system comprising a supercapacitor for powering a load.

Fig. 2 schematically illustrates a system for estimating the remaining lifetime of a supercapacitor according to embodiments.

Fig. 3 schematically illustrates an estimated capacitance value of a supercapacitor as a function of time.

Fig. 4 is a flowchart of a method for estimating the remaining lifetime of a supercapacitor according to embodiments.

Fig. 5 is a flowchart of a method for estimating the remaining lifetime of a supercapacitor according to embodiments.

Detailed description of embodiments

The present invention will now be described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. The systems and devices disclosed herein will be described during operation.

Fig. 1 illustrates a system 100 comprising a load 1 10 and at least one supercapacitor 102. The system 100 may comprise one supercapacitor 102 or a plurality of supercapacitors 102 connected in series or in parallel. The system 100 may be used as a backup system for powering the load 1 10.

Particularly, the supercapacitor 102 may be used as a backup power source for the load 1 10. Normally, the load 1 10 is powered via a power net 1 18.

However, if there is a power fail, the load may be connected to the backup- system comprising the capacitor 102, for example by means of a switch 1 16.

The system 100 has a charging part 1 12 and a discharging part 1 14.

The discharging part 1 14 is associated with a discharging mode of the capacitor 102. Particularly, in the discharging mode, the capacitor 102 is connected to the load 1 10 to power the load 1 10. The capacitor 102 may be connected to the load 1 10 via a voltage regulator 108.

The charging part 1 12 is associated with a charging mode of the capacitor 102. In the charging mode, the capacitor 102 is disconnected from the load 1 10. When in the charging mode, the capacitor 102 is charged by means of a charging voltage which is applied across the capacitor 102. The charging voltage is provided by an adjustable voltage regulator 104. The adjustable voltage regulator 104 regulates the voltage from a power net 120.

Further, the adjustable voltage regulator 104 is connected to a control unit 106. The control unit 106 is arranged to determine an adjusted charge voltage and to send a signal relating to the adjusted charge voltage to the adjustable voltage regulator 104. The control unit 106 is generally of the type disclosed in WO 2013/107497 A1 (see in particular description of Figs 1 , 2 and 6 of WO 2013/107497 A1 ) which is hereby incorporated by reference. In brief, the control unit 106 adjusts the charge voltage on basis of the

(decreasing) capacitance value and sometimes also on basis of the

(decreasing) conductance value of the supercapacitor. There is thus a relationship, e.g. in terms of a function, between the capacitance value and/or the conductance value and the charge voltage applied to the supercapacitor 102. That relationship is used to predict the charge voltage in order to estimate the remaining lifetime as will be described in the following.

Fig. 2 illustrates a system 200 for estimating the remaining lifetime of a supercapacitor. The system 200 may typically form a hardware/software subsystem of and embedded system. The system 200 comprises a

supercapacitor 102, such as the one illustrated in Fig. 1 , an adjustable voltage regulator 104, an arrangement 206 for measuring a capacitance value and a time derivative of the capacitance of the supercapacitor 102 (as indicated by the dashed lines), and a device 208 for estimating the remaining lifetime of the supercapacitor 102.

The device 208 may generally comprise a receiving and a transmitting component for receiving signals from and transmitting signals to the adjustable voltage regulator 104 and the measurement arrangement 206. The device 208 may also transmit an output signal 210 which is indicative of the remaining lifetime of the supercapacitor 102, e.g. to an overlaying system Further, the device 208 comprises a processing component, such as a microcontroller, which may be configured to e.g. control a charge voltage of the supercapacitor 102 as described above (cf. item 106 in Fig. 1 ) and to carry out a method for estimating the remaining lifetime of the supercapacitor 102.

The method may be implemented in software, and in that case the device 206 may comprise a (non-transitory) computer-readable medium for storing instructions to be executed by the processing component in order to estimate the remaining lifetime of the supercapacitor 102.

The arrangement 206 for measuring a capacitance value and a time derivative of the capacitance of the supercapacitor 102 may comprise an arrangement 206a for measuring a charge current, a load 206b, and an arrangement 206c for measuring a discharge current. The capacitance measurement may generally be carried out in the manner described in

WO 2013/107497 A1 with reference to Figs. 7, 4 and 5.

The capacitance measurements described on page 16 in

WO 2013/107497 is based on a linear assumption which requires that the charging current is constant. However, the linear assumption is not always valid. In fact, the supercapacitor 102 typically has a non-linear behavior in that the voltage across the supercapacitor 102 increases non-linearly as a function of time during charging. In order to take the non-linear behavior into account, the capacitance may be measured according to: 2 i{t)v{t)dt

Γ = 7■ ¹ where /(t) is a time-variable charging current, V(t) is the time-variable voltage across the supercapacitor 102, ^ and t₂ (^< t₂) are time points, ^is the voltage across the supercapacitor 102 at time ^ and V₂\s the voltage across the supercapacitor 102 at time t₂.

The derivative of the capacitance may generally be measured by measuring the capacitance at two points in time and estimate the derivative of the capacitance based on the change of the capacitance between these two time points, i.e. as the inclination of a hypothetical straight line fitted to the measured capacitance values at the two points in time.

The operation of the arrangement 200, and in particular a first embodiment of a method for estimating the remaining lifetime of the supercapacitor 102 as performed by the device 208, will now be described with reference to Fig. 1 , Fig. 2, Fig. 3 and the flow chart of Fig. 4.

Fig. 3 illustrates the capacitance of the supercapacitor 102 as a function of time. The supercapacitor 102 has an original capacitance value C which decreases over time due to aging of the supercapacitor 102. As explained above, the aging is influenced by several factors including charge voltage and temperature. The lifetime of the capacitor 102 may be defined as the time until the capacitance has decreased to a certain percentage a of its original capacitance value C. Typically, a may be equal to 50%. There is thus a threshold L = aC which indicates that the life of supercapacitor has come to an end.

The device 208 may regularly, such as once per week, or continuously estimate the remaining lifetime of the supercapacitor 102. For example, the device 208 may estimate the remaining lifetime of the supercapacitor 102 at a first point in time which typically is a current time point, labelled t₀ in Fig. 3. For this purpose, at the first point in time t₀ the device 208 initializes the method, step S02. In more detail, the initialization comprises receiving a measured a capacitance value C₀ of the capacitor 102, a measured time

ddCC

derivative of the capacitance— of the capacitor 102, and a measured or dt to

calculated initial charge voltage V₀ of the capacitor 102. The measured capacitance value C₀ and the time derivative of the capacitance— are dt t₀ typically received from the arrangement 206. The initial charge voltage V₀ of the capacitor 102 may be measured by the adjustable voltage regulator 104 and transmitted to the device 208. Alternatively, the device 208 may calculate the initial charge voltage on basis of the measured capacitance value C₀ of the capacitor 102 as further discussed below. The device 208 further initializes a time indicator 77₀, preferably to zero, and initializes an estimated capacitance value of the supercapacitor 102 to the measured capacitance value C₀ of the capacitor 102.

In step S04 the device 208 checks whether the estimated capacitance value of the supercapacitor 102 is larger than the threshold L which indicates that the life of the supercapacitor has come to an end. In the illustrated example of Fig. 3, the estimated capacitance value C₀ at time t₀ exceeds the threshold L and the device 208 hence proceeds to carry out a first iteration of steps S06 through S12.

In step S06, when iterated the first time, the device 208 calculates a charge voltage V₁ on basis of the estimated capacitance value C₀ of the capacitor 102. The charge voltage V₁ is typically calculated according to a function v₁ = /(C₀) which increases as the estimated capacitance value decreases. As discussed above, the function / reflects the control strategy, i.e. the relationship between the charge voltage and capacitance, used by device 208 (or device 106) when setting the charge voltage of the

supercapacitor 102.

In some embodiments, the device 208 calculates the charge voltage of the supercapacitor 102 based on the estimated capacitance value such that the energy E_stored stored in the supercapacitor 102 is kept essentially constant. The energy E_stored stored in the supercapacitor 102 as a function of the capacitance C and the charge voltage V is given by the relation stored

In order to keep the energy E_stored stored in the supercapacitor 102 essentially constant, the charge voltage V should hence be calculated according to

V

where E_stored is the desired constant energy level. In the first iteration, the device 208 may thus calculate the charge voltage V₁ according to V₁ =

2^'E_stored

Co

In some embodiments, the device 208 calculates the charge voltage of the supercapacitor 102 in a slightly different manner. In more detail, due to surrounding circuits in the system 100, there may be a lower limit E_Umit for the energy that may be discharged from the supercapacitor 102. In such situation, the device 208 may calculate the charge voltage V such that Estored - limit, i-e. the portion of the stored energy exceeding the lower limit, is kept essentially constant.

In step S08 when iterated the first time, the device 208 calculates an estimated decrease in capacitance AC_X during the time interval [t₀, t₀ + At]. The estimated decrease in capacitance AC_X is based on the time derivative of the capacitance— , the length At of the time interval, and a lifetime factor. The lifetime factor is indicative of the aging behaviour of the supercapacitor 102.

According to one embodiment, the lifetime factor depends on a difference between the charge voltage and the initial charge voltage V₀, i.e.

Lifetime factor (V) = g₁ V— V₀).

In the first iteration, the device 208 thus calculates the lifetime factor according to Lifetime factor (l^) = g₁(V₁ - V_Q). The function g₁ may for example be a non-linear function, such as an exponential function on the form

The parameter β varies between different types of supercapacitors 102 and may be derived from long term experiments.

The estimated decrease in capacitance AC_X is then calculated by multiplying the time derivative of the capacitance of the supercapacitor 102 by the length At of the time interval, and the lifetime factor, i.e. according to AC_X =— ^■ Lifetime f actor (V-^ · At.

In step S1 0 of the first iteration, the device 208 updates the estimated capacitance value C₀ by subtracting the estimated decrease in capacitance AC_X from the estimated capacitance value C₀, i.e. Moreover, in step S1 2, the device 208 updates the time indicator 77₀ by an amount which corresponds to the length At of the time interval, i.e.

77-L = TI₀ + At.

The device 208 then turns back to step S04 where the estimated capacitance value C_x is compared to the threshold L. As long as the estimated capacitance value is larger than the threshold L, the device 208 will iterate steps S06-S1 2. Thus, in the i :th iteration, the device 208 will calculate a charge voltage according to:

V_i = f C₁-₁ ,

calculate a lifetime factor according to:

Lifetime factor (V_£) = g^Vi— V_Q), calculate an estimated decrease in capacitance according to:

dC

ACi =— ^■ Lifetime factor{Vi, Γ_£) ^■ At,

dt

update the estimated capacitance value according to:

Ci = Ci-! - ACi , and

increasing the time indicator by an amount corresponding to the length of the time interval corresponding to the current iteration according to:

Tl^ Tl^ + At.

In the example of Fig. 3 steps S06-S1 2 are iterated five times.

However, the sixth time step S04 is carried out, the estimated capacitance value C₅ is below the threshold L. This indicates that the lifetime of the supercapacitor 1 02 has been reached, and the device 208 therefore proceeds to step S1 2 of estimating the remaining lifetime of the supercapacitor 1 02. In more detail, the device 208 estimates the remaining lifetime of the

supercapacitor 1 02 as the value of the time indicator 77 from the last iteration. In the illustrated example, the last iteration was the fifth iteration and the value of the time indicator, and hence the remaining lifetime, is equal to SAt. The device 208 may further generate an output 210 which indicates the remaining lifetime of the supercapacitor 102. For example, the output 210 may include an alarm event which indicates the remaining lifetime of the supercapacitor 102. The output 210 may also be used to update a time gauge or similar which indicates the remaining lifetime of the supercapacitor 102.

In the first embodiment of the method for estimating the remaining lifetime of the supercapacitor 102, the lifetime is estimated by taking the increasing charge voltage into account. According to a second embodiment, also the temperature is taken into account as will be explained in the following.

The second embodiment is similar to the first embodiment. However, in the initialization step S02, which is carried out at the first point in time t₀, the device 208 also receives a measured initial temperature T₀. For example, the arrangement 200 of Fig. 2 could comprise a temperature sensor (not shown) for measuring the temperature and transmitting it to the device 208.

Further, in each iteration of the method (steps S06-S12), a temperature Ti is estimated. The temperature r_£ may for instance be estimated according to a model which takes the temporal variability of the temperature into account, such as variations of the temperature over the day and/or over a season. The temperature T_£ may also be predicted from previously measured values, such as temperature values that were measured before the first point in time t₀.

In step S08, the device 208 then uses the estimated temperature T_£ in order to calculate an estimated decrease in the capacitance value AC_T . In more detail, the lifetime factor may further depend on a difference between the estimated temperature T_£ and the initial temperature T₀ according to:

Lifetime factor (V_f) = - V₀)g₂(T_i - T₀).

The function #₂( i ^{_ T}o) ^m^y be a non-linear function, such as an exponential function. Preferably, the function #₂( i ^{_ T}o) takes the form

Tj-Tp

g₂ (Ti - T₀) = 2 io Thus, in the second embodiment, also the aging of the supercapacitor 102 is taken into account when estimating the remaining lifetime of the

supercapacitor 102.

There may also be examples where the influence of temperature on aging, but not the influence of the charge voltage, is taken into account.

The supercapacitor 102 may further be associated with a conductance G. More precisely, the supercapacitor 102 has an equivalent representation which comprises a capacitive part and a resistive part, the conductance G being the inverse of an equivalent series resistance (ESR) of the resistive part of the supercapacitor 102. The equivalent series resistance is due to imperfections within the material of the supercapacitor 102. Due to aging of the capacitor 102, the capacitance C as well as the conductance G decreases over time.

A third embodiment will now be described with reference to the flowchart of Fig. 5 which takes the decreasing capacitance C as well as the decreasing conductance G into account when estimating the remaining lifetime of the supercapacitor 102. According to this embodiment the lifetime is defined as the time until the capacitance has decreased to a first

percentage (such as 50%) of its original value, thereby defining a first threshold, and the conductance G has decreased to a second percentage (such as 33% or 1 /3) of its original value, thereby defining a second threshold.

The method of Fig. 5 is generally similar in structure to the first and second embodiment described with reference to Fig. 4. However, in addition to iteratively updating the estimated capacitance, the method of Fig. 5 also iteratively updates the conductance in a similar manner.

In more detail, in step T02 the device 208 performs the steps described with reference to step S02 of Fig. 4. In addition, the device 208 receives a measured conductance value, and a measured time derivative of the conductance. The device 208 then initializes an estimated conductance G₀to be equal to the measured conductance value of the supercapacitor 102. The arrangement 206 for measuring a capacitance value may also be used to measure the conductance value and the time derivative of the conductance. This may for example be performed as explained in WO 2013/107497 A1 in conjunction with Figs. 5a and 5b and Fig. 7. In particular, the conductance G may be measured by dividing a charge current (as e.g. measured by the arrangement 206a for measuring a charge current) by a measured ohmic voltage drop across the supercapacitor 102 caused by the ESR in the capacitor 1 02 when a charging or discharging current starts to flow to or from the capacitor 1 02. The time derivative of the conductance may for example be measured (calculated) by measuring the conductance at two points in time and estimate the time derivative of the conductance based on the change of the conductance between the two time points, i.e. as the inclination of a hypothetical straight line fitted to the measured conductance values at the two points in time.

In step S04, the device 208 checks whether the estimated capacitance value is larger than a first threshold (which may be 50% of the original capacitance value) or the estimated conductance is larger than a second threshold (which may be 33% or the original conductance value). As long as not both of the thresholds are trespassed from above, the device 208 iterates steps T06-T1 2.

The ESR is related to an ohmic voltage drop V_ESR of the supercapacitor 1 02. More precisely, as a current is flowing from the capacitor 102, there will according to Ohm's law be a voltage V_ESR across the capacitor being equal to the product of the equivalent series resistance and the current flowing from the supercapacitor 1 02. As a result of the ohmic voltage drop V_ESR of the capacitor 1 02, all of the energy stored in the capacitor E_stored may not be discharged to the load 1 1 0 when the capacitor is in discharging mode since there will be power losses in the resistive component in the supercapacitor 1 02. More precisely, the energy E_backup which may be used to power the load

1 1 0 is given by

E backup ^~ E stored ^~ ^ESR' where E_ESR denotes the energy losses in the ESR of the capacitor 1 02. Due to aging of the capacitor 1 02, the capacitance C as well as the conductance G decreases over time. Accordingly, the stored energy E_stored as well as the useable energy E_backup decrease with time for a constant charge voltage.

In order to compensate for the decrease in capacitance as well as in conductance, the device 208 may in step T06, calculate the charge voltage based on the estimated capacitance value as well as the estimated

conductance value. The device 208 may calculate the charge voltage in step S06 according to a function which increases as the estimated capacitance value and conductance value G of the supercapacitor 1 02 decrease. For example, the device 208 may calculate the charge voltage such that the energy E_backup is kept essentially constant. In this way, the charge voltage of the supercapacitor 1 02 is adjusted based on the estimated conductance value to compensate for the ohmic voltage drop in the supercapacitor 1 02. In more detail, in the above equation, E_stored is a function of the charge voltage V. The charge voltage V may also be thought of as the initial voltage across the supercapacitor 1 02 prior to discharging. The energy losses E_ESR in the capacitor 1 02 depends on the discharge current, i(t) say, which flows from the capacitor 1 02 during discharging. The discharge current i(t) is a function of time since, as the voltage across the supercapacitor 102 decreases during discharging, the discharge current i(t) increases such that a power P provided to the load 1 1 0 is kept constant. The energy losses E_ESR may be expressed as

The discharge current i(t) may be calculated under the assumption that the power P provided to the load 1 10 is kept constant by solving the system of equations:

where v(t) is the voltage across the capacitive part of the equivalent representation of the capacitor 1 02 during discharging, and v(0)=V_c. The above system of equations may for example be solved numerically.

In step T08, the device 208 calculates an estimated decrease in capacitance value as describes with reference to step S08. Moreover, the device 208 calculates an estimated decrease in conductance in a similar manner by multiplying the time derivative of the conductance by the lifetime factor and the length At of the time interval.

In step T1 0, the device 208 updates the capacitance value as set out with respect to step S1 0. In a similar manner, the estimated conductance value is updated by subtracting the estimated decrease in conductance from the estimated conductance value. Next, in step T12, the time indicator is increased as set out with respect to step S12.

Steps T06-T12 are iterated until the estimated capacitance value in step T04 is found to below the first threshold and the estimated conductance value in step T04 is found to be below the second threshold. The device 208 then determines that the life of the supercapacitor 102 has come to an end and sets, in step T14, the remaining lifetime of the supercapacitor to the current value of the time indicator.

It will be appreciated that a person skilled in the art can modify the above-described embodiments in many ways and still use the advantages of the invention as shown in the embodiments above. For example, in an alternative embodiment, there may be no particular threshold for the conductance value. Instead, the threshold for the capacitance value may be a function of the conductance value. In other words, the conductance value is used to adjust the threshold for the capacitance value. In such embodiment, step T04 is modified such that only the capacitance value is compared to a threshold (which depends on the estimated conductance value). In each iteration of steps T06-T12, the updated estimated conductance value is used to adjust the threshold for the capacitance value according to a predefined relationship. Thus, the invention should not be limited to the shown embodiments but should only be defined by the appended claims.

Additionally, as the skilled person understands, the shown embodiments may be combined.

Claims

1 . A method for estimating the remaining lifetime of a supercapacitor, SC, being charged by a charge voltage which is based on a capacitance value of the SC, comprising:

at a first point in time,

initializing a time indicator,

as long as the estimated capacitance value is larger than a threshold value which indicates that the life of the SC has come to an end, iterating the steps of:

2. The method of claim 1 , wherein the charge voltage of the SC is calculated based on the estimated capacitance value according to a function which increases as the estimated capacitance value decreases.

3. The method of claim 2, wherein the charge voltage of the SC is calculated based on the estimated capacitance value such that the energy stored in the SC is kept essentially constant.

4. The method of any claim 2, wherein the SC is only capable of discharging a portion of its stored energy that exceeds a lower limit, and the charge voltage of the SC is calculated based on the estimated capacitance value such that the portion of the stored energy exceeding the lower limit is kept essentially constant.

5. The method of claim 2, wherein the SC is charged by a charge voltage which is based on the capacitance value and a conductance value of the SC, and the charge voltage of the SC is further calculated based on an estimated conductance value of the SC according to a function which increases as the estimated capacitance value and conductance value decrease.

6. The method of claim 5, wherein the conductance value of the SC is associated with a resistive part of the SC which give rise to an ohmic voltage drop, and the charge voltage of the SC is adjusted based on the estimated conductance value to compensate for the ohmic voltage drop.

7. The method of claim 5, wherein the SC is only capable of

discharging a portion of its stored energy that exceeds a lower limit, and the charge voltage of the SC is calculated based on the estimated capacitance value and the estimated conductance value such that the portion of the stored energy exceeding the lower limit is kept essentially constant.

8. The method of claim 1 , wherein the estimated decrease in

capacitance during a time interval corresponding to the current iteration is calculated by multiplying the time derivative of the capacitance of the SC by the length of the time interval, and the factor which is indicative of an aging behavior of the SC.

9. The method of claim 1 , wherein the factor which is indicative of the aging behavior of the SC depends on the difference between the charge voltage and the initial charge voltage according to a non-linear function.

10. The method of claim 7, wherein the non-linear function is an exponential function.

1 1 . The method of claim 1 , further comprising:

at the first point in time, measuring an initial temperature, and during each iteration, estimating a current temperature, wherein the factor which is indicative of the aging behavior of the SC further depends on a difference between the estimated temperature and the initial temperature.

12. The method of claim 1 1 , wherein the current temperature is estimated according to a model which takes the temporal variability of the temperature into account or predicted from previously measured values.

13. The method of claims 1 1 , wherein the factor which is indicative of the aging behavior of the SC depends on the difference between the estimated temperature and the initial temperature according to an exponential function.

14. The method of claim 1 , further comprising generating an output which indicates the remaining lifetime.

15. The method of claim 14, wherein the output includes an alarm event which indicates the remaining lifetime of the SC.

16. The method of claim 1 , wherein the time indicator at initialization is set to zero.

17. The method of claim 1 , wherein the received time derivative of the capacitance of the SC is measured with respect to a time interval prior to the first point in time.

18. A computer program product comprising computer code

instructions stored on computer-readable medium for performing the method according to claim 1 when executed on a device having processing capability.

19. A device for estimating the remaining lifetime of a supercapacitor,

SC, being charged by a charge voltage which is based on a capacitance value of the SC, comprising:

a processing component configured to:

at the first point in time,

initialize a time indicator,

increasing the time indicator by an amount corresponding to the length of the time interval corresponding to the current iteration, and estimate the remaining lifetime of the SC as the value of the time ndicator from the last iteration.

20. A system for estimating the remaining lifetime of a supercapacitor, SC, being charged by a charge voltage which is based on a capacitance value of the SC, comprising:

a device for estimating the remaining lifetime of the SC according to claim 19.