A Deterioration Diagnosis Circuit of a Lithium-Ion Battery Using Microcomputer Implementing Z-Transform Convolution

The deterioration of lithium-ion batteries has been detected by an increase in the battery impedance by means of an alternating current method or a battery capacity test. These methods require an interruption of the operation because the battery has to be removed. A method estimating equivalent circuit parameters of a lithium-ion battery during operation is proposed in this article. The increase in the internal impedance has a close relation to the deterioration of the battery. The circuit is expressed by a resister and an RC parallel circuit connected in series. The parameters can be estimated by applying a convolution technique to the voltage and current fluctuation of the battery during operation. A diagnosis circuit using a microcomputer is developed using a simple algorithm employing z-transformation in the parameter estimation. The estimated parameters depend on its state of charge (SOC) and ambient temperature. The SOC dependency is solved by estimating timing, and the temperature dependency is corrected by a function derived in this article. The deterioration diagnosis of the battery can be applied to a solar power generation system, and the feasibility is discussed in this article.


Introduction
Lithium-ion batteries are used with various apparatus due to their remarkable features, such as high energy density [1]. Recently, sustainable power generations using natural energies such as solar and wind have been introduced to prevent global warming. The lithium-ion battery is used with the power generation systems for stabilizing generated power [2,3]. However, its operating voltage, current, and temperature have to be kept within its specification. Otherwise, the battery is led to thermal runaway or over-charge/discharge, and it may result in a burst [4]. For a safety operation, almost all the lithium-ion batteries are used with BMS (battery management system), which monitors the voltage, current, and temperature of the battery [5,6]. However, most BMSs do not have the deterioration diagnosis feature.
The deterioration has been carried out by a capacity test or by detecting an increase in the battery's internal impedance by means of an alternating current method [7][8][9][10]. The former method requires a charging/discharging system and a long observation time, and the latter method needs an expensive measuring instrument. In addition, both methods require an interruption of the operation of battery-driven equipment because the battery has to be removed [11]. It is not allowed in a system requiring high reliability, such as a large-scale solar power generation system. Therefore, a simple method to detect the deterioration of the battery is required for a safe operation.
This article proposes a deterioration diagnosis method for the lithium-ion battery during its operation. The method is implemented into a microcomputer. It is realized by the simplicity and availability of the proposed algorithm. It indicates that the proposed method can be implemented into BMS.
The proposed method can contribute not only to the safe operation of the batteries but also to the reduction in the environmental load by clarifying the appropriate replacing time of the battery.

Equivalent Circuit of Lithium-Ion Battery
The lithium-ion battery has transient characteristics during charging and discharging. Figure 1 shows a voltage and a current waveform while charging the CGR18650CH cell with a constant current of 1 C (2.25 A). The voltage sharply increases at the beginning of the charging and gradually increases. Similarly, the voltage suddenly decreases at the end of the charging and decreases toward its internal voltage. These transient characteristics can be explained by an equivalent circuit illustrated in Figure 2. It consists of an internal voltage V 0 , a series resistor R B0, and a series-connected RC parallel circuit whose resistance and capacitance are R B1 and C B1 . The voltage across terminals denoted by V B is divided into three parts. Both sudden voltage changes are determined by the series resistance R B0 . The voltage increase during the charging is expressed by the increase in the internal voltage V 0 and the voltage across the internal impedance. The gradual voltage decrease after the charging is due to the discharge of the internal capacitor C B1 .
require an interruption of the operation of battery-driven equipment because the batter has to be removed [11]. It is not allowed in a system requiring high reliability, such as large-scale solar power generation system. Therefore, a simple method to detect the dete rioration of the battery is required for a safe operation.
This article proposes a deterioration diagnosis method for the lithium-ion batter during its operation. The method is implemented into a microcomputer. It is realized b the simplicity and availability of the proposed algorithm. It indicates that the proposed method can be implemented into BMS.
The proposed method can contribute not only to the safe operation of the batterie but also to the reduction in the environmental load by clarifying the appropriate replacin time of the battery.

Equivalent Circuit of Lithium-Ion Battery
The lithium-ion battery has transient characteristics during charging and discharg ing. Figure 1 shows a voltage and a current waveform while charging the CGR18650CH cell with a constant current of 1 C (2.25 A). The voltage sharply increases at the beginnin of the charging and gradually increases. Similarly, the voltage suddenly decreases at th end of the charging and decreases toward its internal voltage. These transient character istics can be explained by an equivalent circuit illustrated in Figure 2. It consists of an internal voltage V0, a series resistor RB0, and a series-connected RC parallel circuit whos resistance and capacitance are RB1 and CB1. The voltage across terminals denoted by VB i divided into three parts. Both sudden voltage changes are determined by the series re sistance RB0. The voltage increase during the charging is expressed by the increase in th internal voltage V0 and the voltage across the internal impedance. The gradual voltag decrease after the charging is due to the discharge of the internal capacitor CB1.   large-scale solar power generation system. Therefore, a simple method to detect the det rioration of the battery is required for a safe operation.
This article proposes a deterioration diagnosis method for the lithium-ion batter during its operation. The method is implemented into a microcomputer. It is realized b the simplicity and availability of the proposed algorithm. It indicates that the propose method can be implemented into BMS.
The proposed method can contribute not only to the safe operation of the batterie but also to the reduction in the environmental load by clarifying the appropriate replacin time of the battery.

Equivalent Circuit of Lithium-Ion Battery
The lithium-ion battery has transient characteristics during charging and discharg ing. Figure 1 shows a voltage and a current waveform while charging the CGR18650CH cell with a constant current of 1 C (2.25 A). The voltage sharply increases at the beginnin of the charging and gradually increases. Similarly, the voltage suddenly decreases at th end of the charging and decreases toward its internal voltage. These transient characte istics can be explained by an equivalent circuit illustrated in Figure 2. It consists of a internal voltage V0, a series resistor RB0, and a series-connected RC parallel circuit whos resistance and capacitance are RB1 and CB1. The voltage across terminals denoted by VB divided into three parts. Both sudden voltage changes are determined by the series r sistance RB0. The voltage increase during the charging is expressed by the increase in th internal voltage V0 and the voltage across the internal impedance. The gradual voltag decrease after the charging is due to the discharge of the internal capacitor CB1.

Deterioration Diagnosis Method Using a Convolution Technique and z-Transformation
A deterioration diagnosis method for lithium-ion batteries should be simple because its algorithm has to be installed into any BMS. Therefore, a deterioration diagnosis method using convolution technique and z-transformation is proposed in this article. The series resistance R B0 and the resistance R B1 in the RC parallel circuit increase with the deterioration of the battery [12]. The parallel resistance R B1 is an effective parameter to diagnose a deterioration of batteries because the series resistance R B0 depends on the contact resistance. A diagnosis of lithium-ion battery can be realized by deriving the parameter R B1 .
The internal impedance Z(s) of the equivalent circuit shown in Figure 2 in a frequency domain is given by Equation (1).
where τ 1 is the time constant given by the product of resistance R B1 and capacitance C B1 . The voltage drop with the internal impedance of the battery in Figure 2 during charge or discharge by current I(t) is given by the convolution of the current and impulse response of the impedance as shown in Equation (2).
where ∆t is sampling time, and n is an arbitrary positive integer. Figure 3 shows the magnified voltage and current waveforms just after starting the charging of the battery shown in Figure 1. The integrated voltage S shown in Figure 3 is given by Equation (3).
where T max is the maximum observation time. Figure 4 shows the integrated voltage waveform S. The parameter R B1 is calculated by applying a nonlinear least-squares method with Equation (3) to the measured integrated voltage S. However, this calculation load for the convolution is heavy, and it needs an initial value for the least-squares method. For these reasons, the method is not suitable from the viewpoint of installation into BMS. Therefore, a simple algorithm using z-transformation is proposed in this article.

Deterioration Diagnosis Method Using a Convolution Technique and z-Transformation
A deterioration diagnosis method for lithium-ion batteries should be simple because its algorithm has to be installed into any BMS. Therefore, a deterioration diagnosis method using convolution technique and z-transformation is proposed in this article. The series resistance RB0 and the resistance RB1 in the RC parallel circuit increase with the deterioration of the battery [12]. The parallel resistance RB1 is an effective parameter to diagnose a deterioration of batteries because the series resistance RB0 depends on the contact resistance. A diagnosis of lithium-ion battery can be realized by deriving the parameter RB1.
The internal impedance Z(s) of the equivalent circuit shown in Figure 2 in a frequency domain is given by Equation (1).
where τ1 is the time constant given by the product of resistance RB1 and capacitance CB1. The voltage drop with the internal impedance of the battery in Figure 2 during charge or discharge by current I(t) is given by the convolution of the current and impulse response of the impedance as shown in Equation (2).
where Δt is sampling time, and n is an arbitrary positive integer. Figure 3 shows the magnified voltage and current waveforms just after starting the charging of the battery shown in Figure 1. The integrated voltage S shown in Figure 3 is given by Equation (3).  reasons, the method is not suitable from the viewpoint of installation into BMS. The a simple algorithm using z-transformation is proposed in this article.  (1) is given by Equations ( (5).

The transfer function H(z) in z-domain in Equation
where Δt is sampling time. The voltage V(z) across the battery's internal impedance z-domain is given by Equation (6).
where I(z) is a charging current in the z-domain. The integrated voltage S(z) by trape rule in z-domain is given by Equation (7) = Equation (8) shows the process of the conversion from z-domain to time-doma The coefficients a0, a1, and b1 can be determined from Equation (8). Equation (9 matrix representation of Equation (8) to be solved.  (1) is given by Equations (4) and (5).

The transfer function H(z) in z-domain in Equation
where ∆t is sampling time. The voltage V(z) across the battery's internal impedance in the z-domain is given by Equation (6).
where I(z) is a charging current in the z-domain. The integrated voltage S(z) by trapezoidal rule in z-domain is given by Equation (7) S(z) = ∆t Equation (8) shows the process of the conversion from z-domain to time-domain.
The coefficients a 0 , a 1 , and b 1 can be determined from Equation (8). Equation (9) is the matrix representation of Equation (8) to be solved.
Energies 2021, 14, 6868 where l is the number of samples. The above equation can be represented by using column vectors (i ), (i ), (s ), and (s ).
The number of equations in Equation (10) is greater than the number of unknowns a 0 , a 1 , and b 1 . For solving the equation, the transposed coefficient matrix is multiplied on both sides of Equation (10).
The coefficients a 0 , a 1 , and b 1 are obtained by solving Equation (11). This corresponds to the solution of Equation (10) by the least square error method. The parameters R B0 , R B1 , and C B1, are given by Equation (12) after transforming Equation (7) from z-domain to time-domain. This calculation method is approximately equal to the convolution in Equation (3). Thus, the diagnosis of a lithium-ion battery for BMS can be realized due to its simplicity.

Diagnosis Method Using Convolution
The effectiveness of the deterioration diagnosis method using the time-domain convolution method shown in Equation (3) is described in this section. The battery CGR18650CH, whose capacity is 2.25 Ah, is used for the experiments. Some deteriorated batteries from 0 to 500 cycles are prepared.  (3) is 100 s, ∆t is 1 s. The efficient parameter R B1 to detect deterioration increases with the deterioration of the battery. The results illustrated in Figure 6a show the method using convolution technique can be diagnosed the deterioration of the lithium-ion battery.  (3) is 100 s, Δt is 1 s. The efficient parameter RB1 to detect deterioration increases with the deterioration of the battery. The results illustrated in Figure 6a show the method using convolution technique can be diagnosed the deterioration of the lithium-ion battery.   Figure 7a shows the SOC dependence of the voltage responses to a new battery. Its initial SOC (at the beginning of the charging) is set from 20% to 80%. The charging current is 1 C (2.25 A), and the charging period is 100 s. The ambient temperature around the battery is 25 °C and the sampling period is 1 s. Figure 7b shows the effect of the initial SOC on the integrated voltage waveforms S(t) obtained from the voltages shown in Figure  7a. The SOC dependence of the integrated voltage is lower than that of the deterioration characteristic. Figure 8 shows the parameters obtained by curve fitting of Equation (3) by means of the least-squares method. Although the RB1 monotonically increases with the deterioration, the value depends on SOC. The practical solution of the SOC dependence is that the diagnostic is scheduled at a specific SOC, which is suitable for each application. Continuous diagnostic is not required; in general, weekly or monthly inspection is enough in most cases.  Figure 6a show the method using convolution technique can be diagnosed the deterioration of the lithium-ion battery.   Figure 7a shows the SOC dependence of the voltage responses to a new battery. Its initial SOC (at the beginning of the charging) is set from 20% to 80%. The charging current is 1 C (2.25 A), and the charging period is 100 s. The ambient temperature around the battery is 25 °C and the sampling period is 1 s. Figure 7b shows the effect of the initial SOC on the integrated voltage waveforms S(t) obtained from the voltages shown in Figure  7a. The SOC dependence of the integrated voltage is lower than that of the deterioration characteristic. Figure 8 shows the parameters obtained by curve fitting of Equation (3) by means of the least-squares method. Although the RB1 monotonically increases with the deterioration, the value depends on SOC. The practical solution of the SOC dependence is that the diagnostic is scheduled at a specific SOC, which is suitable for each application. Continuous diagnostic is not required; in general, weekly or monthly inspection is enough in most cases.  Figure 7a shows the SOC dependence of the voltage responses to a new battery. Its initial SOC (at the beginning of the charging) is set from 20% to 80%. The charging current is 1 C (2.25 A), and the charging period is 100 s. The ambient temperature around the battery is 25 • C and the sampling period is 1 s. Figure 7b shows the effect of the initial SOC on the integrated voltage waveforms S(t) obtained from the voltages shown in Figure 7a. The SOC dependence of the integrated voltage is lower than that of the deterioration characteristic. Figure 8 shows the parameters obtained by curve fitting of Equation (3) by means of the least-squares method. Although the R B1 monotonically increases with the deterioration, the value depends on SOC. The practical solution of the SOC dependence is that the diagnostic is scheduled at a specific SOC, which is suitable for each application. Continuous diagnostic is not required; in general, weekly or monthly inspection is enough in most cases.   Figure 9a shows the temperature dependence of the voltage waveforms. The ambient temperature around the battery is changed from 0 to 40 °C. These waveforms are obtained using a new battery with a sampling period of 1 s. The battery is charged by a current of 1 C (2.25 A) for 100 s, and the initial SOC is 50%. Figure 9b shows the effect of the temperature on the integrated voltage waveforms obtained from the voltages shown in Figure  9a. The equivalent circuit parameters calculated by curve fitting using Equation (3) are shown in Figure 10. The parameter RB1 exponentially decreases with the increase in the ambient temperature. Therefore, a temperature correction of the parameter RB1 is required [12].   Figure 9a shows the temperature dependence of the voltage waveforms. The ambient temperature around the battery is changed from 0 to 40 °C. These waveforms are obtained using a new battery with a sampling period of 1 s. The battery is charged by a current of 1 C (2.25 A) for 100 s, and the initial SOC is 50%. Figure 9b shows the effect of the temperature on the integrated voltage waveforms obtained from the voltages shown in Figure  9a. The equivalent circuit parameters calculated by curve fitting using Equation (3) are shown in Figure 10. The parameter RB1 exponentially decreases with the increase in the ambient temperature. Therefore, a temperature correction of the parameter RB1 is required [12].  Figure 9a shows the temperature dependence of the voltage waveforms. The ambient temperature around the battery is changed from 0 to 40 • C. These waveforms are obtained using a new battery with a sampling period of 1 s. The battery is charged by a current of 1 C (2.25 A) for 100 s, and the initial SOC is 50%. Figure 9b shows the effect of the temperature on the integrated voltage waveforms obtained from the voltages shown in Figure 9a. The equivalent circuit parameters calculated by curve fitting using Equation (3) are shown in Figure 10. The parameter R B1 exponentially decreases with the increase in the ambient temperature. Therefore, a temperature correction of the parameter R B1 is required [12].   Figure 11 shows the parameter RB1 versus temperature with approximated curves expressed by Equation (13).

Temperature Dependence
where T is the ambient temperature, and the parameter RC′ expresses the estimated resistance RB1 at T = 0 °C. Table 1 shows the deterioration characteristic of the corrected parameters RC′ and TC in Equation (13). The parameter RC′ increases by 120% according to the deterioration of   Figure 11 shows the parameter RB1 versus temperature with approximated curves expressed by Equation (13).
where T is the ambient temperature, and the parameter RC′ expresses the estimated resistance RB1 at T = 0 °C. Table 1 shows the deterioration characteristic of the corrected parameters RC′ and TC in Equation (13). The parameter RC′ increases by 120% according to the deterioration of  Figure 11 shows the parameter R B1 versus temperature with approximated curves expressed by Equation (13).
where T is the ambient temperature, and the parameter R C expresses the estimated resistance R B1 at T = 0 • C. the battery, and the coefficient TC can be estimated that it is constant. The corrected parameter RC' can be obtained from the estimated resistance RB1 and ambient temperature T with the averaged coefficient TCA (=57.3 °C) in Table 1.  Figure 12 shows the corrected parameter RC, which shows the resistance RB1 at 25 °C. Because the parameter RC increases with the increase with the number of charging cycles, the deterioration of a lithium-ion battery can be diagnosed using the parameter RC at any temperature.    Table 1 shows the deterioration characteristic of the corrected parameters R C and T C in Equation (13). The parameter R C increases by 120% according to the deterioration of the battery, and the coefficient T C can be estimated that it is constant. The corrected parameter R C ' can be obtained from the estimated resistance R B1 and ambient temperature T with the averaged coefficient T CA (=57.3 • C) in Table 1.  Figure 12 shows the corrected parameter R C , which shows the resistance R B1 at 25 • C. Because the parameter R C increases with the increase with the number of charging cycles, the deterioration of a lithium-ion battery can be diagnosed using the parameter R C at any temperature.   Figure 12 shows the corrected parameter RC, which shows the resistance RB1 at 25 °C. Because the parameter RC increases with the increase with the number of charging cycles, the deterioration of a lithium-ion battery can be diagnosed using the parameter RC at any temperature.  Figure 12. Effect of charging cycle on corrected RC. Figure 12. Effect of charging cycle on corrected R C . Figure 13 shows the block diagram of an automatic diagnosis circuit developed in this article. Arduino Uno is used for realizing the deterioration diagnosis method using a ztransformation proposed in this article. Figure 14 shows the flow chart of the battery-status estimation system.

Automatic Diagnosis Circuit
Energies 2021, 14, x FOR PEER REVIEW Figure 13 shows the block diagram of an automatic diagnosis circuit deve this article. Arduino Uno is used for realizing the deterioration diagnosis method z-transformation proposed in this article. Figure 14 shows the flow chart of the status estimation system.  The deterioration diagnosis circuit developed in this article is shown in Figu first, the voltage, current, and ambient temperature of the battery during its opera measured through the analog input pins of the Arduino Uno. Signal amplitudes justed by operational amplifiers installed in the voltage, current, and ambient temp measurement circuits. The parameters of the equivalent circuit of lithium-ion ba estimated by calculating using Equation (8). Finally, the efficient parameter of d tion of battery RB1 corrected by the ambient temperature is displayed by the lightdiodes (LEDs). The voltage, current, and ambient temperature of the battery are d in the LCD during its operation.  Figure 13 shows the block diagram of an automatic diagnosis circuit deve this article. Arduino Uno is used for realizing the deterioration diagnosis method z-transformation proposed in this article. Figure 14 shows the flow chart of the status estimation system.  The deterioration diagnosis circuit developed in this article is shown in Figu first, the voltage, current, and ambient temperature of the battery during its oper measured through the analog input pins of the Arduino Uno. Signal amplitude justed by operational amplifiers installed in the voltage, current, and ambient tem measurement circuits. The parameters of the equivalent circuit of lithium-ion ba estimated by calculating using Equation (8). Finally, the efficient parameter of d tion of battery RB1 corrected by the ambient temperature is displayed by the lightdiodes (LEDs). The voltage, current, and ambient temperature of the battery are d in the LCD during its operation. The deterioration diagnosis circuit developed in this article is shown in Figure 15. At first, the voltage, current, and ambient temperature of the battery during its operation are measured through the analog input pins of the Arduino Uno. Signal amplitudes are adjusted by operational amplifiers installed in the voltage, current, and ambient temperature measurement circuits. The parameters of the equivalent circuit of lithium-ion battery are estimated by calculating using Equation (8). Finally, the efficient parameter of degradation of battery R B1 corrected by the ambient temperature is displayed by the light-emitting diodes (LEDs). The voltage, current, and ambient temperature of the battery are displayed in the LCD during its operation. The results of the deterioration diagnosis of the lithium-ion battery using t nosis circuit are described below. Because of the difficulty in preparing deterior teries used in a practical system, the effectiveness of the proposed method is conf different circuit conditions provide identical results. A lithium-ion battery us home photovoltaic power generation system is chosen in these experiments. Dou power supply of the DC power distribution system manufactured by Keisoku G shown in Figure 16 is used in this article. This system consists of 10 series of sola (If the insolation is 1 kW/m 2 , the maximum output is 2.084 kW, the open-circuit v 332 V, and the rated operating current is 7.84 A.) The battery used in this artic BT1200B, whose capacity is 24 Ah, composed of 16 series and 8 parallel co US26650FTC1 (3 Ah) cells.   The results of the deterioration diagnosis of the lithium-ion battery using the diagnosis circuit are described below. Because of the difficulty in preparing deteriorated batteries used in a practical system, the effectiveness of the proposed method is confirmed as different circuit conditions provide identical results. A lithium-ion battery used for a home photovoltaic power generation system is chosen in these experiments. Double way power supply of the DC power distribution system manufactured by Keisoku Giken Co. shown in Figure 16 is used in this article. This system consists of 10 series of solar panels. (If the insolation is 1 kW/m 2 , the maximum output is 2.084 kW, the open-circuit voltage is 332 V, and the rated operating current is 7.84 A.) The battery used in this article is NT-BT1200B, whose capacity is 24 Ah, composed of 16 series and 8 parallel connected US26650FTC1 (3 Ah) cells.
The results of the deterioration diagnosis of the lithium-ion battery using the nosis circuit are described below. Because of the difficulty in preparing deteriorate teries used in a practical system, the effectiveness of the proposed method is confirm different circuit conditions provide identical results. A lithium-ion battery used home photovoltaic power generation system is chosen in these experiments. Doubl power supply of the DC power distribution system manufactured by Keisoku Gike shown in Figure 16 is used in this article. This system consists of 10 series of solar p (If the insolation is 1 kW/m 2 , the maximum output is 2.084 kW, the open-circuit volt 332 V, and the rated operating current is 7.84 A.) The battery used in this article i BT1200B, whose capacity is 24 Ah, composed of 16 series and 8 parallel conn US26650FTC1 (3 Ah) cells.     Figure 17 for the investigation. The voltage and current waveforms of a lithium-ion battery in the DC distribution system are shown in Figure 19a-e. The ambient temperature is 21 • C, and the initial SOC is 50%. Figure 20a-c shows the equivalent circuit parameters calculated by the diagnosis circuit. The observation time used for the calculation is 50 s after the beginning of the charging. The effective parameters of deterioration diagnosis R B1 are calculated almost constant in all cases. The differences between the average of R B1 in Case 1-Case 5 and the result of each Case are less than 9%. The resistance difference of R B1 between the new battery and 500-cycle deteriorated battery in Figure 6a is 85.5%, which is far greater than the fluctuation of 9% around the average of R B1 in Case 1-Case 5 shown in Figure 20b. Therefore, the algorithm of estimation of the parameters is adequate, and the deterioration diagnosis of a lithium-ion battery can be realized using the diagnosis circuit developed in this article. The fluctuation determines the resolution of the proposed method. The resolution is enough to indicate the exchange of the batteries. The resistance difference of RB1 between the new battery and 500-cycle deteriorated battery in Figure 6a is 85.5%, which is far greater than the fluctuation of 9% around the average of RB1 in Case 1-Case 5 shown in Figure 20b. Therefore, the algorithm of estimation of the parameters is adequate, and the deterioration diagnosis of a lithium-ion battery can be realized using the diagnosis circuit developed in this article. The fluctuation determines the resolution of the proposed method. The resolution is enough to indicate the exchange of the batteries.  The resistance difference of RB1 between the new battery and 500-cycle deteriorated battery in Figure 6a is 85.5%, which is far greater than the fluctuation of 9% around the average of RB1 in Case 1-Case 5 shown in Figure 20b. Therefore, the algorithm of estimation of the parameters is adequate, and the deterioration diagnosis of a lithium-ion battery can be realized using the diagnosis circuit developed in this article. The fluctuation determines the resolution of the proposed method. The resolution is enough to indicate the exchange of the batteries.

Conclusions
A deterioration diagnosis method for lithium-ion batteries based on circuit parameters of the equivalent circuit is proposed in this article. The parameters are obtained by applying a convolution technique to the voltage and current during its operation. In addition, a diagnosis circuit using a microcomputer is developed. A simple convolution calculation based on z-transformation enables the implementation in the diagnosis circuit.
Lithium-ion batteries can be expressed by an equivalent circuit composed of an internal voltage V0, a series resistance RB0, and a series-connected RB1CB1 parallel circuit, and the circuit expresses the battery's transient characteristic. In other words, the parameters can be obtained from the transient characteristic. The equivalent circuit parameters RB0 and RB1 are increased by the deterioration. However, a series resistance RB0 depends on the contact resistance. Therefore, RB1 is the optimal parameter to detect the deterioration of lithium-ion batteries. The integration of the difference between a voltage across terminals and an internal voltage has close relation with the parameter RB1, which expresses the deterioration of the battery. However, the parameter RB1 depends on not only the deterioration but also the temperature and SOC of the battery. A diagnosis of lithium-ion battery can be realized by the temperature corrected parameter RB1 (Rc). The SOC correction is not practical due to the complicated characteristic. However, the weak point can be avoided by scheduling the deterioration diagnosis at specific SOCs because frequent inspection is not required for the battery diagnostic.
A deterioration diagnosis circuit installed the method based on z-transformation is developed in this article. This circuit measures transient battery voltage, current, and ambient temperature. After that, it derives the effective parameter RC for the battery diagnostic. This circuit can be moved to any microcomputer due to its simple algorithm. The accuracies of the algorithm and circuit are confirmed by a single cell and a power storage

Conclusions
A deterioration diagnosis method for lithium-ion batteries based on circuit parameters of the equivalent circuit is proposed in this article. The parameters are obtained by applying a convolution technique to the voltage and current during its operation. In addition, a diagnosis circuit using a microcomputer is developed. A simple convolution calculation based on z-transformation enables the implementation in the diagnosis circuit.
Lithium-ion batteries can be expressed by an equivalent circuit composed of an internal voltage V 0 , a series resistance R B0, and a series-connected R B1 C B1 parallel circuit, and the circuit expresses the battery's transient characteristic. In other words, the parameters can be obtained from the transient characteristic. The equivalent circuit parameters R B0 and R B1 are increased by the deterioration. However, a series resistance R B0 depends on the contact resistance. Therefore, R B1 is the optimal parameter to detect the deterioration of lithiumion batteries. The integration of the difference between a voltage across terminals and an internal voltage has close relation with the parameter R B1 , which expresses the deterioration of the battery. However, the parameter R B1 depends on not only the deterioration but also the temperature and SOC of the battery. A diagnosis of lithium-ion battery can be realized by the temperature corrected parameter R B1 (Rc). The SOC correction is not practical due to the complicated characteristic. However, the weak point can be avoided by scheduling the deterioration diagnosis at specific SOCs because frequent inspection is not required for the battery diagnostic.
A deterioration diagnosis circuit installed the method based on z-transformation is developed in this article. This circuit measures transient battery voltage, current, and ambient temperature. After that, it derives the effective parameter R C for the battery diagnostic. This circuit can be moved to any microcomputer due to its simple algorithm. The accuracies of the algorithm and circuit are confirmed by a single cell and a power storage system for a 2 kW PV generation. The proposed system may apply to the battery, such as large-scale electric power systems using solar and wind during its operation.