Prediction of State-of-Health for Nickel-Metal Hydride Batteries by a Curve Model Based on Charge-Discharge Tests

Based on charge-discharge cycle tests for commercial nickel-metal hydride (Ni-MH) batteries, a nonlinear relationship is found between the discharging capacity (Cdischarge, Ah) and the voltage changes in 1 s occurring at the start of the charging process (∆Vcharge, mV). This nonlinear relationship between Cdischarge and ∆Vcharge is described with a curve equation, which can be determined using a nonlinear least-squares method. Based on the curve equation, a curve model for the state-of-health (SOH) prediction is constructed without battery models and cycle numbers. The validity of the curve model is verified using (Cdischarge, ∆Vcharge) data groups obtained from the charge-discharge cycle tests at different rates. The results indicate that the curve model can be effectively applied to predict the SOH of the Ni-MH batteries and the best prediction root-mean-square error (RMSE) can reach upto 1.2%. Further research is needed to confirm the application of this empirical curve model in practical fields.


Introduction
Nickel-metal hydride (Ni-MH) batteries have been applied in portable electronics and electric or hybrid-electric vehicles (EVs and HEVs) owing to their relatively good storage and power, higher safety, and excellent environmental acceptability [1][2][3][4].With regard to these applications, the failure of Ni-MH batteries, resulting from irreversible capacity degradation and loss of performance in cycle service [5,6], is a key issue that warrants close attention.The state-of-health (SOH) status of a battery, which defines the current battery performance relative to its unused condition, is a powerful indicator of battery performance [7] and is usually used to predict the end-of-life and aging of batteries [8,9].However, the definition of battery SOH is still somewhat equivocal [10] because different battery parameters can be used as the indicators of battery performance.This ambiguity makes the determination of battery SOH a difficult task.
Generally, three definitions of battery SOH have been reported, including SOH values based on battery impedance [11,12], battery capacity [9], and comprehensive battery parameters such as impedance, capacity, open circuit potential (OCV), charging or discharging current, and temperature (T) [13][14][15][16][17]. Apparently, the SOH based on comprehensive battery parameters reflects the present battery performance more accurately, but it also makes the estimation more complicated.The SOH values based on battery impedance and capacity are considered to reflect the capability of Energies 2015, 8,[12474][12475][12476][12477][12478][12479][12480][12481][12482][12483][12484][12485][12486][12487] the battery to provide a certain power and to store energy, respectively.Both of the SOH values are widely applied in EVs and HEVs.In this case, all the methods for the estimation of battery impedance and battery capacity can be used as a basis for the SOH estimation.However, the estimations of battery impedance and battery capacity are also not easy to do, especially online estimation [18,19].
The reported methods for the estimation of battery SOH can be generally divided into two categories: physics-based model estimations and non-physics-based model estimations.Physics-based model estimations are based on electrical or electrochemical cell models [20][21][22][23][24][25][26][27].In order to improve the estimation accuracy, some algorithms, such as adaptive observers [28], extended Kalman filter (EKF) [29,30], and relevance vector machines (RVMs)-particle filters (PFs) [31], are usually employed.The main problem for this kind of estimation is that the battery electrical or electrochemical models may not be unique and it is difficult to verify their validity.If the used cell model is not appropriate, the estimation accuracy may be lower and difficult to improve.In addition, the used algorithms are relatively complicated, which make it difficult to use them for online estimation.
In the non-physics-based model estimations, some learning algorithms, such as neural networks [32][33][34], support vector machines (SVMs) [35,36], RVMs [37], and fuzzy logic [38,39], are applied, in which the measured impedance parameters [12,32] and other characteristic battery parameters (e.g., discharge current [33], temperature, and state-of-charge (SOC) [34]) are employed as input variables.These methods can learn the battery behavior based on monitored data and thus do not demand battery physics models, but they need lots of training data and depend on the availability of a historic data set.In this case, it is difficult to use them for online estimations.Another kind of non-physics-based model estimation uses various curve equations between the practicable capacity and aging cycles [40][41][42][43][44], in which the parameters of the equations can be obtained by data fitting algorithms, and further they can be adjusted online by using a particle filtering (PF) approach [43] or by combining sets of training data based on Dempster-Shafer theory (DST) and the Bayesian Monte Carlo (BMC) method [44].This method also needs a lot of accelerated aging test data to determine the curve equations, but it does not need complicated mathematic computations.In addition, the number of aging cycles of a used battery may be unknown in practical applications.
The purpose of this work is to find a simple model for the SOH prediction of Ni-MH batteries to avoid the problems mentioned above, such as the uncertainty of battery physics models, complicated algorithms, comprehensive parameters, and unknown cycle numbers.Therefore we tried to construct a simple curve model without electrical or electrochemical battery models and cycle numbers.
In this work, the SOH is defined as Equation (1) due to the presence of charge-discharge efficiency: where C discharge (Ah) is the discharging capacity of fully charged Ni-MH batteries and C rated (Ah) is the rated capacity.As the C rated for one type of Ni-MH battery is a constant value, the SOH prediction is reduced to the prediction of C discharge .Based on the analysis of a lot of charge-discharge cycle test data for commercial Ni-MH batteries, the voltage change in 1 s occurring at the start of the charging process (∆V charge , mV) was selected as the characteristic parameter, and a curve equation between C discharge and ∆V charge was determined to construct a curve model for the C discharge prediction.Then the prediction precision of this curve model was verified by using some typical charge and discharge data for the Ni-MH batteries.

Experimental
Commercial AA-type Ni-MH cells (Pisen , Sichuan, China, C rated = 1.8 Ah and rated voltage = 1.2 V) were used in this study.All charge-discharge tests were conducted using a computer-controlled charge-discharge instrument (BTS-3008-5V3A, Xinwei, Guangzhou, China) at room temperature (20 ˘5 ˝C).The charge-discharge data, including current (I), potential (V), Energies 2015, 8, 12474-12487 time (t), and cycle number (n), were recorded automatically with the sampling frequency of 1 Hz.The charge-discharge rates have significant influence on battery performance, so three different charge-discharge rates (0.5C, 1.0C, and 1.6C) were selected to simulate low, medium, and high charge-discharge rate conditions.The charge-discharge test program is described as follows: The Ni-MH cells with initial SOC at 0% were fully charged at a constant current (I charge : 0.5C, 1.0C, and 1.6C), then after an interval of 600 s, they were discharged at the same constant current (I discharge : 0.5C, 1.0C, and 1.6C) to the cutoff voltage (0.9 V).After an interval of 1800 s, the next cycle of charge-discharge test was started.When the C discharge of the test cells fell to 80% of the initial capacity, the charge-discharge tests were terminated [45].The failure capacity is still defined as 0.8C rated in this paper for convenient applications.

Curve Modeling for C discharge
Figure 1 gives the typical I-t and V-t curves of two cycles for a Ni-MH battery in the charge-discharge test at 1.0C.In each cycle there are four abrupt voltage changes (∆V) occurring at the start/end of the charging and discharging processes, respectively.According to the equivalent circuit of the test cell [8,46], these four ∆V values in each cycle generally reflect the internal resistance (R internal ) of the test cell in different states, and there is little difference among their values.However, the measurement of ∆V is much easier than the test of R internal , so it is selected as studied parameter.Before the start of the charging process, the completely discharged test cell was in a relatively stable state after the recovery of 1800 s, so in this paper we select the ∆V in 1 s at the start of the charging process (∆V charge , mV) as the characteristic parameter to verify its relationship with the aging state of the test cell, as shown in Figure 1.
Energies 2015, 8 4 these four ΔV values in each cycle generally reflect the internal resistance (Rinternal) of the test cell in different states, and there is little difference among their values.However, the measurement of ΔV is much easier than the test of Rinternal, so it is selected as studied parameter.Before the start of the charging process, the completely discharged test cell was in a relatively stable state after the recovery of 1800 s, so in this paper we select the ΔV in 1 s at the start of the charging process (ΔVcharge, mV) as the characteristic parameter to verify its relationship with the aging state of the test cell, as shown in Figure 1.Through the analysis of the I-t and V-t data for each test battery, we obtained Cdischarge and ΔVcharge in each cycle.In order to observe the relationship of these three parameters, we made a three-dimensional (3D) diagram of (Cdischarge-ΔVcharge-n) for all the test Ni-MH batteries.Figure 2a shows a typical 3D diagram for a Ni-MH battery in the charge-discharge test at 1.0C and its two-dimensional (2D) projection diagrams (Cdischarge-n, ΔVcharge-n, Cdischarge-ΔVcharge).
From Figure 2a it can be seen that (Cdischarge; ΔVcharge; n) data points form a curve in the 3D space without obvious discrete point; suggesting that there would be nonlinear relationships between these three parameters.From the different 2D diagrams in Figure 2b-d; it can be seen more clearly that Cdischarge-n; ΔVcharge-n and Cdischarge-ΔVcharge curves all display good nonlinear relationships.The results in Figures 2b,c can be explained appropriately [41] and the nonlinear relationships between Cdischarge-n and ΔVcharge-n can be used to predict the state of Ni-MH batteries.However; sometimes the number of cycles for a Ni-MH battery is unknown.In this case; the nonlinear relationship between Cdischarge and ΔVcharge; as Through the analysis of the I-t and V-t data for each test battery, we obtained C discharge and ∆V charge in each cycle.In order to observe the relationship of these three parameters, we made a three-dimensional (3D) diagram of (C discharge -∆V charge -n) for all the test Ni-MH batteries.Figure 2a shows a typical 3D diagram for a Ni-MH battery in the charge-discharge test at 1.0C and its two-dimensional (2D) projection diagrams (C discharge -n, ∆V charge -n, C discharge -∆V charge ).
From Figure 2a it can be seen that (C discharge ; ∆V charge ; n) data points form a curve in the 3D space without obvious discrete point; suggesting that there would be nonlinear relationships between these three parameters.From the different 2D diagrams in Figure 2b-d; it can be seen more clearly that C discharge -n; ∆V charge -n and C discharge -∆V charge curves all display good nonlinear relationships.The 12476 Energies 2015, 8, 12474-12487 results in Figure 2b,c can be explained appropriately [41] and the nonlinear relationships between C discharge -n and ∆V charge -n can be used to predict the state of Ni-MH batteries.However; sometimes the number of cycles for a Ni-MH battery is unknown.In this case; the nonlinear relationship between C discharge and ∆V charge ; as shown in Figure 2d; is important for a curve model without cycle numbers.If we can obtain the relationship between C discharge and ∆V charge for a Ni-MH battery; we can obtain its C discharge through the measurement of its ∆V charge and then estimate its SOH with Equation (1).
Finally, we identified that Equation ( 2) can be used to describe the nonlinear relationship between ∆V charge (mV) and C discharge (Ah): where a (Ah), k (mV ´1) and c (mV) are constants related to test batteries.We found that k was negative with a small absolute value (|k|) and related to the charge-discharge rate.In the initial cycles, ∆V charge << c and therefore, if |k| is not very small, there is, exp [´k ˆ(∆V charge ´c)] Ñ 0 and a « C discharge .So a can be considered as a parameter related to the initial C discharge of the test battery when |k| is not very small, which may be near its rated capacity.When ∆V charge = c, C discharge = 0.5a.Thus, c can be considered as a parameter related to the ∆V charge when C discharge is at its 50% initial value.
Energies 2015, 8 5 where a (Ah), k (mV −1 ) and c (mV) are constants related to test batteries.We found that k was negative with a small absolute value (|k|) and related to the charge-discharge rate.In the initial cycles, ΔVcharge << c and therefore, if |k| is not very small, there is, exp [−k × (ΔVcharge − c)] → 0 and a ≈ Cdischarge.So a can be considered as a parameter related to the initial Cdischarge of the test battery when |k| is not very small, which may be near its rated capacity.When ΔVcharge = c, Cdischarge = 0.5a.Thus, c can be considered as a parameter related to the ΔVcharge when Cdischarge is at its 50% initial value.The raw data in Figure 2d was fitted with Equation ( 2) by a nonlinear least-squares method, and the fitted curve indicates that it fits the raw data well.In order to verify the validity of Equation ( 2), different (Cdischarge, ΔVcharge) raw data obtained from different Ni-MH batteries with different charge-discharge rates were fitted with Equation ( 2) by the nonlinear least-squares method.Figure 3 shows some typical (Cdischarge, ΔVcharge) raw data at different charge-discharge rates and their fitted curves with Equation (2).Table 1 lists the fitting parameters of a, c, and k for different Ni-MH cells at different rates, in which R 2 is a correlation coefficient and Prange is the selection range of initial values for each fitting parameter when using the nonlinear least-squares method to solve their values.The raw data in Figure 2d was fitted with Equation ( 2) by a nonlinear least-squares method, and the fitted curve indicates that it fits the raw data well.In order to verify the validity of Equation ( 2), different (C discharge , ∆V charge ) raw data obtained from different Ni-MH batteries with different charge-discharge rates were fitted with Equation ( 2) by the nonlinear least-squares method.
Energies 2015, 8, 12474-12487 Figure 3 shows some typical (C discharge , ∆V charge ) raw data at different charge-discharge rates and their fitted curves with Equation (2).Table 1 lists the fitting parameters of a, c, and k for different Ni-MH cells at different rates, in which R 2 is a correlation coefficient and P range is the selection range of initial values for each fitting parameter when using the nonlinear least-squares method to solve their values.
The results in Figure 3 and Table 1 prove that Equation (2) can well describe the relationship between C discharge and ∆V charge .From Table 1, it can be seen that at 0.5C and 1.0C the parameter a is approximate to the rated capacity of the test Ni-MH batteries (1.8 Ah), while at 1.6C it becomes larger than the rated capacity because of the too small |k| value.It should be noted that at 1.6C the too small |k| value makes |´k ˆ(∆V charge ´c)| << 1 and then, there is exp [´k ˆ(∆V charge ´c)] « 1 [k ˆ(∆V charge ´c)].In this case, the nonlinear relationship between C discharge and ∆V charge described with Equation ( 2) becomes an approximate linear relationship especially with an increase in ∆V charge , as shown in Figure 3.     Apparently, the values of the a, c, and k change with the charge-discharge rate.Table 1 shows that when the charge-discharge rate increases from 0.5C to 1.0C, the change of these three parameters is small, while at 1.6C the change of a and k is obvious.It may be speculated that the values of the three parameters just fluctuate slightly when the charge-discharge rate is in a certain range, such as 0.5C-1.0C.If so, this curve model should also be used in practical fields that the charge-discharge rates are not constant during applications.Further research will be conducted to verify the application of this curve model in practical fields.In this paper we just focus on the curve model itself and its SOH prediction accuracy.

State-of-Health (SOH) Prediction of Ni-MH Batteries Based on the Curve Model for C discharge
When the relationship between C discharge and ∆V charge of one battery is built as Equation ( 2), the C discharge can be predicted by measuring the ∆V charge of the battery and then the SOH can be obtained with Equation ( 1).So we used the (C discharge , ∆V charge ) raw date in Table 1 to check whether the curve model for C discharge can be used for the SOH prediction.
Theoretically, only three groups of (C discharge , ∆V charge ) data are needed to solve the three parameters in Equation ( 2).However, the measured raw data usually have some discrete points, as shown in Figures 2 and 3 so more groups of (C discharge , ∆V charge ) data are needed and the prediction error is inevitable.In order to evaluate the prediction error, the root-mean-square error (RMSE) is defined as Equation (3): where C test discharge,j is the test value for sample j; C discharge,j is the predicted value for sample j; and n is the total number of samples.
The prediction method is described as follows: (1) A certain number of (C discharge , ∆V charge ) data groups in continuous charge-discharge test cycles were selected to fit the parameters in Equation (2) using a nonlinear least-squares method to obtain Equation (2) for the test Ni-MH battery; (2) The predicted C discharge values were calculated with Equation (2) using the tested ∆V charge values apart from those used in Step 1; and (3) the predicted RMSE was checked with Equation (3) using the predicted C discharge values and the corresponding tested C discharge values.Through the predicted RMSE we can judge how large the prediction error is and whether the prediction is valid.
Using different numbers of (C discharge , ∆V charge ) data groups in different parts of the C discharge -∆V charge curve, as shown in Figure 4, may result in different prediction results.In this paper, the first cycle C discharge of a selected (C discharge , ∆V charge ) data group (C 0 discharge ) is described with the SOH value of the test battery, and a relative capacity drop (RCd) is defined to describe the variation range of C discharge in the selected (C discharge , ∆V charge ) data group, as presented in Equation ( 4): where C n discharge is the last cycle C discharge of the selected (C discharge , ∆V charge ) data group.Apparently, the RCd value determines the last (C discharge , ∆V charge ) data of the selected data group.Because we used all the (C discharge , ∆V charge ) data in continuous test cycles, a larger RCd value means more data in the selected data group.In this case, the SOH and RCd values for a selected data group can determine its position on the C discharge -∆V charge curve and also indicate the corresponding battery states.
Firstly, the effect of the number of used (C discharge , ∆V charge ) data groups, which is described by RCd, on the prediction of C discharge is investigated.Figure 4 shows a 3D diagram of (C discharge -∆V charge -n) for a Ni-MH battery in the charge-discharge test at 1.0C and the predicted results obtained with SOH = 97.38%and different RCds (0.5%, 1.0%, and 1.5%), in which the Energies 2015, 8, 12474-12487 corresponding cycle numbers of the selected data range and the prediction RMSE values are also listed.In Figure 4, the three data groups are selected from 90-138 cycles (48 data), 90-166 cycles (77 data), and 90-184 cycles (95 data), respectively.The first cycle of the three date groups is the same, so they have the same SOH value (97.38%).Their RCd values are calculated with Equation (4) using the C discharge of the 90th cycle (i.e., C 0 discharge ) and C 138 discharge , C 166 discharge , and C 184 discharge , respectively.It is seen that the prediction RMSE value decreases with an increase in RCd, suggesting that using more (C discharge , ∆V charge ) data groups may decrease the prediction error.
Firstly, the effect of the number of used (Cdischarge, ΔVcharge) data groups, which is described by RCd, on the prediction of Cdischarge is investigated.Figure 4 shows a 3D diagram of (Cdischarge-ΔVcharge-n) for a Ni-MH battery in the charge-discharge test at 1.0C and the predicted results obtained with SOH = 97.38%and different RCds (0.5%, 1.0%, and 1.5%), in which the corresponding cycle numbers of the selected data range and the prediction RMSE values are also listed.In Figure 4, the three data groups are selected from 90-138 cycles (48 data), 90-166 cycles (77 data), and 90-184 cycles (95 data), respectively.The first cycle of the three date groups is the same, so they have the same SOH value (97.38%).Their RCd values are calculated with Equation (4) using the Cdischarge of the 90th cycle (i.e.,  The SOH value of the selected data group, i.e., the position of the starting point of the selected data group, may also influence the prediction result.So we selected data groups at different SOH with various RCds (0.5%, 1.0%, 1.5%, and 2.0%) to fit the parameters in Equation ( 2) and calculated the corresponding prediction RMSE values, as shown in Figure 5.
In Figure 5, the average RMSE value at each SOH is presented and shown with a star symbol, while the general average RMSE value ( RMSE ) is also given and shown with a horizontal dashed line.The RMSE value obtained by using data groups at SOH = 94.89% with RCd = 0.5% deviates significantly The SOH value of the selected data group, i.e., the position of the starting point of the selected data group, may also influence the prediction result.So we selected data groups at different SOH with various RCds (0.5%, 1.0%, 1.5%, and 2.0%) to fit the parameters in Equation ( 2) and calculated the corresponding prediction RMSE values, as shown in Figure 5.
In Figure 5, the average RMSE value at each SOH is presented and shown with a star symbol, while the general average RMSE value (RMSE) is also given and shown with a horizontal dashed line.The RMSE value obtained by using data groups at SOH = 94.89% with RCd = 0.5% deviates significantly from other RMSE values, which may be due to the fluctuation of the raw data groups, so it was eliminated from the calculation of the average RMSE values.From Figure 5, it can be seen that when SOH = 97.82%, the prediction RMSE decreases obviously with an increase in RCd, which is consistent with the result in Figure 4.But when the SOH value decreases to 97.12%, 96.77%, 94.89%, and 94%, at each SOH the prediction RMSE does not decrease with an increase in RCd, suggesting that using more data groups does not necessarily help improve the prediction precision.In addition, the prediction RMSE does not show some regular change with a decrease in the SOH, but generally it seems that when the SOH is in an intermediate range (97.12% and 96.77%), the prediction RMSE is relatively lower.
Figure 6 shows the selected (Cdischarge, ΔVcharge) raw data in Figure 5 and the prediction data with the best RMSE (1.2%) and the RMSE (1.7%) near the RMSE in Figure 5 (1.75%).
From Figure 6, we can clearly see that the prediction data well accords with the raw data, especially when Cdischarge > 1.44 Ah, i.e., before the failure capacity (0.8Crated), indicating that the prediction accuracy of this curve model is satisfied.It should be noted that the initial part of the raw data in Figure 6 (SOH > 97.12%) has some volatility.So using this part of the raw data for the Cdischarge prediction may result in higher prediction error (i.e., large RMSE) and using more data groups (i.e., large RCd) may improve the prediction precision, as shown in Figure 5 at SOH = 97.82%.When the SOH decreases to 97.12%-96.77%, the fluctuation of the raw data decreases, as shown in Figure 6; this may result in lower prediction RMSE, as shown in Figure 5.
Considering the influence of different charge-discharge rates on the prediction results, we selected the (Cdischarge, ΔVcharge) raw data at 0.5C and 1.6C to check the curve model using the same method described above.Figure 7 shows the prediction RMSE for a Ni-MH battery in the charge-discharge test From Figure 5, it can be seen that when SOH = 97.82%, the prediction RMSE decreases obviously with an increase in RCd, which is consistent with the result in Figure 4.But when the SOH value decreases to 97.12%, 96.77%, 94.89%, and 94%, at each SOH the prediction RMSE does not decrease with an increase in RCd, suggesting that using more data groups does not necessarily help improve the prediction precision.In addition, the prediction RMSE does not show some regular change with a decrease in the SOH, but generally it seems that when the SOH is in an intermediate range (97.12% and 96.77%), the prediction RMSE is relatively lower.
Figure 6 shows the selected (C discharge , ∆V charge ) raw data in Figure 5 and the prediction data with the best RMSE (1.2%) and the RMSE (1.7%) near the RMSE in Figure 5 (1.75%).
From Figure 6, we can clearly see that the prediction data well accords with the raw data, especially when C discharge > 1.44 Ah, i.e., before the failure capacity (0.8C rated ), indicating that the prediction accuracy of this curve model is satisfied.It should be noted that the initial part of the raw data in Figure 6 (SOH > 97.12%) has some volatility.So using this part of the raw data for the C discharge prediction may result in higher prediction error (i.e., large RMSE) and using more data groups (i.e., large RCd) may improve the prediction precision, as shown in Figure 5 at SOH = 97.82%.When the SOH decreases to 97.12%-96.77%, the fluctuation of the raw data decreases, as shown in Figure 6; this may result in lower prediction RMSE, as shown in Figure 5.
Considering the influence of different charge-discharge rates on the prediction results, we selected the (C discharge , ∆V charge ) raw data at 0.5C and 1.6C to check the curve model using the same method described above.Figure 7 shows the prediction RMSE for a Ni-MH battery in the charge-discharge test at 0.5C using (C discharge , ∆V charge ) data groups at different SOH with various RCds, in which the average prediction RMSE values are also given as described in Figure 5.The RMSE values at SOH = 98.97% are relative large, which may also result from the fluctuation in the initial data groups, so they are eliminated from the calculation of the RMSE in Figure 7.
Energies 2015, 8, 12474-12487 at 0.5C using (Cdischarge, ΔVcharge) data groups at different SOH with various RCds, in which the average prediction RMSE values are also given as described in Figure 5.The RMSE values at SOH = 98.97% are relative large, which may also result from the fluctuation in the initial data groups, so they are eliminated from the calculation of the RMSE in Figure 7.  From Figure 7, it can be seen that at SOH = 98.97% the prediction RMSE is higher and does not decrease with an increase in RCd, while at lower SOH values (98.02%-94.45%) the prediction RMSE decreases with an increase in RCd.In general, it seems that when the SOH is in an intermediate range Energies 2015, 8 10 at 0.5C using (Cdischarge, ΔVcharge) data groups at different SOH with various RCds, in which the average prediction RMSE values are also given as described in Figure 5.The RMSE values at SOH = 98.97% are relative large, which may also result from the fluctuation in the initial data groups, so they are eliminated from the calculation of the RMSE in Figure 7.  From Figure 7, it can be seen that at SOH = 98.97% the prediction RMSE is higher and does not decrease with an increase in RCd, while at lower SOH values (98.02%-94.45%) the prediction RMSE decreases with an increase in RCd.In general, it seems that when the SOH is in an intermediate range From Figure 7, it can be seen that at SOH = 98.97% the prediction RMSE is higher and does not decrease with an increase in RCd, while at lower SOH values (98.02%-94.45%) the prediction RMSE decreases with an increase in RCd.In general, it seems that when the SOH is in an intermediate range (98.02% and 97.12%), the prediction RMSE is relatively lower, which is consistent with the results in Figure 5.
Figure 8 presents the (C discharge , ∆V charge ) raw data used in Figure 7 and the prediction data with the best RMSE (1.2%) and the RMSE (2.1%) near the RMSE in Figure 7 (2.23%).Similarly, there is fluctuation in the initial part of the (C discharge , ∆V charge ) raw data and the prediction data well accords with the raw data before the failure capacity. Figure 8 presents the (Cdischarge, ΔVcharge) raw data used in Figure 7 and the prediction data with the best RMSE (1.2%) and the RMSE (2.1%) near the RMSE in Figure 7 (2.23%).Similarly, there is fluctuation in the initial part of the (Cdischarge, ΔVcharge) raw data and the prediction data well accords with the raw data before the failure capacity. Figure 9 shows the prediction RMSE for a Ni-MH battery in the charge-discharge test at 1.6C, in which the average prediction RMSE values are also given as described in Figure 5. Similarly, the RMSE values at SOH = 95% are eliminated from the calculation of the RMSE value in Figure 9.
Figure 10 presents the (Cdischarge, ΔVcharge) raw data used in Figure 9 and the prediction data with the best RMSE (1.2%) and the RMSE (1.6%) near the RMSE in Figure 9 (1.53%).It can be seen that there is an obvious increase in the Cdischarge value in the initial charge-discharge period, as shown in Figure 10, so these data groups cannot be used for the prediction.As shown in Figure 9, only when the SOH < 95%, the prediction RMSE decreases to a lower value and the prediction data also accords well with the raw data before the failure capacity, as shown in Figure 10.
The results in Figures 4-10 indicate that the curve model based on Equation ( 2) can be effectively applied to predict the SOH of the Ni-MH batteries, and the best prediction RMSE is around 1.2%.The selected raw data groups for the prediction significantly influence the prediction RMSE.Generally, the data groups in the early period of the charge-discharge test, which depends on the charge-discharge rate, are not suitable for the SOH prediction because of data fluctuations.Increasing the RCd value, i.e., using more data groups for the SOH prediction, does not necessarily help improve the prediction precision, which may also be related to the data fluctuation.Generally, using data groups at the SOH in Figure 9 shows the prediction RMSE for a Ni-MH battery in the charge-discharge test at 1.6C, in which the average prediction RMSE values are also given as described in Figure 5. Similarly, the RMSE values at SOH = 95% are eliminated from the calculation of the RMSE value in Figure 9.
Figure 10 presents the (C discharge , ∆V charge ) raw data used in Figure 9 and the prediction data with the best RMSE (1.2%) and the RMSE (1.6%) near the RMSE in Figure 9 (1.53%).It can be seen that there is an obvious increase in the C discharge value in the initial charge-discharge period, as shown in Figure 10, so these data groups cannot be used for the prediction.As shown in Figure 9, only when the SOH < 95%, the prediction RMSE decreases to a lower value and the prediction data also accords well with the raw data before the failure capacity, as shown in Figure 10.
The results in Figures 4-10 indicate that the curve model based on Equation ( 2) can be effectively applied to predict the SOH of the Ni-MH batteries, and the best prediction RMSE is around 1.2%.The selected raw data groups for the prediction significantly influence the prediction RMSE.Generally, the data groups in the early period of the charge-discharge test, which depends on the charge-discharge rate, are not suitable for the SOH prediction because of data fluctuations.Increasing the RCd value, i.e., using more data groups for the SOH prediction, does not necessarily help improve the prediction precision, which may also be related to the data fluctuation.Generally, using data groups at the SOH in an intermediate range and with the RCd value in the range of 1.5%-2.0%may result in lower prediction RMSE.Apart from good prediction accuracy, this curve model is also simple and easy-to-use.These advantages are important for online applications.For any type of Ni-MH battery, we can use charge-discharge cycle tests to determine the three parameters (a, c, and k) in the curve model at different charge-discharge rates.Then, we can easily detect the SOH of the same type Ni-MH batteries without considering their aging history (e.g., cycle numbers).As shown in Table 1, the three parameters for different batteries change a little in a certain range of charge-discharge rates, such as 0.5-1.0C.In this   Apart from good prediction accuracy, this curve model is also simple and easy-to-use.These advantages are important for online applications.For any type of Ni-MH battery, we can use charge-discharge cycle tests to determine the three parameters (a, c, and k) in the curve model at different charge-discharge rates.Then, we can easily detect the SOH of the same type Ni-MH batteries without considering their aging history (e.g., cycle numbers).As shown in Table 1, the three parameters for different batteries change a little in a certain range of charge-discharge rates, such as 0.5-1.0C.In this Apart from good prediction accuracy, this curve model is also simple and easy-to-use.These advantages are important for online applications.For any type of Ni-MH battery, we can use charge-discharge cycle tests to determine the three parameters (a, c, and k) in the curve model at different charge-discharge rates.Then, we can easily detect the SOH of the same type Ni-MH batteries without considering their aging history (e.g., cycle numbers).As shown in Table 1, the three parameters for different batteries change a little in a certain range of charge-discharge rates, such as 0.5-1.0C.In this case, it is possible to use their average values to construct a general model for the applications with fluctuated charge-discharge rates.Moreover, if more than three (C discharge , ∆V charge ) data can be obtained during applications, the new values of the three parameters (a, c, and k) can be calculated to modify the general curve model.Certainly, this application of the curve model is needed further verification.
One of limitations for this curve model is that it's relatively higher prediction error in a higher SOH range especially at higher charge-discharge rates.In addition, it should be noted that in this

Figure 1 .
Figure 1.Typical I-t and V-t curves in the charge-discharge test at 1.0C for Ni-MH batteries (AA-type, 1.8 Ah, 1.2 V).

Figure 1 .
Figure 1.Typical I-t and V-t curves in the charge-discharge test at 1.0C for Ni-MH batteries (AA-type, 1.8 Ah, 1.2 V).

Energies 2015, 8 6
capacity because of the too small |k| value.It should be noted that at 1.6C the too small |k| value makes |−k × (ΔVcharge − c)| << 1 and then, there is exp [−k × (ΔVcharge − c)] ≈ 1 − [k × (ΔVcharge − c)].In this case, the nonlinear relationship between Cdischarge and ΔVcharge described with Equation (2) becomes an approximate linear relationship especially with an increase in ΔVcharge, as shown in Figure3.

Figure 3 .
Figure 3. (C discharge , ∆V charge ) raw data at different rates and their fitting curves with Equation (2).
It is seen that the prediction RMSE value decreases with an increase in RCd, suggesting that using more (Cdischarge, ΔVcharge) data groups may decrease the prediction error.

Energies 2015, 8 ,
12474-12487Energies 2015, 8 9from other RMSE values, which may be due to the fluctuation of the raw data groups, so it was eliminated from the calculation of the average RMSE values.

Figure 5 .
Figure 5. Prediction root-mean-square error (RMSE) for a Ni-MH battery in the charge-discharge test at 1.0C using (Cdischarge, ΔVcharge) data groups at different SOH with various RCds.

Figure 5 .
Figure 5. Prediction root-mean-square error (RMSE) for a Ni-MH battery in the charge-discharge test at 1.0C using (C discharge , ∆V charge ) data groups at different SOH with various RCds.

Figure 7 .
Figure 7. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 0.5C using (Cdischarge, ΔVcharge) data groups at different state-of-health (SOH) with different RCds.

Figure 6 .
Figure 6.(C discharge , ∆V charge ) raw data used in Figure 5 and the prediction data with the best RMSE (1.2%) and the RMSE (1.7%) near the RMSE in Figure 5 (1.75%).

Figure 7 .
Figure 7. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 0.5C using (Cdischarge, ΔVcharge) data groups at different state-of-health (SOH) with different RCds.

Figure 7 .
Figure 7. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 0.5C using (C discharge , ∆V charge ) data groups at different state-of-health (SOH) with different RCds.

Energies 2015, 8 ,
Figure 5.Figure8presents the (Cdischarge, ΔVcharge) raw data used in Figure7and the prediction data with the best RMSE (1.2%) and the RMSE (2.1%) near the RMSE in Figure7(2.23%).Similarly, there is fluctuation in the initial part of the (Cdischarge, ΔVcharge) raw data and the prediction data well accords with the raw data before the failure capacity.

Figure 8 .
Figure 8. (C discharge , ∆V charge ) raw data used in Figure 7 and the prediction data with the best RMSE (1.2%) and the RMSE (2.1%) near the RMSE in Figure 7 (2.23%).

Energies 2015, 8 ,
12474-12487Energies 2015,8  12    an intermediate range and with the RCd value in the range of 1.5%-2.0%may result in lower prediction RMSE.

Figure 9 .
Figure 9. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 1.6C using (Cdischarge, ΔVcharge) data groups at different SOH with various RCds.

Figure 9 .
Figure 9. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 1.6C using (C discharge , ∆V charge ) data groups at different SOH with various RCds.

Figure 9 .
Figure 9. Prediction RMSE for a Ni-MH battery in the charge-discharge test at 1.6C using (Cdischarge, ΔVcharge) data groups at different SOH with various RCds.

Figure 10 .
Figure 10.(C discharge , ∆V charge ) raw data used in Figure 9 and the prediction data with the best RMSE (1.2%) and the RMSE (1.6%) near the RMSE in Figure 9 (1.52%).

Table 1 .
Fitting parameters of a, c, and k for different Ni-MH cells at different rates.Apparently, the values of the a, c, and k change with the charge-discharge rate.Table1shows that

Table 1 .
Fitting parameters of a, c, and k for different Ni-MH cells at different rates.