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A Fuzzy State-of-Charge Estimation Algorithm Combining Ampere-Hour and an Extended Kalman Filter for Li-Ion Batteries Based on Multi-Model Global Identification
A Fuzzy State-of-Charge Estimation Algorithm Combining Ampere-Hour and an Extended Kalman Filter...
Lai, Xin;Qiao, Dongdong;Zheng, Yuejiu;Zhou, Long
2018-10-23 00:00:00
applied sciences Article A Fuzzy State-of-Charge Estimation Algorithm Combining Ampere-Hour and an Extended Kalman Filter for Li-Ion Batteries Based on Multi-Model Global Identification Xin Lai * , Dongdong Qiao, Yuejiu Zheng * and Long Zhou College of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China; qiaodongdong_usst@126.com (D.Q.); 13917775329@126.com (L.Z.) * Correspondence: laixin@usst.edu.cn (X.L.); yuejiu.zheng@usst.edu.cn (Y.Z.) Received: 25 September 2018; Accepted: 21 October 2018; Published: 23 October 2018 Featured Application: The proposed fuzzy state-of-charge (SOC) estimation algorithm, combining ampere-hour and an extended Kalman filter, can be utilized as an effective SOC estimation algorithm, with high accuracy and robustness, in the whole SOC area. The proposed algorithm helps to overcome the shortcomings of commonly used SOC estimation methods, which are based on ideal conditions of no or very small model and sensor errors, and thus encounter problems in practical applications where such errors are significant, especially due to the ageing of the battery or its subjection to adverse temperatures. Abstract: The popular and widely reported lithium-ion battery model is the equivalent circuit model (ECM). The suitable ECM structure and matched model parameters are equally important for the state-of-charge (SOC) estimation algorithm. This paper focuses on high-accuracy models and the estimation algorithm with high robustness and accuracy in practical application. Firstly, five ECMs and five parameter identification approaches are compared under the New European Driving Cycle (NEDC) working condition in the whole SOC area, and the most appropriate model structure and its parameters are determined to improve model accuracy. Based on this, a multi-model and multi-algorithm (MM-MA) method, considering the SOC distribution area, is proposed. The experimental results show that this method can effectively improve the model accuracy. Secondly, a fuzzy fusion SOC estimation algorithm, based on the extended Kalman filter (EKF) and ampere-hour counting (AH) method, is proposed. The fuzzy fusion algorithm takes advantage of the advantages of EKF, and AH avoids the weaknesses. Six case studies show that the SOC estimation result can hold the satisfactory accuracy even when large sensor and model errors exist. Keywords: state-of-charge; equivalent circuit model; parameter identification; fuzzy fusion; multi-model combination 1. Introduction Due to increasing concerns about global warming, greenhouse gas emissions, and the depletion of fossil fuels, electric vehicles (EVs) have gained massive popularity due to their performances and efficiencies in recent decades [1–3]. Lithium-ion batteries (LIBs) are widely used in EVs for their high-energy density, long service life, and environmental friendliness [4–6]. In actual practice, LIBs need to be well monitored, diagnosed, and controlled by the battery management system (BMS). The accurate state estimation of a battery is one of the most fundamental functions of the BMS. The states of the battery include state-of-charge (SOC), state-of-health (SOH), and state-of-function Appl. Sci. 2018, 8, 2028; doi:10.3390/app8112028 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2028 2 of 19 (SOF) etc. [7,8]. Among these states, SOC in BMS is considered as one of the critical and important factors [9,10], which have been researched in recent decades. Accurate battery state estimation can contribute to reasonable management of batteries to fully exploit them and prolong their lifespan. However, the battery states are non-measurable variables, which can only be indirectly estimated via measurable external characteristic parameters (e.g., voltage, current, and temperature). Owing to the high nonlinearity of the battery system itself, accurate estimation of the battery state is fairly difficult. 1.1. Literature Review With the development of BMS, a large number of SOC estimation methods have been proposed and each one has its own merits and limitations [4,11]. The commonly used methods in the literature can be listed as follows [12,13]: (1) ampere-hour (AH) counting method; (2) open-circuit voltage (OCV) method; (3) model-based method; and (4) neural-network model method. Among these methods, the AH method is most extensively used owing to its ease of implementation. However, it suffers from the drawbacks of accumulated errors. Another classic method is the model-based method, where the Kalman filter (KF) algorithm is generally used [1,14]. Due to its high applicability and accuracy, various forms of the KF algorithm, such as the linear KF (LKF), single KF (SKF), unscented KF (UKF), sigma point KF (SPKF), and extended KF (EKF), have been intensively investigated in recent years [15]. In the practical application of SOC estimation, many factors affect its accuracy. These factors mainly include model and sensor errors [16]. The sensor errors include the voltage and current sensor errors in the BMS, while the model errors include the battery capacity, coulombic efficiency, and ECM errors. The model errors increase with LIB ageing and exposure to adverse temperatures. These errors increase the SOC estimation error. Therefore, the influence of these factors on SOC must be considered to improve the robustness of the SOC estimation algorithm in the actual BMS. The model-based estimator is widely used in the existing SOC estimation. Therefore, an accurate battery model is the prerequisite for precise SOC estimation. The equivalent circuit model (ECM) has the advantages of simple structure, small number of identification parameters, and high accuracy, which have made it become the most popular battery model for existing BMSs [17]. At present, the ECMs mainly include the resistance-capacitance model with different orders (nRC, n = 0, 1, 2, etc.), GNL (general nonlinear) model, and PNGV (the Partnership for a New Generation of Vehicle) model, etc., and the most commonly used model is nRC [18,19]. The characteristics and accuracy of the ECM are affected by the structure and parameters of the model. Therefore, a reasonable model structure and matched model parameters are indispensable for the accurate SOC estimation. It is inappropriate to apply simple parameter identification algorithms to complex models nor complex parameter identification algorithms to simple models. In other words, high model accuracy can only be achieved by using a suitable battery model and matched model parameters at the same time. In this regard, the parameter identification approach and the model structure are equally important. To the best of our knowledge, the parameter identification of ECMs is an optimization problem, and there are many identification algorithms to determine the battery model parameters in the literature, such as the genetic algorithm [20], the particle swarm optimization algorithm [21], and the least-squares method [22]. Previous studies on LIB models have not paid much attention to whether a parameter identification approach is suitable for a certain model. In this paper, we investigated the accuracy of various models in various identification algorithms to determine the most appropriate model structure and parameter identification algorithm for the whole SOC area. 1.2. Main Contributions In order to improve the accuracy and robustness of the SOC estimation algorithm, based on ECMs in the whole SOC area, this paper aims to make three contributions: (1) By comparing and analyzing nine models and five commonly used parameter identification algorithms, the most suitable ECM and parameter identification algorithm are decided. Appl. Sci. 2018, 8, 2028 3 of 19 (2) The whole SOC area is divided into the high SOC area and low SOC area. Different ECMs and parameter identification algorithms are adopted considering SOC distribution. Based on this, a multi-model and multi-algorithm method is developed to fit the battery model. Experimental results show that the proposed composite model has higher model accuracy compared with a single model. (3) According to the error characteristics of EKF and AH, a fuzzy fusion SOC estimation algorithm, combining AH and EKF in the whole SOC area, is proposed, and the accuracy and robustness of the proposed algorithm are verified by six cases. 1.3. Organization of the Paper The remainder of this paper is organized as follows. Section 2 describes the experimental equipment and the curves of current and voltage variation under the New European Driving Cycle (NEDC) working condition in the whole SOC area. Section 3 introduces five ECMs and five parameter identification approaches in five categories. The most suitable model and parameter identification algorithm are selected through comparative study. In Section 4, a fuzzy fusion algorithm, based on the EKF and AH, is proposed. In Section 5, the results of SOC estimation are analyzed and discussed. Finally, Section 6 states the conclusions. 2. Experiments A commercial prismatic pouch lithium-ion cell was tested in this study. The cathode material of the cell was LiNi Co Mn O (NMC). The basic parameters of the cell are listed in Table 1. x y 1 x y 2 The experiments were conducted in a test bench made by DIGATRON, which had a current range of 100 to +100 A and a voltage range of 0 to 20 V. The tested cell was put into a temperature chamber to keep the ambient temperature. The temperature chamber was made by Dongguan Bell Company and the type was TEMI580. Firstly, the capacity test was designed to find the standard capacity of the test cell (Cell#1). The capacity test process was as follows: Cell#1 was placed in the temperature chamber at 25 C for 3 h. Then, Cell#1 was discharged at a constant discharge current (1/3 C) to 2.5 V. After waiting for 1 h, Cell#1 was fully charged using the constant current-constant voltage method. This process was repeated three times, and the average value of the test capacity was taken as the standard capacity of the test cell. Secondly, the hybrid pulse power characterization (HPPC) test was carried out to find the open circuit voltage (OCV). The test steps can be found in References. [2,6], specific steps are not listed here for brevity. Finally, Cell#1 was fully charged at 1/3 C. Then, the discharge process was performed under the NEDC cycles. PC software (BTS-600) was used to record the voltage and current of the cell in real time. Figure 1b,c show these test data, which were used for the model parameter identification and the SOC estimation. Table 1. Main parameters of the test cell (Cell#1). Nominal Nominal Lower Cut-Off Upper Cut-Off Maximum Charge Capacity (Ah) Voltage (V) Voltage (V) Voltage (V) Current (A) 32.5 3.75 2.5 4.15 65 Appl. Sci. 2018, 8, 2028 4 of 20 Appl. Sci. 2018, 8, 2028 4 of 19 Appl. Sci. 2018, 8, 2028 4 of 20 (a) (b) Thermal Chamber (b) (a) Digatron tester Signal lines Battery Digatron tester Thermal Chamber Signal lines Battery T TC CP P/ /I IP P PC T TC CP P/ /I IP P PC Power connect (c) Pow T eherm r connect al C hamber (c) Battery Tester Thermal Chamber Battery Tester PC PC Figure 1. Battery test bench and experimental results under the New European Driving Cycle (NEDC) working cycles. (a) Schematic of the test bench; (b) current profile; and (c) voltage profile. Figu Figure re 1. 1. Battery Batterytest test bench bench and and experimental experimental results results under un the der New the Eur New opean Euro Driving pean Dr Cycle iving (NEDC) Cycle working cycles. (a) Schematic of the test bench; (b) current profile; and (c) voltage profile. (NEDC) working cycles. (a) Schematic of the test bench; (b) current profile; and (c) voltage profile. 3. Model and Parameter Identification 3. Model and Parameter Identification 3. Model and Parameter Identification 3.1. Equivalent Circuit Models 3.1. Equivalent Circuit Models 3.1. Equivalent In this study, Circuwe it Model chose s the popular and widely reported nRC, nRCH, and PNGV models as In this study, we chose the popular and widely reported nRC, nRCH, and PNGV models as battery battery models. The schematics of the three models is shown in Figure 2. In these models, U OCV In this study, we chose the popular and widely reported nRC, nRCH, and PNGV models as models. The schematics of the three models is shown in Figure 2. In these models, U denotes denotes battery voltage source, is the terminal voltage, denotes equivO alC ent V ohmic U R L 0 battery models. The schematics of the three models is shown in Figure 2. In these models, OCV battery voltage source, U is the terminal voltage, R denotes equivalent ohmic resistance, and R and L 0 i resistance, and R and C denote diffusion resistance and diffusion capacitance, respectively. u i i h,k denotes battery voltage source, U is the terminal voltage, R denotes equivalent ohmic L 0 C denote diffusion resistance and diffusion capacitance, respectively. u is the hysteresis voltage, i h,k is the hysteresis voltage, is a decaying factor, and M is the maximum amount of hysteresis resistance, and and de p note diffusion resistance and diffusion capacitance, respectively. R C u k is a decaying factor i , andi M is the maximum amount of hysteresis voltage. C in the PNGV model h,k p b voltage. in the PNGV model is an equivalent capacitance, which describes the changes in the iis s an the equivalent hysteresis capacitance, voltage, which is a de describes caying factor the changes , and M inis the the OCV max caused imum by amount time accumulation. of hysteresis OCV caused by time accumulation. The equations of ECMs are presented in Table 2. The equations of ECMs are presented in Table 2. voltage. in the PNGV model is an equivalent capacitance, which describes the changes in the OCV caused by time accumulation. The equations of ECMs are presented in Table 2. Figure 2. Schematics of equivalent circuit models (ECMs): (a) Resistance-capacitance (nRC) model and Figure 2. Schematics of equivalent circuit models (ECMs): (a) Resistance-capacitance (nRC) model (b) the Partnership for a New Generation of Vehicle (PNGV) model. and (b) the Partnership for a New Generation of Vehicle (PNGV) model. Figure 2. Schematics of equivalent circuit models (ECMs): (a) Resistance-capacitance (nRC) model and (b) the Partnership for a New Generation of Vehicle (PNGV) model. Appl. Sci. 2018, 8, 2028 5 of 19 Table 2. Equations of the ECMs. Models Equations U (k) = U (k) + I(k)R + U (k) < å L OCV 0 i i=1 nRC T T s s t t i i U (k + 1) = U (k)e + I(k)R 1 e i i i U (k) = U (k) + I(k)R + U (k) + u > å L OCV 0 i h,k i=1 T T s s t t i i U (k + 1) = U (k)e + I(k)R 1 e i i i nRCH M I k 0 > ( ) jk I(k)tj > p : u = H 1 e , H = h,k M I k > 0 ( ) > U (k) = U (k) + I(k)R + U (k) + U (k) L OCV 0 1 cb > T T s s t t 1 1 U (k + 1) = U (k)e + I(k)R 1 e 1 1 1 PNGV > s > t U k + 1 = U k + I k 1 e : ( ) ( ) ( ) cb cb 3.2. Optimization Variables and the Objective Function for ECMs In the nRC, nRCH, and PNGV, the model parameters should be identified optimally. The model parameters to be determined by optimization algorithm can be expressed as: " # R R t R t R t R 1 1 2 2 n n > 0 0 (if the model is nRC) | {z } | {z } | {z } 1st RC > 2nd RC n th RC > 2 3 k M V (1) j R R t R t R t R | {z } 4 n n 5 1 1 2 2 0 0 (if the model is nHRC) | {z } | {z } | {z } > hysteresis 1st RC 2nd RC n th RC h i R R t R C (if the model is PNGV) 1 1 b 0 0 where R and R represent the ohmic resistance of charging and discharging, respectively. 0 0 From (1), it was seen that the number of parameters to be identified increased with the order of ECM. For the 4RC model, the number of parameters to be identified can reach up to 10. In this study, the root-mean-square error (RMSE) between the model terminal voltage and the measured terminal voltage was used to evaluate the fitness of model parameters. Correspondingly, the fitness function can be expressed as: M (q) = (u (q) u ˆ (q)) (2) EMSE å i,k i,k k=1 where M represents the RMSE of the ECM, u represents the voltage of ECM, and u ˆ represents RMSE i,k i,k the test voltage. 3.3. Moth-Flame Optimization Algorithm The moth-flame optimization (MFO) algorithm is a nature-inspired optimization proposed by Seyedali Mirjalili in 2015 [23]. Many examples show that the MFO algorithm has the advantages of strong convergence, fast convergence, and wide application, and it is suitable for solving the problem of high dimensional optimization. In the MFO algorithm, it was assumed that the candidate solutions were moths and the problem’s variables were the position of moths in the space. Other key components in the MFO were flames. The set of moths and flames are represented in the following matrices: Appl. Sci. 2018, 8, 2028 6 of 20 Appl. Sci. 2018, 8, 2028 6 of 19 M M M F F F 1,1 1, 2 1,d 1,1 1, 2 1,d 2 3 2 3 M M M F F F M 2M ,1 2 , 2 M 2 ,d 2F ,1 2 ,F 2 2 ,d F 1,1 1,2 1,d 1,1 1,2 1,d M , F (3) 6 7 6 7 M M M F F F 6 7 6 7 2,1 2,2 2,1 2,2 2,d 2,d 6 7 6 7 M = , F = (3) . . . . . . 6 M M M 7 6F F F 7 . n ,1 . n , 2 . n ,d n ,1 . n , 2 . n ,d . 4 . . . 5 4 . . . 5 M M M F F F n,1 n,2 n,d n,1 n,2 n,d where n is the number of moths and flames, and d is the number of variables. For all the moths and flames, we assumed that there were two arrays for storing the where n is the number of moths and flames, and d is the number of variables. corresponding fitness values, as follows: For all the moths and flames, we assumed that there were two arrays for storing the corresponding fitness values, as follows: 2 3 2 3 OM OF 1 1 OM OF 1 1 OM OF 6 7 2 6 2 7 OM OM , OF OF 6 7 6 7 (4) 2 2 6 7 6 7 OM = , OF = (4) . . 6 7 6 7 . . 4 . 5 4 . 5 OM OF n n OM OF n n In the MFO, a random initial solution to calculate the objective function values can be described In the MFO, a random initial solution to calculate the objective function values can be described as follows: as follows: M ub lb rand lb (5) M i= , j (ub i lbi)rand( ) + lbi (5) i,j i i i where ub and lb are the upper and lower bounds of the variables. is a random function. where ub and lb are the upper and lower bounds of the variables. randr(an)dis a random function. i i i i The position of each moth was updated with respect to a flame using the following equation: The position of each moth was updated with respect to a flame using the following equation: bt bt M S M ,F F -M e cos 2t F M = S M , F = F M e cos(2t) + F ((6) 6) i i j j i j i i j j i j where b is a constant, and t is a random number in [−1, 1]. S is the spiral function. where b is a constant, and t is a random number in [ 1, 1]. S is the spiral function. Figure 3 shows the flow chart of the MFO algorithm. The terminating conditions included the Figure 3 shows the flow chart of the MFO algorithm. The terminating conditions included the maximum iteration times and the solution accuracy to meet the setup requirements. maximum iteration times and the solution accuracy to meet the setup requirements. In order to verify the performance of the MFO algorithm and select the most suitable In order to verify the performance of the MFO algorithm and select the most suitable identification identification algorithm, five well-known algorithms in five categories, as shown in Table 3, were algorithm, five well-known algorithms in five categories, as shown in Table 3, were chosen to identify chosen to identify the parameters of the ECMs shown in Table 2. For concise description, the other the parameters of the ECMs shown in Table 2. For concise description, the other algorithms are not algorithms are not described here. described here. Start Update the positions of the moths ... Moth 1 Moth 2 Moth n Store the best positions to the flames Initialization of the moth populations using Eq.(10) Meet the terminating conditions? NO Obtaining the initial fitness value Re-search YES around the Output the global optimal flames Search solution solution with a logarithmic spiral using Eq.(11) End Figure 3. Flow chart of the moth-flame optimization (MFO) algorithm. Figure 3. Flow chart of the moth-flame optimization (MFO) algorithm. Appl. Sci. 2018, 8, 2028 7 of 19 Table 3. Optimization algorithms. Method Type Algorithm Name Inspiration Year of Proposal Nonlinear programming Find minimum of constrained nonlinear (FMIN) N/A 1951 Evolution-based Genetic Algorithm (GA) [20,24] Biological evolution 1992 Physics-based Simulated annealing algorithm (SA) [25] Solid annealing 1983 Swarm-based Particle Swarm optimization (PSO) [26] Bird flock 1995 Nature-inspired Moth-flame optimization (MFO) [23] Moth 2015 Appl. Sci. 2018, 8, 2028 7 of 20 Table 3. Optimization algorithms. Moreover, the whole SOC (0–100%) was divided into 10 subarea, and the parameters in each Method Type Algorithm Name Inspiration Year of Proposal subarea were identified by the above optimization algorithms, respectively. Therefore, 10 groups of Nonlinear programming Find minimum of constrained nonlinear (FMIN) N/A 1951 model parameters, which vary with SOC, were obtained as the model parameters of ECM. Evolution-based Genetic Algorithm (GA) [20,24] Biological evolution 1992 Physics-based Simulated annealing algorithm (SA) [25] Solid annealing 1983 Swarm-based Particle Swarm optimization (PSO) [26] Bird flock 1995 3.4. Comparative Study of Optimization Methods Nature-inspired Moth-flame optimization (MFO) [23] Moth 2015 In this study, the above five optimization methods were used to identify the nine ECMs, Moreover, the whole SOC (0–100%) was divided into 10 subarea, and the parameters in each respectively, then the appropriate models and algorithms were selected by comparing the identification subarea were identified by the above optimization algorithms, respectively. Therefore, 10 groups of accuracy and time cost. model parameters, which vary with SOC, were obtained as the model parameters of ECM. Figure 4a shows the RMSE of model error in the entire SOC area. It can be seen that the PNGV had a great accuracy advantage. For the nRC model, the model accuracy of the MFO increased with 3.4. Comparative Study of Optimization Methods the increase of n, and the accuracy of the other optimization algorithms did not increase or even In this study, the above five optimization methods were used to identify the nine ECMs, decrease when the order was greater than the second. Moreover, MFO had an obvious accuracy respectively, then the appropriate models and algorithms were selected by comparing the advantage for the high-order RC model. The model error distribution in the low SOC area (0–20%) is identification accuracy and time cost. shown in Figure 4b, indicated that the PNGV model had obvious advantages in the low SOC area, Figure 4a shows the RMSE of model error in the entire SOC area. It can be seen that the PNGV and the FMIN, PSO, and MFO had almost the same identification accuracy. The error distribution had a great accuracy advantage. For the nRC model, the model accuracy of the MFO increased with comparison between the PNGV model and the 2RC model in the low SOC area is shown in Figure 5. the increase of n, and the accuracy of the other optimization algorithms did not increase or even It is clear that the PNGV model had a good accuracy in the low SOC area. The error distribution in decrease when the order was greater than the second. Moreover, MFO had an obvious accuracy the high SOC area (20–100%) is shown in Figure 4c. It can be seen that the second- and higher-order advantage for the high-order RC model. The model error distribution in the low SOC area (0–20%) is RC models had high accuracy using PSO and MFO, and the accuracy of MFO was slightly higher. shown in Figure 4b, indicated that the PNGV model had obvious advantages in the low SOC area, The identification and the FMIN, time PSO, of and each MFO identification had almost the algorithm same identifi is shown cation accura in Figur cy. T eh6 e . err Itocan r distri bebseen ution that comparison between the PNGV model and the 2RC model in the low SOC area is shown in Figure 5. the FMIN had the shortest identification time, and PSO had the second shortest identification time It is clear that the PNGV model had a good accuracy in the low SOC area. The error distribution in (approximately one-sixth of the time taken by the other algorithms). Therefore, these two algorithms the high SOC area (20–100%) is shown in Figure 4c. It can be seen that the second- and higher-order are suitable for on-line identification. RC models had high accuracy using PSO and MFO, and the accuracy of MFO was slightly higher. Based on the above analysis, we could see that the PNGV model was the best choice, and the The identification time of each identification algorithm is shown in Figure 6. It can be seen that the matched identification algorithm was FMIN in the low SOC area. In the high SOC area, if it was on-line FMIN had the shortest identification time, and PSO had the second shortest identification time identification, 2RC plus PSO was the best choice for the balance between accuracy and time cost. If it (approximately one-sixth of the time taken by the other algorithms). Therefore, these two algorithms was off-line identification, 4RC plus MFO was the best choice for the highest accuracy. are suitable for on-line identification. (a) Figure 4. Cont. Appl. Sci. 2018, 8, 2028 8 of 20 (b) Appl. Sci. 2018, 8, 2028 8 of 19 Appl. Sci. 2018, 8, 2028 8 of 20 (b) (c) (c) Figure 4. Comparison of identification results. (a) In the entire state-of-charge (SOC) area; (b) in the low SOC area (0–20%); and (c) in the high SOC area (20–100%). Based on the above analysis, we could see that the PNGV model was the best choice, and the matched identification algorithm was FMIN in the low SOC area. In the high SOC area, if it was o Figure n-line 4. identi Comparison fication, 2 of RC identification plus PSO wa results. s the bes (at ) cho In the ice entir for the e state-of-char balance betge ween (SOC) accura area; cy (a bnd ) inti the me Figure 4. Comparison of identification results. (a) In the entire state-of-charge (SOC) area; (b) in the cost. If it was off-line identification, 4RC plus MFO was the best choice for the highest accuracy. low SOC area (0–20%); and (c) in the high SOC area (20–100%). low SOC area (0–20%); and (c) in the high SOC area (20–100%). Based on the above analysis, we could see that the PNGV model was the best choice, and the matched identification algorithm was FMIN in the low SOC area. In the high SOC area, if it was on-line identification, 2RC plus PSO was the best choice for the balance between accuracy and time cost. If it was off-line identification, 4RC plus MFO was the best choice for the highest accuracy. Appl. Sci. 2018, 8, 2028 9 of 20 Figure 5. The error distribution comparison between the PNGV model and the 2RC model. Figure 5. The error distribution comparison between the PNGV model and the 2RC model. FMIN GA SA PSO Figure 5. The error distribution comparison between the PNGV model and the 2RC model. MFO 50 PNGV 1RC 1RCH 2RC 2RCH PNGV 1RC 1RCH 2RC 2RCH 3RC 3RCH 4RC 4RCH Model Figure Figure 6.6Comparison . Comparison of of id identification entification time. time. 3.5. Multi-Model and Multi-Algorithm Combination Based on the above analysis, a multi-model and multi-algorithm combination (MM-MA) method, considering SOC distribution, was developed to improve the global accuracy of the ECM. As shown in Figure 7, the whole SOC area was divided into low SOC area and high SOC area, and different ECMs and optimization algorithms were used in different SOC areas. In the SOC estimation, different ECMs and corresponding model parameters were selected. Low SOC area (0%-20%) Off-line High SOC area (20%-100%) 4RC+MFO On-line PNGV+FMIN 2RC+PSO PNGV+FMIN Identification time Figure 7. Schematic diagram of multi-model and multi-algorithm combinations. Figure 8 shows the identification results of the MM-MA method, which indicated that this method had obvious advantages compared with 2RC, especially in the low SOC area. Moreover, the accuracy of off-line identification was higher than that of on-line identification. It was shown that the MM-MA method could effectively improve the accuracy of the model, and could lay the foundation for accurate SOC estimation. Zoom figure High SOC Low SOC Figure 8. Identification results of the multi-model and multi-algorithm combination (MM-MA) method. Time (s) Identification accuracy Appl. Sci. 2018, 8, 2028 9 of 20 Appl. Sci. 2018, 8, 2028 9 of 20 800 FMIN FMIN GA GA 600 SA SA PSO PSO MFO 400 50 MFO 50 PNGV 1RC 1RCH 2RC 2RCH PNGV 1RC 1RCH 2RC 2RCH PNGV 1RC 1RCH 2RC 2RCH 3RC 3RCH 4RC 4RCH PNGV 1RC 1RCH 2RC 2RCH 3RC 3RCH 4RC 4RCH Appl. Sci. 2018, 8, 2028 9 of 19 Model Model Figure 6. Comparison of identification time. Figure 6. Comparison of identification time. 3.5. Multi-Model and Multi-Algorithm Combination 3.5. Multi-Model and Multi-Algorithm Combination 3.5. Multi-Model and Multi-Algorithm Combination Based on the above analysis, a multi-model and multi-algorithm combination (MM-MA) method, Based on the above analysis, a multi-model and multi-algorithm combination (MM-MA) Based on the above analysis, a multi-model and multi-algorithm combination (MM-MA) considering SOC distribution, was developed to improve the global accuracy of the ECM. As shown in method, considering SOC distribution, was developed to improve the global accuracy of the ECM. method, considering SOC distribution, was developed to improve the global accuracy of the ECM. Figure 7, the whole SOC area was divided into low SOC area and high SOC area, and different ECMs As shown in Figure 7, the whole SOC area was divided into low SOC area and high SOC area, and As shown in Figure 7, the whole SOC area was divided into low SOC area and high SOC area, and and optimization algorithms were used in different SOC areas. In the SOC estimation, different ECMs different ECMs and optimization algorithms were used in different SOC areas. In the SOC different ECMs and optimization algorithms were used in different SOC areas. In the SOC and corr estima esponding tion, differ model ent ECparameters Ms and correwer spondi e selected. ng model parameters were selected. estimation, different ECMs and corresponding model parameters were selected. Low SOC area Low SOC area Off-line (0%-20%) (0%-20%) Off-line High SOC area High SOC area (20%-100%) (20%-100%) 4RC+MFO 4RC+MFO On-line On-line PNGV+FMIN PNGV+FMIN 2RC+PSO 2RC+PSO PNGV+FMIN PNGV+FMIN Identification time Identification time Figure 7. Schematic diagram of multi-model and multi-algorithm combinations. Figure 7. Schematic diagram of multi-model and multi-algorithm combinations. Figure 7. Schematic diagram of multi-model and multi-algorithm combinations. Figure 8 shows the identification results of the MM-MA method, which indicated that this method Figure 8 shows the identification results of the MM-MA method, which indicated that this Figure 8 shows the identification results of the MM-MA method, which indicated that this had obvious method h advantages ad obvious ad compar vantages co ed with mpared wi 2RC, especially th 2RC, espec in the ialllow y in the l SOC ow SOC area. Mor area eover . Moreo , the ver, accuracy the method had obvious advantages compared with 2RC, especially in the low SOC area. Moreover, the accuracy of off-line identification was higher than that of on-line identification. It was shown that the of off-line accuraidentification cy of off-line ide was ntifi higher cation wa than s hithat gher of than on-line that oidentification. f on-line identifiIt catio was n. shown It was show that n the that MM-MA the MM-MA method could effectively improve the accuracy of the model, and could lay the foundation MM-MA method could effectively improve the accuracy of the model, and could lay the foundation method could effectively improve the accuracy of the model, and could lay the foundation for accurate for accurate SOC estimation. for accurate SOC estimation. SOC estimation. Zoom figure Zoom figure High SOC Low SOC High SOC Low SOC Figure 8. Identification results of the multi-model and multi-algorithm combination (MM-MA) Figure 8. Identification results of the multi-model and multi-algorithm combination (MM-MA) method. Figure 8. Identification results of the multi-model and multi-algorithm combination (MM-MA) method. method. 4. SOC Estimation Method 4.1. EKF Method For the second-order RC model, the state variables (x) can be written as: x = [SOC , U , U ] (7) EKF 1 2 Time (s) Time (s) IId de en nttiiffiic ca attiio on n a ac cc cu urra ac cy y Appl. Sci. 2018, 8, 2028 10 of 19 From Table 2, the following expressions can be obtained: 2 32 3 1 0 0 SOC EKF,k 6 76 7 f (x , u ) = 0 exp( Dt/t ) 0 U 4 54 5 k k 1 1,k 0 0 exp( Dt/t ) U 2,k (8) 2 3 hDt/C 6 7 + R (1 exp( Dt/t )) I 4 5 1 1 k R 1 exp Dt/t ( ( )) 2 2 g(x , u ) = U (SOC ) I R U U (9) k k oc EKF,k k 0 1,k 2,k where Dt is the sample period. The form of the state-space expression of the fourth-order RC and PNGV models was similar to (7)–(9), which is not given here for brevity. According to the reference [27], the standard EKF equations for the battery system are listed in Algorithm 1. Many studies have confirmed that the EKF method produces a good convergence and satisfactory estimation accuracy for small model and sensor errors [14,16,28–30]. However, in actual EV operation, the ECM model error increased with battery ageing, temperature, and other adverse factors. Moreover, the sensor measurement error existed objectively, and the random error of the statistical characteristics of the noise is unknown. Under these circumstances, the accuracy of the SOC estimation error determined by the EKF method would decrease. Therefore, model and sensor errors must be considered for designing SOC estimation algorithm to improve the robustness. Algorithm 1. Summary of the extended Kalman filter (EKF) method for SOC estimation. The nonlinear state-space model: x = f (x , u ) + w k+1 k k k y = f (x , u ) + v k k k k where the first equation is the state equation, the second one is the output equation. f (x , u ) is a state k k transition function and g(x , u ) is a measurement function; w and v are independent zero-mean white k k k k Gaussian stochastic processes with covariance matrices å and å respectively. w v Step 1. Initialization. For k = 0, set h i + + + + x ˆ = E[x ˆ ], å = E x x ˆ x x ˆ . 0 0 0 0 x,0 0 0 Step 2. Computation. For k = 1, 2, , compute: + + T (a) Time update: x ˆ = f x ˆ , u + w ; = A A + w. å å å k 1 k 1 k 1 k k 1 x,k x,k 1 k 1 (b) Measurement update: Error innovation: e = y y ˆ = y g x ˆ , u . k k k k k h i T T x x x x x Kalman gain matrix: L = C C X C + å å å k x,k k k k x,k k Measurement update: x ˆ = x ˆ + L [y y ˆ ] k k k k k ¶ f (x ,u ) + x x k k Error covariance measurement update: å = I L C å where A = , x,k k k x,k ¶x + x=x ˆ ¶g(x ,u ) k k C = ¶x x=x ˆ k Appl. Sci. 2018, 8, 2028 11 of 19 4.2. Ampere-Hour Counting Method The AH method can be expressed as follows [31]: h i(x)dx SOC (t) = SOC(t ) + (10) AH 0 where SOC (t) is the SOC value at time t, SOC(t ) is the SOC value at the initial time t , C is the AH 0 0 N nominal capacity, i(x) denotes the current at time x (positive for charging and negative for discharging), and h denotes the columbic efficiency. Although the AH method is very simple and quite useful, it is an open-loop prediction method, thus it suffers from accumulated errors caused by initial SOC value errors, and noise and measurement errors. Moreover, the battery capacity might change in applications, which can lead to SOC error. 4.3. Fuzzy Fusion Algorithm From the above analysis, we can see that the EKF method had high accuracy, but it was easy to be influenced by ECM model parameters and voltage sensor errors. The AH method had relatively low accuracy, but the estimation results were relatively stable because relatively few parameters were affected. Based on the characteristics of the above two algorithms, a fuzzy fusion method, combining AH and EKF, was developed to calculate SOC in this paper. In our algorithm, the SOC was estimated by the AH and EKF methods, respectively. Then, the SOC increments were calculated using the following equations: DSOC (k) = SOC (k) SOC (k 1) (11) AH AH AH DSOC (k) = SOC (k) SOC (k 1) (12) EKF EKF EKF where DSOC (k) and DSOC (k) are the SOC increments for the AH and EKF AH EKF methods, respectively. Then, EKF and AH methods can be fused through the following incremental form: SOC (k) = SOC (k 1) + k DSOC (k) + (1 k )DSOC (k) (13) f f FLC EKF FLC AH where SOC (k) is the SOC value determined by the fusion algorithm, k is ratio coefficient f FLC (0 k 1). FLC The most critical step of the fusion algorithm is the determination of the more credible SOC increments between DSOC (k) and DSOC (k). In other words, the key to SOC estimation in (13) AH EKF is to select suitable k based on the relationship between DSOC (k) and DSOC (k). We assume FLC AH EKF that the incremental ratio of SOC estimated by the two methods at time k is DK (k), and it can be SOC expressed as follows: jDSOC (k)j EKF DK k = (14) ( ) SOC jDSOC (k)j AH In this study, the fuzzy logic control was used to determine the appropriate k based on FLC DK k . The range of DK k was set as 0 to 3 in this paper. In the fuzzy logic control, the input ( ) ( ) SOC SOC and output were divided into fuzzy subsections and expressed by a linguistic variable. As shown in Figure 9, we chose DK k and its derivative DK k as input variables and ( ) ( ) SOC SOC k as output variables. The fuzzy variable of DK (k) was divided into very large (VL), large FLC SOC (L), medium (M), small (S), and very small (VS). DK k was divided into positive (P), zero (Z), ( ) SOC and negative (N). The fuzzy variable of output with respect to k was divided into VL, L, M, S, FLC and VS. The triangular fuzzy membership function was chosen in this paper because the triangular membership function has the characteristics of simplified calculation and good control performance. The membership function of FLC is shown in Figure 10. Appl. Sci. 2018, 8, 2028 12 of 20 Appl. Sci. 2018, 8, 2028 12 of 20 In this study, the fuzzy logic control was used to determine the appropriate k based on FLC In this study, the fuzzy logic control was used to determine the appropriate k based on FLC . The range of was set as 0 to 3 in this paper. In the fuzzy logic control, the K k K k SOC SOC . The range of was set as 0 to 3 in this paper. In the fuzzy logic control, the K k K k SOC SOC input and output were divided into fuzzy subsections and expressed by a linguistic variable. input and output were divided into fuzzy subsections and expressed by a linguistic variable. As shown in Figure 9, we chose K k and its derivative K k as input variables and SOC SOC As shown in Figure 9, we chose K k and its derivative K k as input variables and SOC SOC k as output variables. The fuzzy variable of K k was divided into very large (VL), large FLC SOC k as output variables. The fuzzy variable of K k was divided into very large (VL), large FLC SOC (L), medium (M), small (S), and very small (VS). K k was divided into positive (P), zero (Z), SOC (L), medium (M), small (S), and very small (VS). K k was divided into positive (P), zero (Z), SOC and negative (N). The fuzzy variable of output with respect to k was divided into VL, L, M, S, FLC and negative (N). The fuzzy variable of output with respect to k was divided into VL, L, M, S, FLC and VS. The triangular fuzzy membership function was chosen in this paper because the triangular and VS. The triangular fuzzy membership function was chosen in this paper because the triangular membership function has the characteristics of simplified calculation and good control performance. membership function has the characteristics of simplified calculation and good control performance. Appl. Sci. 2018, 8, 2028 12 of 19 The membership function of FLC is shown in Figure 10. The membership function of FLC is shown in Figure 10. SOC k EKF SOC k EKF Rules K k SOC Rules K k SOC x Eq.(16) Fuzzifier 1 x z FLC Eq.(16) Fuzzifier 1 Eq.(17) z FLC Inference Engine Defuzzifier Eq.(17) K k SOC Inference Engine Defuzzifier Eq.(19) K k SOC Eq.(19) Fuzzifier 2 y Fuzzifier 2 y SOC k AH SOC k AH Figure 9. Block diagram of the fuzzy logic control. Figure 9. Block diagram of the fuzzy logic control. Figure 9. Block diagram of the fuzzy logic control. x x S M L VL VS 1.0 S M L VL VS 1.0 0.5 0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0 (a) (a) y z y z N Z P S M L VL VS 1.0 1.0 N Z P S M L VL VS 1.0 1.0 0.5 0.5 0.5 0.5 -5 0 5 0 0.25 0.5 0.75 1.0 -5 0 5 0 0.25 0.5 0.75 1.0 (b) (c) (b) (c) Figure 10. Membership function. (a) DK (k); (b) DK (k); and (c) k . SOC SOC FLC Figure 10. Membership function. (a) K k ; (b) K k ; and (c) k . SOC SOC FLC Figure 10. Membership function. (a) K k ; (b) K k ; and (c) k . SOC SOC FLC Considering the sensor and model errors, the AH method may not be very accurate, but it had Considering the sensor and model errors, the AH method may not be very accurate, but it had higher reliability increments due to less affected parameters. The EKF method had good accuracy, Considering the sensor and model errors, the AH method may not be very accurate, but it had higher reliability increments due to less affected parameters. The EKF method had good accuracy, but it may not be stable. Therefore, we can choose the SOC increment with high reliability and higher reliability increments due to less affected parameters. The EKF method had good accuracy, but it may not be stable. Therefore, we can choose the SOC increment with high reliability and accuracy based on the relative relationship between DSOC (k) and DSOC (k) to calculate SOC AH EKF but it may not be stable. Therefore, we can choose the SOC increment with high reliability and accuracy based on the relative relationship between SOC (k ) and SOC (k ) to calculate SOC AH EKF according to the value of DK (k). Based on the above characteristics, the following rules were SOC accuracy based on the relative relationship between SOC (k ) and SOC (k ) to calculate SOC AH EKF according to the value of K k . Based on the above characteristics, the following rules were proposed for determining the higher credibility SOC increment between the two methods: (1) If SOC according to the value of K k . Based on the above characteristics, the following rules were SOC DK (k) 1, then DSOC (k) is considered to be more credible (large k should be chosen); (2) if proposed for determining the higher credibility SOC increment between the two methods: (1) If SOC EKF FLC proposed for determining the higher credibility SOC increment between the two methods: (1) If DK (k) > 1, then DSOC (k) is considered to be more credible (small k should be chosen); and K k 1 , then SOC (k ) is considered to be more credible (large k should be chosen); (2) SOC AH FLC EKF FLC SOC K k 1 , then SOC (k ) is considered to be more credible (large k should be chosen); (2) SOC EKF FLC (3) if DK (k) < 0, it means that DK (k) has a decreasing trend, otherwise there is an increasing SOC SOC if , then SOC (k ) is considered to be more credible (small k should be chosen); K k 1 SOC AH FLC if , then SOC (k ) is considered to be more credible (small k should be chosen); K k 1 trend.STher OC efore, the fuzzy rule AH of the fuzzy fusion algorithm can be expressedFLas C follows: and (3) if , it means that K k has a decreasing trend, otherwise there is an K k 0 SOC SOC and (3) if , it means that K k has a decreasing trend, otherwise there is an K k 0 SOC SOC 0 When DK (k) is relatively small and DK (k) is negative, very large k should be chosen SOC SOC FLC increasing trend. Therefore, the fuzzy rule of the fuzzy fusion algorithm can be expressed as follows: increasing trend. Therefore, the fuzzy rule of the fuzzy fusion algorithm can be expressed as follows: to ensure that DSOC (k) is more credible in the fuzzy fusion algorithm. EKF When K k is relatively small and K k is negative, very large k should be When DK SOC(k) is relatively large and DK (SOkC) is positive, very small k should FLC be chosen SOC SOC FLC When K k is relatively small and K k is negative, very large k should be SOC SOC FLC chosen to ensure that SOC (k ) is more credible in the fuzzy fusion algorithm. to ensure that DSOC (k) is more credible in the fuzzy fusion algorithm. AH EKF chosen to ensure that SOC (k ) is more credible in the fuzzy fusion algorithm. EKF When DK (k) is relatively large and DK (k) is negative, small k should be chosen. SOC SOC FLC When DK (k) is relatively small and DK (k) is positive, medium k should be chosen to SOC SOC FLC improve the stability of the control system. Based on the above four rules, 15 control rules for linguistic variables were obtained, shown in Table 4. The fuzzy linguistic output is an unavailable signal for the SOC estimation system. Therefore, the defuzzification is necessary, which is the process of relating the output membership function to a value. The centroid defuzzification method was utilized in this paper, and it can be expressed as follows: zm(z)dz k = (15) FLC m(z)dz Appl. Sci. 2018, 8, 2028 13 of 19 The proposed fuzzy fusion algorithm uses the incremental features of EKF and AH to design fuzzy characteristics, which effectively avoids the shortcomings of the two algorithms and takes advantage of their advantages. Table 4. Rules of the fuzzy logic control for linguistic variable. DK (k) SOC FLC VS S M L VL N VL L M S VS Z L M S VS VS DK (k) SOC P M S S VS VS 5. Results and Discussion 5.1. Estimation Results Based on EKF The SOC estimation error based on EKF is shown in Figure 11. It can be seen that the SOC accuracy based on the PNGV model in the low SOC area was higher than that based on the 2RC model, and that SOC estimation accuracy based on off-line identification was higher than that based on on-line Appl. Sci. 2018, 8, 2028 14 of 20 identification. The results indicate that the proposed MM-MA method can improve the accuracy of the model and the SOC estimation significantly. (a) (b) Figure 11. SOC estimation results using EKF method. (a) Comparison between the PNGV model and Figure 11. SOC estimation results using EKF method. (a) Comparison between the PNGV model and the 2RC model; and (b) SOC error based on MM-MA. the 2RC model; and (b) SOC error based on MM-MA. 5.2. Case Studies for the Fuzzy Fusion Algorithm To verify and evaluate the effectiveness of the proposed fuzzy fusion algorithm, six simulated cases of real LIBs, considering sensor and model errors, were set, as shown in Table 5. The fuzzy fusion algorithm was carried out based on these cases. In addition, the AH and EKF methods were applied to the same cases for comparison. In Table 5, e is the initial SOC error, ECM is the SOC 0 drift ECM error, is the drift current of the current sensor, is the drift voltage of the voltage drift drift sensor, and e is the SOC–OCV curve error. In this study, SOH is expressed as Equation (16), OCV where SOH means the SOH value in the ith cycle, Ci represents the capacity at the ith cycle, C0 represents the initial capacity. With aging of the cell, the cell capacity shows a downward trend. SOH = 100% (16) To evaluate the performance of different SOC estimation methods, the root-mean-square errors (RMSE) of SOC were calculated as follows: RMSE= SOC k SOC k (17) rel Estimated k1 where SOC and SOC denote the real and estimated values of the SOC, respectively. rel Estimated Figure 12 shows the SOC estimation results for Case A. It indicated the influence of initial SOC error (e ) on the AH, EKF, and fuzzy fusion algorithm (fuzzy). It was clear that e had no SOC 0 SOC 0 effect on the EKF method, but significantly affected the AH and the fuzzy methods. Figure 12a,c show that the fuzzy fusion algorithm had a higher estimation accuracy when the initial SOC error Appl. Sci. 2018, 8, 2028 14 of 19 The reason why the accuracy of off-line identification was higher than that of on-line identification is that the more complex model and more suitable identification algorithm (4RC plus MFO) were used for off-line identification. However, greater computing burden and more identification time were needed for off-line identification, which is shown in Figure 6. 5.2. Case Studies for the Fuzzy Fusion Algorithm To verify and evaluate the effectiveness of the proposed fuzzy fusion algorithm, six simulated cases of real LIBs, considering sensor and model errors, were set, as shown in Table 5. The fuzzy fusion algorithm was carried out based on these cases. In addition, the AH and EKF methods were applied to the same cases for comparison. In Table 5, e is the initial SOC error, ECM is the ECM error, SOC0 dri f t I is the drift current of the current sensor, U is the drift voltage of the voltage sensor, and e dri f t dri f t OCV is the SOC–OCV curve error. In this study, SOH is expressed as Equation (16), where SOH means the SOH value in the ith cycle, C represents the capacity at the ith cycle, C represents the initial capacity. i 0 With aging of the cell, the cell capacity shows a downward trend. SOH = 100% (16) To evaluate the performance of different SOC estimation methods, the root-mean-square errors (RMSE) of SOC were calculated as follows: RMSE = (SOC (k) SOC (k)) (17) å rel Estimated k=1 where SOC and SOC denote the real and estimated values of the SOC, respectively. rel Estimated Figure 12 shows the SOC estimation results for Case A. It indicated the influence of initial SOC error (e ) on the AH, EKF, and fuzzy fusion algorithm (fuzzy). It was clear that e had no effect SOC0 SOC0 on the EKF method, but significantly affected the AH and the fuzzy methods. Figure 12a,c show that the fuzzy fusion algorithm had a higher estimation accuracy when the initial SOC error was more accurate ( e < 3%). When e was large, the fuzzy fusion algorithm had a larger estimation j j SOC0 SOC0 error than the other algorithms (Figure 12b). However, the initial SOC correction methods, such as the full-charge calibration method or the OCV method (e.g., with a 3 h rest for the LIBs), were used in the practical EV applications. Therefore, an accurate initial SOC could be ensured, improving the feasibility of the fuzzy fusion method for real EVs. Moreover, Figure 12c shows that the SOC estimation accuracy based on the MM-MA method was higher than that based on the 2RC model, which validates the effectiveness of the proposed MM-MA method to improve the SOC estimation accuracy by improving the model accuracy. Table 5. Case and parameter setting. Case Name Describe Parameters Setting The influence of initial SOC error (e ) SOH = 90%, I = 0.1A, U = 5 mV, SOC0 dri f t dri f t Case A on fuzzy algorithm ECM = 3 mV, e = 5 mV OCV dri f t The influence of model error (ECM ) e = 0.3%, SOH = 95%, I = 0.08A, dri f t SOC0 dri f t Case B on fuzzy algorithm U = 5 mV, e = 5 mV; dri f t OCV The influence of voltage measurement e = 0.3%, SOH = 90%, I = 0.1A, SOC0 dri f t Case C error (U ) on fuzzy algorithm ECM = 3 mV, e = 5 mV OCV dri f t dri f t The influence of current measurement e = 0.5%, SOH = 95%, U = 5 mV, SOC0 dri f t Case D error (I ) on fuzzy algorithm ECM = 5 mV, e = 5 mV dri f t dri f t OCV The influence of the SOH on fuzzy e = 0.3%, I = 0.1A, U = 5 mV, SOC0 dri f t dri f t Case E algorithm. ECM = 5 mV, e = 5 mV dri f t OCV e = 0.3%, SOH = 95%, I = 0.1A, The influence of SOC–OCV curve error SOC0 dri f t Case F ECM = 5 mVU = 5 mV (e ) on fuzzy algorithm. OCV dri f t dri f t Appl. Sci. 2018, 8, 2028 15 of 20 was more accurate ( e 3% ). When e was large, the fuzzy fusion algorithm had a larger SOC 0 SOC 0 estimation error than the other algorithms (Figure 12b). However, the initial SOC correction methods, such as the full-charge calibration method or the OCV method (e.g., with a 3 h rest for the LIBs), were used in the practical EV applications. Therefore, an accurate initial SOC could be ensured, improving the feasibility of the fuzzy fusion method for real EVs. Moreover, Figure 12c shows that the SOC estimation accuracy based on the MM-MA method was higher than that based on the 2RC model, which validates the effectiveness of the proposed MM-MA method to improve the SOC estimation accuracy by improving the model accuracy. Table 5. Case and parameter setting. Case Name Describe Parameters Setting , I 0.1A , U 5 m V , SOH 9 0 % The influence of initial SOC error drift drift Case A (e ) on fuzzy algorithm ECM 3 mV , e 5 m V SOC 0 drift OCV , S O H = 9 5 % , I 0.08A , e 0 .3 % The influence of model error ( ) drift ECM SOC 0 drift Case B , ; U 5 m V e 5 m V on fuzzy algorithm drift OCV , S O H 9 0 % , I 0.1A , The influence of voltage measurement e 0 .3 % SOC 0 drift Case C error (U ) on fuzzy algorithm EC M 3 m V , e 5 m V drift drift OCV The influence of current measurement , S O H 9 5 % , U 5 m V , e 0 .5 % SOC 0 drift Case D error (I ) on fuzzy algorithm EC M 5 m V , e 5 m V drift drift OCV , I 0.1A , U 5 m V , e 0 .3 % SOC 0 drift drift The influence of the SOH on fuzzy Case E algorithm. EC M 5 m V , e 5 m V drift OCV e 0 .3 % , S O H 9 5 % , I 0.1A , The influence of SOC–OCV curve error SOC 0 drift Case F Appl. Sci. 2018, 8, 2028 15 of 19 (e ) on fuzzy algorithm. ECM 5 mV U 5 m V OCV drift drift (a) (b) (c) Figure 12. SOC estimation results for Case A. (a) e = 0.5%; (b) e = 5.0%; and (c) SOC0 SOC0 Figure 12. SOC estimation results for Case A. (a) e =0.5% ; (b) e =5.0% ; and (c) SOC 0 SOC 0 root-mean-square error (RMSE) of SOC in different initial SOC errors. root-mean-square error (RMSE) of SOC in different initial SOC errors. Figure 13 shows the SOC estimation results for Case B, including the relationship between the Figure 13 shows the SOC estimation results for Case B, including the relationship between the model error and the SOC error, for the three methods. Figure 13a shows that the SOC accuracy of model error and the SOC error, for the three methods. Figure 13a shows that the SOC accuracy of the the fuzzy fusion algorithm exceeded that of the EKF when the model error was large. As shown fuzzy fusion algorithm exceeded that of the EKF when the model error was large. As shown in in Figure 13b, the SOC accuracy of the fuzzy fusion algorithm was lower than the EKF when the Figure 13b, the SOC accuracy of the fuzzy fusion algorithm was lower than the EKF when the model model error was very small. Figure 13c shows the relationship between the SOC errors based on three error was very small. Figure 13c shows the relationship between the SOC errors based on three algorithms and model errors of ECM. It was obvious that the accuracy of the fuzzy fusion algorithm was higher than that of both the EKF and AH methods for most conditions, except when the model error was very small ECM < 10 mV . However, even for a very small ECM , the estimation dri f t dri f t accuracy of the fuzzy fusion algorithm was acceptable. Moreover, the accuracy of the SOC estimation based on the MM-MA method was better than that of the 2RC model in the whole model error area. In actual EVs, battery aging and adverse environmental temperatures adversely affect the ECM’s accurate reflection of the LIB characteristics, resulting in increased model error. From the above results, we can see that the proposed fuzzy fusion algorithm significantly outperformed the EKF and AH methods. Figure 14 compares the SOC estimation results of the three algorithms for Case C. Figure 14a,b show the SOC errors of the three algorithms with small and large voltage sensor error (U ). dri f t Figure 14c shows the relationship between SOC error based three algorithms and the U . It can be dri f t seen that the SOC estimation accuracy of the fuzzy fusion algorithm was better than that of the EKF method, except for very small U values. Moreover, the superiority of the fuzzy fusion algorithm dri f t increased with the increase of U . However, even for small U values, the accuracy of the fuzzy dri f t dri f t fusion algorithm was within the acceptable range. Appl. Sci. 2018, 8, 2028 16 of 20 algorithms and model errors of ECM. It was obvious that the accuracy of the fuzzy fusion algorithm was higher than that of both the EKF and AH methods for most conditions, except when the model error was very small . However, even for a very small , the estimation ECM 10 mV ECM drift drift accuracy of the fuzzy fusion algorithm was acceptable. Moreover, the accuracy of the SOC estimation based on the MM-MA method was better than that of the 2RC model in the whole model Appl. Sci. 2018, 8, 2028 16 of 19 error area. (a) (b) (c) Figure 13. SOC estimation results for Case B. (a) ECM 50 mV ; (b) ECM 2 mV ; and (c) RMSE Figure 13. SOC estimation results for Case B. (a) ECM = 50 mV; (b) ECM = 2 mV; and (c) drift drift dri f t dri f t Appl. Sci. 2018, 8, 2028 17 of 20 RMSE of SOC in different model errors. of SOC in different model errors. (a) (b) Real AH EKF Fuzzy In actual EVs, battery aging and adverse environmental temperatures adversely affect the ECM’s accurate reflection of the LIB characteristics, resulting in increased model error. From the -5 -5 Real above results, we can see that the proposed fuzzy fusion algorithm significantly outperformed the -10 -10 AH EKF EKF and AH methods. -15 -15 Fuzzy U 3 mV U 50 mV drift drift Figure 14 compares the SOC estimation results of the three algorithms for Case C. Figure 14a,b -20 -20 0 2 4 6 0 2 4 6 4 Time (s) Time (s) x 10 x 10 show the SOC errors of the three algorithms with small and large voltage sensor error ( ). Figure drift (c) 14c shows the relationship between SOC error based three algorithms and the . It can be seen drift that the SOC estimation accuracy of the fuzzy fusion algorithm was better than that of the EKF method, except for very small values. Moreover, the superiority of the fuzzy fusion algorithm drift U U increased with the increase of . However, even for small values, the accuracy of the drift drift fuzzy fusion algorithm was within the acceptable range. Figure Figure 14. 14. SOC SOC estimation estimation results results for for Case CasC. e C. (a )(aU ) U =3 3 mV mV ;; (( b b )) UU =50 50 mmV V ; ;and and((cc)) RMSE RMSE of of dri f t dri f t drift drift SOC in different drift voltages. SOC in different drift voltages. Figure 15a depicts the relationship between the RMSE of SOC estimation and the error of the Figure 15a depicts the relationship between the RMSE of SOC estimation and the error of the current sensor. We can see that the fuzzy fusion algorithm was better than others in considering the current sensor. We can see that the fuzzy fusion algorithm was better than others in considering the sensor error. Figure 15b shows the relationship between the RMSE of SOC estimation and the SOH. sensor error. Figure 15b shows the relationship between the RMSE of SOC estimation and the SOH. It is clear that the SOC estimation accuracy of the fuzzy fusion algorithm exceeded that of the EKF It is clear that the SOC estimation accuracy of the fuzzy fusion algorithm exceeded that of the EKF when the SOH fell from 100% to 80%. Thus, it is deduced that the Comb algorithm is highly adaptable when the SOH fell from 100% to 80%. Thus, it is deduced that the Comb algorithm is highly to the capacity decay of an LIB. Figure 15c shows the RMSE of SOC estimation for e values of OCV adaptable to the capacity decay of an LIB. Figure 15c shows the RMSE of SOC estimation for OCV 100 to 100 mV. We can see that the fuzzy fusion algorithm was better than the EKF when model and values of −100 to 100 mV. We can see that the fuzzy fusion algorithm was better than the EKF when sensor errors were considered. When the current sensor error existed, the SOC obtained by the AH model and sensor errors were considered. When the current sensor error existed, the SOC obtained had a larger cumulative error. Therefore, our proposed method was better than that of the EKF and by the AH had a larger cumulative error. Therefore, our proposed method was better than that of the AH in cases D, E, and F. EKF and AH in cases D, E, and F. (a) AH 15 EKF Fuzzy -0.2 -0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16 0.2 I (A) drift (b) AH EKF Fuzzy 99 97 95 93 91 89 87 85 83 81 SOH (%) RMSE (%) RMSE (%) SOC error (%) SOC error (%) Appl. Sci. 2018, 8, 2028 17 of 20 (a) Real AH EKF Fuzzy (b) -5 -5 Real -10 -10 AH EKF -15 -15 Fuzzy U 3 mV U 50 mV drift drift -20 -20 0 2 4 6 0 2 4 6 4 Time (s) Time (s) x 10 x 10 (c) Figure 14. SOC estimation results for Case C. (a) U 3 mV ; (b) U 50 mV ; and (c) RMSE of drift drift SOC in different drift voltages. Figure 15a depicts the relationship between the RMSE of SOC estimation and the error of the current sensor. We can see that the fuzzy fusion algorithm was better than others in considering the sensor error. Figure 15b shows the relationship between the RMSE of SOC estimation and the SOH. It is clear that the SOC estimation accuracy of the fuzzy fusion algorithm exceeded that of the EKF when the SOH fell from 100% to 80%. Thus, it is deduced that the Comb algorithm is highly adaptable to the capacity decay of an LIB. Figure 15c shows the RMSE of SOC estimation for OCV values of −100 to 100 mV. We can see that the fuzzy fusion algorithm was better than the EKF when model and sensor errors were considered. When the current sensor error existed, the SOC obtained by the AH had a larger cumulative error. Therefore, our proposed method was better than that of the Appl. Sci. 2018, 8, 2028 17 of 19 EKF and AH in cases D, E, and F. (a) AH 15 EKF Fuzzy -0.2 -0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16 0.2 I (A) drift (b) AH EKF Fuzzy Appl. Sci. 2018, 8, 2028 0 18 of 20 99 97 95 93 91 89 87 85 83 81 SOH (%) (c) Figure 15. RMSE of SOC for cases D, E, and F. (a) Case D: RMSE of SOC in different drift currents; Figure 15. RMSE of SOC for cases D, E, and F. (a) Case D: RMSE of SOC in different drift currents; (b) case E: RMSE of SOC in different SOHs; and (c) Case F: RMSE of SOC in different SOC–OCV (b) case E: RMSE of SOC in different SOHs; and (c) Case F: RMSE of SOC in different SOC–OCV curve errors. curve errors. The above case study indicated that the EKF was slightly better than the fuzzy fusion algorithm The above case study indicated that the EKF was slightly better than the fuzzy fusion algorithm under ideal conditions of no or very small model and sensor errors. However, the results of the under ideal conditions of no or very small model and sensor errors. However, the results of the fuzzy fusion algorithm were also within the acceptable range for these conditions. Under practical fuzzy fusion algorithm were also within the acceptable range for these conditions. Under practical conditions involving significant model, sensor, and SOC–OCV curve errors, the fuzzy fusion algorithm conditions involving significant model, sensor, and SOC–OCV curve errors, the fuzzy fusion significantly outperformed the EKF and AH. Moreover, the proposed MM-MA method not only algorithm significantly outperformed the EKF and AH. Moreover, the proposed MM-MA method improved the accuracy of the model, but also improved the accuracy of the SOC estimation. not only improved the accuracy of the model, but also improved the accuracy of the SOC estimation. In an actual BMS, the computation time of the SOC estimation is as important as the estimation In an actual BMS, the computation time of the SOC estimation is as important as the estimation accuracy. The computation time of various SOC estimation algorithms was performed in MATLAB accuracy. The computation time of various SOC estimation algorithms was performed in MATLAB 2016a on a computer with INTEL Core i5-4440 CPU (3.1 GHz) and 8 GB RAM. Table 6 lists the 2016a on a computer with INTEL Core i5-4440 CPU (3.1 GHz) and 8 GB RAM. Table 6 lists the calculation results under NEDC in the entire SOC area (0–100%). It is noted that the computation time calculation results under NEDC in the entire SOC area (0–100%). It is noted that the computation was the time of SOC estimation in the entire SOC area (0–100%) using our proposed algorithm, based time was the time of SOC estimation in the entire SOC area (0–100%) using our proposed algorithm, on the current and voltage curves obtained from experiments and the identified model parameters. based on the current and voltage curves obtained from experiments and the identified model It can be seen that the computation time of the fuzzy fusion algorithm was only slightly larger than the parameters. It can be seen that the computation time of the fuzzy fusion algorithm was only slightly computation time of EKF. Therefore, the proposed algorithm greatly improved the estimation accuracy, larger than the computation time of EKF. Therefore, the proposed algorithm greatly improved the without increasing the complexity, in the case of large model and sensor errors, and it was highly estimation accuracy, without increasing the complexity, in the case of large model and sensor errors, suitable for practical EV applications. and it was highly suitable for practical EV applications. Table 6. Computing time (s) of various SOC estimation algorithms in the entire SOC area. SOC Estimation Algorithm Time (s) AH 0.0479 EKF 65.8570 Fuzzy fusion 65.8793 6. Conclusions In this study, nine ECMs and five commonly used model parameter identification algorithms were compared in the whole SOC area, and the most suitable ECM and matched parameter identification algorithm, in the high and low SOC area, were obtained, respectively. Based on this, a MM-MA method was proposed. To improve the robustness and accuracy of SOC estimation based on MM-MA in practical application, a fuzzy fusion SOC estimation algorithm based on EKF and AH was proposed. The experimental results show that the satisfactory estimation accuracy can still be maintained even when large model errors and sensor errors exist, and that the accuracy and robustness of the fuzzy fusion algorithm is better than that of EKF and AH. The proposed method did not consider the aging of the battery, and the calculations were carried out using the MATLAB software on a computer. Further works include: (1) Verification and application of the MM-MA method and the fuzzy SOC estimation algorithm in actual BMS; (2) building a new battery model that considers battery aging; (3) verification of the effectiveness of the proposed method under other working conditions. RMSE (%) RMSE (%) SOC error (%) SOC error (%) Appl. Sci. 2018, 8, 2028 18 of 19 Table 6. Computing time (s) of various SOC estimation algorithms in the entire SOC area. SOC Estimation Algorithm Time (s) AH 0.0479 EKF 65.8570 Fuzzy fusion 65.8793 6. Conclusions In this study, nine ECMs and five commonly used model parameter identification algorithms were compared in the whole SOC area, and the most suitable ECM and matched parameter identification algorithm, in the high and low SOC area, were obtained, respectively. Based on this, a MM-MA method was proposed. To improve the robustness and accuracy of SOC estimation based on MM-MA in practical application, a fuzzy fusion SOC estimation algorithm based on EKF and AH was proposed. The experimental results show that the satisfactory estimation accuracy can still be maintained even when large model errors and sensor errors exist, and that the accuracy and robustness of the fuzzy fusion algorithm is better than that of EKF and AH. The proposed method did not consider the aging of the battery, and the calculations were carried out using the MATLAB software on a computer. Further works include: (1) Verification and application of the MM-MA method and the fuzzy SOC estimation algorithm in actual BMS; (2) building a new battery model that considers battery aging; (3) verification of the effectiveness of the proposed method under other working conditions. Author Contributions: All authors contributed to the paper. 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