3. Development and implementation in real-time of SOC Li-Ion battery estimators on MATLAB/SIMULINK platform

In this section we propose for Li-Ion battery EMC SOC estimation three nonlinear on-board real-time estimators integrated in BMS of HEV, based on Kalman Filter (KF) technique, specifically a nonlinear Gaussian Unscented Kalman Filter (UKF), a non-Gaussian nonlinear Particle Filter (PF), and a nonlinear observer estimator (NOE). The simulations results and a comprehensive performance analysis for each proposed SOC estimator are presented in the following subsections of this section.

#### 3.1. Unscented Kalman filter real-time estimator design and robustness analysis

The main aim of this subsection is to build a nonlinear UKF SOC estimator, following the same design procedure described rigorously in [5]. We are motivated by some preliminary results obtained in our research, as you can see in [6, 7]. Technically, UKF estimator is based on the principle that one set of discrete sampled points parameterizes easily the mean and the covariance of a Gaussian random variable, as is stated in [5]. Moreover, the nonlinear estimator UKF yields an equivalent performance compared to a linear extended Kalman filter (EKF), well documented in [3, 4, 8], excluding the linearization steps required by EKF. In addition, the results of UKF real-time implementation for the majority of similar applications are encouraging, and it seems that the anticipated performance of this approach is slightly superior compared to EKF [3, 4, 8]. Furthermore, the nonlinear UKF SOC estimator can be extended to the applications where the distributions of the process and measurements noises are not Gaussian [19]. Concluding, the implementation simplicity and a great estimation accuracy of the proposed UKF SOC estimator recommend it as the most suitable estimator to be used in almost all similar applications. Explicitly, the nonlinear UKF estimator is an algorithm of "predictor– corrector" type applied to a nonlinear discrete - time systems, such as those described in [3–7]:

$$\mathbf{x}(k+1) = f(\mathbf{x}(k), \boldsymbol{\mu}(k), k) + \mathbf{w}(k) \tag{13}$$

$$\mathbf{y}(k) = \mathbf{g}(\mathbf{x}(k), \boldsymbol{\mu}(k), k) + \boldsymbol{\nu}(k) \tag{14}$$

where fð Þ: , gð Þ: are two nonlinear functions of system states and inputs; w kð Þ, v kð Þ are zeromean, uncorrelated process, and measurement Gaussian noise respectively. Since the noise injected in the state and output equations are randomly, also the system state vector and the output become random variables, having the mean and covariance matrices as a statistics. The nonlinear UKF SOC estimator has a "predictor-corrector" structure that propagates the mean and the covariance matrix of a Gaussian distribution for the random state variable x kð Þ in a recursively way, in both prediction and correction phases, as is stated in [3–7]. The propagation of these first two moments is performed by using an unscented transformation (UT) to calculate the statistics of any random variable that undergoes a nonlinear transformation [5–7]. As is stated in the original and fundamental work [5], by means of this UT transformation "a set of points, the so-called sigma points, are chosen such that their sample mean, and sample covariance matrix are x and Pxx, respectively". Also, "the nonlinear function gð Þ: is applied to each sigma points generating a "cloud" of transformed points with the mean y and the covariance matrix Pyy". Furthermore, since statistical convergence is not an issue, only a very small number of sample points is enough to capture high order information about the random state variable distribution. The different selection strategies of the sigma points and the UKF algorithm steps in a "predictorcorrector" structure, as well as its tuning parameters are well documented in the literature, and for details we recommend the fundamental work [5]. Since we follow the same design procedure steps for building a nonlinear UKF SOC estimator as in [5–7], we are focused only to the implementation aspects.

Likewise, the Li-Ion battery OCV for a discharging cycle at 1C-rate (i.e. 6 Ah 1/h = 6A constant input discharging current) is shown in the top of Figure 6, and its corresponding SOC during the same discharging cycle is revealed on the bottom graph of the same Figure 6

Figure 6. The Li-Ion battery EMC OCV curve during a complete discharging cycle at 1C-rate (top) and the corresponding

3. Development and implementation in real-time of SOC Li-Ion battery

In this section we propose for Li-Ion battery EMC SOC estimation three nonlinear on-board real-time estimators integrated in BMS of HEV, based on Kalman Filter (KF) technique, specifically a nonlinear Gaussian Unscented Kalman Filter (UKF), a non-Gaussian nonlinear Particle Filter (PF), and a nonlinear observer estimator (NOE). The simulations results and a comprehensive performance analysis for each proposed SOC estimator are presented in the following

The main aim of this subsection is to build a nonlinear UKF SOC estimator, following the same design procedure described rigorously in [5]. We are motivated by some preliminary results obtained in our research, as you can see in [6, 7]. Technically, UKF estimator is based on the principle that one set of discrete sampled points parameterizes easily the mean and the covariance of a Gaussian random variable, as is stated in [5]. Moreover, the nonlinear estimator UKF yields an equivalent performance compared to a linear extended Kalman filter (EKF),

3.1. Unscented Kalman filter real-time estimator design and robustness analysis

estimators on MATLAB/SIMULINK platform

respectively.

battery SOC (bottom).

70 New Trends in Electrical Vehicle Powertrains

subsections of this section.

The simulation results of the real-time implementation of proposed UKF SOC nonlinear estimator in MATLAB R17a environment are shown in Figures 7 and 8. In Figure 7 are presented the simulation results for EMC SOC true value versus EMC SOC-UKF and ADVI-SOR MATLAB platform estimated values.

Also, at the bottom of Figure 7 is shown the EMC battery output terminal true value DC voltage versus EMC-UKF battery output terminal estimated DC voltage. The simulation results reveal an accurate SOC estimation values, and also a very good robustness of UKF estimator to the changes in initial SOC value (guess value, SOCinit = 20%). In Figure 8 are shown the Li-Ion EMC polarization voltages versus Li-Ion EMC -UKF estimates, and in Figure 9 is represented the robustness of EMC-UKF SOC nonlinear estimator to a gradual

Figure 7. Li-Ion EMC SOC and output terminal battery DC voltage versus Li-Ion EMC-UKF and ADVISOR estimated values for an UDDS cycle current profile test.

The process state estimates are used to predict and smooth the stochastic process, and with the innovations can be estimated the parameters of the linear or nonlinear dynamic model [19]. The basic idea of PF SOC estimator is that any probability distribution function (pdf) of a random variable can be represented as a set of samples (particles) as is described in [19], similar thru sigma points UKF SOC estimator technique developed in subsection 3.1 [5]. Each particle has one set of values for each process state variable. The novelty of this method is its ability to

Figure 9. The robustness of Li-Ion EMC-UKF estimator to the changes in internal battery resistance for an UDDS cycle

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Compared to nonlinear UKF SOC estimator design, the nonlinear PF SOC estimator has almost a similar approach that does not require any local linearization technique, i.e. Jacobean matrices, or any rough functional approximation. Also, the PF can adjust the number of particles to match available computational resources, so a tradeoff between accuracy of estimate and required computation. Furthermore, it is computationally compliant even with complex, non-linear, non-Gaussian models, as a tradeoff between approximate solutions to complex nonlinear dynamic model versus exact solution to approximate dynamic model [6, 19]. In the Bayesian approach to dynamic state estimation the PF estimator attempts to construct the posterior probability function (pdf) of a random state variable based on available information, including the set of received measurements. Since the pdf represents all available statistical information, it can be considered as the complete solution to the optimal estimation problem. More information useful for design and implementation in real-time of a nonlinear PF estimator can be found in the fundamental work [19]. Since we follow the same design procedure steps to build and implement in real-time a nonlinear PF estimator, as is developed in [19], we are focused only on the implementation aspects. The simulation results in real-time

In Figure 10 are shown the simulation results for EMC SOC true value versus EMC SOC-PF and ADVISOR MATLAB platform estimated values. The number of filter particles is set to 1000, a very influent tuning parameter for an accurate SOC estimation performance. At the bottom of Figure 10 are shown the Li-Ion EMC battery output terminal true values of DC instantaneous voltage versus Li-Ion EMC battery output terminal DC voltage estimated by the

represent any arbitrary distribution, even if for non-Gaussian or multi-modal pdfs [5].

MATLAB R17a environment are shown in Figures 10 and 11.

current profile test.

Figure 8. Li-Ion EMC polarization DC voltages versus Li-Ion EMC-UKF estimated values for an UDDS cycle current profile test.

increase in the internal resistance by 1.5 until 2 times of its initial value. The simulation results reveal a significant decrease in Li-Ion EMC UKF SOC estimator performance to an increase in internal resistance, but still remains convergent to EMC measurements after a long transient.

#### 3.2. Particle filter real-time estimator design and robustness analysis

In this subsection we propose a real-time PF SOC nonlinear estimator with a similar" prediction-corrector" structure found to the nonlinear UKF SOC estimator described in the previous subsection 3.1. Consequently, is expected that the proposed nonlinear PF SOC estimator to update recursively an estimate of the state and to find the innovations driving a stochastic process given a sequence of observations, as is shown in detail in the original work [19]. In [19] is stated that the PF SOC estimator accomplishes this objective by a sequential Monte Carlo method (bootstrap filtering), a technique for implementing a recursive Bayesian filter by Monte Carlo simulations.

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Figure 9. The robustness of Li-Ion EMC-UKF estimator to the changes in internal battery resistance for an UDDS cycle current profile test.

The process state estimates are used to predict and smooth the stochastic process, and with the innovations can be estimated the parameters of the linear or nonlinear dynamic model [19]. The basic idea of PF SOC estimator is that any probability distribution function (pdf) of a random variable can be represented as a set of samples (particles) as is described in [19], similar thru sigma points UKF SOC estimator technique developed in subsection 3.1 [5]. Each particle has one set of values for each process state variable. The novelty of this method is its ability to represent any arbitrary distribution, even if for non-Gaussian or multi-modal pdfs [5].

Compared to nonlinear UKF SOC estimator design, the nonlinear PF SOC estimator has almost a similar approach that does not require any local linearization technique, i.e. Jacobean matrices, or any rough functional approximation. Also, the PF can adjust the number of particles to match available computational resources, so a tradeoff between accuracy of estimate and required computation. Furthermore, it is computationally compliant even with complex, non-linear, non-Gaussian models, as a tradeoff between approximate solutions to complex nonlinear dynamic model versus exact solution to approximate dynamic model [6, 19]. In the Bayesian approach to dynamic state estimation the PF estimator attempts to construct the posterior probability function (pdf) of a random state variable based on available information, including the set of received measurements. Since the pdf represents all available statistical information, it can be considered as the complete solution to the optimal estimation problem. More information useful for design and implementation in real-time of a nonlinear PF estimator can be found in the fundamental work [19]. Since we follow the same design procedure steps to build and implement in real-time a nonlinear PF estimator, as is developed in [19], we are focused only on the implementation aspects. The simulation results in real-time MATLAB R17a environment are shown in Figures 10 and 11.

increase in the internal resistance by 1.5 until 2 times of its initial value. The simulation results reveal a significant decrease in Li-Ion EMC UKF SOC estimator performance to an increase in internal resistance, but still remains convergent to EMC measurements after a long transient.

Figure 8. Li-Ion EMC polarization DC voltages versus Li-Ion EMC-UKF estimated values for an UDDS cycle current

Figure 7. Li-Ion EMC SOC and output terminal battery DC voltage versus Li-Ion EMC-UKF and ADVISOR estimated

In this subsection we propose a real-time PF SOC nonlinear estimator with a similar" prediction-corrector" structure found to the nonlinear UKF SOC estimator described in the previous subsection 3.1. Consequently, is expected that the proposed nonlinear PF SOC estimator to update recursively an estimate of the state and to find the innovations driving a stochastic process given a sequence of observations, as is shown in detail in the original work [19]. In [19] is stated that the PF SOC estimator accomplishes this objective by a sequential Monte Carlo method (bootstrap filtering), a technique for implementing a recursive Bayesian

3.2. Particle filter real-time estimator design and robustness analysis

filter by Monte Carlo simulations.

profile test.

values for an UDDS cycle current profile test.

72 New Trends in Electrical Vehicle Powertrains

In Figure 10 are shown the simulation results for EMC SOC true value versus EMC SOC-PF and ADVISOR MATLAB platform estimated values. The number of filter particles is set to 1000, a very influent tuning parameter for an accurate SOC estimation performance. At the bottom of Figure 10 are shown the Li-Ion EMC battery output terminal true values of DC instantaneous voltage versus Li-Ion EMC battery output terminal DC voltage estimated by the

Figure 10. EMC SOC and output terminal voltage versus EMC-PF estimated values during UDDS cycle current profile test.

amongst UKF, PF, and NOE SOC estimators. We follow the same design procedure steps for its design and implementation in a real-time MATLAB R2017a simulation environment as in [1]. The estimator design is based on an important information provided by the linear structure of

Figure 12. The robustness of EMC-PF estimator to the changes in internal battery resistance for an UDDS cycle current

Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles…

According to this structure all three state variables x1ð Þk , x2ð Þk , x3ð Þ¼ k SOC kð Þ change independently. Precisely, the nonlinear, linear and sliding mode observers are most applied in state estimation problems to eliminate state estimation error using deviation feedback, as is mentioned in [1]. Furthermore, for Li-Ion battery SOC estimation, most of existing observers are model based using for structure design the difference between the estimated value of battery output terminal DC instantaneous voltage and its corresponding measured DC voltage value multiplied by the observer gains to correct the dynamics of all estimated states,

where the matrices <sup>A</sup> and <sup>B</sup> are the same as in Eq. (9), <sup>b</sup>x kð Þ is the estimate of the EMC states vector, and Lk is the observer gains vector (similar to the extended Luenberger observer for nonlinear systems). The particular structure of Li-Ion battery EMC reveals that the output estimation error ey is mainly caused by an inaccurate SOC estimated value, as is stated also in [1]. Consequently, only the SOC state estimate from third discrete state Eq. (15) will be affected,

,

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(15)

ð Þ¼ <sup>k</sup> <sup>x</sup>2ð Þ� <sup>k</sup> <sup>b</sup>x2ð Þ<sup>k</sup> , ey <sup>¼</sup> y kð Þ� <sup>b</sup>y kð Þ

<sup>b</sup>x kð Þ¼ <sup>þ</sup> <sup>1</sup> A x\_ ð Þþ <sup>k</sup> Bu kð Þþ Lkð Þ y kð Þ� <sup>b</sup>y kð Þ ,

<sup>b</sup>y kð Þ¼ <sup>V</sup>bbatt <sup>¼</sup> VOC SOC d� � � <sup>b</sup>x1ð Þ� <sup>k</sup> <sup>b</sup>x2ð Þ� <sup>k</sup> Ru kð Þ

ð Þ¼ <sup>k</sup> <sup>x</sup>1ð Þ� <sup>k</sup> <sup>b</sup>x1ð Þ<sup>k</sup> , ex<sup>2</sup> ð Þ¼ <sup>k</sup> eVC<sup>2</sup>

Lk <sup>¼</sup> <sup>l</sup>1<sup>k</sup> <sup>l</sup>2<sup>k</sup> <sup>l</sup>3<sup>k</sup> ½ �T, <sup>b</sup>x kð Þ¼ ½ � <sup>b</sup>x1ð Þ<sup>k</sup> <sup>b</sup>x2ð Þ<sup>k</sup> <sup>b</sup>x3ð Þ<sup>k</sup> <sup>T</sup> <sup>¼</sup> <sup>V</sup><sup>b</sup> <sup>C</sup><sup>1</sup> <sup>V</sup><sup>b</sup> <sup>C</sup><sup>2</sup> SOC <sup>d</sup> h i<sup>T</sup>

the matrix Eq. (9).

profile test.

as follows:

ex<sup>1</sup> ð Þ¼ k eVC<sup>1</sup>

i.e. the observer gains vector becomes:

Figure 11. Li-Ion EMC polarization voltages versus Li-Ion EMC-PF estimated values during UDDS cycle current profile test.

nonlinear PF estimator. In Figure 11 are shown the Li-Ion EMC polarization DC voltages versus EMC PF estimates. In Figure 12 is shown the robustness test of Li-Ion EMC-PF SOC nonlinear estimator to a gradual increase in the internal battery resistance by same values considered for Li-Ion EMC-UKF SOC estimator.

The simulation results reveal a good robustness and convergence of Li-Ion EMC-PF estimator, but with a lot variance in the estimated values. Overall, the simulation results reveal a fast PF estimator convergence, a good SOC filtering, an accurate SOC estimation value, and also a very good robustness of PF estimator to big changes in the initial SOC value (guess value, SOCinit = 20%), and slightly slow behavior to an increase in internal resistance of the Li-Ion battery.

#### 3.3. Nonlinear observer real-time estimator

In this subsection, a nonlinear observer SOC estimator (NOE) is under consideration. It is proposed to have more flexibility for a suitable choice of the best Li-Ion battery SOC estimator Estimation Techniques for State of Charge in Battery Management Systems on Board of Hybrid Electric Vehicles… http://dx.doi.org/10.5772/intechopen.76230 75

Figure 12. The robustness of EMC-PF estimator to the changes in internal battery resistance for an UDDS cycle current profile test.

amongst UKF, PF, and NOE SOC estimators. We follow the same design procedure steps for its design and implementation in a real-time MATLAB R2017a simulation environment as in [1]. The estimator design is based on an important information provided by the linear structure of the matrix Eq. (9).

According to this structure all three state variables x1ð Þk , x2ð Þk , x3ð Þ¼ k SOC kð Þ change independently. Precisely, the nonlinear, linear and sliding mode observers are most applied in state estimation problems to eliminate state estimation error using deviation feedback, as is mentioned in [1]. Furthermore, for Li-Ion battery SOC estimation, most of existing observers are model based using for structure design the difference between the estimated value of battery output terminal DC instantaneous voltage and its corresponding measured DC voltage value multiplied by the observer gains to correct the dynamics of all estimated states, as follows:

nonlinear PF estimator. In Figure 11 are shown the Li-Ion EMC polarization DC voltages versus EMC PF estimates. In Figure 12 is shown the robustness test of Li-Ion EMC-PF SOC nonlinear estimator to a gradual increase in the internal battery resistance by same values

Figure 11. Li-Ion EMC polarization voltages versus Li-Ion EMC-PF estimated values during UDDS cycle current profile test.

Figure 10. EMC SOC and output terminal voltage versus EMC-PF estimated values during UDDS cycle current profile test.

The simulation results reveal a good robustness and convergence of Li-Ion EMC-PF estimator, but with a lot variance in the estimated values. Overall, the simulation results reveal a fast PF estimator convergence, a good SOC filtering, an accurate SOC estimation value, and also a very good robustness of PF estimator to big changes in the initial SOC value (guess value, SOCinit = 20%), and slightly slow behavior to an increase in internal resistance of the Li-Ion

In this subsection, a nonlinear observer SOC estimator (NOE) is under consideration. It is proposed to have more flexibility for a suitable choice of the best Li-Ion battery SOC estimator

considered for Li-Ion EMC-UKF SOC estimator.

74 New Trends in Electrical Vehicle Powertrains

3.3. Nonlinear observer real-time estimator

battery.

$$\begin{aligned} \widehat{\mathbf{x}}(k+1) &= A \cdot \widehat{\mathbf{x}}\,(k) + Bu(k) + L\_k(y(k) - \widehat{y}(k)),\\ L\_k &= \left[l\_{1k}l\_{2k}l\_{3k}\right]^T, \widehat{\mathbf{x}}(k) = \left[\widehat{\mathbf{x}}\_1(k) \,\widehat{\mathbf{x}}\_2(k) \,\widehat{\mathbf{x}}\_3(k)\right]^T = \left[\widehat{V}\_{\mathbb{C}\_1} \,\widehat{V}\_{\mathbb{C}\_2} \,\, \widehat{\mathbf{SOC}}\right]^T, \\ \widehat{y}(k) &= \widehat{V}\_{\text{buff}} = \text{VOC}\left(\widehat{\mathbf{SOC}}\right) - \widehat{\mathbf{x}}\_1(k) - \widehat{\mathbf{x}}\_2(k) - Ru(k) \\ e\_{\mathbf{x}\_1}(k) &= e\_{V\_{\mathbb{C}\_1}}(k) = \mathbf{x}\_1(k) - \widehat{\mathbf{x}}\_1(k), e\_{\mathbf{x}\_2}(k) = e\_{V\_{\mathbb{C}\_2}}(k) = \mathbf{x}\_2(k) - \widehat{\mathbf{x}}\_2(k), e\_{\mathbf{y}} = \mathbf{y}(k) - \widehat{\mathbf{y}}(k) \end{aligned} \tag{15}$$

where the matrices <sup>A</sup> and <sup>B</sup> are the same as in Eq. (9), <sup>b</sup>x kð Þ is the estimate of the EMC states vector, and Lk is the observer gains vector (similar to the extended Luenberger observer for nonlinear systems). The particular structure of Li-Ion battery EMC reveals that the output estimation error ey is mainly caused by an inaccurate SOC estimated value, as is stated also in [1]. Consequently, only the SOC state estimate from third discrete state Eq. (15) will be affected, i.e. the observer gains vector becomes:

$$l\_{1k} = 0, l\_{2k} = 0, l\_{3k} \neq 0 \tag{16}$$

This outcome improves significant the NOE SOC estimation accuracy and simplifies the structural complexity of the proposed nonlinear observer estimator. The dynamics of the nonlinear observer estimation errors can be described by the following differential equations [1]:

$$\begin{aligned} e\_{x\_1}(k+1) &= \left(1 - \frac{T\_s}{T\_1}\right) e\_{x\_1}(k) \\ e\_{x\_2}(k+1) &= \left(1 - \frac{T\_s}{T\_2}\right) e\_{x\_2}(k) \\ e\_{\text{SOC}}(k+1) &= e\_{x\_3}(k) = l\_{\&} e\_{\text{\%}}(k) \end{aligned} \tag{17}$$

In [1] is proved that all three states estimation errors described by the system of Eq. (17) converge asymptotically to zero in steady-state, and the observer gain for the new simplified structure is approximated by an adaptive law:

$$l\_{3k} = l\_{30} + \alpha e^{\beta \left( |e\_y| \right)}, \ l\_{30} > 0, \alpha < 0, \beta < 0 \tag{18}$$

The simulation results in Figure 15 reveal slightly slow robustness to an increase in internal resistance of Li-Ion battery compared to simulation results from Figure 13, but still the convergence is reached in a long transient with much variation in the SOC estimates. Overall, the simulation results from this section reveal a fast NOE estimator convergence, a good SOC filtering, an accurate SOC estimation value, and also a very good robustness of PF estimator to big changes in the initial SOC value (the same guess value as for UKF and PF, SOCinit = 20%), compared to a gradual degrade in SOC estimation performance to an increase in internal

Figure 14. Li-Ion EMC-NOE polarization DC voltages versus Li-Ion EMC-NOE estimated values during UDDS cycle

Figure 13. Li-Ion EMC SOC and output terminal DC voltage versus Li-Ion EMC-NOE estimated values during UDDS

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battery resistance.

current profile test.

cycle current profile test.

that allows the value of l3<sup>k</sup> to change dynamically according to the deviation between the measured battery output DC voltage and battery EMC output DC voltage. In Eq. (18), l30, α and β are tuning parameters designed to adjust the adaptive property of l3k. Amongst them, l<sup>30</sup> determines the convergence rate of the proposed NOE at first "inaccurate" stage, the coefficients α and β are used to adjust observer gain l3<sup>k</sup> when the SOC state estimation also reaches "accurate" stage, as is stated in [1].

Furthermore, three main assumptions are formulated in [1] to tune the values of EMC-NOE parameters l30, α and β: (a) l3<sup>k</sup> ≥ 0 to ensure the stability of the proposed NOE; (b) if SOC state estimation error is large, the value of l3<sup>k</sup> should be big enough to ensure a fast convergence rate; (c) if the voltage estimation error is small, the value of l3<sup>k</sup> should be small enough to avoid SOC estimation "jitter" effect. By extensive simulations performed in a real-time MATLAB R2017a simulation environment the requirements (a), (b), (c) are met if the NOE parameters l30, α and β are tuned for the following values: l<sup>30</sup> = 0.3, α = �0.01, and β = � 1. The simulation results on the estimation performance of Li-Ion EMC-NOE are shown in Figures 13 and 14.

In Figure 13 are shown the simulation results for Li-Ion EMC SOC true value versus Li-Ion EMC SOC-NOE and ADVISOR MATLAB platform estimated values. At the bottom of Figure 13 are presented the Li-Ion EMC battery output terminal true values DC voltage versus Li-Ion EMC battery output terminal DC voltage estimated by the proposed Li-Ion EMC NOE SOC estimator. In Figure 14 are displayed the Li-Ion EMC polarization DC voltages versus Li-Ion battery EMC-NOE estimates.

In Figure 15 is depicted the robustness of Li-Ion EMC-NOE SOC estimator to an increase in internal battery Li-Ion resistance with the same values as used for the previous nonlinear estimators, EMC UKF and EMC PF respectively.

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l1<sup>k</sup> ¼ 0, l2<sup>k</sup> ¼ 0, l3<sup>k</sup> 6¼ 0 (16)

<sup>β</sup>ð Þ <sup>j</sup>ey<sup>j</sup> , l<sup>30</sup> > 0, α < 0, β < 0 (18)

(17)

This outcome improves significant the NOE SOC estimation accuracy and simplifies the structural complexity of the proposed nonlinear observer estimator. The dynamics of the nonlinear

eSOCð Þ¼ k þ 1 ex<sup>3</sup> ð Þ¼ k l3keyð Þk

In [1] is proved that all three states estimation errors described by the system of Eq. (17) converge asymptotically to zero in steady-state, and the observer gain for the new simplified

that allows the value of l3<sup>k</sup> to change dynamically according to the deviation between the measured battery output DC voltage and battery EMC output DC voltage. In Eq. (18), l30, α and β are tuning parameters designed to adjust the adaptive property of l3k. Amongst them, l<sup>30</sup> determines the convergence rate of the proposed NOE at first "inaccurate" stage, the coefficients α and β are used to adjust observer gain l3<sup>k</sup> when the SOC state estimation also

Furthermore, three main assumptions are formulated in [1] to tune the values of EMC-NOE parameters l30, α and β: (a) l3<sup>k</sup> ≥ 0 to ensure the stability of the proposed NOE; (b) if SOC state estimation error is large, the value of l3<sup>k</sup> should be big enough to ensure a fast convergence rate; (c) if the voltage estimation error is small, the value of l3<sup>k</sup> should be small enough to avoid SOC estimation "jitter" effect. By extensive simulations performed in a real-time MATLAB R2017a simulation environment the requirements (a), (b), (c) are met if the NOE parameters l30, α and β are tuned for the following values: l<sup>30</sup> = 0.3, α = �0.01, and β = � 1. The simulation results on the estimation performance of Li-Ion EMC-NOE are shown in Figures 13 and 14.

In Figure 13 are shown the simulation results for Li-Ion EMC SOC true value versus Li-Ion EMC SOC-NOE and ADVISOR MATLAB platform estimated values. At the bottom of Figure 13 are presented the Li-Ion EMC battery output terminal true values DC voltage versus Li-Ion EMC battery output terminal DC voltage estimated by the proposed Li-Ion EMC NOE SOC estimator. In Figure 14 are displayed the Li-Ion EMC polarization DC voltages versus Li-

In Figure 15 is depicted the robustness of Li-Ion EMC-NOE SOC estimator to an increase in internal battery Li-Ion resistance with the same values as used for the previous nonlinear

T1 

T2 

ex<sup>1</sup> ð Þk

ex<sup>2</sup> ð Þk

observer estimation errors can be described by the following differential equations [1]:

ex<sup>1</sup> ð Þ¼ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>1</sup> � Ts

ex<sup>2</sup> ð Þ¼ <sup>k</sup> <sup>þ</sup> <sup>1</sup> <sup>1</sup> � Ts

structure is approximated by an adaptive law:

76 New Trends in Electrical Vehicle Powertrains

reaches "accurate" stage, as is stated in [1].

Ion battery EMC-NOE estimates.

estimators, EMC UKF and EMC PF respectively.

l3<sup>k</sup> ¼ l<sup>30</sup> þ αe

Figure 13. Li-Ion EMC SOC and output terminal DC voltage versus Li-Ion EMC-NOE estimated values during UDDS cycle current profile test.

Figure 14. Li-Ion EMC-NOE polarization DC voltages versus Li-Ion EMC-NOE estimated values during UDDS cycle current profile test.

The simulation results in Figure 15 reveal slightly slow robustness to an increase in internal resistance of Li-Ion battery compared to simulation results from Figure 13, but still the convergence is reached in a long transient with much variation in the SOC estimates. Overall, the simulation results from this section reveal a fast NOE estimator convergence, a good SOC filtering, an accurate SOC estimation value, and also a very good robustness of PF estimator to big changes in the initial SOC value (the same guess value as for UKF and PF, SOCinit = 20%), compared to a gradual degrade in SOC estimation performance to an increase in internal battery resistance.

Figure 15. The robustness of Li-Ion EMC-NOE estimator to the changes in internal battery resistance for an UDDS cycle current profile test.
