**3.2. The proportional integral observer SOC estimation method**

hour method is simple and easy to implement for the calculation of battery SOC for it takes advantage of the definition of SOC. However, it needs the prior knowledge of initial SOC and suffers from accumulated errors of noise and measurement error [30, 31]. The OCV method takes advantage of the certain relationship between SOC and SOC, and is considered to be very accurate. However, to obtain the OCV needs a long rest time, and thus it is difficult be used in real-time applications [30]. Intelligent algorithms, such as fuzzy logic, artificial neural networks, and so forth, have been studied to estimate the SOC [32, 33]. Due to the powerful ability to approximate nonlinear functions, these methods can often obtain a good estimation of SOC. However, the learning process for these methods is often quite computationally demanding and complex, which becomes difficult to be applied in online applications.

Model-based SOC estimation methods are the most popular solutions [34-36]. The main methodology is to take advantage of both the voltage and the current of the battery. Measured currents will be applied to the model and the voltages will be calculated using the present and/ or past states and parameters of the model. The errors between the calculated voltages and the measured voltages are applied to an algorithm to intelligently update the estimation states of the model. The Luenberger observer [37-39], the Kalman filter [40, 41], and the sliding mode

Proposed by D. Luenberger [45] in 1966, the Luenberger observer is now widely used in different applications. Reference [37-39] introduced it to estimate the battery SOC and owned good results. The Kalman filter uses the entire observed input data and output data to find the minimum mean squared error estimation states of the true states of the li-ion batteries [41]. Essentially, the Kalman filter takes advantage of the prior input currents and output terminal voltages to obtain the Kalman gain. This Kalman gain is like the Luenberger gain, which feedbacks to correct the differences between the calculated states and the true states of the li-

Reference [42] introduced the sliding mode observer to estimate the battery SOC. As indicated in the paper, the sliding mode observers inherited the robust properties in SOC estimation. It is robust under modeling uncertainties, but the chatter problem could not be ignored for this

method. The block diagram of these methods is shown in Figure 8.

**Figure 8.** Block diagram of the existing SOC estimation methods.

observer [42-44], etc. could be used in such model-based SOC estimation methods.

ion battery.

96 New Applications of Electric Drives

By an additional integrator, the proportional integral (PI) observer is reported to be more robust with respect to modeling uncertainties [46]. Since modeling errors always exist in a battery model, the PI observer is considered to be able to improve the accuracy and estimation speed of SOC estimation. Thus, the PI observer SOC estimation method is proposed in the section. [47]

A battery model could be considered as follows (referred to as System 1):

$$\begin{cases} \dot{\mathbf{x}} = A\mathbf{x} + Bu\\ y = Cx + Du \end{cases} \tag{13}$$

However, considering the modeling errors, capacity variation, and so forth, System 1 is not sufficient to model the battery. The nonlinear part should be added to the battery model, which could be described as follows (referred to as System 2) [48-50]:

$$\begin{cases} \dot{\mathbf{x}} = A\mathbf{x} + Bu + Ev(\mathbf{x}, u, t) \\ \qquad y = \mathbf{C}\mathbf{x} + Du \end{cases} \tag{14}$$

where *E* describes the influence of the nonlinearities to the different states, and such relation‐ ships could be obtained by experiments and some "trial and error" approaches; *v*(*x*, *u*, *t*) describes the nonlinearities, unknown inputs, and un-modeled dynamics of the plant and may be a nonlinear function of states, inputs, and time; *v*(*x*, *u*, *t*) is referred to as disturbance.

Considering the special applications of the battery for EDVs, the disturbance could be caused by temperature, sensor noise, and so on. Taking temperature as an example, the variation rate could be very slow, and thus *v*˙ ≈0 when the temperature is considered. Meanwhile, the operation temperature range for the battery is limited for the consideration of life cycle and safety. So, *v*(*x*, *u*, *t*) should also be in a small range due to the influence of temperature. For the strict temperature control in EDV applications, the temperature would be stable after a short time, thus lim *t*→*∞ v*(*x*, *u*, *t*) exists for the influence of temperature. It is considered to be Gauss

Noise with zero mean value for sensor noise. Sensor failure could also be considered to be slow changing, and thus the assumption *v*˙ ≈0 could be reasonable. Actually, since the change rate is so small, the sensor drift could be neglected for a certain drive cycle of the EDV as it changes very little for a one-day drive of the vehicle. It is reasonable to assume that lim *v*(*x*, *u*, *t*)=0 for

the influence of current sensor. So it is assumed that the disturbance *v*(*x*, *u*, *t*) is not statedependent, and it is also reasonable to assume that the limitation of disturbance *v*(*x*, *u*, *t*) exists.

According to the definition of the PI observer, the PI observer is designed as follows:

$$\begin{cases} \dot{\tilde{\mathbf{x}}} = A\tilde{\mathbf{x}} + Bu + K\_p(y - \tilde{y}) + K\_{i2}w\\ \dot{w} = K\_{i1}(y - \tilde{y}) \end{cases} \tag{15}$$

*t*→*∞*

Note that variable *w* is defined as the integral of the difference(*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*˜). Vectors *Kp* ∈ℝ2×1 and *Ki*1∈ℝ1×1 *Ki*2∈ℝ2×1 are the proportional and integral gains, respectively. The design block of the PI observer is given in Figure 9.

**Figure 9.** Block diagram of the PI observer SOC estimation method.

The unknown disturbance is considered, which would lead to modeling more accurate battery characteristics. The PI observer is applied to System 2 and when *ex* <sup>=</sup> *<sup>x</sup>*˜ <sup>−</sup> *<sup>x</sup>*, *ev* <sup>=</sup>*<sup>w</sup>* <sup>−</sup>*v*, and *Ki*<sup>2</sup> <sup>=</sup>*<sup>E</sup>* are assumed, the error equations could be obtained as follows:

$$\begin{cases} \dot{e}\_x = Ae\_x - K\_p \mathbf{C} e\_x + K\_{i2} e\_x \\ \dot{w} = -K\_{i1} \mathbf{C} e\_x \end{cases} \tag{16}$$

These equations could be rewritten as:

$$
\begin{pmatrix}
\dot{e}\_x\\ \dot{w}
\end{pmatrix} = \begin{bmatrix}
A - K\_p \mathbf{C} & K\_{i2} \\ -K\_{i1} \mathbf{C} & \mathbf{0} \\ \end{bmatrix} \begin{pmatrix}
e\_x\\ e\_y\\ e\_z
\end{pmatrix} \tag{17}
$$

So,

$$
\begin{pmatrix}
\dot{e}\_x\\\dot{e}\_v
\end{pmatrix} = A\_\epsilon \begin{pmatrix} e\_x\\e\_v \end{pmatrix} - \begin{bmatrix} 0\\I \end{bmatrix} \dot{v} \tag{18}
$$

Since *v*˙ =0 for the certain application as stated above, this equation could be rewritten as follows:

Battery Management System for Electric Drive Vehicles – Modeling, State Estimation and Balancing http://dx.doi.org/10.5772/61609 99

$$\begin{pmatrix} \dot{\boldsymbol{e}}\_{\boldsymbol{x}} \\ \dot{\boldsymbol{e}}\_{\boldsymbol{v}} \end{pmatrix} = \boldsymbol{A}\_{\boldsymbol{e}} \begin{pmatrix} \boldsymbol{e}\_{\boldsymbol{x}} \\ \boldsymbol{e}\_{\boldsymbol{v}} \end{pmatrix} \tag{19}$$

If following matrix pair is observable, *Ae*could be arbitrarily assigned.

$$
\begin{pmatrix}
\begin{bmatrix}
A & E \\
0 & 0
\end{bmatrix}
\end{pmatrix}
\begin{bmatrix}
\mathbf{C} & \mathbf{0}
\end{bmatrix}
\tag{20}
$$

which is equivalent to:

Note that variable *w* is defined as the integral of the difference(*<sup>y</sup>* <sup>−</sup> *<sup>y</sup>*˜). Vectors *Kp* ∈ℝ2×1 and *Ki*1∈ℝ1×1 *Ki*2∈ℝ2×1 are the proportional and integral gains, respectively. The design block of

The unknown disturbance is considered, which would lead to modeling more accurate battery characteristics. The PI observer is applied to System 2 and when *ex* <sup>=</sup> *<sup>x</sup>*˜ <sup>−</sup> *<sup>x</sup>*, *ev* <sup>=</sup>*<sup>w</sup>* <sup>−</sup>*v*, and *Ki*<sup>2</sup> <sup>=</sup>*<sup>E</sup>*

1

*w K Ce* ìï =- +

*x x p x iv i x*

*e Ae K Ce K e*

2

2

<sup>1</sup> 0 *x p i x*

*e A KC K e w K C e* æ ö é ù - æ ö ç ÷ <sup>=</sup> ê úç ÷ ç ÷ è ø ê ú ë û - è ø

> 0 *x x e v v*

*e e A v e eI* æö æö é ù ç÷ ç÷ = - ê ú èø èø ë û

Since *v*˙ =0 for the certain application as stated above, this equation could be rewritten as

*i v*

& (16)

& (17)

& & & (18)

the PI observer is given in Figure 9.

98 New Applications of Electric Drives

**Figure 9.** Block diagram of the PI observer SOC estimation method.

These equations could be rewritten as:

So,

follows:

are assumed, the error equations could be obtained as follows:

í ï = - î &

&

$$
tau \left\{ \begin{bmatrix} A & K\_{l2} \\ C & 0 \end{bmatrix} \right\} = n + r \tag{21}
$$

where *r* is the dimension of *v*, which is assumed to be 1. Since *n* =2 according to the battery model, the rank should be 3.

Substitute the parameters of the battery model into the matrix:

$$\text{rank}\begin{Bmatrix}\begin{bmatrix}A&K\_{i2}\\C&0\end{bmatrix}\end{bmatrix} = \text{rank}\begin{Bmatrix}-\frac{1}{R\_2C\_2}&0&K\_{i2\_1}\\0&0&K\_{i2\_2}\\1&a\_i&0\end{bmatrix}\tag{22}$$

The matrix is full rank, so the conditions are satisfied that *Ae* could be arbitrarily assigned.

By utilizing the pole place method or LQ method, *Kp*and *Ki*<sup>1</sup> could be selected to assure *Ae* is Hurwitz. Since *Ae* is Hurwitz, the system is convergent. Thus, when *t* →*∞*, errors will tend to be zero, which means when *t* →*∞*, *x*˜ would converge to *x*. Take the battery model in this chapter for example, the estimated SOC would converge to the true SOC.
