**3. State of charge estimation**

*2.2.3. Impedance model interpreted by FOC*

94 New Applications of Electric Drives

given in the following equation [20, 24]:

follows:

*β* =1,

semicircle in the Nyquist plot.

A FOC modeling method is introduced to interpret such an element. A fractional element is

<sup>1</sup> ( ) *fractional <sup>r</sup> Z j*

w

when *r* = −1, it is equivalent to an inductor; when *r* =1, it is equivalent to a capacitor.

<sup>1</sup> ( ) *Z j Warburg*

where *α* ∈ℝ (−1≤*α* ≤1) is an arbitrary number, *W* ∈ℝis the coefficient.

Similarly, assume the equation of *CPE*2 as follows:

w

( ) <sup>2</sup> 2

*C j*

where *β* ∈ℝ (0≤*β* ≤1) is an arbitrary number, and *C*2∈ℝ is the coefficient, especially, when

( ) <sup>2</sup> <sup>1</sup> 2

*CPE*2 is a capacitor and the capacitance is *C*2. The paralleling CPE and resistor will form a

<sup>1</sup> ( )| *Z j CPE C j* w b

b

w

w

<sup>1</sup> ( ) *Z j CPE*

w

( )

w

= (8)

= (9)

= (10)

<sup>=</sup> <sup>=</sup> (11)

*Q j*

where *r* ∈ℝ (−1≤*r* ≤1) is the arbitrary order of the fractional element, which can be an integer or a fraction; *Q* ∈ℝ is the coefficient. When *r* =0, the fractional element is equivalent to a resistor;

Furthermore, the phase of the fractional element listed in Eq. (8) is *rπ* / 2, the magnitude is 20*r* dB dec-1, and the Nyquist plot of the fractional element is a straight line. The slope of the line is a constant value *rπ* / 2, which means the fractional element is a CPE. So, a CPE can be expressed by a fractional element, which is also proved in [24]. In this way, a Warburg element can be modeled by a fractional element Zwarburgf as shown in Eq. (9). For the relationship of fractional factor *α* and the slope of the curve in the low-frequency section, *α*can be determined by the curve slope. A Warburg element can be expressed in terms of a fractional element as

( )

w

a

*W j*

As an essential indicator for li-ion batteries used in EDVs, SOC is a key state to estimate the driving range of an EDV. Defined as the ratio of the remaining capacity to the nominal capacity of the battery, SOC of a battery can be described as:

$$\text{SOC} = \frac{\text{Remaining Capacity}}{\text{Normal Capacity}}\tag{12}$$

The usable SOC range could be extended if an accurate SOC can be obtained. Thus, a smaller battery pack will be able to satisfy the demand of an EDV that right now is equipped with a large battery pack. The price will be much lower and it will be a great help for market penetration of EDVs.

## **3.1. Existing methods for state of charge estimation**

Many SOC estimation methods have been reported in previous literature, including the ampere-hour method, the OCV method, the model-based method, and so forth. The amperehour 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 observer [42-44], etc. could be used in such model-based SOC estimation methods.

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 liion battery.

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.
