*3.1.2. Modeling framework delineation*

**Figure 4.** Circuit topology of the electric drive system

Considering comprehensiveness, an ESS including a battery and ultracapacitor is studied. The electric drive with only the battery is analyzed similarly and more easily. The circuit topology of the studied electric drive system in AFV is shown in Figure 4. The EV traction motor is a permanent magnet synchronous motor (PMSM) and the three-phase inverter is comprised of six IGBTs. The UC and battery are connected to the low-voltage (LV) and high-voltage (HV) sides respectively of the bidirectional DC-DC converter, which is constructed in a half-bridge topology.

Individual transient mathematical models (including the PMSM, inverter, DC-DC converter, ultracapacitor, and lithium-ion battery) of all components are built first. For instance, the switching state function is introduced to set up a nonlinear model of the inverter and then it is transferred to a linear model with help of the large-signal disturbance method. The lithiumion battery is modeled using the scheme combining with the equivalent circuit and electro‐ chemical factor-assisted electrical parameter adaptive-adjusting.

The speed *ω* and torque *T* of the traction motor are chosen as unified inputs for the integrated systematic model. The individual model is transformed into expressions by *ω*(*t*) and *T*(*t*). Thus, the multidomain models are related to each other using the unified inputs.

The important variables in all individual models can be finally expressed by *ω*(*s*) and *T*(*s*), where s is a Laplace operator. The time-domain mathematical descriptions for these variables could also be acquired using combination of inverse Laplace transformation and some additional mathematical methods (e.g., residues method). Therefore, the multidomain coupling transient model for the AFV electric drives can be derived and complete frequencyand time-domain expressions of all variables as well.

### **3.2. Design of DC active power filter for ESS chaotic current elimination**

DC-DC converter in the AFV electric drive system. Multiple subjects could be melted including energy storage, electric machine, and power electronics. The entire electric drive can be regarded as a whole without separating relationships between components, and a unified and general mathematic expression is capable of identifying specific numeric relations between transient requirements and parameters of components. In addition, the systematic-, circuit-, and component-class parameter optimization aimed at efficiency improvement in electric

Considering comprehensiveness, an ESS including a battery and ultracapacitor is studied. The electric drive with only the battery is analyzed similarly and more easily. The circuit topology of the studied electric drive system in AFV is shown in Figure 4. The EV traction motor is a permanent magnet synchronous motor (PMSM) and the three-phase inverter is comprised of six IGBTs. The UC and battery are connected to the low-voltage (LV) and high-voltage (HV) sides respectively of the bidirectional DC-DC converter, which is constructed in a half-bridge

Individual transient mathematical models (including the PMSM, inverter, DC-DC converter, ultracapacitor, and lithium-ion battery) of all components are built first. For instance, the switching state function is introduced to set up a nonlinear model of the inverter and then it is transferred to a linear model with help of the large-signal disturbance method. The lithiumion battery is modeled using the scheme combining with the equivalent circuit and electro‐

The speed *ω* and torque *T* of the traction motor are chosen as unified inputs for the integrated systematic model. The individual model is transformed into expressions by *ω*(*t*) and *T*(*t*). Thus,

The important variables in all individual models can be finally expressed by *ω*(*s*) and *T*(*s*), where s is a Laplace operator. The time-domain mathematical descriptions for these variables could also be acquired using combination of inverse Laplace transformation and some additional mathematical methods (e.g., residues method). Therefore, the multidomain

chemical factor-assisted electrical parameter adaptive-adjusting.

the multidomain models are related to each other using the unified inputs.

drives of alternative fuel vehicles could be achieved.

*3.1.2. Modeling framework delineation*

10 New Applications of Electric Drives

**Figure 4.** Circuit topology of the electric drive system

topology.

*3.2.1. Exploration of Interior relationship between traction motor output performance and ESS current ripples*

The output A-phase voltage of the inverter can be expressed as its Fourier series expansions, as follows:

$$V\_{\rm AN} = \sum\_{k=0}^{\ast \alpha} V\_k \sin(k\alpha t) \tag{1}$$

where *Vk* denotes the component of order *k*, *ω*represents the fundamental angular frequency. We can define *α* =2*π* / 3 as the angular difference between two of three phases.

Regardless of the unbalanced currents problem, the nonsinusoidal phase currents can be depicted as follows:

$$\dot{i}\_{\Lambda} = \sum\_{m=0}^{\ast \alpha} I\_m \sin(m\alpha t + \varphi\_m),\\ \dot{i}\_{\mathbb{B}} = \sum\_{m=0}^{\ast \alpha} I\_m \sin(m(\alpha t - \alpha) + \varphi\_m),\\ \dot{i}\_{\mathbb{C}} = \sum\_{m=0}^{\ast \alpha} I\_m \sin(m(\alpha t + \alpha) + \varphi\_m) \tag{2}$$

The DC bus current can be considered as sum of currents through three bridges in the inverter. Meanwhile, one bridge's contribution is composed of the currents from upper and lower branches. Consequently, the DC bus current is obtained as:

$$\dot{I}\_{\rm dc} = \dot{I}\_{\rm dc;\lambda,1} + \dot{I}\_{\rm dc;\lambda,2} + \dot{I}\_{\rm dc;\mathcal{B},1} + \dot{I}\_{\rm dc;\mathcal{B},2} + \dot{I}\_{\rm dc;C,1} + \dot{I}\_{\rm dc;C,2} \tag{3}$$

where the subscript 1 represents the upper branch, and 2 represents the lower branch. As an instance, the upper and lower currents for the Phase-A bridge are described as:

$$\dot{\mathbf{i}}\_{\text{dc},\text{A},1} = \mathbf{K}\_{\text{A},1} \dot{\mathbf{i}}\_{\text{A}} \, ^{\prime} \mathbf{i}\_{\text{dc},\text{A},2} = \mathbf{K}\_{\text{A},2} \dot{\mathbf{i}}\_{\text{A}} \tag{4}$$

where the coefficients are *K*A,1 = *V*AN *V*bat + 1 <sup>2</sup> , and *K*A,2 <sup>=</sup> *V*AN *V*bat − 1 <sup>2</sup> . The Phase-B and Phase-C currents can be computed similarly. With expressions given by equations (1)-(4), the upper and lower currents of three bridges are available to be calculated.

Substituting equation (4) and similar expressions for the other two phases into (3), the DC bus current can be obtained in the form of Fourier series as:

$$\begin{split} i\_{\rm dc} &= \sum\_{m=1}^{\ast \ast} \frac{2I\_m}{V\_{\rm bat}} \left\{ \sum\_{k=1}^{\ast \ast} \frac{V\_k}{2} \cos((k-m)\alpha t - \varphi\_w) \times [2\cos(k-m) + 1] \right. \\ &\left. + \sum\_{k=1}^{\ast \ast} \frac{V\_k}{2} \cos((k+m)\alpha t + \varphi\_w) \times [2\cos(k+m) + 1] \right\} \end{split} \tag{5}$$

Finally, after combining similar terms and sequent calculation, the DC bus current expression is simplified to:

$$\dot{I}\_{\rm dc} = \sum\_{w=1}^{\ast \circ} \frac{\Im I\_{\rm m}}{V\_{\rm bat}} \sum\_{k=1}^{\ast \circ} (V\_{3k \ast w} - V\_{3k \cdot w}) \cos(3kot - \varphi\_{\rm m}) \tag{6}$$

#### *3.2.2. Overall descriptions for the novel DC active power filter*

The operation principle of the proposed DC active power filter (APF) is delineated in Figure 5. The motor controller calculates the synchronous electrical angle and synchronous frequency of phase currents after acquiring rotor position and current/voltage signals. Thus, the funda‐ mental frequency and the fundamental phase for the Fourier Transform of the DC bus current are determined with respect to the theoretical analysis in section 3.2.1. It has to be noted that every Fourier Transform decomposition period could give its information to the current allocation of the next period. So it is available to be predicted that the resulted low-frequency components (closely related to the PMSM speed) cannot change in such a short time. The highfrequency components are derived from the difference of the real-time samplings of the DC bus current and low-frequency components calculated in the previous processing period.

**Figure 5.** Operation principle of the proposed APF strategy with help of motor information.

In the peak power demand process (e.g., acceleration, uphill), the UC is commanded as a large power provider. Thus, the low-frequency components from the DC bus current should be shared reasonably by the UC and battery considering their respective power density. In this paper, the current directly from the UC is considered as a control variable; thereafter, the output current reference *i* con \* on the DC-DC converter high-voltage side has to be transformed to the counterpart *i* cap \* on the low-voltage side, complying with the power conservation law and estimated efficiency. The fixed-boundary-layer sliding mode (FBLSM) control is applied to achieve pulse-width-modulation (PWM) duty cycle for one of two IGBTs (choose T1 or T2 regarding power flow direction) due to consideration of eliminating of control chattering usually caused by frequent control switching in normal sliding mode control. In the meantime, the UC discharging stops when UC SOC reaches the desired lower limit, and then the battery or regenerative braking UC immediately helps UC increase its SOC to a preset value for the further use.

### *3.2.3. Improved fast fourier transform for frequency components' decomposition of ESS currents*

The Discrete Fourier Transform (DFT) and Inverse Discrete Fourier Transform (IDFT) are expressed as the following two equations, respectively:

$$X\{k\} = \sum\_{n=0}^{N-1} \mathbf{x}\{n\} e^{-j\frac{2\pi kn}{N}} = \sum\_{n=0}^{N-1} \mathbf{x}\{n\} W\_{N}^{kn}, \quad 0 \le k \le N-1 \tag{7}$$

$$\mathbf{x}\left(n\right) = \frac{1}{N} \sum\_{k=0}^{N-1} X\left(k\right) e^{\frac{j2\pi kn}{N}} = \sum\_{k=0}^{N-1} \mathbf{x}\left(k\right) W\_N^{-kn}, \quad 0 \le n \le N-1 \tag{8}$$

In order to speed up the computation, the normal FFT is used. The FFT calculation is finally transformed to only two computations by using the following expressions:

$$X\{k\} = G(k) + \mathcal{W}\_N^k H(k), \quad 0 \le k \le \frac{N}{2} - 1 \tag{9}$$

$$X\left(k+\frac{N}{2}\right) = G(k) - W\_N^k H(k), \quad 0 \le k \le \frac{N}{2} - 1\tag{10}$$

where *G*(*k*)= ∑ *r*=0 *N* /2−1 *x*(2*r*)*WN* /2 *rk* and *H* (*k*)= ∑ *r*=0 *N* /2−1 *x*(2*r* + 1)*WN* /2 *rk*

dc

12 New Applications of Electric Drives

is simplified to:

1 1 bat

î

*m k k*

+¥ +¥ = = +¥ =

å å

*V*

*m k*

<sup>2</sup> cos(( ) ) [2cos( ) 1] <sup>2</sup>

w j

<sup>ì</sup> <sup>=</sup> <sup>í</sup> - - ´ -+

*m*

Finally, after combining similar terms and sequent calculation, the DC bus current expression

<sup>3</sup> ( )cos(3 ) *<sup>m</sup> km km m*

The operation principle of the proposed DC active power filter (APF) is delineated in Figure 5. The motor controller calculates the synchronous electrical angle and synchronous frequency of phase currents after acquiring rotor position and current/voltage signals. Thus, the funda‐ mental frequency and the fundamental phase for the Fourier Transform of the DC bus current are determined with respect to the theoretical analysis in section 3.2.1. It has to be noted that every Fourier Transform decomposition period could give its information to the current allocation of the next period. So it is available to be predicted that the resulted low-frequency components (closely related to the PMSM speed) cannot change in such a short time. The highfrequency components are derived from the difference of the real-time samplings of the DC bus current and low-frequency components calculated in the previous processing period.

+ -

*m*

ü

(5)

þ

w j

= -- å å (6)

cos(( ) ) [2cos( ) 1] <sup>2</sup>

*<sup>I</sup> <sup>i</sup> V V kt*

w j

dc 3 3 1 1 bat

**Figure 5.** Operation principle of the proposed APF strategy with help of motor information.

*m k*

= =

+¥ +¥

*V*

*3.2.2. Overall descriptions for the novel DC active power filter*

*<sup>V</sup> kmt km*

+ + + ´ ++ ý

*I V <sup>i</sup> kmt km*

1

*k*

å

To minimize the side lobes, the Blackman-Harris Window as a generalization of the Hamming family, is given by:

$$w(n) = \sum\_{k=0}^{3} (-1)^{k} a\_{k} \cos\left(\frac{2\pi kn}{N}\right) \tag{11}$$

Therefore, the improved FFT expression with the Blackman-Harris Window can be shown as follows based on equations (9) and (10):

$$X\_w\left(k\right) = \sum\_{i=0}^{\beta} (-1)^i \frac{a\_i}{2} \left[ X(k-i) + X(k+i) \right] \tag{12}$$

With use of the improved FFT, the high-frequency and low-frequency components can be decomposed from the load demand, for the power distribution to UC and battery, as shown in Figure 5.
