3. Equivalent circuit model

Figure 2. PWM signal generated using the PRBS method. (a) Photo-coupler circuit for electrical isolation and multiple-

voltage generation. (b) General topology of the isolated DC/DC converter.

44 New Trends in Electrical Vehicle Powertrains

Figure 3. Several topologies for implementing the proposed isolation circuit.

This high-voltage system connected to the insulation monitoring circuit can be modeled as an equivalent circuit, as illustrated in Figure 4, where Vb is the voltage of the high-voltage battery pack, Va is the voltage of the two- or three-phase AC source or AC machine, the inverter/ converter block is the power electronic circuit used to convert power between the AC and DC power stages, and Vg is the output voltage of the isolation circuit. Rp/Rn is the insulation resistor between the ground and the positive/negative terminal of the high-voltage battery pack. Cp and Cn are stray capacitors connected in parallel with Rp and Rn, respectively. Resistor R, which connects the positive terminal of the random voltage sources Vg and the negative terminal of the battery voltage Vb, forms a closed loop between the monitoring circuit and the high-voltage system. We note that the battery voltage Vb, the ground G, and the other insulation resistors and parallel capacitors form the other loop in the circuit.

Therefore, we estimate the insulation resistances Rn and Rp online by an algorithm that is based on the adaptive control law. According to Kirchhoff's circuit laws, the equivalent circuit shown in Figure 4 can be described as follows:

Figure 4. Equivalent circuit model for the insulation monitoring system.

$$I\_P = \mathbb{C}\_P \dot{V}\_P + \frac{V\_P}{R\_P} \tag{1}$$

4. Proof of adaptive algorithm

the dynamic model in an estimated formation as follows:

<sup>e</sup>\_ <sup>¼</sup> <sup>V</sup>\_ <sup>N</sup> � \_

<sup>S</sup>\_ <sup>¼</sup> ee\_ <sup>þ</sup> \_

θeT

¼ �ð Þ θ<sup>1</sup> þ λ e

¼ �ð Þ θ<sup>1</sup> þ λ e

provided that the adaptation law is as follows:

insulation resistance as follows:

description of the process is as follows:

\_ <sup>θ</sup><sup>b</sup> <sup>¼</sup> <sup>Σ</sup>�1=<sup>2</sup>

\_

If we suppose all the actual parameter values and the voltage VN are unknown, we can write

Adaptive Control for Estimating Insulation Resistance of High-Voltage Battery System in Electric Vehicles

where Yb denotes the estimation of Y, and u is one part of the adaptation law that lets all

�θk < ε. We define the estimated error for VN and the parametric vector as e ¼ VN � Vb <sup>N</sup> and

<sup>V</sup><sup>b</sup> <sup>N</sup> <sup>¼</sup> �V<sup>b</sup> <sup>N</sup>VG � VB � <sup>V</sup>\_ <sup>B</sup>

<sup>2</sup> <sup>þ</sup> <sup>θ</sup>e<sup>T</sup>

h ieθ<sup>e</sup> � <sup>θ</sup>1<sup>e</sup>

h i<sup>e</sup> � \_

, u ¼ λe 2

where Σ could be a positive diagonal matrix for design simplicity. We can compute the

<sup>R</sup> and <sup>R</sup><sup>b</sup> <sup>N</sup> <sup>¼</sup> <sup>1</sup>

Figure 5 shows a calculation flowchart for estimating the insulation resistance. A detailed

θ^1 θ^2 � 1 � � <sup>1</sup>

<sup>R</sup> � <sup>1</sup> R^ P

� �Σ<sup>1</sup>=<sup>2</sup>

Invoking the Lyapunov stability criteria shows that the positive-definite function:

<sup>Σ</sup>θ<sup>e</sup> <sup>¼</sup> �V<sup>b</sup> <sup>N</sup>VG � VB � <sup>V</sup>\_ <sup>B</sup>

�VbNe VGe �VBe �V\_ Be

<sup>2</sup> <sup>þ</sup> �V<sup>b</sup> <sup>N</sup>VG � VB � <sup>V</sup>\_ <sup>B</sup>

S ¼ e

h iθ<sup>b</sup> <sup>þ</sup> u, (7)

h iθ<sup>e</sup> � <sup>θ</sup>1<sup>e</sup> � <sup>u</sup>: (8)

� �

http://dx.doi.org/10.5772/intechopen.75468

Σθe (9)

θe

, and λ > 0, (11)

: (12)

<sup>2</sup> � ue � \_ θb T Σθe

θb T Σ<sup>1</sup>=<sup>2</sup> � <sup>&</sup>lt; <sup>δ</sup> and lim<sup>t</sup>!<sup>∞</sup> <sup>θ</sup>bð Þ<sup>t</sup>

� � � 47

(10)

� � �

<sup>V</sup><sup>b</sup> <sup>N</sup> <sup>¼</sup> �V<sup>b</sup> <sup>N</sup>VG � VB � <sup>V</sup>\_ <sup>B</sup>

the estimated values approach their true values, i.e., lim<sup>t</sup>!<sup>∞</sup> <sup>V</sup><sup>b</sup> <sup>N</sup>ð Þ� <sup>t</sup> VNð Þ<sup>t</sup>

θe ¼ θ � θb, respectively. If we differentiate the estimated error, we have:

will approach zero for the negative semi-definite of its derivative; that is:

<sup>2</sup> < 0,

<sup>R</sup><sup>b</sup> <sup>P</sup> <sup>¼</sup> <sup>θ</sup>b<sup>2</sup> θb3

$$I\_N - I\_P = \mathbb{C}\_N \dot{V}\_N + \frac{V\_N}{R\_N} \tag{2}$$

Substituting Eq. (1) into Eq. (2), together with

$$V\_P = V\_B + V\_{N'}$$

yields:

$$\begin{split} I\_N &= \mathbf{C}\_N \dot{V}\_N + \frac{V\_N}{R\_N} + I\_P \\ &= \mathbf{C}\_N \dot{V}\_N + \frac{V\_N}{R\_N} + \mathbf{C}\_P \dot{V}\_P + \frac{V\_P}{R\_P} \\ &= \mathbf{C}\_N \dot{V}\_N + \frac{V\_N}{R\_N} + \frac{V\_P}{R\_P} + \mathbf{C}\_P \left( \dot{V}\_B + \dot{V}\_N \right) \\ &= \left( \mathbf{C}\_N + \mathbf{C}\_P \right) \dot{V}\_N + \mathbf{C}\_P \dot{V}\_B + \frac{V\_N}{R\_N} + \frac{V\_P}{R\_P} \end{split}$$

which can be rewritten as follows:

$$\begin{split} \dot{V}\_{N} &= -\frac{1}{R\_{N}(\mathbb{C}\_{N} + \mathbb{C}\_{P})} V\_{N} - \frac{1}{R\_{P}(\mathbb{C}\_{N} + \mathbb{C}\_{P})} V\_{P} + \frac{1}{\mathbb{C}\_{N} + \mathbb{C}\_{P}} I\_{N} - \frac{\mathbb{C}\_{P}}{\mathbb{C}\_{N} + \mathbb{C}\_{P}} \dot{V}\_{B} \\ &= -\left(\frac{1}{R\_{N}} + \frac{1}{R} + \frac{1}{R\_{P}}\right) \frac{1}{\mathbb{C}\_{N} + \mathbb{C}\_{P}} V\_{N} + \frac{1}{R(\mathbb{C}\_{N} + \mathbb{C}\_{P})} V\_{C} - \frac{1}{R\_{P}(\mathbb{C}\_{N} + \mathbb{C}\_{P})} V\_{B} \\ &- \frac{\mathbb{C}\_{P}}{\mathbb{C}\_{N} + \mathbb{C}\_{P}} \dot{V}\_{B} \end{split} \tag{3}$$

Let us define the parametric vector as follows:

$$\begin{aligned} \boldsymbol{\Theta}^{T} &= \left[ \frac{1}{\mathbf{C}\_{N} + \mathbf{C}\_{P}} \left( \frac{1}{R} + \frac{1}{R\_{N}} + \frac{1}{R\_{P}} \right) \right. \frac{1}{R(\mathbf{C}\_{N} + \mathbf{C}\_{P})} \left. \frac{1}{R\_{P}(\mathbf{C}\_{N} + \mathbf{C}\_{P})} \right. \frac{\mathbf{C}\_{P}}{\mathbf{C}\_{N} + \mathbf{C}\_{P}} \right] \\ &= \begin{bmatrix} \boldsymbol{\theta}\_{1} & \boldsymbol{\theta}\_{2} & \boldsymbol{\theta}\_{3} & \boldsymbol{\theta}\_{4} \end{bmatrix} \end{aligned} \tag{4}$$

and the variable vector as:

$$\begin{aligned} \mathbf{X}^T = \begin{bmatrix} -V\_N & V\_G & -V\_B & -\dot{V}\_B \end{bmatrix} \end{aligned} \tag{5}$$

such that the dynamics of the insulation monitoring system are formulated as follows:

<sup>V</sup>\_ <sup>N</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup>θ, (6)

where the parametric vector includes all the resistance and capacitance values that must be known and the variable vector is composed of the variables that can be evaluated from all the measurements in the system, i.e., VG, VN, and VP.
