A Comparative Study Between a Perturb and Observe Based Passivity and a Classical Perturb and Observe Based PI for the Thermoelectric Generator

Thermoelectric energy is renewable source of energy, are able to convert the heat energy to electrical power. The thermoelectric generator consists of a number of thermocouples that are connected electrically in series to increase the voltage and thermally in parallel [1], as example, the HZ-20 module which it consists of 71 thermocouples [2, 3]. If a load is connected with the TEM HZ-20, a direct current flows and its power output of this device varies with temperature difference.

To extract maximum power from thermoelectric system, the Boost converter with MPPT controller is used. Many MPPT algorithms have been proposed in the literature [4]. The method of Perturb and observe (P&O) is the most commonly used because its ease and simple implantation. The main disadvantage of this algorithm that when the MPP is reached, the output power oscillates around the MPP [5]. Therefore, resulting in a power loss in the thermoelectric system.

In this paper, we propose an MPPT Perturb and observe algorithm based on a PI controller to achieve better TEM output performance. The main problem of using the PI controller is tuning its parameters to achieve optimum performance. However, the problems of stability get avoided in Perturb and observe based passivity control. The main purpose of this paper is to present a comparative analysis between the conventional Perturb and Observe (P&O) algorithm, the P&O method using a PI and the Perturb and observe based passivity for extracting the maximum power from the TEM system. The results show the Perturb and observe based passivity has much faster tracking time than the P&O method using a PI and the oscillation can be set to near zero.

This paper is organized as follows: section 1 presents the basic system configuration of the thermoelectric power generation system, operation characteristics, section 2 presents the theory of passivity (modeling of the Boost converter by Euler Lagrange) and a configuration of the algorithm perturb and observe based passivity approach (P&O/EL-PBC) MPPT. Section 3 discuss the simulation results with comparisons between a perturb and observe based passivity and a classical perturb and observe based PI in various situation, using MATLAB/SIMULINK tools and without it ending with the conclusions presented in section 4.

Figure 3 shows the output power generated as a function of the current for the HZ-20 TEG module. We can see that the output power increases with rising temperature difference.

figure_3.jpg

Figure 3. P-I Characteristic of the module or different temperature gradients

The TEM will not automatically work at the MPP because it changes over the changes different temperature gradients. Maximum Power Point Tracking (MPPT) is used to track the MPP and makes TEM module work at the MPP all the time.

Perturb and Observe based passivity use to have a faster controller response and to increase system stability once reached the MPP. This technique demonstrates its effectiveness and makes it among the best control for the tracking MPP. In this article a Lagrangian dynamics is presented.

5.1 The basic principles of Lagrangian

The Lagrangian of the system is represented by the differential equation as follows [10-11]:

$\frac{d}{d t} \frac{\partial L(q, \dot{q})}{\partial \dot{q}}-\frac{\partial L(q \dot{q})}{\partial q}=-\frac{\partial F(\dot{q})}{\partial \dot{q}}+Q$  (4)

where:

L: The Lagrangian of the system, represented as the magnetic co-energy of the inductive elements and the electric field energy of the capacitive elements:

$L(q, \dot{q})=T(\dot{q}, q)-V(q)$   (5)

T: is kinetic energy.

V: is potential energy.

F: the Rayleigh dissipation function is given by:

$F_{q}=\frac{1}{2} \dot{q}^{T} R \dot{q}$  (6)

Q: exogenous inputs to the system.

5.2 Modeling of Boost converter by Lagrangian

Figure 5 shows basic circuit of a DC-DC boost converter circuit. The DC-DC boost converter will boost up the output voltage to be higher than input voltage [7].

figure_5.jpg

Figure 5. Boost converter

The subsystem’s Euler-Lagrange model is established as follows:

$\left\{\begin{array}{c}{T_{\mu}(\dot{q})=\frac{1}{2} L q_{L}^{2}} \\ {V_{\mu}(q)=\frac{1}{2 C} q_{C}^{2}} \\ {F_{\mu}(\dot{q})=\frac{1}{2} R_{c h}\left((1-\mu) \dot{q}_{L}-\dot{q}_{C}\right)^{2}} \\ {Q_{\mu}=\left(\begin{array}{c}{V} \\ {0}\end{array}\right)}\end{array}\right.$  (7)

where q_L and q_C are the quantity of charge in the inductance and the output capacitance respectively. Moreover, the switch position takes the value 0 or 1. Now, Lagrangian can be formulated as:

$L_{\mu}(q, \dot{q})=T_{\mu}(\dot{q})-V_{\mu}(q)=\frac{1}{2}\left(L \dot{q}_{L}^{2}-\frac{q_{C}^{2}}{C}\right)$(8)

Then, by replacing (7) and (8) in (4) it yields to:

$\left(\begin{array}{cc}{L} & {0} \\ {0} & {C}\end{array}\right) \dot{x}=\left(\left(\begin{array}{cc}{0} & {-(1-\mu)} \\ {1-\mu} & {0}\end{array}\right)-\left(\begin{array}{cc}{0} & {0} \\ {0} & {\frac{1}{R_{c h}}}\end{array}\right))x+\left(\begin{array}{c}{V} \\ {0}\end{array}\right)\right.$ (9)

That is, the input current defined as $x_{1}=\dot{q}_{L}$ and the output voltage capacitor is represented as $x_{2}=q_{c} / C$. Thus, the model of our system (equation 9) is described in Lagrangian form;

$M \dot{x}+(1-\mu) J x+R x=E$ (10)

where M the diagonal matrix, and where J=-J^Tantisymmetric and its element are given by (-1, 0, 1). Then the vector E=[V 0] represents the exogenous inputs.

$M=\left(\begin{array}{cc}{L} & {0} \\ {0} & {C}\end{array}\right) \quad J=\left(\begin{array}{cc}{0} & {1} \\ {-1} & {0}\end{array}\right) R=\left(\begin{array}{cc}{0} & {0} \\ {0} & {\frac{1}{R_{c h}}}\end{array}\right)$

Thus, we introduce the damping injection function in the system to ensure the asymptotic stability.

The damping injection function consists in modifying the Rayleigh dissipation function as follow:

$R=R_{d}-R_{B}$    (11)

where:

R_d: the desired dissipation.

R_B: the injected damping matrix.

The injected damping matrix can be described through the following equation:

$R_{B}=\left(\begin{array}{cc}{R_{1}} & {0} \\ {0} & {0}\end{array}\right)$   (12)

We consider that a storage function of the system can be chosen according to the form:

$H(x)=\frac{1}{2} x^{T} M x$   (13)

So, we can propose as a desired storage function:

$H_{d}(x)=\frac{1}{2} \widetilde{x}^{T} M \widetilde{x}$(14)

where:

$\breve{\mathcal{X}}$ ; Then, x_d desired value of x.

If we replace the error dynamics $\breve{\mathcal{X}}$  in the system equation (10), we find:

$M \dot{\tilde{x}}+(1-\mu) J \tilde{x}+R \tilde{x}=$

$E-M \dot{x}_{d}-(1-\mu) J x_{d}-R x_{d}=\Phi$ (15)

We ensure the stabilization of our system, if we set the Φ=0.

$M \dot{\tilde{x}}+(1-\mu) J \widetilde{x}+R \widetilde{x}=0$  (16)

The error behavior is asymptotically stable to zero; it means that is independently of µ.

In order to satisfy (16), one must demand:

$M \dot{x}_{d}+(1-\mu) J x_{d}+R x_{d}-R_{1 B} \widetilde{x}=E$   (17)

Finally the scalar form:

$\left\{\begin{array}{c}{L \dot{x}_{1 d}+(1-\mu) x_{2 d}-R_{1}\left(x_{1}-x_{1 d}\right)=V} \\ {C \dot{x}_{2 d}-(1-\mu) x_{1}+\frac{1}{R_{c h}} x_{2 d}=0}\end{array}\right.$ (18)

Then, the solving of equation (18) results:

$\mu=1-\frac{1}{V_{d}}\left(V+R_{1}\left(I_{L}-\frac{V_{d}^{2}}{V \cdot R_{c h}}\right)\right)$ (19)

where:

V_d: is the output MPPT P&O voltage delivered to the passivity bloc control system.

figure_6.jpg

Figure 6. P&O/PBC MPPT algorithm

This paper presents the comparison of P&O MPPT, P&O besed PI and P&O/EL-PBC in relation to their performance. The results indicate that P&O/PI method has low efficiency because of its lack of speed in tracking the MPPT. But, the P&O/EL-PBC method shows the effectiveness of the last to eliminate the oscillations and achieve less response time successfully.