1. Introduction

2. Bond-graph modeling of the energy system

2.1. Modeling of the photovoltaic source

The photovoltaic generator is a special finite energy source of non-linear current-voltage characteristic.

In order to take into account physical phenomena such as contact resistances and leakage current through the edges of the solar cells as well as cell ageing, the model is completed by two series resistances Rs and shunt Rsh as shown in the equivalent electrical diagram of the photovoltaic cell in Figure 1 [13,14,15].

For the bond-graph representation of the photovoltaic diode, we model it by a nonlinear resistor whose current-voltage relationship is denoted by a nonlinear function, whose current is:

$$I_D=I_S\left(\exp \left(\frac{v_p}{v_t}\right)-1\right)$$

(1)

The model is reduced to the ideal model and the electrical characteristic of the PV generator is described by the following equation:

$$I_p=I_{ph}-I_s\left[\exp \left(\frac{V_p}{V_T}\right)-1\right]$$

(2)

The bond-graph model of a photovoltaic source coupled to a RL load is given in Figure 2. The capacitor Ca is often used in the PV system for start-up assistance during low sunlight [7,18].

To simulate the characteristic of the photovoltaic panel I=f(V), we use the BG model represented by Figure 3. In this simulation the receiver is a capacitor initially discharged.

The parameters related to the PV source of the pilot unit are identified following an experimental characterization of the PV source, performed on the pilot unit. We present some results of this experimentation during one day.

Figure 4. External view of the water treatment pilot unit.

Figure 4. External view of the water treatment pilot unit.

The results of the PV field parameter identification used are:

Table 1. Parameters identified for a PV field.

Table 1. Parameters identified for a PV field.

Iph(A)	Is(mA)	Rs(Ω)	Rsh	VT (V)
15.8	0.0497	0.107	190.67	2.95

The simulated photovoltaic panel characteristic I = f (V) is shown in the following Figure 5:

The experimental results are shown in the following Figure 6, Figure 7 and Figure 8:

2.2. Modeling of energy storage batteries

For an ideal model, the battery is represented by a simple voltage source Eb in series with a resistance rb (represents the internal resistance of the battery) and a storage capacity Cb to model the effect of charge and discharge of the battery [19].

Figure 9. Equivalent electrical diagram of a battery.

Figure 9. Equivalent electrical diagram of a battery.

This model has an advantage over other models. Indeed, it does not require direct or real time measurements (battery state of charge, acid concentration, battery current...).

The storage batteries used in the pilot water treatment unit studied can be modeled by a voltage source in series with a capacitor and a resistor (internal resistance of the battery), this is the simplified model of lead batteries.

Figure 10 shows schematically the simplest connection between a series-parallel grouping of batteries and a series-parallel grouping of photovoltaic modules. A protection diode Ds avoids the discharge of the accumulators through the photovoltaic cells during the non-illuminating periods.

The bon-graph model related to this electric circuit is represented by the following Figure 11:

3. Modeling of the water treatment reactor

3.1. Bond graph modeling of the ballast and the UV lamp

The electronic power supply (ballast) of a UV lamp can be broken down into three parts:

₋

The DC/AC converter: inverter;

₋

The transformer;

₋

The resonant circuit (oscillating);

3.1.1. Modeling of the DC/AC converter: inverter

The UV water treatment system is powered by a photovoltaic panel through a storage battery, so it has a continuous power supply of 24V. The resonant circuit requires a square voltage ±E, which is obtained with an inverter [7,20].

In our system, we use a single-phase inverter with its bond graph model is represented by the following Figure 12:

The simulation of the bond-graph model in the previous figure is shown as follows:

Figure 13. Simulation of the output voltage of the inverter.

Figure 13. Simulation of the output voltage of the inverter.

The output voltage varies between 220V and -220V, with a fixed frequency depending on the operation of the electronic ballast.

3.1.2. Modeling of the resonant circuit

To start the lamp, we must create a short duration overvoltage at its terminals. This overvoltage is obtained thanks to a resonant circuit (RLC). The square voltage delivered by the DC/AC converter, drives a transformer to a resonant system formed by a resistor Rr in series with an inductance L and two capacitors Cr1 and Cr2. The receiver is a discharge lamp (UV) which can be assimilated to infinite impedance before the ignition and low value impedance after the ignition.

Taking into account the high impedance presented by the lamp before the ignition, the current chooses the shortest path and crosses Cr2. The overvoltage created by the resonant circuit, ignites the lamp, which by its low impedance value, short-circuits Cr2 [7,20].

Figure 14. Electrical diagram of the power supply UV lamp.

Figure 14. Electrical diagram of the power supply UV lamp.

Figure 15. Bond-graph modeling of the resonant circuit.

Figure 15. Bond-graph modeling of the resonant circuit.

After the application of an overvoltage to the lamp terminals (400V ignition voltage), it takes at least 2 to 3 ms (overvoltage duration) for the lamp to ignite.

Figure 16. Simulation of the resonant circuit.

Figure 16. Simulation of the resonant circuit.

After having modeled the resonant circuit and the UV lamp, we can then replace the square voltage source by a DC voltage source (delivered by the photovoltaic panel) through an inverter (DC/AC converter).

The bond graph model and the simulation are shown in the following Figure 17 and Figure 18:

4. Bond-graph modeling of the hydraulic system

Figure 19. Schematic diagram of the UV treatment system.

Figure 19. Schematic diagram of the UV treatment system.

4.1. Modeling of the DC Motor

The hydraulic system consists of a DC motor connected to a centrifugal pump.

The bond-graph model of the DC machine takes into account the theory of electrical engineering and shows the electrical, electromagnetic and mechanical equations as in the classical model [20,22].

The dynamics of the DC motor is described by the following equations:

$$U_a=R_a{I}_a+L_a\frac{d{I}_a}{dt}+K_{b}w$$

(3)

$$C_m=J_m\frac{dw}{dt}+Fw+C_r$$

(4)

This figure represents the bond graph model of a DC motor:

Figure 20. Bond graph model for a DC motor.

Figure 20. Bond graph model for a DC motor.

4.2. Modeling of the centrifugal pump

The water treatment process consists of circulating the water through a UV reactor to kill viruses and bacteria. Turning on the centrifugal pump causes the impeller to rotate produces a pressure and a speed that determines the flow of the fluid in the hydraulic system [7,23].

The equations that define the operation of the centrifugal pump such as: the theory of elevation 'He', the pressure 'Pe' and the torque transmitted by the motor 'Te' are represented respectively as follows:

$$H_e=\frac{w^2{r_2}^2}{2g}$$

(5)

The pressure developed by the wheel is directly proportional to the angular velocity w. The shaft speed is proportional to the flow rate Q. The developed pressure and the transmitted torque have the following expressions respectively:

$$P_e=\rho g{H}_e=\frac{\rho r_2^2{w}^2}{2}$$

(6)

$$T_e=\frac{\rho r_2^{2}w}{2}Q$$

(7)

With:

$P_e$ : pressure developed by the pump, $T_e$ : motor torque, r2, b2 : pump size, β2 : angle of exit of the vane,

The relation between the pressure supplied by the pump and the rotation speed (ω) can be modeled in bond-graph by a modulated gyrator element: “MGY” transforms the mechanical flow (motor torque) into hydraulic force (pressure).

The pressure losses in the pump are due to:

- Friction of the liquid threads between them and against the walls of the machine such as:

$$T_{fp}=f_{mp}w$$

(8)

This friction is modeled in bond-graph by a resistive element R: fmp

The moment of inertia of the rotating parts of the pump having the expression:

$$T_{ip}=J_{pt}\frac{dw}{dt}$$

(9)

This moment of inertia is modeled in bond-graph by an inertial element I : Jpt

The bond graph model associated with the centrifugal pump is shown in the following Figure 21:

The height difference between the two tanks is modeled by a negative force source as shown in the following equation:

$$Se=-\delta g{H}_e$$

(10)

For a stationary regime (constant speed) the theoretical characteristic He = f(Q) of a pure centrifugal vacuum pump (isolated from the hydraulic network) is of the following form:

Figure 22. Simulation of the He (Q) characteristic from the developed bond-graph model.

Figure 22. Simulation of the He (Q) characteristic from the developed bond-graph model.

5. Structural analysis

5.3. Commandability - structural observability

A system modeled by bond-graph is strictly controllable (respectively observable) if and only if the following conditions are realized.

o

There is a causal path linking a source (respectively a sensor) to each element I and C in integral causality when we put the bond graph in integral causality.

o

All the elements I and C admit a derived causality when we put the B.G. in derived causality. If not dualize the sources (respectively the sensors) to put the remaining I and C in integral, in derived causality.

According to the global bond graph, the system is composed of 11 elements “C” and 5 elements “I” in integral causality admitting derivative causality when we put the bond-graph in derivative causality. The system is of order 16 where no conflict is encountered when we put “I” and “C” in causality integral.

Figure 27. Global derived causal graph model of the pilot water treatment unit.

Figure 27. Global derived causal graph model of the pilot water treatment unit.

Figure 28. Global bond-graph model of the pilot water treatment unit and causal chains.

Figure 28. Global bond-graph model of the pilot water treatment unit and causal chains.

6. Application of status feedback and sliding mode control of the DC motor pump speed

6.1. Control of a motor with contained current by the robust controller in sliding mode

The sliding mode control is divided into two parts [25,26,27]:

₋

Determining a region of state space such that once the system is in that region, it has the desired behavior.

₋

Finding a control law that drives the system into this region of state space.

In order to design a sliding mode control, it is necessary to define a sliding surface S (in the error space x), which guarantees the global stability of the system.

The switching function is a linear vector function of the state of the form:

$$\mathrm{S}\left(\mathrm{x}\right)=\mathrm{C}\mathrm{x}$$

(13)

x(t) ε ℝn ; It is a matrix of appropriate dimension.

6.1.1. Configuration and bond-graph model

The motor pump for water circulation in the UV reactor is of the continuous type. This pump is supplied from the storage battery (Ub). The corresponding bond-graph model is shown in the following Figure 29:

6.1.2. Putting into equation

The bond graph model can be considered as a junction structure connected to four field’s sources, Sensors, dissipative field and storage field.

The vectors associated with this representation are:

X: the state vector composed of the variables p (magnetic flux) on the elements I and q (load on the C elements) split into xi and xd , the vectors associated with the components in integral and derivative causality.

$\dot{\mathit{X}}$: the derivative with respect to time of the state vector.

Z: the complementary of x splits into Zi and Z_d .

u: input vector (sources)

y: output vector (sensors)

Din: input of the dissipative field "R".

Dout: output of the dissipative field "R".

Figure 30. Decomposition of the bond-graph model into junction structure and four fields.

Figure 30. Decomposition of the bond-graph model into junction structure and four fields.

This representation allows building the structure matrix; we are interested in the control of the subsystem constituted by the DC motor pump and the batteries. The vectors X, $\dot{\mathit{X}}$, Z, Din and Dout are given respectively by:

$$\begin{array}{ll}& \mathrm{X}={\left[\begin{array}{ll}{\mathrm{P}}_6& {\mathrm{P}}_2\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{ll}{\mathrm{P}}_{\mathrm{J}}& {\mathrm{P}}_{\mathrm{a}}\end{array}\right]}^{\mathrm{T}}\\ & \dot{\mathrm{X}}={\left[\begin{array}{ll}{\mathrm{e}}_6& {\mathrm{e}}_2\end{array}\right]}^{\mathrm{T}}\\ & {\mathrm{Z}}_1={\left[\begin{array}{ll}{\mathrm{f}}_2& {\mathrm{f}}_6\end{array}\right]}^{\mathrm{T}}\\ & \\ & {\mathrm{D}}_{\mathrm{in}}={\left[\begin{array}{ll}{\mathrm{f}}_3& {\mathrm{f}}_7\end{array}\right]}^{\mathrm{T}}\\ & {\mathrm{D}}_{\mathrm{out}}={\left[\begin{array}{ll}{\mathrm{e}}_3& {\mathrm{e}}_7\end{array}\right]}^{\mathrm{T}}\end{array}$$

(14)

Knowing that: ${\mathrm{Z}}_1=\mathrm{T}\cdot \mathrm{X}$

$$Z_1=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]=\left[\begin{array}{cc}\frac{1}{J}& 0\\ 0& \frac{1}{L_a}\end{array}\right]\left[\begin{array}{l}{P}_6\\ P_2\end{array}\right]$$

(15)

And, ${\mathrm{D}}_{\mathrm{out}}=\mathrm{E}{\mathrm{D}}_{\mathrm{in}}$

$$\left[\begin{array}{l}{e}_3\\ e_7\end{array}\right]=\left[\begin{array}{ll}{r}_a& 0\\ 0& f\end{array}\right]\left[\begin{array}{l}{f}_3\\ f_7\end{array}\right]$$

(17)

From the properties of the bond graph models we can write:

• Junction 1

$$\begin{array}{c}\left\{\begin{array}{l}{e}_1=e_2+e_3+e_4\\ f_1=f_2=f_3=f_4\end{array}\right.\\ \left\{\begin{array}{l}{e}_5=e_6+e_7\\ f_5=f_6=f_7\end{array}\right.\end{array}$$

(18)

• Gyrator

$$\left\{\begin{array}{l}{\mathrm{e}}_4={\mathrm{k}}_{\mathrm{b}}{\mathrm{f}}_5\\ {\mathrm{e}}_5={\mathrm{k}}_{\mathrm{b}}{\mathrm{f}}_4\end{array}\right.$$

(19)

$$\left[\begin{array}{l}{\mathrm{e}}_2\\ {\mathrm{e}}_6\\ {\mathrm{f}}_3\\ {\mathrm{f}}_7\end{array}\right]=\left[\begin{array}{ccccc}0& -{\mathrm{k}}_{\mathrm{b}}& -1& 0& 1\\ {\mathrm{k}}_{\mathrm{b}}& 0& 0& -1& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\end{array}\right]\left[\begin{array}{l}{\mathrm{f}}_2\\ {\mathrm{f}}_6\\ {\mathrm{e}}_3\\ {\mathrm{e}}_1\end{array}\right]$$

(20)

The causality being integral the structural relationship is then:

$$\left[\begin{array}{l}\mathrm{X}\\ {\mathrm{D}}_{\mathrm{in}}\end{array}\right]=\left[\begin{array}{lll}{\mathrm{S}}_{11}& {\mathrm{S}}_{12}& {\mathrm{S}}_{13}\\ {\mathrm{S}}_{21}& {\mathrm{S}}_{22}& {\mathrm{S}}_{23}\end{array}\right]\left[\begin{array}{l}{\mathrm{Z}}_1\\ {\mathrm{D}}_{\mathrm{out}}\\ \mathrm{U}\end{array}\right]$$

(21)

Then

$$\left\{\begin{array}{l}{\dot{P}}_{\mathrm{a}}={\mathrm{e}}_2=\mathrm{u}-{\mathrm{r}}_{\mathrm{a}}\frac{{\mathrm{P}}_{\mathrm{a}}}{{\mathrm{L}}_{\mathrm{a}}}-{\mathrm{k}}_{\mathrm{b}}\frac{{\mathrm{P}}_{\mathrm{J}}}{\mathrm{J}}\\ {\dot{\mathrm{P}}}_{\mathrm{J}}={\mathrm{e}}_6={\mathrm{k}}_{\mathrm{b}}\frac{{\mathrm{P}}_{\mathrm{a}}}{{\mathrm{L}}_{\mathrm{a}}}-\left(\mathrm{f}+{\mathrm{k}}_{\mathrm{T}}\right)\frac{\mathrm{P}}{\mathrm{J}}\end{array}\right.$$

(22)

From the state equation deduced from the bond graph model we can deduce the often-used differential equation where the state vector is:

$$\left[\begin{array}{c}\omega \\ I_a\end{array}\right]=\left[\begin{array}{c}\frac{P_J}{J}\\ \frac{{\mathrm{P}}_{\mathrm{a}}}{{\mathrm{L}}_{\mathrm{a}}}\end{array}\right]$$

(23)

$$\left\{\begin{array}{c}{\dot{I}}_a=-\frac{r_a}{L_a}{I}_a-\frac{k_b}{L_a}\omega +u\\ \dot{\omega }=\frac{k_b}{J}{I}_a-\frac{f+k_T}{J}\omega \end{array}\right.$$

(24)

$$\Rightarrow \left[\begin{array}{c}{\mathrm{X}}_1\\ {\mathrm{X}}_2\end{array}\right]=\left[\begin{array}{cc}-\frac{\left(\mathrm{f}+{\mathrm{k}}_{\mathrm{T}}\right)}{\mathrm{J}}& \frac{{\mathrm{k}}_{\mathrm{b}}}{\mathrm{J}}\\ -\frac{{\mathrm{k}}_{\mathrm{b}}}{{\mathrm{L}}_{\mathrm{a}}}& -\frac{{\mathrm{R}}_{\mathrm{a}}}{{\mathrm{L}}_{\mathrm{a}}}\end{array}\right]\left[\begin{array}{c}{\mathrm{X}}_1\\ {\mathrm{X}}_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{{\mathrm{L}}_{\mathrm{a}}}\end{array}\right]\left[\begin{array}{c}U\end{array}\right]$$

(25)

With:

$$X_1=\omega ,X_2=I_a\mathrm{a}\mathrm{n}\mathrm{d}X=\left[\begin{array}{c}\omega \\ I_a\end{array}\right]$$

(26)

The equation of state found from the bond graph model can be obtained by performing the Laplace transform of the characteristic equations of the current motor.

6.1.3. Determination of the control law

Let's recall the sliding mode control law to be implemented:

$$U=Lx+\rho \frac{Nx}{\parallel Mx\parallel +\delta }$$

(27)

With $\mathit{\delta }$ is moothing parameter.

The state variables of the system can be defined as the error:

₋

The speed: $x1={\omega }_d-\omega$

₋

The current: $x_2=I_d-I$

The equation of state of the system is of the form:

$$\dot{X}=Ax\left(t\right)+Bu\left(t\right)$$

(28)

With:

$$A=\left[\begin{array}{cc}-\frac{\left(\mathrm{f}+{\mathrm{k}}_{\mathrm{T}}\right)}{\mathrm{J}}& \frac{{\mathrm{k}}_{\mathrm{b}}}{\mathrm{J}}\\ -\frac{{\mathrm{k}}_{\mathrm{b}}}{{\mathrm{L}}_{\mathrm{a}}}& -\frac{{\mathrm{R}}_{\mathrm{a}}}{{\mathrm{L}}_{\mathrm{a}}}\end{array}\right]\mathrm{et}B=\left[\begin{array}{c}0\\ \frac{1}{{\mathrm{L}}_{\mathrm{a}}}\end{array}\right]$$

(29)

In sliding mode, after the performed transformation, the dynamics of the system became:

$$A=\dot{x}=-F{X}_1$$

(30)

The determination of F is obtained by placing the eigenvalues of (A11 - A12F) in the left complex half-plane.

The order of the system is reduced, corresponding to a stable pole: P = -1, with state feedback: F = 0.247.

The sliding surface $C=\left[\begin{array}{ll}\mathrm{F}& {\mathrm{I}}_{\mathrm{m}}\end{array}\right]\mathrm{T}$ is deducted:

$$C=\left[\begin{array}{ll}-0.0541& -0.124\end{array}\right]$$

(31)

The matrices L, M and N obtained are:

$$\begin{array}{rr}\mathrm{L}& =\left[\begin{array}{ll}0.293635& 1.116298\end{array}\right]\\ \mathrm{M}& =\left[\begin{array}{ll}0.0717857& -0.25000\end{array}\right]\\ \mathrm{N}& =\left[\begin{array}{ll}0.001723& -0.006\end{array}\right]\end{array}$$

(32)

7. Experimental Simulations and Results

8. Conclusion

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