Variable structure MPP controller for photovoltaic pumping system

Abstract

This work is oriented to research performance in terms of optimal transfer of energy from a photovoltaic generator to a permeant magnet synchronous motor type used as a centrifuge pump driver. The performance improvement is done using a variable structure controller with sliding mode as a maximum power point tracker (MPPT). A stability analysis for the proposed MPPT is developed for a photovoltaic (PV) pumping system. The global stability of the variable structure algorithm is demonstrated by means of Lyapunov’s approach. The tracking algorithm leads the PV generator coordinates to the maximum power point by changing the pulse width modulation (PWM) signal frequency of the boost converter. The steady-state behaviour of the PV pumping system with variable structure control is characterized by a stable oscillation around the maximum power point. The effectiveness of the proposed MPPT scheme is demonstrated under internal and external disturbances.

Keywords

PV MPPT battery PMSM pump boost variable structure

Introduction

Nowadays, it seems that no one can doubt the importance of the constant need for water and energy in modern life. With advances in technology, energy requirements are increasing. This energy problem is particularly critical in remote areas where the use of a conventional resource is very expensive and even impossible (David et al., 2002). Indeed, several constraints are found in these types of sites, such as fuel transportation, electrified lines and so on. Renewable energy, such as photovoltaic energy, represents an excellent alternative, often contributes to a peaceful environment and is increasingly used today. These types of energy are becoming less expensive and inexhaustible. Moreover, we often speak of a ‘green’ energy, since renewable energy avoids the pollution produced by traditional or alternative sources (Luiela, 2009).

Pumping water in agricultural and rural sites, which are often isolated, is a vital task. Pumping the water based on solar energy uses electric storage elements in most of its system structure. In this work the main goal is to improve photovoltaic (PV) pumping system performances in order to ensure proper operation and increase the system lifespan.

In this work a control study and implementation of the PV pumping system is achieved. The system efficiency improvement was done by extracting maximum power from the PV generators (PVG); this extraction is a prime necessity whatever the dynamics of entry imposed. Therefore, the high performance of the tracking controller of maximum power point tracking (MPPT) is a high priority target for optimizing the efficiency of PV applications (Bendib et al., 2015). The MPPT control is a challenge facing the impinging solar flux to the PVG. In fact, the PVG is considered a complex nonlinear system (Mahmud et al., 2014). For these reasons, the design and implementation of an appropriate MPPT controller is difficult (Farhat et al., 2015a).

Many methods have been developed to determine the maximum power point (MPP). MPPT methods vary in complexity, implementation hardware, popularity, convergence speed and sensed parameters; for example, P&O (Femia et al., 2005), Hill climbing (Al-Atrash et al., 2010), Inc.Cond (Cipriani et al., 2015) and the fractional open circuit and short circuit algorithm (Sher et al., 2015). These techniques are easily implemented and have been widely adopted for low-cost applications. Algorithms such as Fuzzy Logic and artificial neural network (ANN) are more complex and less often used (Farhat and Sbita, 2015).

Nowadays, more research work has been focused on control systems using sliding mode controls. Based on the PV module nonlinear characteristics, the MPP locus may be approximated by a nonlinear method (Gonzalez-Montoya et al., 2016). Thus, a nonlinear controller was designed which drives the PVG to the MPP. This method proposes a fast and robust MPPT control scheme based on the theory of variable structure systems (VSS) (Haroun et al., 2015).

The reliability of a PV application depends on the efficiency of power electronic devices to operate the PVG at its highest efficiency even under varying climatic conditions and the presence of internal disturbances (Farhat et al., 2015c). So, in order to have a PV system that operates with higher efficiency, the MPPT controller will drive a matching stage that adapts the best impedance to the PVG (resistive, machine, batteries, etc.) (El-Khateb et al., 2015). In this work the permanent magnet synchronous motor (PMSM) is selected because it is the most efficient in many applications such as the pumping system. Its superior advantages compared with other motors, such as induction, DC and reluctance motors, the high efficiency, low inertia, high torque to current ratio, high power factor, almost no need for maintenance and, especially, its smaller size make this motor type the most well adapted for high performance applications (Patel et al., 2012).

Energy conversion

PV model

The PV cell can be substituted to an equivalent electric circuit that includes a power supply and a diode as showed in Figure 1 (Farhat and Sbita, 2015; Farhat et al., 2014).

Figure 1.

Simplified PV cell equivalent circuit.

By exploiting the node law

I_{c} = I_{ph} - I_{d} - I_{sh}

(1)

The current $I_{ph}$ can be evaluated as

I_{ph} = \frac{G}{G_{ref}} (I_{rs_ref} + K_{SCT} (T_{c} - T_{c_ref}))

(2)

I_{d} = I_{rs} (\exp \frac{q (V_{c} + R_{s} I_{c})}{α kT} - 1)

(3)

I_{sh} = \frac{1}{R_{p}} (V_{c} + R_{s} I_{c})

(4)

The reverse saturation current can be approximately obtained as

I_{rs} = \frac{I_{rs_ref}}{\exp [\frac{q V_{oc}}{n_{s} . n . β . T_{c}}] - 1}

(5)

Finally, the cell current $I_{c}$ can be given by

I_{c} = I_{ph} - I_{rs} (\exp \frac{q (V_{c} + R_{s} I_{c})}{α kT} - 1) - \frac{1}{R_{p}} (V_{c} + R_{s} I_{c})

(6)

The model of a PVG depending on number of series and parallel cells, respectively N_s and N_p, is as shown in Figure 2 (Zhang and Bai, 2008)

{\begin{matrix} I_{p} = N_{p} I_{c} \\ V_{p} = N_{s} n_{s} V_{c} \end{matrix}

(7)

Figure 2.

PVG modules association; N_s stands for numbers in series and N_p in parallel.

Finally the PVG current can be given by

I_{p} = N_{p} I_{ph} - N_{p} I_{rs} (\exp \frac{q}{α k T} (\frac{V_{p}}{n_{s} N_{s}} + \frac{R_{s} I_{P}}{N_{p}}) - 1) - \frac{N_{p}}{R_{p}} (\frac{V_{p}}{n_{s} N_{s}} + \frac{R_{s} I_{P}}{N_{p}})

(8)

The terms containing R_s and R_p parameters are not be considered (using simplification and assumption R_p >> R_s). Here the model with R_s= 0 and R_p= ∞

I_{p} = N_{p} I_{ph} - N_{p} I_{rs} (\exp \frac{q}{n β T_{c}} (\frac{V_{p}}{n_{s} N_{s}}) - 1)

(9)

The highly nonlinear characteristics are shown in Figures 3 and 4. It is efficient to operate in the third zone.

Figure 3.

PV module (a) I-V curves at a various irradiation levels from 100W/m² to 1000W/m² and at fixed temperature of 25°C, (b) P-V curves at a various irradiation levels from 100W/m² to 1000W/m² and at fixed temperature of 25°C.

Figure 4.

(a) I-V curves at various temperature levels from 10°C to 25°C and at fixed irradiation 1000W/m², (b) P-V curves at various temperature levels from 10°C to 25°C and at fixed irradiation 1000W/m².

Figure 5 shows three specific operating zones of the GPV:

The voltage zone, which is characterized by the open circuit voltage V_OC.

The current zone, which is characterized by the short circuit current I_SC.

The P_max Zone, which is characterized by the maximum of the power P_max.

Figure 5.

I-V and P-V characteristics at fixed irradiation (1000W/m²) and temperature values (25°C).

Matching stage

The operation of the PVG at its MPP is ensured by using a matching stage. This stage adapts the load to the PVG by facing changes in the atmospheric conditions and internal disturbances. Figure 6 shows a PVG connected to a matching stage that consists of a boost converter, MPPT controller and a storage battery (Farhat et al., 2015b).

Figure 6.

Matching stage incorporating a MPPT boost converter in a PV pumping system.

In a continuous conduction mode (CCM) (Valenzuela and Guemez, 2015), the output voltage of the boost converter is related to the input one as follows

V_{out} = \frac{1}{1 - D} V_{in}

(10)

where V_in corresponds to the input voltage power (V_in), where (V_out) is the converter output voltage and D is the converter duty cycle as a switching command signal, i_p is the controller inputs, i_L is the inductance current iout is the controller output (El-Khateb et al., 2015).

S_W represents the switch state command

S_W = 0 indicates that the switch is off

S_W = 1 indicates that the switch is on.

The dynamic model of the boost converter circuit is (Farhat et al., 2015b; Valenzuela and Guemez, 2015)

{\begin{matrix} \frac{d V_{p}}{dt} = \frac{1}{C_{1}} (i_{p} - i_{L}) \\ \frac{d i_{L}}{dt} = \frac{1}{L} V_{p} + \frac{1}{L} (S_{W} - 1) V_{out} \\ \frac{d V_{out}}{dt} = \frac{1}{C_{2}} (i_{L} - i_{out}) - \frac{1}{C_{2}} S_{W} i_{L} \end{matrix}

(11)

with

\begin{array}{l} x = [\begin{array}{l} V_{p} \\ i_{L} \\ V_{o u t} \end{array}]; f (x) = [\begin{array}{l} \frac{1}{C_{1}} (i_{p} - i_{L}) \\ \frac{1}{L} (V_{p} - V_{o u t}) \\ \frac{1}{C_{2}} (i_{L} - i_{o u t}) \end{array}]; \\ g (x) = [\begin{array}{l} 0 \\ \frac{1}{L} V_{o u t} \\ \frac{1}{C_{2}} i_{L} \end{array}] \dot{x} = f (x) + g (x) S_{W} \end{array}

(12)

The MPPT controls the PV system and applies the optimal duty cycle to the boost converter. This paper is focused on the sliding mode MPPT method based on its robust and high performance dynamic controller (Asim et al., 2009).

The battery model is based on the electrical diagram given in Figure 7, the batteries are described by just two elements (whose characteristics depend on a set of parameters): a voltage source and an internal resistance. For the ‘n_b’ battery cells in series, the output voltage V_bat can be expressed as

V_{bat} = n_{b} \cdot E_{b} + n_{b} \cdot R_{i} \cdot I_{bat}

(13)

where V_bat and I_bat are, respectively, the battery voltage and current (according to receptor convention), E_b is the electromotive force and R_i is the internal resistance.

Figure 7.

Equivalent electrical diagram of n_b battery elements in series.

When the V_out voltage is lesser then, V_bat the battery discharge operation occurs. On the other hand, when the V_bat voltage is lesseer, the battery goes into charging mode (Du and Lu, 2011).

Modelling of the three-phase voltage inverter

A three-phase voltage inverter, as shown in Figure 8, is used to interface the PVG with the moto-pump by converting the power. It allows the conversion of DC power to AC.

Figure 8.

Voltage source inverter topology.

The function of an inverter is to transform a DC input voltage to a three-phase symmetric AC output voltage of a desired magnitude and frequency. The DC input voltage could be fixed or variable. As noted in a previous paper, the use of a directly coupled pump to PVG gives a poor result performance when the DC input voltage varies (Farhat et al., 2012). The present work focuses on the use of a constant DC-bus input voltage by the use of a battery fed by an MPPT boost converter.

The inverter provides a three-phase voltage system, which is generated according to the three pulse width modulation (PWM) command reference voltages. The a, b and c output voltages feed the PMSM coupled to the centrifugal pump. These output voltages can be expressed as indicated in (16) (Wenyi et al., 2014)

[\begin{matrix} Va \\ Vb \\ Vc \end{matrix}] = \frac{V_{DC}}{3} [\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}] [\begin{matrix} g_{1} \\ g_{2} \\ g_{3} \end{matrix}]

(14)

with V_DC is the boost output voltage. g₁, g₂ and g₃ are the PWM gates control signals.

A DC-bus is used to ensure an energy balance between the power generated by the PVG and the power feeding the PMSM pump. This is done by simply charging or discharging the capacitor.

The inverter input voltage V_DC is given by

i_{C} = \frac{d V_{DC}}{dt}

(15)

and the node law gives

I_{out} = I_{bat} + i_{e} + i_{C}

(16)

The relationship that connects the input current and the output current of the inverter is given by the following expression

i_{e} = i_{a} g_{1} + i_{b} g_{2} + i_{c} g_{3}

(17)

with i_a, i_b and i_c are the inverter current as shown in Figure 8.

Moto-pump modelling

PMSM modelling

In this part we propose the PMSM mathematical modelling as: the electrical equations presented in equation (18), the electromagnet torque as described in equation (19) and the mechanical expression in equation (20) (Soreshjani, 2015). R_ss and L_s are, respectively, the stator resistance and inductance, λ_m is the permanent magnet flux produced by the magnet inserted in the rotor, P is the poles number, J is the rotor inertia and T_l is the load torque

{\begin{matrix} v_{d} = R_{ss} i_{d} + L_{s} \frac{d i_{d}}{dt} - ω L_{s} i_{q} \\ v_{q} = R_{ss} i_{q} + L_{s} \frac{d i_{q}}{dt} + ω L_{s} i_{d} + ω λ_{m} \end{matrix}

(18)

{\begin{matrix} T_{e} = (\frac{3}{2}) (\frac{P}{2}) (λ_{d} i_{q} - λ_{q} i_{d}) \\ λ_{d} = L_{s} i_{d} + λ_{m} and λ_{q} = L_{s} i_{q} \end{matrix}

(19)

(T_{e} - T_{l}) = (\frac{P}{2}) (J \frac{d ω}{dt} + f ω)

(20)

Centrifugal pump

The centrifugal pump is designed for a given total head (TH). The flow rate of the pump varies in proportion to the rotational PMSM shaft speed, so the torque increases very rapidly with the speed and height of the outlet, which is square based of the motor speed. The rotational speed of the motor must be very fast to ensure a good flow. The power consumption is proportional to flow rate Q. A typical centrifugal pump is used for high flow rates and low or medium depths. The aerodynamic load is characterized by the following equation (Wenyi et al., 2014)

T_{l} = k \cdot ω^{2}

(21)

where k is the pump constant. Centrifugal pump head to flow rate characteristics are given by the following expression

H = a_{1} \cdot ω^{2} + a_{2} \cdot ω \cdot Q + a_{3} \cdot Q^{2}

(22)

Where a₁, a₂ and a₃ are the coefficients generally given by the manufacturers. Figure 9 shows the characteristics of the head versus capacity (h = f (Q)). Those characteristics are shown for different speed values.

Figure 9.

Pump characteristics.

The hydraulic power load torque and the mechanic power are given as

p_{H} = ρ gQH

(23)

Assuming that the efficiency of the coupling between the drive machine and the pump is equal to 1, then the mechanical power of the PMSM is equal to the power absorbed by the pump. The mechanical power of the PMSM is

P_{m} = k \cdot ω^{3}

(24)

The pump efficiency is defined as the ratio of the power transmitted by the hydraulic pump and mechanical power. It is expressed by the following (Farhat et al., 2015a)

η = \frac{P_{H}}{P_{m}}

(25)

η = \frac{ρ gQH}{k ω^{3}}

(26)

MPPT control algorithms

The maximum power of the PVG is obtained by solving equation 27 (Farhat et al., 2015a) and by considering that the PVG output power is P = V_pI_p. Figure 10 show variations of power to voltage.

\frac{\partial P}{\partial V_{p}} = 0

(27)

\frac{\partial P}{\partial V_{p}} = I_{p} + \frac{\partial Ip}{\partial V_{p}} V_{p}

(28)

Figure 10.

Variations of power to voltage.

The used MPPT is a variable structure control system. These control systems are characterized by a suite of feedback control laws and a decision rule. The decision rule, named the switching function, selects a particular feedback control in accordance with the system’s behaviour. The switching surface is S = 0.

The state space is divided into two parts S < 0 and S > 0 (Yau and Chen, 2012). This algorithm is desirable to switch the function

S = \frac{\partial P}{\partial V_{p}} = I_{p} + \frac{\partial ip}{\partial V_{p}} V_{p}

(29)

based on the two states of PVG in Figure 10. The boost power gate drive signal control function is

S_{W} {\begin{matrix} 1 & S < 0 \\ 0 & S > 0 \end{matrix}

(30)

Let

\overset{\cdot}{S} = \frac{\partial S}{\partial x^{T}} \overset{\cdot}{x} = \frac{\partial S}{\partial x^{T}} f (x) + \frac{\partial S}{\partial x^{T}} g (x) S_{W_{eq}} = 0

(31)

Proof. Based on the Lyapunov function, the variable structure mode control requires that (Costabeber et al., 2015)

V = \frac{1}{2} S^{2} > 0, \overset{\cdot}{V} = S \frac{dS}{dt}, S \overset{\cdot}{S} < 0

(32)

If the expression in equation (30) is adopted for the system, it could make the system stabilizing from any initial state.

{\begin{matrix} S = \frac{\partial P}{\partial V_{p}} = I_{p} + \frac{\partial Ip}{\partial V_{p}} V_{p} \\ with I_{p} = N_{p} I_{ph} - N_{p} I_{rs} (\exp \frac{q}{n β T_{c}} (\frac{V_{p}}{n_{s} N_{s}}) - 1) \end{matrix}

(33)

\begin{matrix} S = \frac{\partial P}{\partial V_{p}} = I_{p} + \frac{\partial Ip}{\partial V_{p}} V_{p} \\ = N_{p} I_{ph} - N_{p} I_{rs} (\exp (A V_{p}) - 1) - N_{p} I_{rs} A \exp (A V_{p}) V_{p} \end{matrix}

(34)

with $A = \frac{q}{n β T_{c}} \frac{1}{n_{s} N_{s}}$

When S > 0

Based on Figure 10, the system is operating on the left side, the switch function S_W = 0. The switch command is at a low level; this implies that the duty cycle will decrease and, as a result, V_p is increasing

\frac{d V_{p}}{dt} > 0

(35)

\frac{dS}{dt} = N_{p} {[(I_{ph} - I_{rs}) - I_{rs} (1 + A V_{p}) \exp (A V_{p})]}^{'}

(36)

\frac{dS}{dt} = - N_{p} I_{rs} A [\exp (A V_{p}) + (1 + A V_{p}) \exp (A V_{p})] {\overset{\cdot}{V}}_{p}

(37)

Using equation (35) in equation (37) gives

\frac{dS}{dt} > 0

and finally $S \frac{dS}{dt} < 0$

When S < 0

The system operates on the right side, the switch function S_W= 1. The switch command is at a high level, this implies that the duty cycle will increase and, as a result, V_p is decreasing

\frac{d V_{p}}{dt} < 0

(38)

so $\frac{dS}{dt}$ < 0 and $S \frac{dS}{dt} < 0$

Finally the system reaches the global stability.

Simulation results of PV pumping system

In this section we propose the global control pumping system via a solar power source as presented in Figures 11 and 12. The PMSM is used as the pumping rotational engine. The PMSM parameters used in these applications are given in Table 1.

Figure 11.

Power line of the PV pumping system.

Figure 12.

Command line.

Table 1.

Specifications.

Parameters	Rated characteristics
R_ss 5.2Ω	ω 2500 r/min
L_d=L_q=L_s 21.5 m H	Torque 1.5 N m
λ_m 0.24 Wb	Current I=2.5 A
J 85×10⁻⁶ kg m²	Maximum Current 5A
f 0.00036 N m s/rad	Rated characteristics
ρ 1000 Kg/m²	ω 2500 r/min
g 9.81 m /s²	Torque 1.5 N m
N_p 5	Current I=2.5 A
N_s 5	Maximum Current 5 A
n_s 36
q 1.6×10⁻¹⁹
Short circuit current I_sc 3.7
Open circuit voltage 20.5
Solar cell’s ideal factor 1.2
Band gap energy: E_g1.11 eV
T_{c_ref} 25°C

The proposed control scheme illustrated in Figure 11 shows the power line of the whole PV pumping system. Figure 12 shows first the boost converter control scheme, and then shows the field oriented vector control strategy of the PMSM. This technique regroups many essential blocs. Effectively, after comparing the shaft and the desired (ω*) speeds, the proportional integral (PI) controller is added in the outer loop to generate the transversal stator current as a reference current. This last current signal is also compared with the measured one as an inner loop. The obtained error is injected in the PI controller, which generates the transversal voltage as a reference voltage. In the conventional field oriented control (FOC) strategy, the direct reference stator current is clamped to zero. Therefore, for the direct currents, just an inner control loop is built around a PI controller.

To increase the PVG available power, the PV current and voltage must be increased as well, this is achieved by increasing the number of parallel and series-mounted modules. The achieved PVG in such a study case consists of N_p = 5 and N_s = 5, as shown in Figure 13 and based on the Figures 2 and 3 data.

Figure 13.

I-V and P-V PVG curves.

In order to prove the good performance of the proposed MPPT algorithm, an abrupt and instantaneous irradiation, temperature and boost capacitor variation signal is applied to the PVG, as shown in Figures 14, 15 and 16. For the pump load case, the load torque applied to the motor is that of the centrifugal pump.

Figure 14.

Temperature variation.

Figure 15.

Boost capacitor variation

Figure 16.

Irradiation variation.

In Figure 17, the given results demonstrate the obtained performances of variable structure MPPT tracker controller.

Figure 17.

Power to voltage variations ratio (Surface).

Figure 18 shows the direct PWM signal generated by the variable structure MPPT algorithm. This last has the benefit avoiding the use of a PWM commutation signal (Saw signal). It permits us to build directly a PWM output signal toward the Insulated-gate bipolar transistor (IGBT) gate.

Figure. 18.

Boost tracker command signal.

Figures 19, 20 and 21 show, respectively, the dynamic evolution of the PVG power, current and voltage signals for the same irradiation appliance (see Figure 13). One can easily see that the PVG under the proposed MPPT tracker meets the maximum power operating points. It is clearly shown that the variable structure controller is suitable for good rapid and dynamic PVG signal responses.

Figure 19.

PV power.

Figure 20.

PV current.

Figure 21.

PV voltage.

Figure 22 presents the battery voltage used as a DC link. This last remains at a constant level as long as the MPPT trackers fit the maximum power operating region.

Figure 22.

The battery voltage signal.

Figures 23 to 28 show, respectively, the electromagnetic torque, load torque, speed, id, iq d- and q-axis stator, and finally the pump flow.

Figure 23.

Electromagnetic torques dynamic response.

Figure 24.

Load torques dynamic response.

Figure 25.

Speed.

Figure 26.

Stator current (d).

Figure 27.

Stator current (q).

Figure 28.

The pump flow.

Conclusion

A variable structure control strategy has been presented. This proposed algorithm is formulated and applied to the PV pumping system. The effectiveness of the proposed MPPT is proven over simulation results. This new algorithm is applicable in several industries; eventually the economic benefits would be enormous because it is an economic controller that does not require a lot of sensors. This control law could be implemented by means of standard operational amplifiers, analogue multipliers and digital devices in an experimental platform, or using an acquisition and control dSPACE (DSP) board.

Footnotes

Appendix

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

The authors are very grateful to the UPV/EHU for its support through the projects GIU13/41 and UFI11/07, and to the Tunisian Ministry of Higher Education and Scientific Research for their support of the research unit code UR11ES82.

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