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The paper analyses extremum-seeking control technique for maximum power point tracking circuits in PV systems. Specifically, the paper describes and analyses the sinusoidal extremum-seeking control considering stability issues by means a Lyapunov function. Based on this technique, a new architecture of MPPT for PV generation is proposed. In order to assess the proposed solution, the paper provides some experimental measurements in a 100 W prototype which corroborate the effectiveness of the approach.

Photovoltaic panels require adapting the output-port voltage to extract the maximum deliverable power when weather conditions change. Such energy adaptation should have a good performance; that is, the operation point should be ensured to be near to the maximum and the adapting electronic system should behave reliably and efficiently. A reliable behavior means that the adaptation mechanism is dynamically stable and does not induce great stress to its components. Efficiency of the adapting electronic system depends on the losses in switching converters which adapt the voltage levels between the PV panels and the loads [

There exist several approaches to implement maximum power point tracking (MPPT) in PV systems. Remarkable surveys on this subject can be found in [

Direct methods take measurements of PV panel current and voltage and their corresponding time derivatives at the operating point in order to drive PV systems toward the maximum. Given that direct methods do not carry out abrupt changes in the operating point to make measurements, measuring frequency can be higher, and therefore these MPPT methods can faster track the optimum power point.

The most common direct MPPT methods are the P&O algorithm [

In parallel, with the development of MPPT methods in renewable energy field, some researchers have studied the technique named extremum seeking control (ESC) in the field of automatic control. This technique is concerned with algorithms that seek the maximum or the minimum of a nonlinear map. A system governed by ESC autooscillates around the optimum or the oscillation is forced by a sinusoidal signal. Autooscillating ESC algorithm is reported by Morosanov [

Nevertheless, sinusoidal ESC is still in an incipient stage in PV systems, despite the research of Brunton et al. [

Stability issues about sinusoidal ESC can be found in [

In the paper, we review the sinusoidal ESC technique. Also, we present a new stability demonstration based on Lyapunov analysis where we only assume that the nonlinear map is concave. This approach ensures the stability and therefore the reliability for all the range of operation of the PV system. The paper also presents a 100 W prototype which allows us to evaluate the effectiveness of the approach. The prototype differs from that of [

The paper is organized as follows. In Section

The objective of ESC is to force the operating point to be as close as possible to the optimum for a system described by an unknown nonlinear map with an only extremum (i.e., a maximum or a minimum).

Sinusoidal ESC principle, shown in Figure

Sinusoidal ESC principle.

The method can be implemented by the schema of Figure

Sinusoidal ESC schema.

In the following sections, we describe the detection block function and analyze the stability condition of the schema shown in Figure

Detection block output

Given a signal

Considering that the sinusoidal perturbation is small, namely, given the condition

Thus, using the trigonometric identity

Now, considering that the low-pass filter attenuates completely the first and second harmonics, the expression of the filter output

Therefore, under the assumptions of small amplitude of the sinusoidal perturbation and enough attenuation of the harmonics of multiplier output, it can be stated that the output signal of the detection block

The following analysis is carried out by means of averaged signals. Averaged signals and real signals differ only in a magnitude which depends on

Averaged signals that take part in the sinusoidal extremum-seeking circuit are

Averaged-signal extremum-seeking block diagram.

Therefore, denoting the constants terms as

We assume that

It can be observed that the time derivative of the gradient corresponds to

Now, renaming the state variables as

Then, we consider the following candidate Lyapunov function to prove the stability of the nonlinear system (

First, in order to verify the positive definiteness of

Now taking the function

In addition the time derivative of

And substituting the state variables derivatives according to (

Hence, given that

We should remark that the only assumption is that the nonlinear map has an only máximum, that is,

Moreover, we remark this prove ensures the stability and therefore the reliability for all the range of operation of the PV system. An analysis that only considers the stability in front of small signal disturbances can be found in [

The solar panel model, as that depicted in Figure

Proposed MPPT PV generator schema.

We adapt the PV panel output voltage to the load (in our case a battery) by means of a boost DC-DC switching converter, as shown in Figure

In this section, we describe an electronic implementation of the schema depicted in Figure

Figure

Prototype of PV generator with MPPT based on sinusoidal ESC.

Besides the PV module, the dc-dc converter and its PWM circuit, the battery, and the power calculating multiplier, there exists also an adder block which sums the sinusoidal dithering before the PWM block. The key parameters of the sinusoidal dithering generator are its amplitude and frequency. The generator amplitude must be much larger than the switching ripple but small enough since MPPT efficiency depends on it. In the prototype, the generator amplitude and frequency are, respectively, 0.5 Vpp and 2 kHz, whereas the PWM sawtooth amplitude is 7 Vpp. The demodulating or detecting multiplication is also carried out by the IC AD633. A first-order filter is located after the detecting multiplier whose cut-off frequency is 200 Hz aprox. A tradeoff appears in the election of the crossover frequency since a higher value allows a faster behavior; nevertheless a lower value filters undesired harmonics better. The integrator block values must ensure that no saturation appears, and the controller gain must ensure stability and a suitable degree of damping. Their values are shown in Figure

In this subsection, we explain the behavior of the prototype in steady state, also when a 3 V pulse is put in series with photovoltaic panel, and when a 6 V pulse is put in series with the battery. Figure

Steady-state waveforms.

Figure

(a) Transient waveforms for voltage-pulsed signal in series with the PV panel. (b) Detail of transient for voltage pulse added to panel voltage.

Figure

(a) Transient waveform when voltage pulse signal is added to battery voltage. (b) Detail of the transient waveforms during a decrease step of the battery voltage.

The paper reviews the control technique named sinusoidal ESC. A novel architecture for MPPT photovoltaic generation is proposed based on this technique. Such technique provides a high efficient method to track the maximum power point, since it results in a small oscillation around the MPP. This oscillation depends on the amplitude of the sinusoidal modulator signal and is proportional to the slope of the power-voltage curve of the PV panel. Therefore, given that the slope is small near optimum, then the oscillation is small near the maximum and the MPPT has a very good performance. The paper also analyzes the stability of sinusoidal ESC for PV systems by means a Lyapunov function. In addition, in order to verify the effectiveness of the approach, we describe the implementation a prototype for a 100 Wp solar generator, which provides accurate measures which are in good agreement with previous derivation.