Fixed-frequency implementation of sliding-mode controllers for photovoltaic systems

The solar irradiance is a renewable and clean energy source that is attracting interest due to its low environmental impact, high sustainability and availability at almost any place [1]. Hence, photovoltaic (PV) generators have become a stronger option to address the continuous increment of the power demand and to support the initiative of reducing the use of fossil fuels [1]. Moreover, the PV systems could be used to provide ancillary services for supporting the grid [2, 3] but, at the same time, the introduction of PV systems generates changes in the power-system dynamics that must be modeled to avoid instabilities [4]. One of the challenges in controlling PV systems concerns the variability of the power production, which is mainly caused by the variation on the irradiance level, or by partial shading produced by surrounding objects [5], causing the activation of bypass diodes. The most commonly adopted solutions to face the partial shading problem are: reconfiguration of the electrical connections forming the PV array [6], global maximum power point tracking techniques [7] and distributed maximum power point (DMPPT) architectures [8]. The DMPPT architectures are the solutions that produce the highest power, since bypass diodes are not activated. Therefore, micro-inverters have been developed to isolate one PV panel from the others, forming in this way a DMPPT architecture. Such a solution improves the power production of small PV plants by reducing the effect of partial shading conditions [7], but introducing the same problems of any grid-connected PV system.

A typical grid-connected PV system is composed by a PV array, a dc/dc converter to transform the power provided by the PV source, a controller to mitigate disturbances, and a grid-connected inverter as it is illustrated in Fig. 1. The PV array exhibits a non-linear behavior that changes significantly with the operating conditions (irradiance, temperature, and voltage at the array terminals), which makes difficult to predict the optimal PV voltage and current to guarantee the maximum power production [9, 10].

The design of a controller capable of rejecting disturbances occurring at both the photovoltaic array and the load represent one of the main challenges in the implementation of this kind of systems [11, 12]. Usually, grid-connected PV systems are controlled by a cascade connection of a maximum power point tracking (MPPT) algorithm and a PV voltage controller, where linear controllers (PI, PID or lead–lag) are a widely adopted alternative [9, 11‐13]. Moreover, the Perturb and Observe (P&O) is a widely adopted MPPT algorithm, which provides a satisfactory trade-off between complexity and performance for the optimization of the operating point of renewable power sources [14].

To guarantee a correct tracking of the reference generated by the MPPT algorithm, and to mitigate the disturbances in both the load and the environmental conditions, the linear controllers are commonly designed from the model of the PV system linearized around a given specific operating point, e.g., the maximum power point (MPP) at the lowest irradiance condition [9, 10]. Thus, to guarantee the same performance in all the operation ranges it is necessary to design non-linear controllers based on non-linearized models. In this way, the sliding-mode control (SMC) is an effective non-linear approach with several advantages: robustness against parametric tolerances, global stability, and a binary control signal matching the requirements of the dc/dc converter [9]. Therefore, the regulation of grid-connected PV system using SMC has been extensively addressed in the literature as reported in [9, 15‐20]. However, those solutions provide the classical implementation based on a variable-switching frequency that depends on the dc/dc converter parameters and on the operating conditions imposed by both the environment and load. Such a variable-switching frequency is a major problem for practical circuits based on SMC ([15, 21]) due to the operational restrictions imposed to the semiconductor devices, making the implementation of filters required for measuring the signals used for processing both the controller and MPPT algorithm also difficult [22]. Moreover, some SMC implementations use digital devices [23], hence the switching frequency cannot exceed the sampling frequency of the analog-to-digital-converters (ADC) used to measure the PV voltage and current [24]. Therefore, a constant switching frequency is desired to simplify the implementation of SMC for PV systems: accurate filters design, correct selection of both ADC and control microprocessors, precise selection of the semiconductors depending of both Turn-ON and Turn-OFF times, among other advantages that will decrease the implementation cost and complexity.

This topic, in the context of dc/dc converters, was addressed in [21] using the equivalent control value of the SMC as duty cycle. This approach results in a duty cycle regulation following the theoretical sliding-mode surface provided to a traditional PWM circuit. Another approach was proposed in [25], which reports a fixed-frequency SMC to improve the dynamic response of a single-phase inverter subjected to a sudden fluctuation of the load. This solution uses a flip-flop to generate the control signal with an additional constant frequency clock to impose a constant Turn-ON time. However, those solutions were not analyzed considering the PV source.

This paper proposes a methodology for implementing sliding-mode controllers with constant switching frequency for the Maximum Power Point Tracking of PV systems. The proposed method is aimed at implementing first-order sliding-mode controllers due to the large amount of those solutions reported in literature [9, 15, 16, 19, 20]. The paper is organized as follows: Section 2 presents the background of the proposed design procedure, then Sect. 3 describes the proposed methodology based on an adaptive hysteresis. Section 4 illustrates the application of the proposed solution to SMCs reported in literature; this section demonstrates the performance of the proposed implementation method using detailed simulations. Moreover, Sect. 5 provides experimental verifications of the fixed-frequency provided by the proposed solution. Finally, the conclusions close the paper.

The solution proposed in this paper is aimed at implementing sliding mode controllers, operating at fixed frequency, to regulate the PV voltage and to mitigate the low frequency voltage oscillations caused at the bulk capacitor \(C_{\text{b}}\), reported in Fig. 2, by the inverter operation in grid-connected PV systems [19].

The PV system depicted in Fig. 2 is based on a boost converter, which is widely adopted to develop grid-connected PV systems because it provides a satisfactory trade-off between simplicity, cost and voltage-boosting factor [26]. Therefore, the boost converter is the topology adopted in this paper for the dc/dc stage of the grid-connected PV system. The simplified circuital scheme of the PV system, presented in Fig. 2, includes a voltage source modeling the dc-link and the grid-connected inverter. Such a voltage source model is an accurate representation because the dc/ac stage regulates the dc-link voltage (\(C_{\text{b}}\) capacitor voltage) as it is reported in [9, 11, 26].

Equations (1) and (2) provide the dynamic model of the PV system [26], where \(v_{{\text{pv}}}\) and \(i_{{\text{pv}}}\) represent the voltage and the current of the PV array, respectively, \(i_{{\text{L}}}\) represents the inductor current, \(v_{\text{b}}\) is the load voltage, while L and \(C_{{\text{in}}}\) represent the inductor and input capacitor values. Moreover, u represents the control signal that defines the MOSFET and diode status: \(u = 0\) open the MOSFET (close the diode) and \(u=1\) close the MOSFET (open the diode).The PV array is modeled using the ideal single diode model (ISDM) reported in [9], which is extensively used to design controllers due to the satisfactory balance between accuracy and simplicity [26]. From that model, the PV current \(i_{{\text{pv}}}\) is given in (3). In the ISDM \(i_{{\text{sc}}}\) represents the short-circuit current, which is almost proportional to the irradiance, B is the diode saturation current and A represents the inverse of the thermal voltage, which depends on the temperature [26]. Those parameters can be calculated from datasheet values or experimental measurements [26], hence the model can be updated depending on the changes occurring on the environmental conditions.Figure 2 also depicts the connection between the MPPT algorithm and the main controller. This paper adopts the P&O algorithm due to its efficiency and simplicity. The operation principle of the P&O algorithm is the following: the PV voltage is perturbed in the direction that increases the PV power, hence if the power is decreased, the PV voltage is perturbed in the opposite direction [14]. This behavior enables to detect the optimal PV voltage that maximizes the power production. The flowchart of the P&O algorithm is given in Fig. 3, in which k and \(k-1\) represent the present and previous processing cycles of the algorithm, respectively, which are executed each disturbance period \(T_{\text{a}}\). The variable Sign codifies the direction in which the PV voltage is perturbed (Sign = 1 for increment and Sign = − 1 for decrement), while term \(\varDelta v_{{\text{pv}}}\) corresponds to the disturbance magnitude that will be introduced into the PV voltage. This flow-chart put into evidence the discrete nature of the P&O algorithm, hence it must be implemented using a digital microprocessor.

There are reported in literature multiple SMC aimed at regulating the PV voltage to follow the reference generated by a P&O algorithm, i.e., following the structure given in Fig. 2. From those solutions, three approaches with different relations between complexity and performance have been selected to illustrate the application and usefulness of the proposed fixed-frequency implementation technique. The selected SMCs are: first, the approach published in [15], which is based on the sliding-mode control of the inductor current, it requiring simple analytical expressions; however, the design of an additional cascade voltage control requires an equivalent model parameterized in a given operating point. The second approach, published in [16], is based on the sliding-mode control of the input capacitor current, which requires more complex analytical expressions but the resulting cascade voltage control is independent of the operating point. The third approach, published in [9], is based on a sliding surface depending on both the input capacitor current and PV voltage, which avoids the requirement of cascade voltage controllers. This feature increases the bandwidth of the controller at the expense of much complicated mathematical analyses. In conclusion, applying the proposed fixed-frequency implementation technique to those control approaches will put into evidence the practical usefulness of the solution.

The following subsection describes the basic tools required to analyze sliding-mode controllers, which are used to introduce the proposed fixed-frequency implementation technique. Moreover, Sects 2.2, 2.3 and 2.4 provide the main equations of each selected case, which are required to apply the implementation technique.

Section 2.1.2 analyzed the two conditions causing the switching frequency variation in presence of disturbances: the hysteresis band is constant but the derivatives of the switching function change, hence the time required by the switching function to travel between the limits of the hysteresis band also changes. Since the switching period \(T_{{\text{sw}}}\) depends on the time required by the switching function to travel between the limits of the hysteresis band, the switching frequency \(F_{{\text{sw}}}\) changes.

From Fig. 4 the practical control law is extracted, including the effect of the hysteresis band, which modifies the theoretical reachability conditions (5) and (6) as follow:Combining (28) and (29) with the reachability conditions for a positive transversality value, given in (7), leads to the practical control law for systems with a positive transversality value given in (30).Similarly, combining (28) and (29) with (8) leads to the practical control law for systems with a negative transversality value given in (31).To avoid the frequency variation due to changes in the operating point, the width h of the hysteresis band must be adapted: if the derivative of the switching function increases, h must be increased to keep constant the time needed to reach the band limit; similarly, if the derivative of the switching function decreases, h must be decreased. This concept is illustrated in Fig. 8, in which the hysteresis band is reduced to keep the switching frequency constant.

The implementation technique proposed in this paper is aimed for first-order sliding surfaces, hence the switching function must have first-order terms only. This decision is based on the large amount of first-order SMC designed for PV systems reported in literature [9, 15, 16, 18]. The first-order condition ensures a triangular waveform of the switching function, as it is depicted in both Figs. 4 and 8, which enables a fast calculation of the time required by the switching function to reach the hysteresis band limits.

Figure 8 also illustrates the method used to calculate the time \(t_{\text {up}}\) required by the switching function \(\varPsi\) to travel from the lower band limit to the upper band limit, hence traveling a distance equal to the band width h. Similarly, the time required by \(\varPsi\) to travel from the upper limit to the lower limit is \(t_{\text{dw}}\). From Fig. 8 it is concluded that \(t_{\text{up}}\) depends on the positive derivative of the switching function \(\frac{{\text{d}} \varPsi _{\text {up}}}{{\text{d}}t} > 0\) as follows:From the figure it is also concluded that \(t_{{\text{dw}}}\) depends on the negative derivative of the switching function \(\frac{ d\varPsi _{{\text{dw}}}}{{\text{d}}t} < 0\) as it is given in (33). In that expression the derivative is negative, hence the vectorial quantity changed in \(\varPsi\) is negative (\(-h\)).Figure 8 also shows that the switching period \(T_{{\text{sw}}}\) corresponds to the sum of both \(t_{\text {up}}\) and \(t_{{\text{dw}}}\), which leads to the following expression for the switching frequency \(F_{{\text{sw}}}\):Replacing (32) and (33) into (34) leads to:Finally, the expression for adapting h to ensure a fixed switching frequency \(F_{{\text{sw}}}\) is given in (36). It must be highlighted that such an expression depends on the switching function derivatives, therefore the SMC to be implemented must be analyzed as described in the examples reported in Sect. 2. In any case, those analyses are required to demonstrate global stability and to design the dynamic behavior of the closed-loop system.To simplify the design of the circuital implementation of the fixed-frequency SMC, it is helpful to transform the practical control law, given in (30) and (31) as inclusive conditions, into the exclusive conditions given in (37) and (38) for both positive and negative transversality values, respectively.Those control laws are implemented using two comparators, a Flip-Flop S-R and a digital microprocessor (or analog circuitry) to calculate \(h\left( v_{{\text{pv}}}, i_{{\text{pv}}}, v_{\text{b}}\right)\) online. Figures 9 and 10 show the electrical scheme of the proposed fixed-frequency implementation for SMC with both positive and negative transversality values, respectively. In such schemes it is assumed a digital microprocessor for calculating \(h\left( v_{{\text{pv}}}, i_{{\text{pv}}}, v_{\text{b}}\right)\) since a wide range of commercial devices are available for this purpose. Moreover, such schemes can be also emulated in the power electronics simulator PSIM using a C block. In any case, analog circuitry could be also used, but such an implementation will require analog multipliers and dividers.

This section illustrates the application of the proposed implementation technique using the three cases described in Sect. 2: inductor current control (case 1), input capacitor current control (case 2) and direct PV voltage control (case 3) for a grid-connected PV system.

The simulations of those SMC consider the following parameters: a BP585 PV module with parameters \(A=0.703 \; \mathsf {V^{-1}}\) and \(B=0.894 \; \mathsf {\mu A}\), a MPP voltage between \(16.39 \; \mathsf {V}\) and \(18.13 \; \mathsf {V}\), a dc/dc converter with \(L=330 \; \mathsf {uH}\) and \(C_{{\text{in}}} = 22 \; \mathsf {\mu F}\), an irradiance operation range from \(S = 100 \; \mathsf {W/m^2}\) to \(S = 1000 \; \mathsf {W/m^2}\), and a desired switching frequency \(F_{{\text{sw}}} = 60 \; \mathsf {kHz}\).

The simulations of the implementation technique were performed using the power electronics simulation PSIM, and the calculation of \(h\left( v_{{\text{pv}}}, i_{{\text{pv}}}, v_{\text{b}}\right)\) was implemented using C language. This procedure enables to tests the proposed solution by emulating the behavior of a Digital Signal Processor (DSP).

This section presents the experimental validation of the proposed implementation methodology. The proof-of-concept prototype reported in Fig. 18 enables to validate the performance of the fixed-frequency implementations of Case 1 and Case 3. In particular, Fig. 18a describes the prototype setup. The selected PV panel is a BP585 module, while the dc-link and the grid-connected inverter are emulated using a four quadrant electronic source/load BOP 50-20GL from Kepco. Such a device enables to implement the sinusoidal voltage disturbances considered in the simulation. Moreover, a P&O MPPT algorithm was designed with \(\varDelta v_{{\text{pv}}}=1\)V and \(T_{\text{a}}=0.1\)s following the procedure reported in [14]. The implemented surfaces \(\varPsi\), the adaptive hysteresis and the MPPT algorithm are calculated using a DSP F28335 controlCARD from Texas Instruments and converted to analog voltages with a DAC MCP4822; those signals are provided to the SMC Analog circuit. In addition, the switching circuits of Figs. 9 and 10 are implemented with a set of amplifiers, comparators and a TS555 integrated circuit as it is explained in [9]. The hysteresis comparators are implemented to enable the modification of the analog value h, which provides the variable-switching circuit presented in Fig. 9. Moreover, an amplification circuit is used to scale \(\varPsi\) to the required TS555 offset. Subsequently, the binary control signal u is generated by the TS555, which is provided to the MOSFET driver A3120.

The boost dc/dc converter is implemented using a 2218-H-RC inductor from Bourns Inc with \(L = 330 \; \mu H\), two MKT1813622016 capacitors from Vishay BC with \(C = 22 \; \mu F\) for \(C_{{\text{in}}}\) and \(C_{{\text{b}}}\), and two IRF540N MOSFETs from International Rectifier. In addition, the inductor and the input capacitor currents are measured using shunt-resistors \(R_{\text{s}}= 5 \; m \varOmega\) and AD8210 amplifiers. Those current and voltage measurements are acquired by the DSP using the onboard ADCs. Finally, the physical setup of the experimental platform is shown in Fig. 18b, where the PV module, the dc/dc converter, the DSP and the SMC analog circuit are observed.

The solution is evaluated with two experiments; the first one validates the proposed implementation technique with the positive transversality SMC based on \(\varPsi _L\) (Case 1), while the second experiment validates the proposed implementation technique with the negative transversality SMC based on \(\varPsi _V\) (Case 3).

In the first experiment, the SMC based on \(\varPsi _L\) was programed into the DSP. A comparison between both the variable-frequency and the fixed-frequency implementations of the SMC is reported in Fig. 19, which is in agreement with the simulation results previously presented in Fig. 13. The experimental validation illustrates in Fig. 19a the performance of the classical approach, which exhibits frequency variations when the disturbance in the load voltage occurs. The frequency variation is observed in the signal density of \(\varPsi\): higher density means higher frequency and viceversa. The switching frequency measurement provided by the oscilloscope in this experiment is always changing, hence it is not provided in the figure.

In contrast, Fig. 19b presents the validation of the fixed-frequency implementation, which provides the three-point behavior of the PV voltage and a constant switching frequency of \(59.78 \; \mathsf {kHz}\), which is a \(0.36 \; \%\) error with respect to the desired switching frequency of \(60 \; \mathsf {kHz}\). In addition, this experiment shows the hysteresis band is adapted to compensate the load disturbances and to avoid changes in the switching frequency. Comparing the experimental results given in Fig. 19a, b it is noted that both solutions reach the same optimal PV voltage, hence the proposed fixed-frequency implementation provides the same performance, in terms of power production, in comparison with the classical variable-frequency approach. In conclusion, the results of this first experiment puts into evidence the satisfactory performance of the proposed solution to guarantee a fixed switching frequency to the SMC.

The second set of experiments considers the implementation of the SMC based on \(\varPsi _V\) presented in Case 3. Similar to the previous validation, the experimental platform was evaluated considering the comparison between both the variable-frequency and the fixed-frequency implementations shown in Fig. 20. The variable-frequency implementation, based on a fixed hysteresis band, is evaluated in Fig. 20a. In that experiment the PV system is producing the maximum power (denoted by the three-point behavior), but the switching frequency is not constant when the load disturbances are present. Figure 20b shows the fixed-frequency implementation, which ensures a constant switching frequency of \(60.65 \; \mathsf {kHz}\), hence a \(1.08 \; \%\) error with respect to the desired switching frequency of \(60 \; \mathsf {kHz}\), without degrading the PV power production. Therefore, this second set of experiments shows the effectiveness of the proposed solution also in the implementations of SMC with direct regulation of the PV voltage.

Finally, the experimental results presented in this section demonstrate the correctness of the proposed fixed-frequency implementation for SMC applied to PV systems, in both positive and negative transversality cases.

This paper has presented a novel implementation methodology for sliding-mode controllers featuring constant switching frequency, which is aimed for SMC applied to photovoltaic systems. The proposed solution does not affect the performance of the SMC, hence there is no difference in the power produced by the PV system in comparison with the classical implementation based on variable frequency. Such a performance was validated using both detailed simulations and experimental measurements on a real prototype under different atmospheric conditions and load disturbances.

Therefore, the proposed fixed-frequency implementation methodology enables to precisely design filters for removing the switching noise from current and voltage signals, and to design the elements of the dc/dc converter without accounting for a worst-case scenario as in the case of variable frequency. However, this method is only applicable to SMC based on first-order surfaces, which limits the control systems able to be implemented of this solution. In any case, a large amount of PV systems are based on first-order surfaces, as it was discussed in the paper.

Finally, the proposed methodology could be extended to enable the implementation of high-order surfaces, e.g., maximum power point tracking strategies based on non-linear equations. Such an approach will require to extend the equations used to calculate the dynamic hysteresis band depending on each particular surface. This work is under development to increase the spectrum of sliding-mode controllers able to be implemented with the proposed solution.