Article
Sliding-Mode Controller for Maximum Power Point Tracking in Grid-Connected Photovoltaic Systems Paula Andrea Ortiz Valencia 1, * and Carlos Andres Ramos-Paja 2 Received: 11 September 2015 ; Accepted: 19 October 2015 ; Published: 2 November 2015 Academic Editor: Tapas Mallick 1 2
*
Instituto Tecnológico Metropolitano (ITM), Calle 73 No 76A-354 Vía al Volador, Medellín 05001000, Colombia Universidad Nacional de Colombia, Carrera 80 No 65-223-Block M8-113, Medellín 050041, Colombia;
[email protected] Correspondence:
[email protected]; Tel.: +57-4581-3957 or +57-4460-0727 (ext. 5587)
Abstract: The maximum power point tracking (MPPT) of photovoltaic systems must be as fast and accurate as possible to increase the power production, which eventually increases the PV system profitability. This paper proposes and mathematically analyses a sliding-mode controller to provide a fast and accurate maximum power point tracking in grid-connected photovoltaic systems using a single control stage. This approach avoids the circular dependency in the design of classical cascade controllers used to optimize the photovoltaic system operation, and at the same time, it reduces the number of controllers and avoids the use of linearized models to provide global stability in all the operation range. Such a compact solution also reduces the system cost and implementation complexity. To ensure the stability of the proposed solution, detailed mathematical analyses are performed to demonstrate the fulfillment of the transversality, reachability and equivalent control conditions. Finally, the performance of the proposed solution is validated using detailed simulations, executed in the power electronics simulator PSIM, accounting for both environmental and load perturbations. Keywords: DC/DC converter; maximum power point tracking (MPPT); photovoltaic systems; sliding mode control; grid-connection
1. Introduction The world energy demand is projected to more than double by 2050, and more than triple by the end of the century [1]. Incremental improvements in existing energy networks will not be adequate to supply this demand in a sustainable way. Hence, it is necessary to find sources of clean energy with a wide distribution around the world. The energy generation with photovoltaic (PV) systems is inexhaustible, hence it is a suitable candidate for a long-term, reliable and environmentally friendly source of electricity. However, PV systems require specialized control algorithms to guarantee the extraction of the maximum power available, otherwise the system could be unsustainable. The PV generator, also known as PV array, produces DC power that depends on the environmental conditions and operating point imposed by the load. To provide a high power production, the PV system includes a DC/DC converter to isolate the operating point of the generator (voltage and current) from the load, where such a power converter is regulated by an algorithm that searches, online, the maximum power point (MPP, i.e., the optimal operation condition) known as Maximum Power Point Tracking (MPPT) algorithm. The classical structure of a grid-connected photovoltaic system is presented in Figure 1, in which the PV generator interacts with a DC/DC converter controlled by a MPPT algorithm [2,3]. Such a structure enables the PV system to modify
Energies 2015, 8, 12363–12387; doi:10.3390/en81112318
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Tracking (MPPT) algorithm. The classical structure of a grid-connected photovoltaic system is presented in Figure 1, in which the PV generator interacts with a DC/DC converter controlled by a MPPT algorithm [2,3]. Such a structure enables the PV system to modify the operation conditions in agreement with the environmental circumstances (mainly changed by the irradiance and temperature) Energies 2015, 8, 12363–12387 so that a maximum power production is achieved [4,5]. Figure 1 also illustrates the gird-connection side of the PV system, which is formed by a DC-link (capacitor ) and a DC/AC converter (inverter). the operation conditions in agreement with the environmental circumstances (mainly changed by The inverter is controlled to follow required power factor, provideissynchronization and protect the irradiance and temperature) so athat a maximum power production achieved [4,5]. Figure 1 against islanding, among others. Moreover, the inverter must to regulate the DC-link voltage at the also illustrates the gird-connection side of the PV system, which is formed by a DC-link (capacitor bulk Cb ) and, where a DC/AC inverter is controlled to follow a required power capacitor twoconverter cases are(inverter). possible:The first, the inverter regulates the DC component of factor, voltage, provide synchronization and protect against islanding, among others. Moreover, the inverter must but due to the sinusoidal power injection into the grid, voltage experiments a sinusoidal perturbation at to regulate the DC-link voltage at the bulk capacitor Cb , where two cases are possible: first, the twice inverter the grid regulates frequencythe and a magnitude proportional the capacitance [3]. In the second DCwith component of Cbinversely voltage, but due to thetosinusoidal power injection into case, the the grid, DC component of voltage is not properly regulated, which produces multiple harmonic Cb voltage experiments a sinusoidal perturbation at twice the grid frequency and with a magnitude inversely proportional to the capacitance [3]. In the second case, the DC component of components with amplitude inversely proportional to the capacitance [3]. In both cases, the DC/DC Cb voltage is not properly regulated, which produces multiple harmonic components with amplitude converter output terminals are exposed to voltage perturbations that could be transferred to the PV inversely proportional to the capacitance [3]. In both cases, the DC/DC converter output terminals generator terminals, thus perturbations degrading the process. DC/DC are exposed to voltage thatMPP could tracking be transferred to theConcerning PV generator the terminals, thuspower converter, the boost topology the most widelythe used due power to theconverter, low voltage levels exhibited degrading the MPP tracking is process. Concerning DC/DC the boost topology is by the most widely used[6]. due to the low voltage levels exhibited by commercial PV modules [6]. commercial PV modules
Figure 1. Typical structure of a photovoltaic (PV) system. Figure 1. Typical structure of a photovoltaic (PV) system.
Multiple types of MPPT solutions are reported in the literature, which differ in complexity, number of sensors needed for operation, convergence speed, cost-effective range, etc. [7]. Two of the most commonly used MPPT techniques are Perturb and Observe (P & O) and Incremental Conductance (IC); the reason for this popularity is its implementation simplicity and its relatively good performance [4,8]. Other MPPT techniques are based on using fractional values of the open circuit voltage and short circuit current, i.e., the Fractional Open Circuit Voltage and the Fractional Short-Circuit Current. The main advantages of those solutions are the low cost and implementation simplicity since they only require a single (voltage or current) sensor [9,10]; But their efficiency is low compared with the P & O and IC algorithms. In contrasts, techniques based on computational intelligence, such as neural networks and fuzzy logic, offer speed and efficiency in tracking the MPP [11–13]; however its complexity and implementation costs are high compared with the P & O and IC algorithms, which make them costly solutions. The main problem of using traditional MPPT algorithms acting on the duty cycle of the DC/DC converter associated to the PV generator, i.e., Figure 1, concerns the large disturbances caused by irradiance transients in the system operating point, which generates a slow tracking of the MPP. This condition is also present at the system start-up, in which the MPPT algorithm takes a large amount of time to reach the MPP [3]. To mitigate such disturbances and speed-up the MPPT procedure, a two-stage control structure is usually adopted: it generally involves an algorithmic MPPT controller in cascade with a conventional voltage regulator (e.g., based on lineal or nonlinear control) as depicted in Figure 2. Moreover, such a structure is also needed to increase the reliability of double-stage grid-connected PV systems: the sinusoidal oscillation on the DC-link caused by the inverter operation must be mitigated, otherwise the MPPT procedure could be inefficient as reported in [3]. Such mitigation is traditionally performed by using large electrolytic capacitors for Cb , however the electrolytic technology introduces reliability problems due to its high failure rate [14]. Then, the
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Figure 2. Moreover, such a structure is also needed to increase the reliability of double-stage grid-connected PV systems: the sinusoidal oscillation on the DC-link caused by the inverter operation must be mitigated, otherwise the MPPT procedure could be inefficient as reported in [3]. Such Energies is 2015, 8, 12363–12387 performed by using large electrolytic capacitors for Cb, however the mitigation traditionally electrolytic technology introduces reliability problems due to its high failure rate [14]. Then, the voltage regulator in Figure 2 enables voltageoscillations oscillations in the DC-link produced voltage regulator in Figure 2 enablestotomitigate mitigate the the voltage in the DC-link produced by by small non-electrolytic capacitors, thus improving the system reliability [3,14]. small non-electrolytic capacitors, thus improving the system reliability [3,14].
2. of Structure of a PV system with apulse classical pulse width modulation (PWM)-based Figure 2. Figure Structure a PV system with a classical width modulation (PWM)-based cascade control. cascade control.
However, the design process of such a voltage regulator depends on the MPPT parameters, while However, the design process of such a voltage regulator depends on the MPPT parameters, while the MPPT controller stability also depends on the cascade controller performance [3]: for example, to the MPPT controller stability also depends on the cascade controller performance [3]: for example, to ensureensure the stability of aofPa& OOalgorithm, perturbationperiod periodmust must larger the settling the stability P& algorithm, its its perturbation be be larger thanthan the settling time time thevoltage, PV voltage, which depends thesystem system operating operating point thethe cascade voltage controller. of theofPV which depends ononthe pointand andonon cascade voltage controller. However, the voltage controller design requires performance criteria, e.g., the settling time as in [3],in [3], However, the voltage controller design requires performance criteria, e.g., the settling time as which are usually imposed in terms of the P & O parameters. Such a circular dependency in the two controllers design makes difficult to guarantee both system stability and desired performance in all the operating conditions. When the voltage regulator is implemented using conventional linear control techniques it is necessary to linearize the system model around a given operation point, which is usually the MPP at some irradiance condition [3]. However, due to the nonlinear nature of the PV module and DC/DC converter, the performance (and even stability) of the linear controller is limited to the neighborhood around the MPP [15]. This constraint puts at risk the system performance since the operating point changes with the unpredictable and unavoidable environmental perturbations. To address this problem, the work in [16] uses a sliding mode controller (SMC) to regulate the inductor current of a boost converter associated to the PV module, which enables to guarantee global system stability at any operating point. The solution proposed in that work considers three controllers in cascade as follows: the SMC that generates the activation signal for the MOSFET, a PI controller designed to provide the SMC reference depending on the command provided by a P & O algorithm, which is in charge of optimizing the power. However, the design of the PI controller requires a linearized model of the system around the MPP, hence it cannot guarantee the same performance in all the range of operation. In fact, a wrong design of such a PI controller could make the P & O unstable, hence both PI and P & O controllers have a circular dependency on their parameters. Similarly, the work in [17] uses a SMC to regulate the input capacitor current of the boost converter. This solution has a major advantage over the work reported in [16]: the solution in [17] does not require a linearized model since the transfer function between the capacitor current and voltage is linear and it does not depend on the irradiance or temperature conditions. Therefore, such a solution is able to guarantee the desired performance in all the operating range. However, as in the previous work, the three controllers are designed separately, which makes difficult to perform the system design: again, a wrong P & O perturbation period could lead to an unstable system operation. To avoid the circular dependency between voltage controllers and MPPT algorithms, a single controller in charge of both MPPT and voltage control operations is required. This problem has been addressed in [18–21] by using the sliding mode control technique. However, those works 12365
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are based on two considerations difficult to apply to grid-connected PV systems: First, there is not considered a capacitor linking the PV array and the DC/DC converter; second, the load impedance is considered constant. Regarding the first consideration, PV systems commonly consider a capacitor between the PV source and the DC/DC converter to stabilize the PV voltage, which in turns stabilizes the power produced, otherwise the current ripple at the DC/DC converter input will produce undesired oscillations that will degrade the MPPT procedure. The worst case of that current ripple is exhibited by DC/DC converters with discontinuous input current, e.g., buck or buck-boost topologies, while DC/DC converters with continuous input current only inject the inductor current ripple, e.g., boost, Sepic or Cuk topologies. In any case, a capacitor in parallel with the PV source is almost always considered; in fact, the experimental scheme of [21] includes such a capacitor despite it is not taken into account for the SMC analysis. It must be noted that this capacitor defines the dynamic behavior of the PV voltage and power, therefore it must be considered in the PV controller design. Regarding the second consideration, which assumes constant the load impedance, it is not applicable to grid-connected inverters since, as reported in [22], such inverters are controlled to provide a constant average value in the DC-link, with sinusoidal oscillations at twice the grid frequency due to the injection of single-phase AC power. Therefore, the best representation of such a load (from the DC/DC converter side) is a voltage source and not a constant impedance. This model will be further justified in Section 2. Therefore, in the design of a PV controller for grid-connected applications it is desirable to account for loads with non-constant impedance. Other interesting sliding-mode controller for PV systems is presented in [23], it considering a buck converter. Despite this work considers the capacitor linking the PV source and the power stage, it is not applicable to grid-connected PV systems since the DC/DC converter provides an output voltage lower than the one provided by the PV source. Such a condition is incompatible with the high input-voltage required by classical buck inverters for grid-connection [22,24]. Moreover, this work also assumes constant the load impedance, and it does not analyze the switching frequency, which is an important parameter for implementation. On the basis of those previous works and considerations, this paper presents the analysis and design of a SMC aimed at performing a fast MPPT action on grid-connected PV systems using a single control stage. This approach avoids the circular dependency among cascade controllers, reduces the number of controllers and avoids the use of linearized models to provide global stability in all the operation range. In such a way, even the use of a PWM is not needed. Such a compact design also reduces the system cost and complexity. The paper is organized as follows: Section 2 presents the non-linear mathematical model representing the PV system, then Section 3 introduces the proposed sliding surface and the controller structure. Section 4 presents the mathematical analysis of the transversality, reachability and equivalent control conditions to demonstrate the global stability of the proposed controller. Finally, Section 5 illustrates the performance of the proposed solution using detailed simulations executed in a standard power electronics simulator. The conclusions close the paper. 2. System Model A simplified circuital scheme of the PV system is presented in the Figure 3, which considers a boost converter due to the widely adoption of such a step-up DC/DC structure in PV systems, however the analysis presented in this paper can be extended to other DC/DC topologies. The scheme includes a voltage source as the system load, this to model the DC-link of double-stage structures in commercial PV inverters, in which the DC/AC stage regulates the DC-link voltage (Cb capacitor voltage) [25]. This voltage source model is widely used to represent the closed-loop grid-connected inverters due to its satisfactory relation between accuracy and simplicity, which is confirmed in references [3,17,25–29].
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The scheme considers the SMC acting directly on the MOSFET by means of the signal u, which is a discontinuous variable. Hence, no linearization or PWM are needed. This condition reduces the implementation cost, and circuit complexity, in comparison with SMC solutions implemented with Energies 2015, 8 e.g., [21]. PWM circuits,
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3. Circuital scheme of the sliding mode controller (SMC) loop. FigureFigure 3. Circuital scheme of the sliding mode controller (SMC) loop.
The dynamic behavior of the DC/DC converter is modeled by the switched Equations (1) and The dynamic behavior of the DC/DC converter is modeled by the switched Equations (1) and (2) [30], (2) [30], where i L represents the inductor current, v pv is the PV voltage, i pv represents the PV module wherecurrent, represents the inductor current, is the PV voltage, represents the PV module current, v is the load voltage and L and C represent the inductor and capacitor values: b
is the load voltage and
and
in
represent the inductor and capacitor values: v pv ´ vb p1 ´ uq di L “ 1 dt L
(1)
dv pv i pv ´ i L “ dt Cin
(2)
(1) (2)
The current of the PV module is modeled with the simplified single diode model [17] given in Equation model isc the short-circuit current thatdiode is almost proportional The current(3). of In thesuch PV amodule is represented modeled with the simplified single model [17] given in to the irradiance [17], B is the diode saturation current and A represents the inverse of the Equation (3). In such a model represented the short-circuit current that is almost proportional to the thermal voltage that depends on the temperature [25]. Such a model parameters are calculated
lnp1´i MPP {istc qof the thermal voltage that Bstc and A represents the irradiance thep´B diode saturation current , where istc and Tstc and Bstc “ pvinverse as A [17], “ Bistcis.exp stc .voc q, B “ 1`a . T ´T MPP ´voc q v p pv stc q depends temperature [25].and Such a model of parameters areunder calculated as test conditions . exp (STC),. , are on the the short-circuit current temperature the module standard ⁄ ln 1 respectively. vand and i MPP to the PV oc represents the open-circuit voltage, while ,v MPP andcorrespond are the short-circuit where
. voltage and current, respectively, at the MPP for the given operating conditions. Finally, av is the voltage temperature of coefficient [31]: under standard test conditions (STC), respectively. current and temperature the module represents
´ ¯ PV voltage and current, respectively, at the open-circuit voltage, while and correspond to the i pv “ isc ´ B e Av pv ´ 1 (3) the MPP for the given operating conditions. Finally, is the voltage temperature coefficient [31]: In conclusion, the non-linear equations system formed 1 by Equations (1)–(3) describes the PV system dynamic behavior in any operation condition.
(3)
In conclusion, the non-linear equations system formed by Equations (1)–(3) describes the PV 3. Sliding Surface and Controller Structure system dynamic behavior in any operation condition. The non-linear relation between the PV current and voltage, given in Equation (3), produces a non-linear relation between theStructure PV power and voltage (or current), which exhibits a maximum as 3. Sliding Surface and Controller depicted in Figure 4. In such a maximum, i.e., the MPP, the derivative of the power with respect to the voltage (or current) is zero as given in Equation (4). Therefore, the sliding-mode controllers proposed The non-linear relation between the PV current and voltage, given in Equation (3), produces a in [18–21,23] are based on Equation (4). In a similar way, this paper proposes a sliding surface based non-linear relation on Equation (4):between the PV power and voltage (or current), which exhibits a maximum as dp pvthe derivative of the power with respect to the pv the MPP, depicted in Figure 4. In such a maximum,dpi.e., “ 0 and “0 (4) dv pv di pv voltage (or current) is zero as given in Equation (4). Therefore, the sliding-mode controllers proposed
in [18–21,23] are based on Equation (4). In a similar way, this paper proposes a sliding surface based on Equation (4): 12367 0
0
(4)
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7
90
90
80
80
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70
(d ppv/d vpv) = 0
50
pv
40
60
p [W]
ppv [W]
60
40
30
30
20
20
10
10
0
0
5
10
15
(d ppv/d ipv) = 0
50
0
20
0
1
2
vpv [V]
ipv [A]
3
4
5
Figure4.4.Power Power curves curves of Figure ofaaPV PVmodule. module.
Taking into account thatthat p pv = v pv . . i pv , the holds at at thethe MPP: Taking into account , thefollowing followingrelation relation holds MPP: ˘ dp pv dv pv d ` . 0 v pv .i pv “ v pv ` i pv “ “0 di pv di pv di pv
(5) (5)
Then, the following relation defines the maximum power condition, i.e., at the MPP:
Then, the following relation defines the maximum power condition, i.e., at the MPP:
0 for
and
(6) dv pv v pv ` “ 0 for v pv “ v MPP and i pv “ i MPP (6) di pv Equation i pv To correlate the MPP condition (4) with the time-varying signals of the DC/DC converter, the condition in Equation (6) condition is expressed in terms(4) of with the time of both the of PVthe voltage and To correlate the MPP Equation the derivatives time-varying signals DC/DC current, obtaining expression Equation (7): converter, the condition in Equation (6) is expressed in terms of the time derivatives of both the PV voltage and current, obtaining⁄ expression Equation (7): ⁄ dv pv {dt
0 for
and
(7)
v pv
“ 0 for v pv “ v MPP and Ψ i pv and “ i MPP From such an expression, the `proposed switching function sliding surface Φ , given (7) in di pv {dt i pv Equation (8), are defined: From such an expression, the proposed switching function Ψ and sliding surface Φ, given in ⁄ Equation (8), are defined: Ψ ∧ Φ Ψ 0 (8) dv pv⁄{dt v pv ` ^ Φ “ tΨ “ 0u (8) Ψ“ di pv {dt i pv
Then, the state-space model of both the PV system and SMC is formed by the simultaneous Then,(1)–(3) the state-space model of bothwithin the PV system and SMCmodel is formed by the simultaneous Equations and (8). The relations such a state-space are illustrated in the block Equations (1)–(3) and (8). The relations within such a state-space model are illustrated in block diagram of Figure 5, which put in evidence the signals processing and variables exchangedthe between diagram of Figure 5, which put in evidence the signals processing and variables exchanged between the PV system and the SMC: the PV system disturbances are the short-circuit current (defined by the the PV system and the SMC: the PV system disturbances are the short-circuit current (defined by the irradiance and temperature) and the load voltage; the control signal is generated by the SMC on the irradiance and temperature) and the load voltage; the control signal u is generated by the SMC on the basis current and derives both both the the PV PVvoltage voltageand and current current to to construct construct basisof of the the PV PV current and voltage; voltage; and and the the SMC SMC derives the includes aa comparator comparator to to implement implement the the sign sign function function that that theswitching switchingfunction. function. Finally, Finally, the the SMC SMC includes triggers such aa comparator comparator is is described described in in the the following following triggersthe thechanges changes on on the the signal signal u.. The The design design of of such section in terms of the transversality and reachability conditions. section in terms of the transversality and reachability conditions.
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Figure 5. Block diagram of the SMC and PV system. Figure 5. Block diagram of the SMC and PV system.
4. Analysis of the Sliding-Mode Controller
4. Analysis of the Sliding-Mode Controller
The SMC must fulfill three conditions to guarantee stability and a satisfactory performance: transversality, and conditions equivalent control [32]. The transversality the controllability The SMC mustreachability fulfill three to guarantee stability andanalyses a satisfactory performance: of the system, the reachability analyses the ability of the closed-loop system to reach the surface, and transversality, reachability and equivalent control [32]. The transversality analyses the controllability the equivalent control grants local stability. of the system, reachability analyses theofability of the closed-loop system to the reach the surface, Those the conditions grant the existence the sliding-mode, which also imposes conditions defined in Equation (9)grants [33]: local stability. and the equivalent control dΨ Those conditions grant the existence of the sliding-mode, defined “ 0 which also imposes the conditions (9) Ψ“ 0 ^ dt
in Equation (9) [33]: Such expressions provide information concerning the surface and its derivative. In that way, the Ψ (10): derivate of the switching function is given in Equation Ψ
dΨ
0Λ
d2 v
0
d2 i
ˆ
(9) dv
di
˙
pv pv pv pv “ 2 i pv ` 2 v pv ` 2 ˚ (10) Such expressions provide information concerning the surface dt dt dtand its derivative. In that way, d t d t the derivate of the switching function is given in Equation (10):
To analyze expression Equation (10) it is required to also derive expressions Equations (2) and (3), it leading to expressions Ψ Equations (11)–(13), which are components of Equation (10):
2
∗
(10)
ˆ ˙ d2 v pv dv pv Av pv di L 1 disc “ ´ BA e ´ Cinis required dt d2 t(10) it Equation todtalso derivedtexpressions
(11)
To analyze expression Equations (2) and (3), it leading to expressions Equations (11)–(13), aredvcomponents of Equation (10): d i pv which di pv 1dt
“
sc
dt
´ BA
dt
e Av pv
d2 i pv d2 v pv d2 isc “ 2 ´ BA 2 e Av pv ´ BA2 2 d t d t d t
ˆ
(12)
dv pv dt
(11)
˙2 e Av pv
(13)
Then, the small-signal admittance of the PV module y “ ´BAe Av pv is introduced into the(12) previous equations to provide more compact expressions:
d2 v pv 1 “ Cin d2 t
ˆ
dv pv di L disc `y ´ dt dt dt
Then, the small-signal admittance of the PV module y equations to provide more compact expressions: 12369
˙
(13) (14)
is introduced into the previous
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d i pv dv pv disc “ `y dt dt dt d2 v pv d2 i pv d2 isc “ ` y ` Ay d2 t d2 t d2 t
(15) ˆ
dv pv dt
˙2 (16)
Replacing Equations (1) and (14)–(16) into Equation (10), and performing some mathematical manipulations, the following expression for the switching function derivate is obtained: dΨ dt
“
´
¯
´
¯
i pv `y v pv v ´v p1´uq i pv `y v pv ´ pv Lb C Cin ´in ´ ¯ i pv `y v pv dv pv dv pv ` dt y ` yAv pv dt ` 2 didtsc Cin 2 ` v pv dd2isct
disc dt
` 2y
dv pv dt
¯
(17)
This new expression is used to analyze the transversality, reachability and equivalent control conditions in the following subsections. 4.1. Transversality Condition To ensure the ability of the controller to act on the system dynamics, the transversality condition given in Equation (18) should be granted [33]: d du
ˆ
dΨ dt
˙ ‰0
(18)
Deriving Equation (17) with respect to the signal u leads to expression Equation (19): d du
ˆ
dΨ dt
˙
v “´ b L
ˆ
i pv ` y v pv Cin
˙ (19)
The analysis of Equation (19) requires to review the condition in which the MPP occurs: taking into account that p pv = v pv ¨ i pv , condition Equation (4) corresponds to expression Equation (20). Therefore, expressions Equations (18) and (19) lead to the transversality condition Equation (21): ´ ¯ dp pv “ v pv ¨ ´BAe Av pv ` i pv “ v pv .y ` i pv “ 0 dv pv d du
ˆ
dΨ dt
(20)
˙ ‰ 0 i f v pv ‰ v MPP
(21)
Figure 6 shows the simulation of Equation (19) considering the following conditions: a BP585 PV module with parameters B = 0.894 µF and A = 0.703 V´1 , and a DC/DC converter with L = 100 µH, Cin = 44 µF and vb = 24 V. The simulation illustrates the transversality condition provided by Equation (21): ‚ At the left of the MPP (voltages lower than v MPP ) the transversality condition is fulfilled, hence the SMC is able to act on the PV system to drive it towards the MPP. ‚ At the right of the MPP (voltages higher than v MPP ) the transversality condition is also fulfilled. ‚ At the MPP (v pv = v MPP ) the transversality condition is not fulfilled. This is not a problem because the PV system is already at the MPP. Moreover, if the system diverges from the MPP the transversality condition is fulfilled.
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At the right of the MPP (voltages higher than ) the transversality condition is also fulfilled. At the MPP ( ) the transversality condition is not fulfilled. This is not a problem because PV system Energies 2015, the 8, 12363–12387
is already at the MPP. Moreover, if the system diverges from the MPP the transversality condition is fulfilled.
ppv [W]
80 60
MPP
40 20 0
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No transversality
20 0 -20
Positive transversality
Negative transversality 0
2
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22
Voltage [V]
Figure 6. Simulation of the transversality condition.
Figure 6. Simulation of the transversality condition. The previous conditions impose a behavior similar to the classical hysteresis implementation
Theofprevious conditions impose a behavior similar to the classical hysteresis implementation of the the SMC for DC/DC converters: the system oscillates around the surface forming a hysteretic SMC trajectory. for DC/DC converters: system oscillates around the surface a hysteretic trajectory. In the followingthe subsections it will be demonstrated that suchforming hysteresis band is imposed by the peak-to-peak amplitude of the voltage ripple at the PV module terminals. In the following subsections it will be demonstrated that such hysteresis band is imposed by the peak-to-peak amplitude ofCondition the voltage ripple at the PV module terminals. 4.2. Equivalent Control The equivalent control condition analyzes the ability of the system to remain trapped inside 4.2. Equivalent Control Condition
the surface [31,32]. This condition is analyzed in terms of the equivalent analog value ueq of the discontinuous control signal u: if such an equivalent value is constrained within the operational The equivalent control condition analyzes the ability of the system to remain trapped inside the limits the system will not be saturated and its operation inside the surface is possible. Since the surface [31,32]. condition tois the analyzed terms ofsignal, the equivalent analog of the control signalThis u corresponds MOSFETinactivation its operational limits value are 0 and 1. dΨ “ 0 holds.limits Moreover,control since the analysis system inside surface, the condition discontinuous signal : ifconsiders such an the equivalent valuethe is constrained within thedtoperational Then, matching expression Equation (17) to zero, and replacing u = u , enables to obtain the ueqsignal eq the system will not be saturated and its operation inside the surface is possible. Since the control value given in Equation (22):
u corresponds to the MOSFET activation signal, its operational limits are 0 and 1. Moreover, since the C v dv pv y dv pv in analysis considers the system inside surface, 0 holds. Then, matching expression ueq “ 1the ´ vpv ` v1 didtscthe ` vcondition dt ` dt b
b
b
vb pi pv ¯`y v pv q
dv pv dv pv di Equation (17) to zero, and replacing ˚ yAv pv, enables tosc obtain the dt ` 2 dt ` 2y dt
´
1
1
`
Cin v pv d2 isc vb pi pv `yv pv q d2 t
value given in Equation (22): (22)
˙
Such an expression can be rewritten as Equation (23), where u1 and u2 are given in Equations(22) ∗ 2 2 (24) and (25), respectively: ueq “ u1 ` u2 (23) v pv Such an expression can be rewritten as Equation (23), where u1 “ 1 ´ vb and (25), respectively: „ ˆ u2 “
1 vb
dv pv dv pv disc Cin `y ` dt dt i pv ` y v pv dt
yAv pv
and
dv pv dv pv disc `2 ` 2y dt dt dt
1
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are given in Equations (24) (24) ˙ `
Cin v pv d2 isc i pv ` y v pv d2 t
(25)
(23) (24)
Energies 2015, 8, 12363–12387
The analysis of Equation (23) must be addressed by writing the derivative of Equation (8) in a different way as given in Equation (26): ˆ dΨ “ dt
i pv
2
d2 v pv di pv dt d2 t
´
d2 i pv dv pv d2 t dt
˙
i pv 2 ˚
´ ` ´
di pv dt
di pv dt
¯2 ´
dv pv dt i pv
´
di pv dt .v pv
¯ (26)
¯2
Since the solar irradiance does not exhibit fast changes in comparison with the switching frequency of the converter, it is assumed constant ( didtsc “ 0) to simplify the analysis of Equations (17), (25) and (26), obtaining the simplified expressions Equations (27) and (28), respectively: dv pv dΨ “ dt dt u2 “
y dv pv vb dt
˜
ˆ 1`
¸ ` ˘ i pv y ´ i pv A ´ y2 v pv y ˚ i pv ˘ dv pv ` Cin Av pv ` 2 i pv ` y v pv dt
(27) ˙ (28)
Taking into account that i pv “ 0 is a realistic condition since i pv = 0 does not occur at any MPP, the sliding-mode condition dv pv dt
dΨ dt
“ 0 is reached in Equation (27) for
dv pv dt
“ 0. Then, replacing those
“ 0 and i pv “ 0 into Equation (28) leads to u2 = 0. conditions Therefore, at low irradiance derivatives, expression Equation (23) for ueq is approximately equal to u1 . To evaluate the validity of such an approximation, ueq from Equation (23) was simulated at different irradiance derivatives and contrasted with u1 from Equation (24): the first test considers a change from the highest irradiance possible on earth to a complete shade in 1 second, i.e., 1 sun per second or dS/dt = 1 kW/(m2 ¨ s). The second test considers a derivative of 10 suns per s (dS/dt = 10 kW/(m2 ¨ s)), and the third test considers a derivative of 100 suns per s (dS/dt = 100 kW/(m2 ¨ s)), which is very large. The tests results are presented in the Table 1, where it is observed that the error generated by assuming ueq « u1 is less than 1% for the case with the largest derivative, while the errors for the other cases are less than 0.1% and 0.01%. Table 1. Error presented at different irradiance changes. dS/dt
Error (%) = ((ueq ´ u1 )/ueq ˆ 100%
1 kW /(m2 . s) 10 kW /(m2 . s) 100 kW /(m2 . s)
0.0075% < e < 0.008% ´0.0912% < e < 0.04% 0.989% < e < 0.6883%
Thus, the equivalent control analysis is based on expression Equation (29), which does not introduce a significant error: v pv ueq « 1 ´ (29) vb Finally, since the equivalent control condition for a DC/DC converter application is 0 < ueq < 1, such an expression is analyzed using the ueq value given in Equation (29), which leads to expression Equation (30): v pv 0 ă 1´ ă 1 Ñ 0 ă v pv ă vb (30) vb The condition in Equation (30) is always fulfilled since the analyzed PV system considers a boost converter. Therefore, the equivalent control is fulfilled.
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Energies 2015, 8, 12363–12387
4.3. Reachability Conditions The reachability conditions analyze the ability of the system to reach the desired condition Ψ = 0. The work in [32] demonstrated that a system that fulfills the equivalent control condition also fulfills the reachability conditions. That work also shows that the sign of the transversality condition imposes the value of u for each reachability condition. A positive value of the transversality condition imposes the following reachability conditions: ˇ ˇ dΨ ˇˇ dΨ ˇˇ “ ą0 lim ˇ dt ˇu“1,Ψ“0 ΨÑ0´ dt u“1
(31)
ˇ ˇ dΨ ˇˇ dΨ ˇˇ “ ă0 dt ˇu“0 dt ˇu“0,Ψ“0
(32)
lim
Ψ Ñ0`
Instead, a negative value reachability conditions:
of
the transversality
condition imposes
the
following
ˇ ˇ dΨ ˇˇ dΨ ˇˇ lim “ ą0 ˇ dt ˇu“0,Ψ“0 ΨÑ0´ dt u“0 ˇ ˇ dΨ ˇˇ dΨ ˇˇ lim “ ă0 ˇ dt ˇu“1,Ψ“0 ΨÑ0` dt u“1
(33) (34)
Moreover, the reachability conditions define the implementation of the switching law [32]. However, Section 4.1 shows that the transversality condition in this SMC is both positive and negative depending on the operation condition. In such a way, from Figure 6 it is observed that the voltage range for negative transversality (v pv < v MPP ) is much larger than the voltage range for positive transversality (v pv > v MPP ); therefore the implementation and analysis of the proposed SMC is performed in such a negative transversality condition, which imposes the following control law: {Ψ < 0 Ñ u = 0 ^ Ψ > 0 Ñ u = 1}. Then, the reachability of the surface is fulfilled in the following conditions: dΨ dt ă 0 with u = 1 dΨ and dt ą 0 with u = 0. Moreover, from Equation (27) it is observed that the sign of dΨ dt is the same sign of
dv pv dt .
Therefore, the analyzed reachability conditions are fulfilled in the following states:
‚ The MOSFET is ON (u = 1) and the voltage is decreasing, i.e., ‚ The MOSFET is OFF (u = 0) and the voltage is increasing, i.e.,
dv pv dt dv pv dt
ă 0. ą 0.
It must be noted that a DC/DC converter always exhibits a periodic voltage ripple around an average value [34]. In fact, the PV voltage corresponds to the voltage at the input capacitor Cin in Figure 3, which voltage ripple is defined by the second order filter formed by Cin and the inductor L. From the first differential equation of the DC/DC converter, i.e., Equation (1), it is observed that the inductor current exhibits an almost triangular waveform as in any other second-order filter; hence in steady-state the capacitor current has a triangular waveform centered in zero, again as in any other second order filter [30]. Therefore, the capacitor current increases with the MOSFET OFF (u = 0). This behavior is observed in the simulation of Figure 7, which includes the proposed SMC with the control action implemented as u = 1 for Ψ < 0 and u = 0 for Ψ > 0, and adopting the same electrical parameters previously described in Section 4.1. The implementation of the SMC is described in Section 5. The simulation also confirms that the capacitor current decreases with the MOSFET ON (u = 1). Moreover, it is noted that with a constant value for u (u = 0 or u = 1) the capacitor current exhibits positive, negative and zero values.
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This behavior is observed in the simulation of Figure 7, which includes the proposed SMC with the control action implemented as 1 for Ψ 0 and 0 for Ψ 0, and adopting the same electrical parameters previously described in Subsection 4.1. The implementation of the SMC is described in Section 5. The simulation also confirms that the capacitor current decreases with the MOSFET ON (Energies 1).2015, Moreover, it is noted that with a constant value for ( 0 or 1) the capacitor current 8, 12363–12387 exhibits positive, negative and zero values. vpv [V]
18.6 18.4
d vpv /dt = 0
d vpv /dt > 0
d vpv /dt < 0
18.2
iL [A]
10
0.0204
0.0204
0.0204
0.0204
0.0204
0.0203
0.0204
0.0204
0.0204
0.0204
0.0204
0.0203
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0.0204
0.0204
0.0204
0.0204
0.0203
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0.0204
0.0204
0.0204
0.0204
5
iC [A]
0 2 0 -2 1
u [-]
0.0203
0.5 0
Time [s]
Figure7.7.PV PV system system dynamic Figure dynamicbehavior. behavior.
This means that, DC/DC converter, i.e., This means that,asasreported reportedbybythe thesecond seconddifferential differential equation equation of of the the DC/DC converter, i.e., Equation (2), the voltage ripple is a series of quadratic waveforms generated from the integral of Equation (2), the voltage ripple is a series of quadratic waveforms generated from the integral of the the capacitor current = iipv ´ such i L . Ina way, such the a way, PV voltage and ipv − PV the voltage exhibits exhibits positive positive and negative capacitor current since since iCin =iCin L. In negative derivatives for both states of u as reported in Figure 7. Therefore, since the switching derivatives for both states of as reported in Figure 7. Therefore, since the switching function derivative function derivative has the same sign of the PV voltage derivative, as reported in Equation (27), the has the same sign of the PV voltage derivative, as reported in Equation (27), the reachability conditions reachability conditions are fulfilled in a fraction of the time intervals in which the MOSFET is in ON are fulfilled in a fraction of the time intervals in which the MOSFET is in ON and OFF states, which and OFF states, which means that in the other fractions of the time intervals the system diverges from means that inSuch the conditions other fractions time behavior intervals in thewhich system from the surface. Such the surface. createofa the periodic thediverges system converges and diverges conditions create a periodic behavior in which the system converges and diverges to/from the surface. to/from the surface. illustrate periodic behavior the reachability conditions, the resulting ToTo illustrate the the periodic behavior of theofreachability conditions, and theand resulting periodic periodic behavior behavior of the SMC, Figuretwo 8 shows two diagrams: A state flow describing the convergence and of the SMC, Figure 8 shows diagrams: A state flow describing the convergence and divergence divergence states in Figure 8(a), and the associated time-depending signal of the switching function states in Figure 8(a), and the associated time-depending signal of the switching function in Figure 8(b). in Figure 8(b). For the sake of illustration, the system is considered at the beginning of the analysis For the sake of illustration, the system is considered at the beginning of the analysis underdΨ the surface, i.e., under the surface, i.e., Ψ < 0, and with a negative switching function derivative, i.e., dt ă 0, which Ψ
0 , and with a negative switching function dvderivative, i.e., pv
corresponds to a negative PV voltage derivative
dt dv Cin dtpv
0 , which corresponds to a
ă 0 as reported in Equation (27). Due to the
differential equation governing the capacitor “ iCin , the condition dΨ dt ă 0 also corresponds to a negative input capacitor current, i.e., iCin ă 0. Those initial conditions are observed at the top of Figure 8(a). Then, due to the control action implemented for the SMC, the control signal is set to u = 0 and the system enters in the first state of operation: 1. Diverge from the MPP: u = 0, Ψ < 0, dΨ dt ă 0 (iCin ă 0). In this first stage (block 1 in Figure 8(a)) the reachability conditions are not fulfilled, hence the system diverges from the MPP, which is observed in Figure 8(b). Such a condition is caused by the negative value of the switching function derivative, which reduces even more the value of Ψ, hence the desired condition Ψ = 0 is not achievable. However, the control action u = 0 forces the increment of iCin , it driving the dΨ system to iCin “ 0 ( dΨ dt “ 0) and eventually to iCin ą 0 ( dt ą 0), which corresponds to the second stage of the SMC operation. 2. Converge to the MPP: u = 0, Ψ < 0, dΨ dt ą 0 (iCin ą 0). In this second stage the reachability condition Equation (33) is fulfilled, hence the system converges to the MPP, which is also observed in Figure 8(b). In this case, the positive value of the switching function derivative increases the value of Ψ, it driving the system towards the desired condition Ψ = 0. However, when the condition Ψ = 0 is reached the switching function derivative is still positive, i.e., dΨ dt ą 0 (iCin ą 0), hence Ψ becomes positive. Moreover, when Ψ ą 0 the SMC imposes u “ 1, which corresponds to the third stage of the SMC operation. 12374
Energies 2015, 8, 12363–12387
3. Diverge from the MPP: u = 1, Ψ ą 0, dΨ dt ą 0 (iCin ą 0). In this third stage the reachability conditions are not fulfilled, hence the system diverges from the MPP. Such a condition is caused by the positive value of the switching function derivative, which increases even more the value of Ψ, hence the desired condition Ψ “ 0 is not achievable. However, the control action u “ 1 forces the decrement of iCin , it driving the system to iCin “ 0 ( dΨ dt “ 0) and eventually to iCin ă 0 dΨ ( dt ă 0), which corresponds to the fourth stage of the SMC operation. 4. Converge to the MPP: u “ 1, Ψ ą 0, dΨ dt ă 0 iCin ă 0). In this fourth stage the reachability condition Equation (34) is fulfilled, hence the system converges to the MPP. In this case, the negative value of the switching function derivative decreases the value of Ψ, it driving the system towards the desired condition Ψ “ 0. However, when the condition Ψ “ 0 is reached the switching function derivative is still negative, i.e., dΨ dt ă 0 (iCin ă 0), hence Ψ becomes negative. Moreover, when Ψ ă 0 the SMC imposes u = 0, which corresponds to the first stage of the SMC operation. 15
Limit cycle: Repeated periodically generating the hysterical behavior
Energies 2015, 8
(a)
(b) 8. behavior Periodic behavior of the reachability conditions. (a) Convergence divergence Figure 8. Figure Periodic of the reachability conditions. (a) Convergence and and divergence conditions; conditions; (b) Hysteretic behavior. (b) Hysteretic behavior. The previous four stages of the system operation are continuously repeated, they form a limit-cycle [35] that imposes a hysteretic behavior to the SMC switching function around the 12375 surface Ψ 0. Moreover, from Figure 8 it is noted that the hysteresis band is defined by the condition 0, which corresponds to 0 and 0 as reported in Equation (27). This dependency is
Energies 2015, 8, 12363–12387
The previous four stages of the system operation are continuously repeated, they form a
Energies 2015, 8[35] that imposes a hysteretic behavior to the SMC switching function around the limit-cycle
16
surface Ψ “ 0. Moreover, from Figure 8 it is noted that the hysteresis band is defined by the condition dv pv
dΨ which 7, corresponds to dt by “ 0Figure and icin9,“which 0 as reported Equation This dependency depicted in0,Figure and confirmed presentsinΨ, and(27). u generated by the same dt “
is depicted in Figure 7, and confirmed by Figure 9, which presents Ψ,
dΨ
and u generated by the
simulation producing producing the signalsthe presented in Figurein7.Figure In addition, Figure dt9Figure also confirms the limit-cycle same simulation signals presented 7. In addition, 9 also confirms the of thelimit-cycle switchingof function. Finally, it is noted that the hysteresis band of the SMC is defined the switching function. Finally, it is noted that the hysteresis band of the SMC is definedby the by ripple the voltage the input capacitor, which can be modified by changing the input capacitance. voltage at theripple inputatcapacitor, which can be modified by changing the input capacitance. 0.2
Divergence Condition
[-] 0
Reachabality Condition
-0.2 -0.4 0.0203 5
0.0203
0.0203
0.0203
0.0203
0.0204
0.0203
0.0203
0.0203
0.0203
0.0203
0.0204
0.0203
0.0203
0.0203
0.0203
0.0203
0.0204
x 10
4
d /dt [-] 0 -5 1
u [-] 0.5 0
Time [s] Figure 9. SMC hysteretic behavior. Figure 9. SMC hysteretic behavior.
4.4. Switching Frequency
4.4. Switching Frequency
The switching frequency is an important parameter in the operation of DC/DC converters, hence it must be calculated in order provide practical guidelines the SMCof implementation. The switching frequency is antoimportant parameter in thefor operation DC/DC converters, hence it The switching frequency is calculated from the DC/DC converter differential equations and must be calculated in order to provide practical guidelines for the SMC implementation. ripple magnitudes as reported in [30]. In such a way, in a boost converter the ripple magnitude for The frequency is calculated from the DC/DC converter differential equations andTsripple theswitching inductor current and capacitor voltage are given in Equations (35) and (36), respectively, where represents switching the duty andconverter Fsw =1/Ts is theripple switching frequency: magnitudes as the reported in period, [30]. Ind represents such a way, in acycle, boost the magnitude for the
inductor current and capacitor voltage are given v pv in Equations v pv (35) and (36), respectively, where Ts ∆i L “ dTs , d “ 1 ´ (35) 2L duty cycle, and vb Fsw =1/Ts is the switching frequency: represents the switching period, d represents the ∆
∆i ∆v pvd“ , L Ts1 2 8Cin
(36)
(35)
Figure 10 presents a simulation of the closed-loop ∆ SMC system highlighting the time intervals (36) in which the MOSFET is turned on (Ton ) ∆and turned 8 off (Toff ). In such a figure it is observed that the magnitude of the voltage ripple ∆v pv is mesured from the average PV voltage v pv to the peak Figure 10 presents ofpvthe highlighting the time intervals in (maximum) voltagea vsimulation “ vclosed-loop Then, system approximating the average PV voltage pv´ max as ∆v pv´max ´ v pv . SMC to the MPP voltage, the voltage ripple magnitude becomes ∆v « v ´ v , where the MPP pv pv ´ max MPP which the MOSFET is turned on (Ton) and turned off (Toff). In such a figure it is observed that the
LambertW pexpp1qp B`isc qq´1 voltageofis the calculated from Equation as v MPP “from the average the LambertW magnitude voltage ripple ∆ (20) is mesured PV using voltage to the peak A function [36]. (maximum) voltage as ∆ . Then, approximating the average PV voltage , where the MPP to the MPP voltage, the voltage ripple magnitude becomes ∆
voltage
is
calculated
from
Equation
(20)
as
LambertW function [36]. 12376
using
the
Energies 2015, 8, 12363–12387
vpv [V]
Energies 2015, 8
17
18.5
vpv-max
18.4
v
vpv
18.3 18.2 0.0197
0.0197
0.0197
0.0197
0.0197
0.0198
0.0198
0.0198
0.0198
0.0197
0.0197
0.0198
0.0198
0.0198
0.0198
u [-]
1
0.5
Ton
Toff
0 0.0197
0.0197
0.0197
Time [s] Figure 10. Principle for calculating the switching frequency.
Figure 10. Principle for calculating the switching frequency. dv
The maximum PV voltage v pv´ max is achieved when dtpv “ 0, which also corresponds to dΨ “0 Theand maximum PV voltage is achieved when 0, which also corresponds to dt 0 and u = 1 as demonstrated in the previous subsection. Then, replacing those values in Equation (17), isc constant, (37)subsection. is obtained: Then, replacing those values in Equation (17), = 1and as assuming demonstrated in theEquation previous
u and assuming
˙ ˆ constant, Equation (37) is v pvobtained: i pv ` y v pv´max ´max 0“´
0
L
(37)
Cin
(37)
Finally, replacing expressions Equations (35) and (37) into Equation (36) with v pv « v MPP and ∆v pv «replacing v pv´max ´ expressions v MPP enables Equations to calculate T on as:and (37) into Equation (36) with Finally, (35)
∆
g ˆ as: ˇ enables to calculate f Ton “
f f 16LCin e
´
16
i pv ˇ y ˇv “ MPP pv
and
˙ ´ v MPP (38)
v MPP d
(38) dv pv dt
Similarly, taking into account that the minimum PV voltage v pv´ min is achieved when “ 0, “ 0 and u = 0, the voltage ripple measured from v pv´min to the average PV voltage can be Similarly, takingas into the . minimum PV voltageas in Equation is achieved when approximated ∆v pv account « v pv´maxthat ´ v MPP Then, To f f is calculated (39), where d1 = 1´d represents the complementary duty cycle: dΨ dt
0 and u = 0, the voltage ripple measured from
approximated as ∆ represents the complementary
0,
to the average PV voltage can be
g ˙ f . Then,ˆ ˇ f i ˇ is calculated as in ´ v f 16LCin ´ pv MPP y ˇv “ MPP e pv duty To f fcycle: “ v MPP d1
Equation (39), where d′ = 1−d (39)
Finally, the switching frequency16 imposed by the SMC is given in Equation (40). In such an expression the duty cycle is evaluated as reported in Equation (35):
(39)
′
1 Fsw “ “d ˆ ` To f f switchingTonfrequency imposed
1
ˇ i pv ˇ ´ by y ˇ
(40)
Finally, the the SMC given in Equation 16LCin ´v MPP is ´b ¯ (40). In such an b v pv “MPP vb vb expression the duty cycle is evaluated as reportedv MPP in Equation (35):vb ´v MPP ` v MPP ˙
1 switching frequency of the PV system The accuracy 1 of expression is illustrated by calculating the simulated in Figure 10: Equation (40) predicts a switching frequency equal to 27.4 kHz, while the simulation reports a switching16 frequency equal to 28.41 kHz, which correspond to an acceptable prediction error of 3.5%. 12377
(40)
Energies 2015, 8, 12363–12387
Moreover, from expression Equation (40) it is evident that the switching frequency depends on the inductance and capacitance values, PV module characteristics and system operating point. Therefore, if Cin or L change due to aging or by other effects such as inductor saturation, the switching frequency will also change. However, the previous subsections demonstrate the robustness of the proposed SMC to changes in Cin and L: the system will remain stable for any positive values of Cin and L. In conclusion, tolerances in the converter parameters due to aging or practical saturations will not compromise the system stability. 4.5. Sliding-Mode Dynamics The sliding-mode dynamics provide information concerning the averaged voltage and current behavior, which is useful to predict important time-based performance criteria such as settling time. The sliding-mode dynamics consider the system within the sliding surface, hence expressions Equation (7) hold, which leads to Equation (41): v pv di pv dv pv “´ . dt i pv dt
(41)
Similarly, from the second differential equation of the DC/DC converter, i.e., Equation (2), the relation i pv “ Cin
dv pv dt
` i L is obtained, which derivative is given in Equation (42): di pv d2 v pv di “ Cin 2 ` L dt dt d t
(42)
Then, substituting Equations (1) and (41) into Equation (42), and naming the instantaneous PV v pv , the relation Equation (43) is obtained: module impedance R pv “ i pv R pv Cin L
d2 v pv dv pv `L ` R pv v pv “ R pv vb d1 dt d2 t
(43)
Expressing Equation (43) in Laplace domain leads to the model reported in Equation (44): R pv vb v psq “ d1 psq R pv Cin L s2 ` L s ` R pv
(44)
Such a second-order model describes the closed-loop system behavior around a given operation a condition, in a which the natural frequency is wn “ 1{ pCin Lq and the damping ratio is ξ “ 1{p2R pv C 1{ pCin Lqq. From the classical second order system analysis [37], the settling time of the PV voltage is, approximately, ts “ 6.4R pv Cin . To test such an estimation, the PV system controlled by the SMC, considering the parameters previously described in Section 4.1, was simulated in both step-up and step-down irradiance transients. The first test considers a step change in the irradiance from 600 W/m2 to 1000 W/m2 , which corresponds to a change in isc from 3 A to 5 A, obtaining the results presented in Figure 11(a): in such conditions the average PV module impedance is 5.1 Ω, which leads to ts = 1.4 ms. To provide a more clear calculation of the settling time from the DC/DC converter waveforms, the PV voltage is also filtered to remove the switching ripple, which enables to verify the accuracy of the proposed approximation. Similarly, the second test considers a step change in the irradiance from 1000 W/m2 to 600 W/m2 , obtaining the results presented in Figure 11b, where the settling time estimation is the same, i.e., ts = 1.4 ms. Both simulations confirm the validity of the approximation. It must be noted that the settling time will change if Cin or L change, e.g., due to aging or practical saturations.
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enables to verify the accuracy of the proposed approximation. Similarly, the second test considers a step change in the irradiance from 1000 W/m2 to 600 W/m2, obtaining the results presented in Figure 11b, where the settling time estimation is the same, i.e., ts = 1.4 ms. Both simulations confirm the validity of the approximation. It must be noted that the settling time will change if or change, e.g., due to aging Energies 2015, 8, 12363–12387 or practical saturations. 19
vpv [V]
18.5
Unfiltered signal Filtered signal
18
Predicted ts ts = 1.4 ms
17.5 17 0.028
0.0285
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2
0.032
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(a) Transient from 600 W/m to 1000 W/m
19
vpv [V]
0.0315 2
18 17
Predicted ts ts = 1.4 ms
16 0.028
0.0285
0.029
0.0295
0.03
0.0305
0.031
Time [s] 2 2 (b) Transient from 1000 W/m to 600 W/m Figure Settling-time estimation. Figure 11.11.Settling-time estimation.
5. Numerical Results and Performance Evaluation 5. Numerical Results and Performance Evaluation The proposed SMC was implemented in the power electronics simulator PSIM using C language. procedure to tests, realistically, the performance of the SMC by emulating the TheThis proposed SMCenables was implemented in the power electronics simulator PSIM using C language. implementation of the controller in a Digital Signal Processor (DSP). In addition, the ANSI C code This procedure enables to tests, realistically, the performance of the SMC by emulating the implementation used in the PSIM simulation (within a C block) can be directly executed in any DSP without of the major controller in a Digital Signal Processor (DSP). In addition, the ANSI C code used in the PSIM modifications. simulation Figure (within12a presents C block)the cansimulation be directly executed anyimplementation, DSP without major modifications. scheme and in SMC which uses four devices to implement the controller: a C block scheme emulating theSMC DSP, two classical comparators a S-R flip-flop Figure 12 presents the simulation and implementation, whichand uses four devices to to store the u signal. Such a structure is widely adopted in literature to implement the switching implement the controller: a C block emulating the DSP, two classical comparators and a S-R flip-flop function due to its simplicity and reliability [32]. However, since in this work the SMC performs to store the u signal. Such a structure is widely adopted in literature to implement the switching both MPPT and voltage regulation actions, there is no need of cascade (or any other) controllers or function due to itsFigure simplicity reliability [32].C However, since in this the SMC performs modulators. 12 alsoand presents the ANSI code used to calculate thework switching function, which both provided by using an emulated Digital-to-Analog to the circuit driving MPPTis and voltage regulation actions, there is no Converter need of (DAC) cascade (orswitching any other) controllers or the MOSFET. To test the SMC in the same conditions used to illustrate the mathematical analysis, this modulators. Figure 12 also presents the ANSI C code used to calculate the switching function, which is simulation scheme considers a DC-link voltage near to 30 V imposed by the standard voltage source provided by representing using an emulated Digital-to-Analog Converterscheme (DAC)will to be theused switching circuit driving model the inverter. An additional simulation afterwards to test the the MOSFET. testhigh theboosting SMC inconditions, the sameand conditions used the mathematical analysis, this SMC, To under interacting withtoa illustrate detailed grid-connected inverter. 13 considers shows twoatests performed in the first scheme: 13(a) presents simulationFigure scheme DC-link voltage near to simulation 30 V imposed by Figure the standard voltagethe source dynamic response of the system for a change in the irradiance from 600 W/m2 to 1000 W/m2 , model representing the inverter. An additional simulation scheme will be used afterwards to test the while Figure 13(b) presents the dynamic response of the system for a change in the irradiance from SMC, 1000 underW/m high2 to boosting conditions, interacting with a the detailed grid-connected inverter. 2 . In bothand 600 W/m cases the SMC drives PV system to the optimal operation Figure 13 shows two performed in time. the first simulation Figure condition within thetests estimated settling Moreover, suchscheme: simulations also13(a) show presents the pointsthe in dynamic the 2 2 condition; then I-V curve in which the PV system operates: (1) corresponds to the initial steady-state response of the system for a change in the irradiance from 600 W/m to 1000 W/m , while Figure 13(b) condition (2) is triggered by an irradiance perturbation; and (3) is the new MPP detected by2 the SMC. 2
presents the dynamic response of the system for a change in the irradiance from 1000 W/m to 600 W/m .
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In In both both cases cases the the SMC SMC drives drives the the PV PV system system to to the the optimal optimal operation operation condition condition within within the the estimated estimated settling settling time. time. Moreover, Moreover, such such simulations simulations also also show show the the points points in in the the I-V I-V curve curve in in which which the the PV PV system system operates: operates: (1) (1) corresponds corresponds to to the the initial initial steady-state steady-state condition; condition; then then condition condition (2) (2) is is triggered triggered by by an an Energies 2015, 8, 12363–12387 irradiance perturbation; and (3) is the new MPP detected by the SMC. irradiance perturbation; and (3) is the new MPP detected by the SMC.
Power[W] [W] Power
Power[W] [W] Power
Voltage[V] [V] Voltage
Figure 12. Simulationscheme scheme and implementation. Figure 12. and SMC implementation. Figure 12. Simulation Simulation scheme andSMC SMC implementation. 19 19 18 18 17 17 16 16
15 15 14 14 0.02 0.02 100 100
(1) (1) (2) (2)
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(b) Figure 13. System Response to irradiance perturbations. (a) Transient from 600 W/m2 to 2 2 2 2 to 1000 W/m2 ; 2 1000Figure W/m ; (b) Transient from W/m to 600 (a) W/m . (a) from System Response to irradiance perturbations. Transient 600 W/m Figure 13.13.System Response to1000 irradiance perturbations. Transient from 600 W/m to (b) Transient from 1000 W/m2 to 600 W/m2 .
1000 W/m2; (b) Transient from 1000 W/m2 to 600 W/m2. Taking into account that grid-connected PV systems exhibit perturbations at the DC-link, the SMC Taking into in account that grid-connected PV systems exhibit perturbations atathe DC-link, the must be also presence of sinusoidal voltage oscillations at Cb. Such test is presented in Taking intotested account that grid-connected PV systems exhibit perturbations at the DC-link, the SMC SMC must be also tested in presence of sinusoidal voltage oscillations at Cb . Such a test is presented Figurebe14, where a large 20 V peak-to-peak hasatbeen superimposed topresented a 29 V DC must also tested in apresence sinusoidalvoltage voltageoscillation oscillations Csuperimposed in b. Such a testtois in Figure 14, where large 20 Vofpeak-to-peak voltage oscillation has been a 29 V DC component, this corresponding to a 69% perturbation. The simulation also considers an irradiance corresponding to a 69% perturbation. The simulation also considers an irradiance Figurecomponent, 14, where this a large 20 V peak-to-peak voltage oscillation has been superimposed to a 29 V DC 2 2 22 perturbation from 1000 W/m to 600 W/m at t = 0.03 s, and even under the perturbation in thein load perturbation from 1000 W/m to 600 W/m at t = 0.03 s, and even under the perturbation the load component, this corresponding to a 69% perturbation. The simulation also considers an irradiance voltage the SMC is able to impose the expected behavior to reach the new MPP. 2 2 voltage the SMC ableW/m to impose theW/m expected reach theunder new MPP. perturbation from is1000 to 600 at t behavior = 0.03 s,toand even the perturbation in the load
vb [V]v [V] b
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voltage the SMC is able 20to impose the expected behavior to reach the new MPP. 20 18 18 16 16 0 90
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Figure 14. System response with load perturbation. Figure 14. System response with load perturbation. 12381
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As anticipated anticipated before, before, aa second second simulation simulation scheme scheme is is used used to to test test the the SMC SMC under under high high boosting boosting As Energies 2015, 8, 12363–12387 conditions and and interacting interacting with with aa detailed detailed grid-connected grid-connected inverter. inverter. Such Such aa scheme scheme is is presented presented in in conditions Figure 15 15 [24], [24], which which includes includes aa closed-loop closed-loop inverter inverter with with two control control objectives: provide provide a given given power power Figure As anticipated before, a second simulation scheme istwo used to test objectives: the SMC under highaboosting factor and and regulate regulate the the DC-link DC-link voltage. voltage. The The inverter inverter isis designed designed to to operate operate at at 110 110 VAC@60 VAC@60 Hz Hz with an an factor conditions and interacting with a detailed grid-connected inverter. Such a scheme is presented with in average DC-link voltage equal to 220 220 V. The The DC-link DC-link is formed by objectives: 14 F, F, which which ata1000 1000 W/m22 Figure 15 [24], whichequal includes a closed-loop inverter is with two control provideat givenW/m average DC-link voltage to V. formed by 14 power factor and 120 regulate the DC-link voltage. The oscillation inverter is designed to operatetoat the 110 VAC@60 Hz experiment large V peak-to-peak peak-to-peak voltage superimposed DC component, component, experiment aa large 120 V voltage oscillation superimposed to the DC with an average DC-link voltage equal to 220 V. The DC-link is formed by C “ 14 µF, which b this corresponding corresponding2 to to aa 55% 55% perturbation. perturbation. Moreover, Moreover, the the input input capacitor was was changed changed to 150 FF this to 150 at 1000 W/m experiment a large 120 V peak-to-peak voltagecapacitor oscillation superimposed to the DC to avoid avoid large PV PVthis voltage oscillations due perturbation. to the the high high voltage voltage conversion ratio imposed imposed to the the DC/DC DC/DC to large voltage oscillations due to conversion to component, corresponding to a 55% Moreover, the input ratio capacitor was changed to converter. Therefore, the SMC SMCPV switching frequency,due calculated from Equation (40),ratio is approximately approximately Cin “Therefore, 150 µF to avoid large voltage oscillations to the high voltage conversion imposed converter. the switching frequency, calculated from Equation (40), is to theIn DC/DC converter. Therefore, the SMC switching frequency, calculated from Equation (40),W/m is 22 to 100 kHz. addition, the simulation also considers an irradiance perturbation from 1000 100 kHz. In addition, the simulation also considers an irradiance perturbation from 1000 W/m to approximately 100 kHz. In addition, the simulation also considers an irradiance perturbation from 2 300 W/m 2 at t = 0.65 s to illustrate the MPPT operation of the SMC. The results are presented in 2 at t = 0.65 at t 2=to0.65 s to illustrate thes toMPPT operation of operation the SMC.of The results presented in 300 W/m 1000 W/m 300 W/m illustrate the MPPT the SMC. Theare results are Figure 16, which confirms that even under high DC-link perturbations the SMC is able to reach the in Figure 16, which confirms that evenDC-link under high DC-link perturbations SMCto is reach able the Figurepresented 16, which confirms that even under high perturbations the SMC the is able new MPP. It must be point out that the inverter also imposes a transitory behavior in the DC-link to reach the new MPP. It must be point out that the inverter also imposes a transitory behavior in new MPP. It must be point out that the inverter also imposes a transitory behavior in the the DC-link DC-link due toin the change in at PV power t which = 0.65 s,iswhich is also mitigated the SMC. voltage due to tovoltage the change change PV power 0.65at also mitigated mitigated by the the by SMC. voltage due the in PV power at tt == 0.65 s,s, which is also by SMC.
vbv [V] [V] b
pv
ppv p [W] [W]
pv
vpv v [V] [V]
15. Circuital schemeof of the the detailed grid-connected PV system. FigureFigure 15. Circuital Circuital scheme detailed grid-connected PV system. system. Figure 15. scheme of the detailed grid-connected PV 20 20 18 18
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16. SMC behaviorinteracting interacting with a grid-connected inverter. FigureFigure 16. SMC SMC behavior with grid-connected inverter. Figure 16. behavior interacting with aa grid-connected inverter.
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Energies 2015, 8 23 Energies 2015, 8 23 Energies 2015, 8, 12363–12387 With the aim of further evaluate the proposed SMC, the performance of this controller is also contrasted solutionsthe based on the SMC, P & O the algorithm. The P of & this O method is based on With thewith aimtwo of classical further evaluate proposed performance controller is also With the aim of further evaluate the proposed SMC, the performance of this controller is also contrasted classical solutions P& O algorithm. The P &observe O method is based on perturb thewith inputtwo variable of the system,based e.g., on thethe duty cycle of the converter, the variation in contrasted with two classical solutions based on the P & O algorithm. The P & O method is the output and increase decrease thethe perturbed variable to converter, increase the power.the The flowchart perturb thepower, input variable of theorsystem, e.g., duty cycle of the observe variation in based on perturb the input variable of the system, e.g., the duty cycle of the converter, observe the of P & power, O algorithm is shown Figure 17. thethe output increase orin decrease theorperturbed variable to increase theincrease power.the The flowchart variation in theand output power, and increase decrease the perturbed variable to power. The flowchart of the P & O algorithm is shown in Figure 17. of the P & O algorithm is shown in Figure 17.
Figure 17. Flowchart of the Perturb and Observe (P & O) algorithm. 17. Flowchart thePerturb Perturb and (P &(P O)& algorithm. FigureFigure 17. Flowchart ofofthe andObserve Observe O) algorithm. The structure of the two classical solutions based on the P & O algorithm are presented in Figure 18: structure ofdefining the two the classical P algorithm &O O+ algorithm are presented in 18: of the two classical solutions based on theonP the & are presented in MOSFET; Figure (a) The a P structure &The O algorithm duty solutions cycle of based a PWM (P O& PWM) driving the Figure 18: (a) a P & O algorithm defining the duty cycle of a PWM (P & O + PWM) driving the and & algorithm O algorithm defining voltage of a (P PID&controller, which in turns the (a) a(b) P a&P O defining thetheduty cyclereference of a PWM O + PWM) driving the defines MOSFET; MOSFET; and (b) a P & O algorithm defining the voltage reference of a PID controller, which in turns duty cycle a duty PWM (P of& + PID Such aSuch figure put in thethe of the and (b) a P of &theO algorithm defining the&+voltage of a aPID controller, which inadvantages turns defines defines cycle aO PWM (P OPWM). + PID reference + PWM). figure putevidence in evidence advantages of thesolution: proposed solution: requires a control single the classical solutions duty cycle of a PWM (P & the O +SMC PIDonly + PWM). a control figure put ininstead evidence the advantages of the proposed the SMC only requires a singleSuch block,block, instead the classical solutions require require two and three control blocks. two and three control proposed solution: theblocks. SMC only requires a single control block, instead the classical solutions require two and three control blocks. Cin
PV
Cin
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P&O P&O (b) (b)
(a) (a)
dc/dc dc/dc
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It requires the use of cascade control It requires the use of cascade control (a) P&O + PWM (a) P&O P&O ++ PID PWM (b) + PWM (b) P&O + PID + PWM
Figure 18. Structure of a conventional Maximum Power Point Tracking (MPPT) architecture. Figure 18. Structure of a conventional Maximum Power Point Tracking (MPPT) architecture. Figure 18. Structure of a conventional Maximum Power Point Tracking (MPPT) architecture. & O algorithms were designed followingthe the guidelines guidelines given [3,38], which are aimed to The P The & OP algorithms were designed following giveninin [3,38], which are aimed to guarantee a perturbation period larger than the PV voltage settling time. Concerning the stand-alone guarantee perturbation larger than the PV the voltage settlinggiven time.inConcerning theare stand-alone The P &a O algorithmsperiod were designed following guidelines [3,38], which aimed to guarantee a perturbation period larger than the PV voltage settling time. Concerning the stand-alone
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P & OEnergies controller (a), the parameters are: perturbation period Ta = 0.5 ms and perturbation amplitude 2015, 8, 12363–12387 Δd = 0.2. TheP design of the(a), PID cascade period P & OTasolution (b) perturbation was based on the system & O controller thecontroller parametersfor are:the perturbation = 0.5 ms and amplitude 2 Equations linearized at the lower irradiance condition (100 W/m ) as suggested in [3]. Then, ∆d =(1)–(3) 0.2. design between of the PIDthe controller for the cascade P &cycle O solution was based system (45). is (b) calculated as on in the Equation the transferThe function PV voltage and the duty Equations (1)–(3) linearized at the lower irradiance condition (100 W/m2 ) as suggested in [3]. Then, changes with the solar irradiance and temperature, hence the settling v the transfer function between the PV voltage and the duty cycle dpv is calculated as in Equation ts(45). of the voltage to constantly changes [39]: with the solar irradiance and temperature, hence the It isPV important remark that R pv changes settling time ts of the PV voltage constantly changes [39]:
It is important to remark that time
(45)
v pv psq R pv vb “´ d psq L Cin R pv s2 ` L s ` R pv
(45)
Therefore, conventional linear controllers for PV systems must be designed at the longest settling 2 controllers be designed the longest time value,Therefore, which is conventional obtained forlinear the larger valuefor ofPV systems (lowermust irradiance, 100atW/m in thissettling example). (lower irradiance, 100 W/m2 in this example).
time value, which is obtained for the larger value of R
pv Then, using the root-locus placement technique, the controller in Equation (46) was designed. Then, using the root-locus placement technique, the controller in Equation (46) was designed. The perturbation period of the is is designed scenario(longest (longest ) The perturbation period of Pthe&PO& algorithm O algorithm designedfor forthe theworst worst case case scenario following the guidelines given given in [3],in it[3],leading to to TaTa== 0.5 ms and and∆v Δv 0.25 V (perturbation ts ) following the guidelines it leading 0.5 ms 0.25 V (perturbation refref= = amplitude applied to the PID reference): amplitude applied to the PID reference):
10.1302 4ps ` 1423q 1423 10.1302 ps `2.25 2.25e4q s ps `8.453 8.453e5q5
(46) (46)
Gc psq “ ´
vpv[ V]
to provide a fair comparison, & O + PWM & O + PID were + PWM were Finally, Finally, to provide a fair comparison, both P &both O +PPWM and P &and O +PPID + PWM implemented implemented with a switching frequency equal to 100 kHz, which is the same switching frequency with a switching frequency equal to 100 kHz, which is the same switching frequency exhibited by the SMC. exhibited by the SMC. A first test was considering a much larger capacitanceCb “ 200200 to reduce A first testperformed was performed considering a much largerDC-link DC-link capacitance µF to F reduce the voltage oscillations generated by inverter. the inverter. test also includes start-up of the the voltage oscillations generated by the This This test also includes the the start-up of the PVPV system and an irradiance perturbation 0.65simulation s. The simulation are presented in Figure and ansystem irradiance perturbation at t = 0.65ats.t =The resultsresults are presented in Figure 19,19,where where the fast tracking of the MPP provided by the proposed SMC is observed. The simulation also the fastreports tracking of the MPP provided by the proposed SMC is observed. The simulation also reports the small voltage oscillations exhibited due to the large value of Cb by the three systems, byFinally, the three which enables the small voltage oscillations to Pthe valuesolution. of which enables the correctexhibited operation due of the & large O + PWM the systems, fast response of the SMCoperation is translated intoPa& higher energysolution. harvestedFinally, from the array, whichof eventually the into the correct of the O + PWM thePVfast response the SMCincreases is translated system profitability. a higher energy harvested from the PV array, which eventually increases the system profitability. SMC
20 18.5 18 17.5
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Figure 19. Performance comparison between the SMC and conventional P & O-based solutions
Figure 19. Performance comparison between the SMC and conventional P & O-based considering small DC-link voltage oscillations. solutions considering small DC-link voltage oscillations.
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A second test was performed accounting for large voltage oscillations at the DC-link, i.e., using a A second test was performed accounting for large voltage oscillations at the DC-link, i.e., using smaller simulation results are reported in Figure 20, where again again the SMC exhibits a a smaller14 Cb F. “ The 14 µF. The simulation results are reported in Figure 20, where the SMC faster MPPT procedure higher energy production. Moreover, the simplest O faster exhibits MPPT aprocedure with higherwith energy production. Moreover, the simplest P & PO&solution solution (P & O + PWM) is unstable since the DC-link voltage oscillations are transferred to the PV (P & O + PWM) is unstable since the DC-link voltage oscillations are transferred to the PV array array terminals, which confuses the P & O algorithm. In contrast, the more complex P & O solution terminals, which confuses the P & O algorithm. In contrast, the more complex P & O solution (P & O + PID + PWM) is stable, but it requires a long time to reach the MPP in comparison with (P & O + SMC. PID + PWM) is stable, but it requires a long time to reach the MPP in comparison with the SMC. the vpv [V]
25 20 15
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Figure 20. Performance comparison between the SMC and conventional P & O-based solutions Figure 20. Performance comparison between the SMC and conventional P & O-based considering large DC-link voltage oscillations. solutions considering large DC-link voltage oscillations.
6. Conclusions
6. Conclusions
A detailed analysis and design of a MPPT solution based on sliding-mode control theory has been introduced in this paper. This solution is aimed at performing a fast MPPT action on PV systems A detailed analysis and design of a MPPT solution based on sliding-mode control theory has been using a single control stage. Moreover, since it is based on a non-linear model, no linearization process introduced in thisTherefore, paper. This is aimed a fast MPPT action on PVcontrollers, systems using is needed. thissolution approach avoids at theperforming circular dependency among cascade reduces the stage. number of controlsince devices provides stability in allnothe operation range. a single control Moreover, it isand based on a global non-linear model, linearization process is Such characteristics are major improvements over classical MPPT solutions based on cascade needed. Therefore, this approach avoids the circular dependency among cascade controllers,and reduces linear controllers. the numberThe of performance control devices and provides global stability in all the operation range. Such characteristics of the proposed solution was tested using realistic simulations accounting for are major improvements classical MPPT based on cascade and linear controllers. perturbations in bothover the irradiance and thesolutions load voltage, obtaining satisfactory results. Moreover, the sliding-mode controller was contrasted with classical MPPT solutions based on the P & O algorithm, The performance of the proposed solution was tested using realistic simulations accounting for where the SMC provides a much faster tracking of the MPP, it increasing the produced energy. perturbations in both the irradiance and the load voltage, obtaining satisfactory results. Moreover, the Therefore, the proposed SMC enables to increment the profitability of PV installations; hence the sliding-mode controller was return-of-investment timecontrasted is reduced. with classical MPPT solutions based on the P & O algorithm, where theFinally, SMC aprovides a much faster ofbethe MPP, itin increasing the produced energy. further improvement to thetracking SMC could performed a future work—a different, but more complex, switching circuit could to be increment designed to the avoid the sign change of the transversality—to Therefore, the proposed SMC enables profitability of PV installations; hence the remove the divergence stages in the SMC operation. This new development will enable to constraint return-of-investment time is reduced. the hysteresis band of the SMC, which will eventually enable to reduce, even more, the settling time Finally, a further to the SMC could be performed in a future work—a different, but of the PV power improvement to achieve a faster MPPT procedure.
more complex, switching circuit could be designed to avoid the sign change of the transversality—to Acknowledgments: This work was supported by the Instituto Tecnológico Metropolitano, the Universidad remove the divergence stages in the SMC operation. new development will laenable to constraint Nacional de Colombia and Colciencias (Fondo nacional This de financiamiento para ciencia, tecnología y la innovación Francisco José de Caldas) under the projects MicroRENIZ-25439 (Code 1118-669-46197), P14215 the hysteresis band of the SMC, which will eventually enable to reduce, even more, the settlingand time of P14220, and under the study commission-agreement number 25 from Instituto Tecnológico Metropolitano. the PVAuthor power to achieve a faster MPPT procedure. Contributions: All the authors contributed equally to the work.
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