Maximum Power Point Tracking Employing Sliding Mode ... - CiteSeerX

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Maximum Power Point Tracking Employing Sliding Mode Control Yoash Levron and Doron Shmilovitz, Member, IEEE

Abstract—A fast and unconditionally stable maximum power point tracking scheme with high tracking efficiency is proposed for photovoltaic generators. The fast dynamics and all range stability are attained by a sliding mode control and the high tracking efficiency by a maximum power point algorithm with fine step. In response to a sudden change in radiation, our experiments show a typical convergence time of 15 ms. This is the fastest convergence time reported to date. In addition we demonstrate stable convergence all across the photovoltaic curve, from short-circuit to open-circuit. The theory is validated experimentally. Index Terms—Maximum power point tracking, MPPT, photovoltaic generator, sliding mode control.

I. INTRODUCTION

A

T GIVEN atmospheric conditions (mainly dependent on temperature [1] and insolation level), photovoltaic (PV) cells supply maximum power at a particular operating point—the maximum power point (MPP). Unlike conventional (fueled) power sources, it is desirable to operate PV systems at their MPP [1]–[13]. However, the MPP locus varies over a wide range, depending on a PV array’s temperature and insolation intensity [1], [2]. Instantaneous shading conditions and ageing of PV cells also affect the MPP locus. The problem is further complicated should the load’s electrical characteristics also vary [3]. Hence, in order to achieve operation at the MPP, a time varying matching network which interfaces the varying source and the potentially varying load is required. The role of this matching network, called the maximum power point tracking network (MPPT), is to ensure operation of the PV generator (PVG) at its MPP, in the face of changing atmospheric conditions and load variations. Typically, the MPPT power stage is implemented by means of a dc-dc converter at the front end, most commonly pulse width modulated (PWM). In the case of standalone PVG this PWM controlled dc-dc converter may constitute the entire power stage, whereas in the case of a grid connected PVG, it would be followed by a dc-link capacitor and an inverter. Since the matching network must be adaptive [2], [3], the MPPT network must include a self-tuning mechanism which governs the power stage and drives the system to operate at the MPPT (eventually through the converters duty cycle). Many

Manuscript received November 17, 2011; revised February 14, 2012; accepted March 11, 2012. Date of publication February 13, 2013; date of current version February 21, 2013. This paper was recommended by Associate Editor E. Alarcon. The authors are with the School of Electrical Engineering, Tel-Aviv University, Tel Aviv, 69978, Israel (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSI.2012.2215760

MPPT algorithms have been proposed [4], some exhibit better performance in steady state [4], [5], while others are superior during transitions [6]–[15]. The “perturb and observe” (P&O) and the Hill-Climbing algorithms are probably those most widely used. The MPPT algorithm operation principle is quite similar with both, the voltage and current at the PVG (i.e., at the power stage input) are sensed and the power is calculated. Then, the MPP is sought iteratively. These algorithms imply a tradeoff in choosing the increment value by which the controlled parameter, such as duty cycle or reference voltage, is adjusted; small values decrease the losses in steady state due to small perturbations around the MPP, while large values improve the dynamic behavior in situations involving quickly changing irradiation conditions or load characteristics, such as in portable applications [6]–[8]. Recently, effort was made to implement MPPT algorithms with adaptive step size. For instance, [6] suggested a sophisticated algorithm where an intermediate control variable— —was employed. Reference [6] showed that in a first phase may be tracked rather than the power, but the fast tracking in the first phase has not compromised tracking accuracy in steady state. In order to attain high steady-state tracking-efficiency, a second phase consisting of a conventional MPPT scheme was required. Sophisticated algorithms present two main shortcomings: increased computational load which may require costly hardware, and a slow dynamic response. The latest is an intrinsic limitation resulting from the iterative search nature requiring successive sampling of voltage and current. The time intervals between algorithm iterations should be short to allow faster tracking, but on the other hand they must be longer than the settling time of the PV current and voltage for reliable signal measurement. Therefore, the analysis of a PV system dynamics is required to determine the P&O time intervals and controller parameters [9]. Employing a converter that is optimized for interfacing with the PVG has an important effect on the dynamics and stability of the system [10]–[13]. The analysis presented in [13] shows that commonly employed voltage fed converters are actually not suitable for PVG due to its nonlinear characteristics. This incompatibility may result in reduced performance and even instability. Based on the PV module characteristics, the MPP locus may be approximated by a linear relation [11], [12]. Thus, a linear controller was designed which drives the PVG to its approximate MPP. Hardware implemented, this action is performed much faster than the MPPT algorithm. The MPPT algorithm may then apply small steps to drive the operation point to the exact MPP. It has become widely accepted that sliding mode control is very adequate for controlling switched mode converters [16],

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[17]. This control exhibits high stability and fast dynamics (typically superior to linear controllers), and in many cases is simple to implement (depending on the sliding surface). In particular, canonical power processors suitable for interfacing PVGs were facilitated in [18]. In contrast to PWM based MPPT, we propose a sliding-mode controlled MPPT. Such a controller presents two major advantages: first, by a proper choice of the switching surface, the response to variations in radiation is accelerated by an order of magnitude. In addition, the sliding-mode control facilitates operation as either a voltage-source or a current source, thus, it guarantees stability all across the photovoltaic curve—from short-circuit to open-circuit. The implementation of the control is simple and requires inexpensive hardware. This work validates these results by theory, simulations, and experimentation. This work may be viewed as an extension of the work by Bianconi et al. [19]. This group suggested an MPPT controller based on an inner sliding-mode loop, which uses the input capacitor current as the main state variable. By comparison, in this work the inductor’s current is taken as the main state variable. This approach is generally more robust and stable, as the energy stored in the converter is a direct outcome of the inductor’s current. Consequently, the converter is inherently over-current protected, even during start-up or transient conditions, and is stable over a very wide range of voltages and currents. This is confirmed by the theoretical stability analysis, presented in Section V. Moreover, by combining the inductor’s current with the input voltage, it is possible to design an optimal switching surface, which results in faster transient response. These principles are validated both experimentally and theoretically. II. PRINCIPLE OF OPERATION The PVG i-v characteristics are nonlinear. In addition, the power to voltage (or current) relation is non-monotonic and exhibits a maximum. Therefore, tracking of the MPP necessitates a nonlinear, and in most cases iterative, technique, typically involving sampling of the PVG voltage and current. The time intervals between successive samples must not be shorter than the converter settling time, otherwise the sampled data would be false. The converter interfacing the PVG is at least a second order system consisting of reactive components with significant values. This results in long settling times and consequently in long intervals between the MPPT’s successive samples [9]. Consequently, as the MPP search requires multiple iterations, the MPPT convergence speed is limited and may be regarded as a low bandwidth control loop. This is an intrinsic limitation of the system which cannot be overcome by a faster processor in the MPPT unit. The transient response is improved by adding an inner control loop with a much wider bandwidth. Typically the inner controller regulates the PVG voltage [2]–[5] or current [9]–[13], see Fig. 1. With the proposed system, the MPPT algorithm unit provides the sliding mode inner controller with an adaptive reference, which results in an adaptive switching surface. This in a way resembles sliding mode based inverters [20]. The inner feedback (sliding mode) regulates a linear combination of the PVG voltage and current to attain its operation along a line in close vicinity to the MPP loci (see Fig. 3). This is accomplished in a very rapid manner, at least an order of magnitude faster than

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Fig. 1. Dual loop MPPT.

Fig. 2. Sliding mode based MPPT implementation.

the MPPT reaction. This allows using a small step increment in the MPPT algorithm (resulting in a high tracking efficiency in steady state) since most of the tracking is carried out by the inner loop, and the MPPT requires very few iterations (as in our experiments 1 or 2). III. CONTROL DESIGN AND IMPLEMENTATION In typical dc-dc converter applications, it is desirable to regulate the output voltage to a constant value and the switching surface is chosen according [16], [17]. In other cases, such as sliding-mode controlled inverters, the output voltage needs to follow a sinusoidal shape resulting in a time varying switching surface [20]. Similarly, in the case of gyrators [18]–[23], an adaptive surface is required. For MPP tracking, we define the switching surface by a linear combination of the PVG voltage and current. The system is illustrated in Fig. 2. Assuming a large output capacitor (or battery load) the converter output voltage may be assumed to be semi-constant. Thus, the PVG voltage and current constitute all the converter’s state variables, resulting in the switching surface given by (1). (1) where and are the inductor’s current and input voltage, and , , and ref define the switching surface. and set the slope in the PVG i-v plane, and are chosen as non-negative, and ref sets the offset. When (on state), the low transistor is on,

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Fig. 3. PVG characteristics under different insolations, and the optimal . switching surface (blue)

and the inductor charges. The current increases, and increases. When (off state), the low transistor is off, and the stored energy flows to the load. The current decreases, and decreases. As a result, is kept in a “switching band” around . This controller constricts the system’s operation to the sliding switching surface. The sliding mode circuitry consists of an op-amp and a comparator, which directly toggles the transistor gates as shown in Fig. 2. The operating point of the PV module results from the intersection of the switching surface, , with the PV curve (Fig. 3). To extract maximum power, the switching surface is adjusted by the MPPT algorithm, to intersect with the exact MPP at any given time. This algorithm is implemented in the micro-controller. It samples the input power, and adjusts the switching surface by updating the constant ref, which is fed to the comparator. The proposed approach is indifferent to the MPPT algorithm selected; we chose P&O due to its simplicity. IV. OPTIMAL SELECTION OF THE SWITCHING SURFACE SLOPE Many switching surfaces provide both stability and adequate dynamics. We choose a linear surface, as in (1), due to the ease of implementation (a single operational amplifier). The offset constant, ref, is set by the MPPT controller. However, the slope of the surface, represented by constants and , is a free parameter. A good choice of slope can significantly shorten the MPPT’s convergence time. In steady state, the switching surface intersects the MPP point. A change in radiation causes a shift of the MPP point, so the MPPT should readjust ref to (iteratively) drive the switching surface to the new MPP. If the new MPP point is already in close proximity to the switching surface, the required update of ref is minimized as the new MPP is reached in very few steps (even if the step size is fine), thus, the MPPT tracking is accelerated. Therefore, a natural choice of slope is made by applying LSE (least square estimate) to the set of MPPs which corresponds to the various radiation levels. The constants and are chosen so as to minimize the error: (2) where are the set of MPPs to be estimated. Fig. 3 shows a set of MPPs, and the corresponding LSE switching surface. Fig. 4 shows a conceptual example of the system trajectory due to an abrupt step of the insolation.

Fig. 4. Conceptual trajectory following a sudden change of insolation. The system “slides” to the new maximum power point.

V. STABILITY ANALYSIS The state vector is defined in (3), where voltage, and the inductor’s current.

is the input

(3) To assure stability, the switching surface must comply with the converter dynamics. If the switching surface is not properly designed the system may enter a limit cycle, which can incite harmful jitter and noise. Intuitively, stability is guaranteed if the state-space trajectories evolve towards the switching surface, . This is visualized in Fig. 5(b), which shows the system trajectories with stable and instable regions. In a stable region, the trajectory velocity points toward the switching surface, while in an instable region, it points away from the switching surface. The state space trajectory velocity, , results directly from the converter dynamics in the “on” and “off” states according to (4).

(4) is the PVG current, is the input capacitance, is the inductance, and is the output voltage. The criterion for stability is that “the velocity points toward the switching surface.” This is the sliding-mode “existence condition,” which is given by (5):

(5) (the gradient of ) is normal to the switching surface , where and are the constants that define the surface slope. The stability condition (6) in the “on” state is reached by substitution. The “or” statement indicates that “on” state stability

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Solving for the current : (9) Then, combining (7) and (9):

(10) Equation (10) is the eventual stability condition, which defines a “stability region” in state-space, see Fig. 5. Stable points, i.e., points that are attracted toward the switching surface, are those that satisfy (10). Stability is guaranteed as long as the trajectories are bounded within this region. VI. SWITCHING FREQUENCY As a first order analysis, the switching surface was assumed to be a line (of zero width). Switching occurs over a theoretical line, , as in (1). However, in practice, the switching surface must have a small but finite width, to prevent chattering. This width is denoted by , and the actual switching borders are given by (11). is set by adjusting the positive feedback resistor across the comparator (Fig. 2). The switching frequency is directly associated with the width, , thus, this width must be well chosen to assure an adequately low switching frequency and a resulting high efficiency, under all possible operation points. Equation (12) relates the switching frequency, , to the width . Both sides of the equation equal the switching period, and are therefore equal to each other. (11) (12)

Fig. 5. (a) Stability region and optimal switching surface in the state-space. (b) Typical system trajectories.

applies only if vant.

. Thus, if

Applying the gradient of S, and the state space velocity of (4) . and (5), results in an explicit expression for

, the condition is irrele(13)

(6) Solving for the current , results in (7), which is an explicit stability criterion for the “on” state:

The following analysis is limited to steady-state conditions, after convergence to the MPP. At this operation point, some approximations can be assumed, resulting in a much simpler expression for :

(7) (14)

Similarly, (8) is the “off” state stability condition. Rearranging (14) yields (15). (8)

(15)

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Fig. 6. Experimental setup.

This last expression presents the switching frequency, , in steady-state, with respect to the MPP of the PV panel. With a larger width , the frequency decreases. VII. EXPERIMENTAL RESULTS A. System Structure The experimental setup (Fig. 6) includes a PV emulator, a synchronous boost converter, gate drive circuitry, sensors, sliding-mode controller implementation, and a micro-controller for MPPT execution. The PV emulator is implemented by means of a laboratory power supply operated in the current limit mode, set to 3.5 A, in parallel connection with a diodes’ string. The diodes string consists of 30 diodes of type “MUR820” connected in series. An electronic load, configured as a current source, is connected in parallel to the power supply and diodes, thus operating as a programed current sink. Radiation changes are emulated by stepping the current sink, in the range of 0.5 A ( , 1 sun) to 3.5 A ( , 0 sun). The emulated PV curves are measured, as shown in Fig. 3. It may be noted that the emulated PV characteristics are similar to those exhibited by actual PVGs. Thus, this emulation method is quite effective for verifying the MPPT & the sliding mode controller performance. The boost converter consists of an inductor of 200 ( saturation, series resistance), an input capacitor of 100 , and an output capacitor . The load is resistive, 28 . Two complementary of 200 N-ch FETs of type “STP60NF10” are used. Each FET (high and low side) is driven by floating gate circuitry, based on the “MIC4423,” 3 A gate driver. A delay circuit synchronizes the on/off transitions of the high and low FETs, to prevent shoot-through currents. Turn-off is immediate, and turn-on is delayed by 20 nS A high-side current sensor (MAX4173) is positioned in series with the inductor on the input side. The sensed input voltage and inductor’s current are fed to both the sliding-mode controller and the micro-controller internal A/D. The sliding-mode circuitry is based on a fast op-amp

Fig. 7. MPPT algorithm used with the sliding-mode controller. This being the ordinary “perturb & observe” (P&O) algorithm, outputting ref to the D/A.

(AD8605) and a fast comparator (MAX941). The MPPT is implemented on a dsPIC30F2020 evaluation board (microchip). It runs a simple perturb and observer (P&O) MPPT algorithm (see Fig. 7). The micro-controller drives an external D/A (AD5060), which outputs ref back to the sliding-mode control. We also run a “PWM based MPPT” test mode, in which the micro-controller generates a PWM signal with a varying dutycycle, which is fed directly to the gate drive (no sliding-mode). This being the ordinary “PWM” based MPPT. We use it as a test case to compare the sliding-mode performances.

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Fig. 9. Stability verification. Convergence from an initial “open-circuit”: the current climbs monotonically. Yellow trace -voltage (5 V/div), pink trace—output power (10 W/div), Blue trace -current (0.5 A/div), Time base: 40 ms/div.

Fig. 8. Convergence comparison of sliding-mode MPPT and PWM based MPPT following an insolation step. (a) Sliding-mode, reaction to a positive step. (b) PWM MPPT, reaction to a positive step. (c) Sliding-mode, mode reaction to a negative step. (d) PWM MPPT, reaction to a negative step. Yellow trace—input voltage (2 V/div), blue trace—input current (0.5 A/div), pink trace—output power (10 W/div). Time base is 40 ms/div.

B. Controller Design Procedure The optimal switching surface is selected by applying (2) to the MPP loci, see Fig. 3. The resulting slope (normalized by ) is , . ref is set in real-time by the MPPT algorithm. Its theoretical optimal value, according to (2) is . The switching surface width, , is designed for a switching frequency of . Applying (15), with , , results in . A graphical stability analysis with these values is presented in Fig. 5, showing stability all across the PV curve, with adequate margins. The actual implementation is shown in Fig. 6. C. Measurements The sliding-mode MPPT, with optimal switching surface, is compared with an ordinary “PWM based MPPT.” We use the PV emulator to generate a sudden step in input “radiation,” and measure the step response of both controllers. Fig. 8 shows the results. The PV emulator’s current source is switched instantaneously from 1.5 A to 3.0 A to emulate a 100% positive insolation step (on the left), and from 3 A to 1.5 A to emulate a 50% negative insolation step (on the right). The MPPT algorithm step size is for the sliding mode MPPT, and 2% for PWM based MPPT. The step time is identical for both controllers—20 ms per step. The sliding mode MPPT is seen to converge within 15 ms, whereas the PWM based MPPT converges within 210 ms. It should be noted that the power shown in Fig. 8 is the converter’s output power while the PVG’s power rises even faster, see Fig. 11 for a comparison of the input and output powers following an insolation step. In this experiment, the current source is switched instantaneously from 1.5 A to 3.0 A and the input power is seen to rise within 5 ms. Figs. 9 and 10 display the results of the sliding mode stability tests. The system is initialized far from the MPP, in order to cause it to stroll across the PV curve, thus testing stability through a variety of operation points. In Fig. 9, the system con-

Fig. 10. Stability verification. Convergence from an initial “short-circuit”: The current is steady while the voltage climbs monotonically. Blue trace -current (0.2 A/div), pink trace—output power (10 W/div), yellow trace -voltage (5 V/div), Time base is 100 ms/div.

verges from an initial “open-circuit.” The initial state is determined by setting an initial high ref value of 19.4 (compared to about 13.7 at MPP). This initializes the system on the right side of the PV curve, where the PV module exhibits voltage source characteristics. The initial current is low (0.85 A) and converges to the MPP (2.73 A). The voltage is almost steady throughout the transient. In this experiment, ref step and the MPPT step time was 20 ms. Fig. 10 shows the convergence from an initial “short-circuit.” ref is initially low , setting the system on the left side of the PV curve, where the module operates as a current source. The initial voltage is low (3.6 V) and converges to the MPP (20.2 V). The current is seen to be nearly constant throughout the transient. As in the previous experiment, step and the MPPT step time was 20 ms. VIII. DISCUSSION This work presents a sliding-mode control based MPPT applying an optimally matched switching surface. The slidingmode MPPT results in two major advantages over PWM based MPPT: fast tracking in response to a radiation change, and stability across the entire photo-voltaic curve. A. Fast Tracking With sliding-mode, the tracking response to a radiation change is highly accelerated, as seen in Fig. 8. This was

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verified by comparing the dynamic tracking of the proposed sliding-mode based MPPT with that of a reference PWM based MPPT. In order to compare dynamic performance, a radiation step was emulated. Although such a radiation step is quite extreme (and maybe not realistic); it provides a good measure for testing the convergence speed. The PWM based controller converges within 210 ms, while the sliding-mode controller converges within 15 ms (14 times faster).Moreover, the converter’s input power follows the radiation change almost immediately, as seen in Fig. 11. It should be noted that, although the converter’s input power raises in some 5 ms. (even this is much faster than previously reported PWM based MPPT performance), it can converge even faster, but we cannot demonstrate it since the PV emulator’s speed is limited to about 1 A/ms. The fast convergence is attained by an optimal selection of the switching surface slope (according to (2)). Following a radiation change, if the switching surface is in proximity to the new MPP, most of the convergence (typically over 90%) is performed by the high band-width hardware-implemented inner control loop. The system “slides” along the switching surface from the previous PV characteristics to the new one, in close proximity to the new MPP, as shown in Fig. 4. Thus, the output power has risen close to its final value, before the MPPT unit (and software) even reacted. Then, convergence to the exact MPP slowly continues by means of the software implemented MPPT algorithm. This is seen in Fig. 8(a), where the current still increases by a few percentages following the initial response. In comparison, PWM based MPPTs are slower, because the convergence is performed mainly in software, by the MPPT unit. A comparison of the convergence speed is displayed in Fig. 11. The proposed system is compared with a software based MPPT, using an inner voltage loop, proposed by Sera et al. [15]. This MPPT algorithm is optimized for speed, yet its convergence time is on a scale of seconds. In contrast, our proposed topology converges to maximum power within 15 mS. Since they are also fully hardware implemented, extremum seeking based MPPT are also very fast [24], [25]. However, it seems the sliding mode based MPPT is simpler to implement and requires fewer components. To conclude, the sliding-mode based MPPT acts through a hardware implemented inner feedback loop with very high bandwidth, while PWM based MPPTs converge mainly through the MPPT unit involving successive sampling and software based feedback. As a result, sliding-mode convergence is faster by an order of magnitude. B. Stability In contrast to PWM based MPPT which requires careful design and accurate knowledge of the PVG and converter parameters in order to ensure stability (and even so the stability is only attained within a certain range of PVG voltages) [9], [10], [13], stability is easily attained with the sliding-mode control and it is guaranteed over the entire PVG voltage range. After the optimal slope is estimated, the stability of the corresponding switching surface should be analyzed. The sliding-mode controller can operate across the entire photovoltaic curve. Stability is maintained when the photovoltaic source is a current source (on the left) or a voltage source (on the right). This can be confirmed per case with the theoretical stability analysis (10), and by plotting the stability regions. Fig. 5 shows the stability region for the

Fig. 11. Input power and output power following an insolation step. (a) Measured results of the proposed system. Blue trace—input power (10 W/div), Pink trace—output power (10 W/div), Time base: 4 ms/div. (b) Results from Sera et al. [15], demonstrating the difference in convergence speed.

values used in the experiment ( , , , , , , ). The photovoltaic curve, is contained by in the stability region, verifying the stability of all operation points along the PV curve. Figs. 9 and 10 show the experimental stability tests. The system is forced to converge (in software) across the right side and the left side of the PV curve, thus verifying the stability at multiple operation points. The system is stable at all the points on the PV curve, confirming the theoretical stability analysis. C. Voltage or Current Feedback Instead of applying the optimal slope, which requires two sensors, the sliding-mode control may be implemented by using a single voltage sensor, or a single current sensor. The resulting switching surface is not optimal, and the tracking speed would depend on software. With only a voltage sensor, the slope constant is zeroed (1), and with only a current sensor, is zeroed. Fig. 12 shows typical stability regions, which are evaluated by (10). With a voltage feedback, the system is quasi-stable, as the instability boundaries overlap the PV curve. However, with a current feedback the system is stable at all points, and under any conditions. This is in compliance with other works [9], [11], [13]. However, it should be noted that it is not recommended to operate the system purely with a current feedback, as it might loss stability if the target current is set higher than the PV shortcircuit current. It is best to operate the system with an optimal slope, combining both the voltage and current feedbacks. D. Switching Frequency A low switching frequency reduces the switching loss, and is important for high efficiency. The higher the width, , the lower the switching frequency, as in (15). We designed the system for a switching frequency of , resulting in (a lower frequency would have required a larger and bulky inductor). The resulting efficiency was 93% near MPP, at 1 sun . The actual frequency measured 119.9 kHz at

LEVRON AND SHMILOVITZ: MAXIMUM POWER POINT TRACKING EMPLOYING SLIDING MODE CONTROL

Fig. 12. Stability regions and typical trajectories with single sensed parameters: (a) a voltage feedback

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or (b) a current feedback

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scenario is demonstrated by simulation, in Fig. 13. The system rapidly slides to a point near a new MPP. However it turns out that in this case the final MPP reached is a local one. Under heavy shading conditions, the PV module’s i-v curve may become sporadic. Consequentially, the selection of sliding surface has no effect on the convergence of the system, and the transient response will be an outcome merely of the software based MPPT algorithm, similarly to duty-cycle based systems. IX. CONCLUSION This work introduces a sliding-mode based MPPT method. In comparison to PWM based MPPTs, convergence to maximum power is accelerated by an order of magnitude. This is accomplished by an optimal selection of the switching surface. In addition, the controller can operate either as a voltage source or a current source, maintaining stability all across the photo-voltaic curve. Stability is analyzed using the “stability region” method, which is easy to comprehend and intuitive. REFERENCES Fig. 13. System performance under partial shading (simulation results).

MPP, a 20% error in comparison with the theoretical value. As a first degree approximation, only the MPP voltage affects the switching frequency. The current at MPP only slightly affects the switching frequency. Note that this current does not appear in (15). The MPP voltage changes a little with radiation, thus, the switching frequency is quite constant under radiation shifts. E. The Effect of Shading With partially shaded modules, the estimated sliding-surface is no longer optimal. However, the system is likely to slide in close vicinity to a local or global maximum, as a typical shaded i-v curve usually preserve a maximum power point (local or global) with voltage similar to the MPP of a non-shaded profile. As a result, the system will converge fast; however, it may reach a local maximum point, missing the global maximum. Such a

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Yoash Levron received his B.Sc. from the Technion, Haifa, Israel, and an M.Sc. from the Tel-Aviv University, Israel, in 2001 and 2006, respectively. He is currently pursuing his Ph.D. at Tel-Aviv University. His research focuses on the control of power processing systems.

D. Shmilovitz (M’98) received his B.Sc., M.Sc., and Ph.D. from Tel-Aviv University, Tel-Aviv, Israel, in 1986, 1993, and 1997, respectively, all in electrical engineering. During 1986–1990 he worked in R&D for the IAF, where he developed programmable electronic loads. During 1997–1999, he was a Postdoctorate Fellow at the New York Polytechnic University, Brooklyn. Since 2000, he has been with the Faculty of Engineering, Tel-Aviv University. His research interests include switched-mode converters, topology, dynamics and control, and circuits for alternative energy sources and for powering of sensor networks and implanted medical devices, and general circuit theory.