The quantum-mode regulated power point tracking in a photovoltaic array for application under the quantised converter duty ratio

The objective of this work is to develop a realistic control technique for stably regulating the power output of a photovoltaic array. The associated process is referred to as the regulated power point tracking (RPPT). Ideally, the converter duty ratio is expected to be a continuous variable so that the photovoltaic power output can be freely adjusted. How-ever, such a continuous-mode RPPT is possible only with the help of an analogue pulse-width-modulation (PWM) control. In contrast, the modern digital PWM control causes a duty ratio quantisation. As a result, the photovoltaic power output can, practically, be adjusted only at some discrete power levels. Therefore, existing RPPT control techniques, which optimistically assume a continuous power production range, may lead to persistent low-frequency power oscillations because of the digital PWM. To overcome the particular problem, a quantum-mode RPPT control technique is proposed in this paper. The quantum-mode RPPT basically limits its search only to the practically attainable discrete power levels by suitably modifying the given power reference command. Thus, the power tracking can always reach a converged state. For the faster RPPT, the perturbation step size is dynamically adapted. The proposed RPPT control technique is veriﬁed via both simulations and hardware experiments.


INTRODUCTION
The sustainability of the electric power generation is one of the foremost concerns of the modern heavily electricity-dependent society, given the fact that the fuel sources of conventional power plants are on the verge of extinction. The deployment of alternative and renewable energy sources for electricity production is increasing to create a solution for meeting the energy need of the mankind in the upcoming years. The sunlight is one such major renewable energy source, which is ubiquitously available and can provide electricity through the photovoltaic (PV) energy conversion. Large-scale photovoltaic plants have already been installed in many countries, such as China, India, Germany, Mexico, United States and so on, for feeding power directly to the main grid [1,2]. The convenience of the installation and maintenance of the photovoltaic system has also made it attractive for use in microgrids [3] or in the dedicated load applications [4][5][6]. The versatile and proliferating utilization of the photovoltaic power generation makes it relevant to have a flexible control over the power output of a photovoltaic array. Typically, the power output of a photovoltaic array is controlled by means of the maximum power point tracking (MPPT) so as to extract as much photovoltaic power as possible. Extensive works have been reported in the literature to develop the MPPT controller [7][8][9][10][11]. The MPPT control techniques can be broadly classified into four categories, such as the estimative method (e.g. the fractional open/short circuit technique) [12][13][14], the enumerative method [15][16][17][18][19][20], the soft-computational method [21][22][23][24][25][26] and the error-integration method [27][28][29]. The perturb-and-observe and the incremental conductance are the popular enumerative MPPT control techniques because of their ability to stably, quickly and accurately track the maximum power point (MPP). The power tracking can further be made faster by suitably initialising the operating state of the photovoltaic array before assuming the enumerative MPPT [30,31]. In order to reduce steady-state power oscillations without slowing down the power tracking speed, variable perturbation step size is used in the enumerative MPPT. The perturbation step size is typically adapted either as a function of an MPP proximity index [32][33][34] or by employing a fuzzy logic [35][36][37][38]. In addition, an interesting idea of the MPP proximity zone was also proposed in [39]. An MPP proximity zone is, basically, defined by the smallest voltage range, in which the MPPT dynamics finally settle down for a specific perturbation step size. Whenever the MPPT dynamics enters into an MPP proximity zone, the perturbation step size is reduced in a stepped manner and a higher (i.e. with a narrower voltage range) level of MPP proximity is targeted. The number of MPP proximity levels to be traversed is determined by a lower limit on the perturbation step size.
The MPPT-controlled operation of a photovoltaic array is the most appropriate choice when there is ample local energy storage capacity to absorb the excess power generated at any point in time. Deficiency in the local energy storage capacity may cause a photovoltaic array to flush out the entire generated power into the grid or into its dedicated load. This, in turn, restricts the deployment of a photovoltaic array since the network or the load element may, at times, get overloaded because of the excessive power generation by the photovoltaic array [40][41][42]. The network congestion caused owing to the excessive power injection into the grid also has a strong impact on the energy prices in the power market, which may be as severe as negative locational marginal prices threatening the revenues of photovoltaic power producers in the long run [43]. In order to ensure the continuous service of an MPPT-controlled photovoltaic array under the limited energy storage capacity, auxiliary load units may be used to burn the surplus power. However, the use of auxiliary load units also increases the system cost. Moreover, carrying a higher amount of power without having its full utilisation either in short term or in long term unnecessarily degrades the life of a converter [44].
The major disadvantages of constantly drawing maximum power from a photovoltaic array without the support of an abundant local energy storage capacity have been highlighted in the above discussion. At the same time, the MPPT control undermines the capability of a photovoltaic plant to provide the frequency regulation and other reserve services [45][46][47][48]. Consequently, it is very important to upgrade the energy harvesting mechanism for a photovoltaic array with the incorporation of the provision of regulating its power output as per the situation and the necessity. This, in turn, requires the replacement of the MPPT mechanism with a regulated power point tracking (RPPT) mechanism. The RPPT mechanism essentially adjusts the power output of a photovoltaic array by following an input power reference command, provided the power reference is set below the MPP. In case the MPP falls below the power reference line, the power output of the photovoltaic array settles down at the MPP only. Limited works have been carried out so far on the RPPT control of a photovoltaic array. Techniques proposed in [44,[49][50][51] combine an MPP tracker and a lower power point (LPP) tracker to implement the RPPT control. Only one tracker acts at a time and the tracker switching is dynamically carried out by means of a power comparison [44,49] or voltage comparison [50] or current comparison [51] based upon the real-time signals. The LPP tracker directs the power output of the photovoltaic array when the power reference command is lower than the maximum available power. Otherwise, MPP tracker takes charge of controlling the photovoltaic array power output. Alternative methodologies proposed in [52][53][54] use an integrated power tracker to implement the RPPT control without involving the effort of the tracker switching.
In comparison to the MPPT, the RPPT involves a higher level of complexity. Although some RPPT control techniques have been reported in the literature, those essentially assume that the power output of a photovoltaic array is finely adjustable at any point in the continuous range between zero and the maximum power. The RPPT for finely adjusting the power output of a photovoltaic array can be referred to as the continuous-mode RPPT. The continuous-mode RPPT is possible only by having the provision of executing a finely tuned converter duty ratio as per the power reference command. This, ideally, requires an analogue implementation of the pulse-width-modulation (PWM) control. However, the modern implementation of the PWM control basically makes use of a digital processor that can update a result only at discrete time instants. This, in turn, renders a natural quantisation of the converter duty ratio, restricting the feasible set of power levels that the photovoltaic array can output. In other words, the quantisation of the converter duty ratio also leads to the quantisation of the photovoltaic array power output. Although the particular quantisation phenomenon also takes place in the MPPT control, it does not cause any severe problem there since the slope of the power-voltage (i.e. P-V) curve is typically very small near the MPP. On the other hand, the slope of the P-V curve may be very high near an LPP. Consequently, there can be a large difference between the power levels of two successive feasible operating points on the P-V curve. Thus, any attempt to perform the RPPT in the continuous mode with the digital PWM may lead to significant lowfrequency power oscillations in the steady state because of the duty ratio quantisation.
The general contribution of this work is to devise a suitable enumerative control algorithm for flexibly regulating the power output of a photovoltaic array with due consideration for the above mentioned practical limitation of the RPPT. In specific, the following innovative works are carried out.
1. The nature of the converter duty ratio quantisation problem is precisely investigated. 2. The RPPT is performed in a quantum mode for assuring a non-oscillating and robust steady-state power tracking, while FIGURE 1 Complete layout of a two-stage grid-connected photovoltaic system without local energy storage obeying the specified power reference command as closely as possible. 3. The idea of the equilibrium proximity, as was originally proposed in [39] for adapting the perturbation step size, is reexplored for its upgradation to take into account the phenomenon of the equilibrium shift before reaching the convergence.
The quantum-mode power tracking basically refers to exploring only a finite set of power levels to set up the system equilibrium. The methodology proposed here to execute the RPPT in the quantum mode is referred to as the oscillation-guided adaptivestep enumeration (OGASE) algorithm. Only the core power tracking module (which can work alone under the uniform irradiance) is presented in this paper. In order to work under the partial shading condition, the OGASE controller should be supported by a separate front-end multi-modality processing adapter (MMPA) module. The objective of an MMPA is basically to perform an operating state reinitialisation for avoiding the local peak trap under the partial shading condition. The use of the MMPA concept in the context of the MPPT can be found in [55] and [56]. An elementary version of the MMPA for the RPPT control was also proposed in [57]. However, designing a complete MMPA for the OGASE controller is beyond the scope of this paper. The rest of the paper is organised as follows. The P-V curve discretisation phenomenon observed during the power output control of a photovoltaic array is thoroughly discussed in Section 2. The OGASE technique proposed to perform the RPPT in the quantum mode is explained in Section 3. Simulation and experimental results for verifying the effectiveness of the OGASE controller are presented in Sections 4 and 5, respectively. Finally, the paper is concluded in Section 6.

DETAILED INVESTIGATION OF THE DUTY RATIO QUANTISATION PROBLEM
The photovoltaic power conversion from the array voltage to the load/grid voltage can be carried out either in a single stage or via two stages. In the case of the single-stage power conversion, the photovoltaic power is transferred to the load/grid via a single converter. On the other hand, the two-stage power conversion requires a DC-DC array voltage converter with fixed output voltage at the first stage. In the second stage, there should be another converter so that the power transfer to the load/grid may appropriately take place. Although the single-stage power conversion is more economic, the two-stage power conversion provides more flexibility to choose the array dimensions. This is because the DC-DC power conversion at the first stage helps in boosting up a low photovoltaic array output voltage near the level required in the final power conversion to meet the load/grid voltage. In this paper, attention is paid only to the two-stage power conversion.
The power circuit and control system arrangements of a gridconnected two-stage photovoltaic system is shown in Figure 1. For the sake of simplicity, no local energy storage is considered here. In Figure 1, the static and dynamic reference commands are indicated by 'ref' and '*', respectively, in subscripts and superscripts. The user specified power reference command to the RPPT controller is indicated by P sp re f . Symbol D cm represents the duty ratio commanded to the DC-DC converter. The modulation index vector of the DC-AC voltage source converter (VSC) is represented by m abc , and all other symbols are self-explanatory. Figure 2 shows the ideal DC-side equivalent circuit representation of a two-stage photovoltaic system. The voltage source  The duty ratio quantisation problem basically makes a difference between the commanded duty ratio and the effective duty ratio. The commanded duty ratio is the value that is fed as an input to the PWM controller. On the other hand, the actual duty ratio executed by the PWM controller is referred to as the effective duty ratio. The process of the digital PWM control of a DC-DC converter is explained via Figure 3. The carrier waveform is drawn with thin green lines and the commanded duty ratio is shown by the horizontal thick red line. The period of the carrier signal is indicated by T cs . The digital PWM controller samples signals at a finite rate with the sampling period of T pw seconds. The respective sampling time instants are marked by t s1 , t s2 , …, on the time axis.
Ideally, the converter switch should be turned on and turned off at time instants t A and t B , respectively, satisfying the follow-ing relationship.
As mentioned earlier, D cm indicates the commanded duty ratio. However, as per Figure 3, no sampling of the duty ratio command and the carrier wave signal takes place at time instants t A and t B . Therefore, the PWM controller cannot issue commands to change the switch status at those time instants. The situations for turning on and turning off the switch can be recognised only after collecting samples at time instants t s2 and t s7 , respectively. There is a finite time requirement for the processing of a particular sample vector, which may be indicated by Δt pr . Therefore, based upon the sampling at time instant t s * , the updated switching command is issued by the PWM controller at the time instant t s * + Δt pr . Thus, the effective duty ratio (indicated by D e f ) of the converter switch appears as follows.
The factor k e f is always an integer number indicating that the effective duty ratio can only be an integer multiple of the ratio T pw ∕T cs . For example, if the carrier wave frequency is 10 kHz and the sampling period of the PWM controller is 1 s, the number of practically possible duty ratios is only 101. The respective duty ratios are 0, 0.01, 0.02, 0.03 and so forth. On the other hand, k cm can have a fractional component. As a result, the difference between the commanded and effective duty ratios becomes apparent. The effective duty ratio is obtained from the commanded duty ratio in a certain manner. In general, the following relations hold true, which can be easily observed from Figure 3.
According to (5), the following relationship happens between k e f and k cm .
Factors k e f and k cm have been defined in the first line after Equation (2). Equation (6) can be interpreted as follows.
1. The value of k e f is exactly the same as the value of k cm iff k cm is an integer. This essentially means that the commanded duty ratio must be an integer multiple of the ratio T pw ∕T cs to yield the same value as the effective duty ratio. ing integer of k cm if the latter is a fractional number. Therefore, the effective duty ratio gets adjusted to the previous or next closest integer multiple of T pw ∕T cs around the commanded duty ratio.
The situation explained in Point 2 above is exactly what is referred to as the duty ratio quantisation problem. Whether the effective duty ratio gets quantised at a higher value or at a lower value depends upon the specific value of the commanded duty ratio. In fact, the nature of the quantisation also varies with time unless the period of the carrier waveform is an integer multiple of the PWM sampling period. In such a case, the effective duty ratio may keep on oscillating even though the commanded duty ratio is kept constant. Therefore, in order to ensure a unique D e f for a given D cm , the T cs ∕T pw ratio is taken as an integer in this paper. The effect of the duty ratio quantisation on the RPPT is illustrated with the help of the P-V curve presented in Figure 4. The operating points corresponding to practically feasible duty ratios for T cs = 0.1 ms and T pw = 1 s are marked by red dots on the P-V curve. The RPPT can be performed either on the left side or on the right side of the MPP. The feasible operating points are very closely placed on the left side of the MPP. On the other hand, the power gap between two successive feasible operating points may be quite large on the right hand side because of the sharp slope of the P-V curve in this zone. However, tracking an LPP on the left side of the MPP has the major disadvantage of having the possibility of facing a significantly low voltage level. Operating the photovoltaic array at a lower voltage level ultimately leads to lowering the power conversion efficiency of the DC-DC array voltage converter because of the increased current value. It is also difficult to adequately boost up a very low voltage level because of the practical limitations regarding the converter gain. In addition, some stability problems can be noticed if it is attempted to perform the RPPT on the left side of the MPP. Therefore, the right side of the MPP is the preferable zone to perform the RPPT. For the given power reference line, the photovoltaic array output power and voltage should ideally settle at Point B on the P-V curve. However, Point B does not correspond to a feasible duty ratio of the DC-DC converter and none of the nearest feasible operating points (i.e. A and C) is very close to B. Thus, any attempt to precisely track Point B ultimately leads to significant fluctuations (say, between A and C) of the photovoltaic array power output. Therefore, instead of trying to reach exactly Point B, it is more appropriate to track Point A or Point C so that the photovoltaic array can have a steady power output.The particular principle is the basis of the quantum-mode RPPT.

THE OGASE ALGORITHM FOR THE RPPT
The general block diagram of the OGASE-based RPPT control module is presented in Figure 5 by using the format of a discrete-time dynamical system. The photovoltaic array output voltage or current always contains a high-frequency oscillatory component because of the effect of converter switching. In order to eliminate the particular high-frequency oscillatory component along with noises, measurements are first processed through low-pass filters (LPFs). In this paper, the filtered measurements are represented by putting bars on symbols of respective raw measurements. Symbol n is the index for the sampling time instant of the RPPT control module. It is to be noted that the RPPT control module is a formal discrete-time dynamical system. Therefore, the sampling frequency of the RPPT control module is much lower than that of the digital PWM controller. Outputs of the OGASE controller are a perturbation direction factor (PDF) and a step adjustment factor (SAF) . The PDF essentially indicates whether the duty ratio should be increased or decreased or kept unaltered. It takes a value out of 1, -1 and 0 accordingly. The SAF is an integer number to ensure that the duty ratio is perturbed by an integer multiple of T pw ∕T cs . The value of the SAF is adapted according to an assessment of the equilibrium proximity. A limiter block is used to restrict the generated duty ratio command within an upper limit (i.e. D max ) and a lower limit (i.e. D min ). As usual, inputs to the OGASE controller are the photovoltaic array voltage, the photovoltaic array current and the user specified power reference command (i.e. P sp re f ). The user specified power reference command is suitably converted to an effective power reference command P e f re f in due course. The effective power reference command is prepared by choosing the closest practically attainable power level around the specified power reference. The RPPT ultimately follows P e f re f instead of P sp re f .

Internal block diagram of the OGASE controller
The OGASE controller comprises nine building blocks. The names and general functionalities of different blocks are discussed below.

Horizontal operating position observer (HOPO):
The HOPO is used to determine whether or not the operating point at a The CDC is used to get the time that has been elapsed so far in walking towards the convergence after observing the equilibrium proximity. 9. Equilibrium shift observer (ESO): The ESO is used to monitor if the target equilibrium point has been modified after the RPPT dynamics attained a convergence or got into the proximity of the previous equilibrium.
The internal block diagram arrangement of the OGASE controller, depicting the interactions of different blocks, is shown in Figure 6. Symbols used to indicate the outputs of different blocks are also marked in the figure. The external inputs and final outputs of the OGASE controller are highlighted by red circles and red squares, respectively. Three different types of arrows are used to show the signal flows. The solid and thin black arrows carry the latest values of variables, whereas, a broken red arrow carries the value of a variable at the previous time instant. A solid and thick red arrow that emanates from aẑ −m block basically carries a vector of (m + 1) most recent samples (including the value at the present time instant) of a variable. The sequence of the process flow is determined only by solid black and red arrows. The detailed dynamics of different blocks are explained in the next subsection.

Formulation of the dynamics of OGASE building blocks
The dynamics of the OGASE building blocks are described by using the symbols defined in Figure 6. The outputs of HOPO, VOPO, PTCO, EPO and ESO blocks (i.e. h , v , pp , and ) can be either 1 or 0 indicating "true" and "false", respectively.
The possible values and the implications of those values of the PDM output (i.e. ) have already been explained in the description of Figure 5. Outputs of remaining blocks are simple arithmetic variables.

3.2.1
Horizontal operating position observer The HOPO block primarily uses the principle of monitoring the slope of the P-V curve. The slope-based criterion for the horizontal position indication appears as follows.
Here, Δ(. .The sign function "sgn" returns 1 for a positive number and 0 for a non-positive number. There is, however, a certain problem [58] in making the horizontal position indication merely based upon the slope observation. This is because the slope of the P-V curve is theoretically zero both at the MPP voltage as well as at voltages higher than the open circuit voltage. Therefore, with due consideration for the measurement noise and imprecision, a very small slope value can be ambiguous. In order to avoid such ambiguity, an additional criterion needs to be enforced to correctly locate the horizontal position of the present operating point.    The PTCO flowchart since the photovoltaic array power output may, in such a case, remain disturbed most of the times. Therefore, once the power tracking converges on the basis of the prevailing P-V curve, any further perturbation to the converter duty ratio should be prevented until the environmental condition changes decently. In order to achieve the particular goal, the convergence is first achieved with a smaller power mismatch tolerance (namely, in ) by looking only into the residual measurement ripple. Subsequently, a larger value of the power mismatch tolerance (namely, out ) is used to reject small changes in environmental conditions. This, in turn, leads to the following dynamics for [n].
The flowchart of a PTCO iteration is presented in Figure 9.

Perturbation direction moderator
The PDM output is set to zero if convergence is achieved (i.e.  [n] should become 1 if any of these conditions is met. After reaching the first equilibrium proximity zone, the photovoltaic operating state continues to be at the equilibrium proximity. However, the oscillation series may get discontinued because of

FIGURE 11
The EPO flowchart reducing the perturbation step size. Therefore, [n] should, in general, be set equal to [n − 1] if no -oscillation is observed. All the above rules for updating [n] is applicable only if no disturbance is presently identified by the ESO (i.e. [n] = 0 ). In the case [n] = 1, the power tracking has to be started afresh by setting [n] to 0. Revoking the equilibrium proximity status of RPPT iterations by observing a disturbance is one of the salient features of the OGASE algorithm. It is also to be noted that the three-point power comparison technique proposed in [39] cannot be used in RPPT. The flowchart of an EPO iteration is presented in Figure 11.

3.2.6
Power reference conditioner The PRC block also observes an oscillatory phenomenon to appropriately set the value of P

Perturbation step size moderator
The PSSM dynamics is governed by the following equation.
According to the above equation, a power tracking round begins with a large value of the perturbation step size by using the SAF value of max . As mentioned earlier, a new round of power tracking takes place after every identification of the equilibrium shift. The perturbation step size is immediately reduced to the smallest possible value once the operating point enters into the first equilibrium proximity zone. By looking into the discrete nature of the converter duty ratio, only one high value and one low value for should be useful. The flowchart of a PSSM iteration is presented in Figure 13.

Convergence delay counter
The CDC output is governed by the following equation.
Thus, the CDC output has a non-zero and continuously growing value only during the phase of perceiving the equilibrium proximity until the convergence is achieved. It can be shown that c d [n] cannot exceed max + 2 unless there is a disturbance before reaching the target equilibrium of a power tracking round. This is because, after reaching the first equilibrium proximity zone, the RPPT should not take more than max + 2 iterations to converge if the target equilibrium point remains unaltered. The flowchart of a CDC iteration is presented in Figure 14.

Equilibrium shift observer
The identification of the alteration of the target equilibrium is relevant only if the slower RPPT iteration process (i.e. with the smaller perturbation step size) has been started. After the RPPT iterations are pushed to the slower pace, any shifting of the target equilibrium point may significantly delay the achievement of the convergence unless the faster pace with the larger per- Note that, as per Condition 2 and the PTCO rule, the ESO does not report any small shift of the target equilibrium after the RPPT attains a convergence. Also, the CDC upper limit is taken one more than the ideal value to ignore small equilibrium shifts for changing the pace of the power tracking. The basis of the third condition has been explained in the discussion of CDC. The flowchart of an ESO iteration is presented in Figure 15.

Variable initialisation and parameter tuning
Most of the blocks of the OGASE controller need some past values of different variables. Therefore, a variable initialisation work is involved before starting the OGASE controller. The power tracking usually starts from a state when the photovoltaic array is effectively in the open circuited condition because of the zero duty ratio. An improper initialisation of the past values of the PDM output may falsely indicate equilibrium proximity even if the photovoltaic unit has just started. Therefore, since the photovoltaic array may remain in the open circuited condition with almost the same power output (which is close to zero) for a few RPPT iterations, a false convergence may happen at the very beginning as per the PRC and PTCO rules. As a result the photovoltaic array power output may fail to pick up. In order to avoid such false convergence, all the initial values The parameters that need to be tuned for the OGASE controller are v pv,th in HOPO, in and out in PRC, and max in PSSM. All these parameters can be tuned through simulation studies under a set of credible environmental conditions. The value of v pv,th should be set according to the lowest observed open circuit voltage under the uniform irradiance. Parameter in is to be determined based upon the worst case residual highfrequency ripple in the measured power after filtering. A smaller value of in can be attained by increasing the LPF time constant and compromising with the speed of power tracking. The extent to which the LPF time constant needs to be increased depends upon the power range over which the RPPT should be performed. This is because the slope of the P-V curve increases with the decline in the array power output, which, in turn, increases the magnitude of the power ripple for the same capacitor voltage ripple. The value of out should be the direct choice of the user and can be set by ensuring that the power tracking process is not restarted very frequently owing to the typical irradiance variation over a day. Finally, the max can be chosen in a way so that, starting from the open circuit voltage, the MPP is reached in minimum number of steps via the OGASE approach under the standard test condition. Thus, the parameter tuning of the OGASE controller needs very simple procedures, which is another advantage of the OGASE controller.

SIMULATION RESULTS
Simulation studies are performed by using the detailed model of a 24 kW grid-connected photovoltaic system. As mentioned earlier, the power conversion is carried out in two stages without any local energy storage. The DC-side voltage controller (DCVC) and the AC-side current controller (ACCC) are taken from [60]. The phase locked loop (PLL) is taken from [61]. As is indicated in Figure 5, input LPFs are used in the RPPT control to clean the high-frequency oscillatory components from photovoltaic current and voltage measurements. The complete details of photovoltaic array, DC-DC boost converter, DC-AC converter and AC grid parameters are provided in Table 1. The inductance, resistance and capacitance values correspond to the positive sequence network. The photovoltaic cell open circuit voltage and short circuit current values mentioned in Table 1 correspond to the standard test condition (i.e. 25 0 C junction temperature and 1 kW/m 2 irradiance). The information of controller parameters is provided in Table 2. For the given system, the DC-DC converter duty ratio can effectively be varied as integer multiples 0.01. Four different cases are studied by maintaining the temperature constant at 25 0 . The DC-AC converter is always operated at the unity power factor.

Case 1: Response to the large variation of irradiance during the mixed power tracking
In this case, the specified power reference command is kept fixed, but the irradiance on the photovoltaic array is significantly varied. However, the power reference is set in a way so that the both the LPP tracking and the MPP tracking can be observed. The specific value of the input power reference is taken as 18 kW. The initial irradiance level is set to 1 kW/m 2 . During 10-12 s, the irradiance is reduced to 0.6 kW/m 2 at the ramp rate of 0.2 kW/m 2 s. At the same ramp rate, the irradiance level is later raised to 1.2 kW/m 2 during 20-23 s. The maximum power availabilities at those different irradiance levels are 24.003, 14.192 and 28.883 kW, respectively. The corresponding RPPT results obtained by deploying the OGASE controller are plotted in Figure 16. It can be observed that, excepting 0.6 kW/m 2 irradiance, the photovoltaic power output could closely follow the specified power reference command without any low-frequency oscillation. The absence of the low-frequency oscillation is indicated by the attainment of a steady duty ratio after a disturbance. As are observed through the zoomed window in Figure 16a, some high-frequency oscillations remain present in the  photovoltaic power output because of the DC-DC converter switching effect. In the same way, the DC-AC converter switching causes the presence of high-frequency oscillations in the VSC output power. However, those do not affect the output voltage of the DC-DC converter much (i.e. the DC-link voltage). Under 0.6 kW/m 2 irradiance, the maximum power availability falls short of the power reference specification; therefore, the photovoltaic power output settles near the MPP. The power tracking delays are observed to be only 0.28 and 0.12 s, respectively, after the first and second transitions. Although there are negligible power oscillations in the steady state, some large power oscillations can be observed in Figure 16 during the power tracking transient. The reason behind such large power oscillations is again the fact that the slope of the P-V curve is very high on the right side of the MPP. Moreover, the irradiance has been varied in ramp, which has triggered the ESO pulse several times. Those oscillations usually sustain as long as the irradiance (or any other external factor) is varying. The respective power oscillations can be reduced by decreasing the duty ratio perturbation step size. The same can be achieved by reducing either the converter switching frequency or the PWM controller sampling period or the maximum value of the SAF. However, as the duty ratio perturbation step size is reduced, the power tracking also becomes slower. Furthermore, at a lower converter switching frequency, the converter transient gets prolonged because of larger inductance and capacitance values. This, in turn, increases the RPPT controller sampling step size making the power tracking further slow. Overall, the power oscillations during the power tracking transient can be reduced only at the cost of the power tracking speed.

Case 2: Response to the large variation of irradiance during purely MPP tracking
The scenario in Case 2 is identical to that in Case 1 except for the fact that the input power reference command is set to a very high value so that the OGASE controller always tracks the MPP. Results for this case, by setting the input power reference command to 30 kW, are produced in Figure 17. Therefore, the OGASE controller is also useful to perform only the MPPT under the varying irradiance condition.

Case 3: Response to the small variation of irradiance
The scenario in Case 3 is also identical to that in Case 1. However, the magnitude of the irradiance variation is kept  Figure 18 that the DC-DC converter duty ratio does not respond to such small irradiance variation, although the photovoltaic power output slightly changes because of the modified P-V characteristic.    Figure 19. The photovoltaic array power output can be seen to change according to the specified power reference command. The power tracking error is very small when the input power reference command is at 17 kW. This because of a very small difference between the particular power reference command and the nearest practically attainable power level. The respective difference increases at other values of the input power reference command leading to larger power tracking errors. The error in power tracking is, in general, inevitable in the case of the quantum-mode RPPT. But, the error magnitude depends upon the input power reference command. It is, therefore, necessary to take due care while deriving the input power reference command for the RPPT (i.e. while scheduling the photovoltaic array power output) so that the actual power output of the photovoltaic array can be maintained close to the scheduled level.

HARDWARE EXPERIMENTATION
Because of the limited resource availability at the laboratory, a full experimental setup for the grid-connected photovoltaic system could not be prepared. Instead, the experiment is performed on the basis of the DC-side equivalent circuit representation of a two-stage photovoltaic system as is shown in Figure 2. As per the simulation results in Figure 16, the ripple in the DC-link voltage remains very small even during the power  Figure 2 seems to be a reasonably accurate representation of a two-stage grid-connected photovoltaic system on the DC side. The experiment is, as well, carried out at a low power level. As a result, the signal-to-noise ratio of the measured current or voltage always happens to be very low.
In order to eliminate the noise quantities from the current and voltage measurements under such low signal-to-noise ratios, the LPF time constant needs to be set at a very high value. Under such a situation, the OGASE controller sampling period also greatly increases. In the present experiment, the OGASE controller sampling period is set to 1 s. Although the OGASE controller sampling period in the hardware experimentation is much higher than the corresponding value in the simulation study, it is to be noted that simulation results were produced at high power levels. Pertaining to the PWM controller sampling period, it was difficult to set this value lower than 2 in the simulation study. This is because of both the memory issue and the long time requirement for executing the simulation. On the other hand, the PWM controller sampling period is reduced to 1 in the hardware experimentation for a smoother power tracking. The converter switching frequency is as usual taken as 5 kHz.
The photovoltaic array considered in the hardware experimentation comprises 2 parallel strings with 54 series connected cells in each string. The cell parameters are the same as those in Table 1. The DC-link voltage is set around 35.5 V. The converter and OGASE parameters are scaled down or scaled up accordingly. The voltage source on the output side of the DC-DC converter is implemented by means of a battery bank. The photovoltaic array is emulated by using the ECOSENSE programmable current source. The OGASE-based RPPT control system is developed in LabVIEW and is implemented in real time by using a PXI device. The analogue and digital signal transfer between the DC-DC converter and the PXI device takes place by means of an NI SCB-68 box. The photograph of experimental setup prepared in the laboratory is shown in Figure 20.
It is difficult to create a scenario of the dynamic irradiance variation in the above experimental setup. This is because the photovoltaic emulator used here does not provide this particular option. Therefore, experimental results are produced only for the case of the power reference command variation. As before, the photovoltaic cell junction temperature is taken as 25 0 C. However, the irradiance on the photovoltaic array is set to a very low value, which is 270 W/m 2 . The maximum available power from the photovoltaic array and its open circuit voltage at this irradiance level are 101.1 W and 30.54 V, respectively.
The input power reference command to the OGASE controller is varied between two values, such as 80 and 110 W. The 80 W power level is associated with a theoretically reachable point on the P-V curve. On the other hand, the power reference command of 110 W exceeds the maximum power that can be produced by the photovoltaic array. In the first experiment, the power reference command is varied from the low value to the high value. The reverse transition is made in the other experiment. The exact battery voltages during the first and second experiments are observed to be 36 and 35.4 V, respectively. Since the ramp transition has already been considered in the simulation study, only the step transition is applied in the hardware experimentation.
The results of the above experiments are produced in Figures 21 and 22, respectively. Five quantities are captured by oscilloscopes in each case. Those are the unfiltered voltage mea- surement, unfiltered current measurement, unfiltered power measurement, converter duty ratio and the power tracking error. The power tracking error is computed based upon the filtered power measurement. Both the power tracking error and the converter duty ratio are obtained from the analogue output ports of the PXI device via a port interfacing box.
The results produced in the above plots confirm that the OGASE controller is able to follow the power reference command in the desired fashion. The power tracking error in less than 2 W when the power reference command is at 80 W. For the power reference command of 110 W, the power tracking error is found to be around 9 W. This effectively indicates that the array power output is correctly settled near the MPP. Although the power tracking transient lasts for nearly 10 s in both the cases, the same has happened mainly because of the very low sampling frequency of the OGASE controller. As mentioned earlier, the sampling of the OGASE controller had to be made slower because of using LPFs of large time constants so as to suppress the voltage and current measurement noises. Furthermore, there is a difference between the power tracking transients in simulation and experimental results. The power tracking dynamics is much smoother in the results obtained from hardware experiments. This has happened because of two reasons. First of all, the PWM sampling period in experiments is taken to be the half of the corresponding value in simulation studies. Second, the power reference is varied in a stepped manner in experiments in contrast to its ramped variation in simulations. However, it can be easily checked that a simulation of the same case as in the hardware experiments produces identical results as those in Figures 21 and 22.

CONCLUSION
A novel control technique has been proposed in this paper to regulate the power output of a photovoltaic array in the quantum mode. The dual-stage configuration of the photovoltaic unit has been deployed. It has been recognised that the power output of a photovoltaic array can, practically, be adjusted only at some discrete values because of the natural quantisation of the converter duty ratio under the digital PWM. It has also been understood that the power gap between two successive feasible operating points on the P-V curve may not be negligible. Therefore, instead of attempting to exactly track the user specified power reference command, the power tracking has been automatically guided to the nearest feasible power level. As a result, the presence of a large low-frequency oscillation in the final settlement of the photovoltaic array power output could be got rid of. The proposed OGASE controller performs a constrained enumerative update of the converter duty ratio by considering only the practically feasible duty ratio values. The perturbation step size in the OGASE control has been toggled between a low value and a high value depending upon the beginning of a new power tracking round and the possible attainment of equilibrium proximity. The attainment of equilibrium proximity has been identified on the basis of some oscillatory phenomena. An effective power reference command has been duly derived from the specified power reference command so as to ensure the achievement of an oscillation-free steady state. Simulation and experimental results have demonstrated the effectiveness of the OGASE algorithm to perform smooth power tracking under both power sufficiency and power deficiency. The capability of the OGASE controller to overlook small variations in environment conditions has also been demonstrated. At the same time, the speed of the power tracking without very noisy measurements has been found to be quick enough to meet the practical requirement. In the end, studies of the present paper have been performed by considering only the uniform irradiance scenario. Extension of the OGASE control to the partial shading condition through the development of a suitable MMPA is the future scope of this work.