Abstract
To obtain efficient photovoltaic (PV) systems, optimum maximum power point tracking (MPPT) algorithms are inevitable. The efficiency of MPPT algorithms depends on two MPPT parameters, i.e., perturbation amplitude and perturbation period. The optimization of MPPT algorithms affect both the tracking speed and steady-state oscillation. In this paper, optimization methods of MPPT parameters are reviewed and classified into fixed and variable methods. The fixed MPPT parameters are constant during MPPT performance, and a trade-off should be made between the tracking speed and steady-state oscillation. However, the variable MPPT parameters will be changed to improve both the tracking speed and the steady-state oscillations. Moreover, some of them are simulated, compared, and discussed to evaluate the real contributions of the optimization methods to the MPPT efficiency. Furthermore, significant features of the optimization methods, i.e., noise immunity, robustness, and computation effort, are investigated.
PHOTOVOLTAIC (PV) systems have recently gained popularity since they are environmentally friendly and can be installed in residential and remote areas. In PV systems, the output power reaches the maximum at a point called maximum power point (MPP), and its location continuously changes based on the irradiance level and temperature. As a result, to extract the maximum power from the PV generator (PVG), an operation point should be located at MPP. Therefore, MPP must continuously be tracked by maximum power point tracking (MPPT) algorithms. Generally, MPPT algorithms are classified into two categories due to uniform irradiance and partial shading conditions [
MPPT algorithms are usually applied in either single-loop or multi-loop MPPT structures. In the single-loop MPPT structure, the MPPT algorithm directly changes the duty cycle D of interfacing power converter (IPC), as shown in

Fig. 1 Different PV system structures. (a) Single-loop. (b) Multi-loop.
The efficiency of MPPT highly depends on the MPPT parameters, i.e., perturbation amplitude ΔD and perturbation period Tp. The perturbation amplitude is a change either to the duty cycle of IPC in the single-loop structure or to the reference voltage ΔVref and reference current ΔIref in the multi-loop structure. The perturbation period is the time interval between two consecutive perturbations. The characteristics of the MPPT perturbation which lead to the optimum efficiency of MPPT is depicted in

Fig. 2 ΔD and Tp on duty cycle of IPC leading to optimum efficiency of MPPT.
In the literature, different types of MPPT algorithms are studied ranging from the conventional algorithms to smart ones [
According to
(1) |
(2) |

Fig. 3 I-V and P-V curves of dynamic and static resistors.
where and are the DC parts of the PV voltage and the PV current, respectively.
According to
The PV system shown in

Fig. 4 PV system consisting of a PV panel, a boost converter, and a resistive load.
Therefore, the PV voltage dynamic characteristics can be derived by (3) and (4) in response to the perturbation ΔD [
(3) |
where is the natural frequency; is the zero frequency induced by the input capacitor equivalent series resistance; is the damping factor; is the DC gain; and , , , , , , , and are the circuit parameters of
The response of the PV voltage to as in (4) can be calculated by (3). Consequently, by substituting (4) into (5)-(7), the PV power response in different regions can be achieved [
(4) |
(5) |
(6) |
(7) |
Moreover, when the MPPT structure is multi-loop, (8) can be derived by the reference voltage to the PV voltage transfer function [
(8) |
where is the open-loop PV voltage loop plant and C(s) is the transfer function of the IVC. By substituting (8) into (9)-(11), the PV power for the multi-loop structure can be achieved in different regions [
(9) |
(10) |
(11) |
where is the inverse Laplace transform operator. Finally, the parameters of PV system performance such as the PV power settling time in both the single-loop and multi-loop structures can be achieved in different regions by (5)-(7) and (9)-(11), respectively.
In this section, different values of MPPT parameters are compared to show how they affect the efficiency of PV system. The single-loop PV system in
The perturbation amplitude is a change which is applied by MPPT algorithm to the duty cycle of IPC. According to

Fig. 5 Effect of ΔD on oscillation amplitude of operation point around MPP.
The optimum value of Tp leads to the three-point oscillation in the steady state. According to

Fig. 6 Effect of Tp which is smaller than system settling time on oscillation amplitude of operation point around MPP.

Fig. 7 Effect of Tp which is greater than system settling time on oscillation amplitude of operation point around MPP.
The steady-state efficiency described by (12) shows how the above observations affect the PV system efficiency.
(12) |
where NL and NR are the numbers of oscillation points in the left and right sides of MPP, respectively [
The proper selection of leads to fast and accurate tracking of MPP. Therefore, the knowledge of optimization methods of is of importance to reach high efficiency in PV systems. According to

Fig. 8 Optimization methods of perturbation amplitude.
In the fixed ΔD methods, ΔD is calculated in the design stage and is constant during the MPPT performance. Large ΔD shortens the transient time while it increases oscillation amplitude in the steady state. Small ΔD extends the transient time while it decreases oscillation amplitude in the steady state. Therefore, a trade-off should be made between the transient speed and the steady-state efficiency.
The main idea of this kind of method is to select ΔD (13) in a way that the MPPT algorithm can distinguish the variations of the PV power caused by the duty cycle modulation from those caused by the irradiance variation. As a result, the MPPT algorithm does not fail to track MPP when the irradiance level changes with the rate of [
(13) |
where G0 is the DC gain of (3); VMPP is the PV voltage at MPP; is the rate of the irradiance change; and and H are the parameters.
In grid-connected PV systems shown in

Fig. 9 Dual-stage grid-connected PV system.
(14) |
where Cb, Vb, and fac are the bulk capacitor, the voltage across Cb, and the grid frequency, respectively.
Achieving the maximum efficiency is a desire for all the mentioned methods. It is directly used in the form of a formula. For instance, the maximum efficiency which is a function of is given by (15). With different values of , the optimum one that leads to the highest efficiency can be achieved [
(15) |
where id,0 is the current through the p-n junction of a PV cell; q is the electron charge, ; k is Boltzmann’s constant, ; A is the diode quality factor; and T is the absolute temperature. According to
(16) |

Fig. 10 Choosing optimum ΔD based on power difference between point A and point B.
where is the maximum admissible loss; and and are the current and voltage of point B, respectively.
A bifurcation diagram shows the effect of a bifurcation parameter on the system performance and can be used to determine the appropriate range of the parameter in the design stage. Therefore, this diagram is used to determine a proper ΔD. According to

Fig. 11 Bifurcation diagram of PV voltage-step size for P&O algorithm.
Variable has been proposed to improve not only the steady-state behavior but also the dynamic performance of MPPT algorithms. Several important methods are introduced as follows.
1) Slope of P-V, P-D, and Io-D Curves
In some special curves such as P-V, P-D, and Io-D, the slope has an increasing trend. In fact, according to

Fig. 12 P-D and Io-D curves. (a) P-D curve. (b) Io-D curve.
(17) |
where N is the scaling factor; is the fixed maximum perturbation amplitude; and and are the differences in the PV power and voltage between two consecutive samples, respectively.
2) -based Method
In the previous method, a small value of ΔV leads to a large value of , thus the operation point moves away from MPP. In addition, it needs a division operation. Therefore, one possible solution to address the problem is to use only the difference in (18) [
(18) |
where is the scaling factor.
3) Auxiliary-functions-based Method
Auxiliary functions are mostly derived from the P-V and P-D curves and are used to determine variable or the area in which MPP is located. In fact, in some methods, the limit of MPP is firstly predicted either by simple methods like fractional open-circuit voltage [

Fig. 13 Auxiliary function used for predicting limit of MPP.
Afterwards, based on the location of the operation point than the predicted area, small and big values are assigned to , seperately. Auxiliary functions are used to determine the variable . However, according to

Fig. 14 Auxiliary functions used for determining variable ΔD. (a) and . (b) . (c) . (d) .
For example, the variable is determined by the two curves in
4) Zero Oscillation
In this method, the steady state is firstly identified either by the recognition of the three-point oscillation around MPP [
5) Controller-based Method
According to

Fig. 15 Block diagram of controller-based method.
The optimized value of Tp leads to the three-point oscillation in the steady state and the fast tracking in the transient state. Therefore, the knowledge of the optimization methods of Tp is of vital importance to improve the efficiency of MPPT. According to

Fig. 16 Optimization methods of perturbation period.
In this method, Tp is calculated in the design stage and is constant with MPPT performance. Large Tp extends the transient time while it guarantees the three-point oscillation in the steady state. On the contrary, small Tp leads to the fast tracking while it increases oscillations around MPP to more than three points.
By setting Tp to be the settling time of PV system, MPPT waits enough before applying the next perturbation so that the fluctuation of the PV voltage and current is restricted to a band of ±. Therefore, the sampled PV voltage and current lead to the correct tracking of MPP. The settling time of PV system can either be calculated in CPR by (19) [
(19) |
where is the relative magnitude of settling band. Furthermore, it is proven that the settling time of PV system in CCR (20) is longer than CPR and CVR [
(20) |
When the MPPT structure is multi-loop, the effect of the PV panel dynamic resistor on the system settling time is negligible [
(21) |
(22) |
where , , and are the DC gain of P(s) in (8), the nominal value of , and a desired loop gain crossover frequency, respectively; is the damping factor; and is the natural frequency.
When Tp is equal to the IPC switching period or the digital-to-analog conversion period, the tracking speed substantially increases [
The settling time of PV system is affected by environmental factors, system components, and ΔD. Therefore, fixed Tp cannot be a valid value under different weather conditions. Thus, system identification techniques and experimental equations have been applied to provide an adaptive Tp for PV systems.
System identification techniques are used to identify important parameters of system performance which can be used for system monitoring or controller tuning. System identification techniques are classified into two categories: non-parametric and parametric [
Recently, the system identification techniques have been used in PV systems to identify the online value of Tp. According to (19), and determine the settling time and the value of Tp. However, and depend on rpv, weather conditions, and IPC parameters. Furthermore, the aging phenomenon of PV systems affects the values of and . As a result, with the online system identification, the optimized value of Tp can be achieved independent of PV system component values, weather conditions, and the aging phenomenon.
According to

Fig. 17 Flowchart of non-parametric system identification of PV system.
The overall procedure in the parametric system identification is similar to the non-parametric one. Nevertheless, the objective of the parametric system identification is to identify the parameters of the defined PV system model. The Kalman filter [
In this method, an experimental equation which is a function of is derived to determine Tp. In fact, different values of lead to different values of the settling time. Therefore, according to

Fig. 18 Settling time of PV system versus ΔD.
(23) |
(24) |
The contribution of the mentioned optimization methods to the PV system efficiency is quantified by conducting simulations in MATLAB/Simulink software. Afterwards, simulation results are compared and discussed. Since the methods are simulated under the same simulation condition and selected from different mentioned categories, it is possible to draw a fair comparison. To this end, the PV system in
The simulation results are given in
(25) |
Note: a indicates current sensorless method with Auto-Modulation; b indicates adaptive scaling factor Beta method; c indicate

Fig. 19 Output power and voltage of PV systems of Table II. (a) Irradiance profile. (b) Method No. 4. (c) Method No. 5. (d) Method No. 6. (e) Method No. 7. (f) Method No. 8. (g) Method No. 9. (h) Method No. 11.

Fig. 20 Variable ΔD in different methods of Table II. (a) Irradiance profile. (b) Method No. 4. (c) Method No. 5. (d) Method No. 6. (e) Method No. 7. (f) Method No. 8. (g) Method No. 9. (h) Method No. 11.

Fig. 21 Output power and voltage of PV systems of Table II. (a) Irradiance profile. (b) Method No. 9. (c) Method No. 10. (d) Method No. 11.

Fig. 22 Tp in different methods of Table II. (a) Irradiance profile. (b) Method No. 9. (c) Method No. 10. (d) Method No. 11.
where Pmpp is the theoretical maximum power.
The importance of on the efficiency of MPPT is revealed in
The method of variable further improves the efficiency of MPPT although the results are not desirable in some cases. For instance, according to methods No. 4 and No. 5 of
Method No. 6, which indicates the optimized based on , is computationally simple. Moreover, it is not efficient since the first steady state is missed according to Figs.
Furthermore, method No. 7 speeds up the transient states at the cost of wild fluctuations in the steady states according to Figs.
The optimization of is also considered. For example, the variable ΔD and Tp depicted in Figs.
Moreover, according to
The output power and voltage of method No. 10 are shown in
The following outcomes are deduced.
1) When is optimized based on the slope of the P-V and P-D curves, different scaling factors must be allocated to .
2) Tp has different behaviors in the steady and transient states, thus each state needs a different optimization method regarding Tp.
3) The zero-oscillation technique greatly improves the MPPT efficiency. However, to suppress the steady-state oscillations successfully, the steady state should firstly reach the three-point oscillation. Moreover, system identification techniques guarantee the three-point oscillation by the online optimization of Tp. Thus, the system identification techniques can be used to improve the reliability of the zero-oscillation method.
To achieve a desired tracking speed and efficiency of MPPT, and Tp should be optimized accordingly. In this paper, various optimization methods of the MPPT parameters are surveyed, classified, simulated, compared, and discussed. With the discussion and comparison based on
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