# A New Hybrid MPPT Based on Incremental Conductance-Integral Backstepping Controller Applied to a PV System under Fast-Changing Operating Conditions

Authors: Ambe Harrison; Njimboh Henry Alombah; Jean de Dieu Nguimfack Ndongmo
Journal: International Journal of Photoenergy
 (2023)

---

## Abstract

Maximum power point tracking (MPPT) is becoming more and more important in the optimization of photovoltaic systems. Several MPPT algorithms and nonlinear controllers have been developed for improving the energy yield of PV systems. On the one hand, most of the conventional algorithms such as the incremental conductance (INC) demonstrate a good affinity for the maximum power point (MPP) but often fail to ensure acceptable stability and robustness of the PV system against fast-changing operating conditions. On the other hand, the MPPT nonlinear controllers can palliate the robust limitations of the algorithms. However, most of these controllers rely on expensive solar irradiance measurement systems or complex and relatively less accurate methods to seek the maximum power voltage. In this paper, we propose a new hybrid MPPT based on the incremental conductance algorithm and the integral backstepping controller. The hybrid scheme exploits the benefits of the INC algorithm in seeking the maximum power voltage and feeds a nonlinear integral backstepping controller whose stability was ensured by the Lyapunov theory. Therefore, in terms of characteristics, the overall system is a blend of the MPP-seeking potential of the INC and the nonlinear and robust potentials of the integral backstepping controller (IBSC). It was noted that the hybrid system successfully palliates the conventional limitations of the isolated INC and relieves the PV system from the expensive burden of solar irradiance measurement. The proposed hybrid system increased the operational efficiency of the PV system to 99.94% and was found better than the INC MPPT algorithm and 8 other recently published MPPT methods. An extended validation under experimental environmental conditions showed that the hybrid system is approximately four times faster than the INC in tracking the maximum power with better energy yield than the latter.


---

## Body

## 1. Introduction

The energy sector has a significant influence on our climate and environment as it contributes to roughly two-thirds of all harmful greenhouse gas emissions [1]. There is a need for these emissions to be reduced significantly, which can only be achieved by phasing out fossil fuels. Therefore, a sustainable revision of the energy sector should be an important consideration in relieving the world of the problems of climate change and global warming. The 2015 Paris Agreement on climate change mandates the transition to renewable energy generation to limit the rise of global temperatures to well below 2 degrees Celsius and ideally below 1.5 degrees Celsius above preindustrial levels [2]. During the recent decade, several renewable energy sources have been sought, amongst which solar energy is one of the most reliable and sustainable. The prominent nature of solar energy is attributable to its environmental friendliness and global abundance [3–7]. Consequently, global solar photovoltaic (PV) generation has been on the rise, with a recent increase from 823.8 terawatt-hours (TWH) to 1002.9 TWh [8]. However, according to the International Energy Agency (IEA), an annual global PV generation of approximately 7400 TWh by 2030, aligning with the net zero greenhouse gas scenario, from the current 1002.9 TWh requires annual average generation growth of about 25% during 2020-2030. Despite the high drift towards solar PV systems, there are two main challenges to their exploitation: low efficiency and high financial cost of installation. To avoid ambiguity with the term efficiency, it is worthy to distinguish the two types of efficiency related to the PV: conversion efficiency which is related to the technology of the solar cells and operational efficiency which is related to the electrical operation of the PV system. This work focuses on operational efficiency. There are two ways to improve the cost and compensate for the operational efficiency of PV systems: on the one hand, the structure of the PV generator is changed at the implementation stage or the PV system is forced to operate at its maximum energy potential [9]. The latter approach is the maximum power point tracking algorithm and focus of this paper.In the recent decade, numerous maximum power point tracking algorithms have been deployed in the literature [10–22]. These algorithms differ in cost, complexity, and performance. A retrospective survey shows that perturb and observe is one of the most popular algorithms for MPPT in PV systems. However, this algorithm suffers from several limitations: oscillations under fast-changing operating conditions, loss of tracking direction under fast-changing irradiance, and the trade-off between dynamic response and steady-state oscillations. Given the ease of implementation and simplicity of this algorithm, numerous efforts are being vested to alleviate its aforementioned problem. The incremental conductance (INC), which is an improved variant of perturb and observe (P&O), can perceive that the maximum power point (MPP) has been attained, hence overcoming the disadvantages of P&O [23]. Even though the INC provides better accuracy than P&O under fast-changing irradiance conditions, it still cannot guarantee finite stability of the operating point around the neighborhood of the maximum power point. The adaptive variable step-size algorithm P&O in [24, 25] improves the decision in making the P&O perturbation step; however, it does not solve the problem of stabilization of the operating point at the MPP, especially under fast-changing operating conditions. The beta algorithm proposed in [26–28] addressed the trade-off problem in the conventional P&O; however, its complete dependency on the latter is a serious limitation of the method.Numerous other schemes such as the intelligent and optimization algorithms including artificial neural networks [29], fuzzy logic [29–32], particle swarm optimization [33], ant colony optimization [34], and grey wolf optimizer [35, 36] have recently received attention with regard to MPPT. The optimization algorithms provide a better solution to the tracking performance of the conventional P&O and INC; however, the problem of the stability of the MPP is still recurrent. The ANN MPPT approach on the other hand is considered one of the most successful approaches for MPPT. This is because a trained intelligent model has better recognition capability of the MPP under any environmental conditions. However, the ANN method always fails to ensure satisfactory performance under varying load scenarios.It is a fact that the maximum power point stability problem especially under fast-changing operating conditions presented by the above MPPT algorithms is due to their lack of nonlinear capacity. This is even obvious given that the PV generator is a nonlinear system, coupled to the nonlinear time-variant nature of power electronic converters used for their control. It is therefore a better choice to seek nonlinear techniques that can ensure the stability of the PV system under changing operating conditions and at the same time provide good efficiency in the tracking mechanism.In the light of nonlinear control of the PV generator, some controllers have been proposed in the literature. The sliding mode controller (SMC) in [37–39] provides robust performance under changing operating conditions; however, the discontinuous nature of the SMC nonlinear control law introduces chattering problems [40], defined in [41] as “finite amplitude oscillations.” The synergic controller (SC) which is an improved nonlinear control strategy that has recently been applied to the maximum power point control of the PV system [40, 42] presented better performance than the SMC. The prominence of the SC lies in the continuity of its control law, hence solving the chattering problem [42].Another class of advanced nonlinear controllers superior to the SC is the backstepping controller (BSC), based on the recursive Lyapunov design that has been applied to the MPPT control of PV systems [43]. Though a comparison between SC and BSC in an MPPT context has not been well established, the work in [44] shows that in a class of nonlinear systems out of the MPPT context, the BSC outclassed the SC. Though exceptional in performance, however, unknown dynamics are usually not modeled in the BSC controller design; consequently, the controller usually presents steady-state errors. To allow good robustness of the controller against unmodeled dynamics and perturbations, an integral action was added to form the integral backstepping controller [45, 46].From the presented literature survey and trend of nonlinear controllers, it can be seen that the backstepping and its integral improvement are the most prominent within the context of MPPT. However, as in every closed-loop control system topology, the controller has to be properly fed with an accurate reference. This is where the above-presented controllers show serious limitations. In all the above nonlinear techniques, the reference (maximum power voltage) is generated using either an offline-trained regression plane [45] or offline-trained neural network-based techniques [46]. The offline-trained network requires frequent maintenance once deployed in the field because it will be subject to wear and tear. However, the main limitation of these offline-trained models for reference generation is the fact that when used in the PV system, it requires a very expensive solar irradiance sensor (pyranometer).In this paper, we present a novel hybrid controller based on incremental conductance-integral backstepping (INC-IBSC). The hybrid MPPT controller exploits the benefit of the incremental conductance algorithm in seeking the maximum power point and the nonlinear stability and robustness of the integral backstepping controller. Firstly, the proposed system alleviates the problem of poor tracking performance under fast-changing operating conditions, exhibited by the conventional INC algorithm, and finally, it relieves the MPPT system from the cost burden of the irradiance sensor. In-depth numerical investigations proved the good robust performance of the controller under fast-changing operating conditions. The proposed hybrid system was found better than the conventional INC MPPT algorithm and eight other recently published methods in the literature.The rest of this paper is organized as follows: in Section2, a general description of the proposed system is presented. The incremental conductance algorithm is overviewed in Section-3. The mathematical modeling of the boost converter is described in Section 4 while the nonlinear controller is designed step by step in Section 5. Numerous investigations and numerical simulations are discussed in Section 6 while the paper is concluded in Section 7.


## 2. General Description of the Proposed System

The synoptic diagram of the proposed system is presented in Figure1. It consists of an MSX-60 PV whose mathematical model is obtained according to [23], a DC-DC boost converter, a DC load, and the hybrid incremental conductance-integral backstepping controller (INC-IBSC).Figure 1
Synoptic diagram of the proposed hybrid MPPT system.The 60-watt PV is the main generator in the system that we desire to control. The incremental conductance algorithm, in its conventional form, generates a reference voltage that is fed into the INC-IBSC. The controller in its nonlinear capacity generates the nonlinear control law that modulates a pulse width signal which in turn drives the system via the boost converter coupled to the DC-variable load.


## 3. Incremental Conductance Algorithm (INC)

The incremental conductance (INC) algorithm is one of the most popular MPP algorithms [15, 47]. The method derives its name from the fact that the conductance of the PV is used to determine the movement towards the maximum power point (MPP). The basic principle of the algorithm is presented in Figure 2, while its mathematical description is as follows.
(1)Vref=V+δV,ΔIpvΔVpv>−IpvVpv,V+0,ΔIpvΔVpv=−IpvVpv,V−δV,ΔIpvΔVpv<−IpvVpv,where ΔIpv, ΔVpv, and ΔVpv are the change in current, voltage, and power due to voltage step size δV.Figure 2
Graphical demonstration of the INC algorithm.The operation of the INC is summarized in Equation (1) and Figure 2. It holds that when the operating point is to the left of the MPP, the reference voltage, Vref is increased by a step δV to the MPP point. To the right of the MPP, the reference voltage is reduced by the same step. Continuous adjustment of the operating point according to Equation (1) moves the PV point at the MPP. To this end, the benefit of the INC as presented shall be exploited in a hybrid scheme by the nonlinear controller.


## 4. Mathematical Modeling of the Boost Converter

The boost converter is a switched-mode step-down DC-DC converter. A DC-DC converter is the key component of every MPPT system as it ensures impedance matching between the PV source and the load. The boost converter as shown in Figure3 is made of an inductor (L) carrying current iL, input capacitor C1 carrying current ic1 with the PV voltage VPV at its terminals, a transistor switch (k), diode D, and output storage capacitor C2 in parallel with the load R. In this paper, the converter is considered to operate in continuous conduction mode (CCM). Furthermore, it is assumed that parasitic components in the inductor and capacitors are zero. In CCM, the converter operates in two modes. In mode 1, the switch k is on while the diode D is off. In the second mode of operation, the switch is off while diode D conducts.Figure 3
Electrical model of the DC-DC boost converter.The dynamic models for the two operating modes described above can be obtained using Kirchhoff’s laws and are given in Equations (2) and (3) for mode 1 and mode 2, respectively [48]:The state-space average model in Equation (4) is obtained from the dynamics of the two operating modes and is obtained as described in [48]:
(2)ic1=iPV−iL,vL=vPV,ic2=−vSR,(3)ic1=iPV−iL,vL=vPV−vS,ic2=iL−vSR.(4)x1̇=iPV−x2C1x2̇=x1L−x3L1−ux3̇=x2C21−u−x3C2Rwhere x1x2x3T=vc1iLvsT=x are the average state variables, with vc1,iL, and vs denoting the input PV voltage, inductor current, and output voltage, respectively. The control law is denoted u.Equation (4) which is the dynamical equation can then be exploited by a nonlinear controllerAssumption 1.
All the state variables of the model are continuous and bounded.Assumption 2.
iPV, x1, x2, and x3 are accessible for measurement.


## 5. Nonlinear Control Design

The nonlinear controller is synthesized in this section. The controller receives a reference voltage from the INC algorithm and exploits the dynamics of the converter to drive the PV system at the MPP. The controller proposed in this work is the integral backstepping controller (IBSC), whose design is based on Lyapunov theory. Let the voltage fed to the controller beVref.We recall that Vref.is generated by the INC algorithm. Since this variable is the first state of the system, we can write that
(5)x1ref=Vref.Lete1 be the first error considered in the Lyapunov design; we define it as the difference between the actual PV voltage and the reference from the INC algorithm, written mathematically as
(6)e1=x1−x1ref.We have to label out the dynamics of this error, by taking its derivative written as(7)ė1=ẋ1−ẋ1ref=iPVC1−x2C1−ẋ1ref.Introducing the integral actionp to this error results in a macrovariable:
(8)e=e1+p,(9)p=∫ote1dt=∫otx1−x1refdt.It is worth noting that the integral action is introduced to allow good robustness of the controller against unmodeled dynamics and perturbations.To ensure the stability of the controller, Lyapunov functions are introduced. Therefore, letV1 be a positive definite Lyapunov function for ensuring stability, tracking, and convergence to the reference voltage from the INC algorithm. According to the Lyapunov theory, e1 converges to zero if the first derivative of the Lyapunov function is negative definite. To this end, let the Lyapunov function be
(10)V1=12e12+k2p2,where k>0.Taking the first derivative of Equation (10) results to
(11)V1̇=e1ė1+kpṗ.Inserting Equation (7) and the derivative of Equation (9) into (11) gives
(12)V1̇=e1iPVC1−x2C1−ẋ1ref+kpx1−x1ref.However,e1=x1−x1ref.Under this setting, Equation (12) is simplified as
(13)V1̇=e1iPVC1−x2C1−ẋ1ref+kp.We must ensure thatV1̇ is negative definite. To this end, the first Lyapunov parameter, K1, is introduced such that
(14)iPVC1−x2C1−ẋ1ref+kp=−K1e1,K>0.Makingx2, subject in Equation (14) results to
(15)x2=C1K1e1+iPVC1−ẋ1ref+kp.Consider Equation (15) as the reference virtual input. To this end, a deviation e2 from the desired virtual input is introduced. Let φ be this reference. Therefore, Equation (15) is reformulated as follows:
(16)φ=C1K1e1+iPVC1−ẋ1ref+kp.The errore2 is now written as
(17)e2=x2−φ.Rearrange Equation (17) such that
(18)x2=e2+φ.Inserting Equation (18) into Equation (7) results to
(19)ė1=ẋ1−ẋ1ref=iPVC1−e2+φC1−ẋ1ref.Combining Equations (16) and (19) results to
(20)ė1=iPVC1−e2+C1K1e1+iPV/C1−ẋ1ref+kpC1−ẋ1ref.Simplifying Equation (20), we have
(21)ė1=−K1e1−e2C1−kp.Under this setting, Equation (11) becomes
(22)V1̇=e1ė1+kpṗ=e1−K1e1−e2C1−kp+kpṗ.Further simplifying Equation (22) with the fact that p=∫ote1dt and ṗ=e1, we have that
(23)V1̇=−K1e12−e1e2C1.Sincee1 or e2 can be either positive or negative, we have no idea of their state (sign) at any moment in the control loop, so we cannot conclude on the sign of e1e2/C1. However, the first term −K1e12 will always be negative definite. Exploiting the derivatives of Equations (16) and (17), we have
(24)ė2=ẋ2−φ̇,(25)φ̇=C1K1ė1+i̇̇PVC1−x¨1ref+kṗ.Simplifying Equation (25) using Equations (9) and (21), we have
(26)φ̇=C1K1−K1e1−e2C1−kp+i̇̇PVC1−x¨1ref+kx1−x1ref.Inserting Equation (26) into (24) results to
(27)ė2=ẋ2−C1K1−K1e1−e2C1−kp+i̇̇PVC1−x¨1ref+kx1−x1ref.Further simplification of Equation (27) leads to
(28)ė2=ẋ2−C1−K12e1−K1e2C1−kK1p+i̇̇PVC1+C1x¨1ref−ke1.Under this setting, to guarantee the convergence of both errors to zero, a new Lyapunov functionVf is introduced. Vfis an augmented Lyapunov function. Its derivative must be negative. Therefore, let Vf be defined as follows
(29)Vf=V1+12e22.Therefore, taking the time derivative of Equation (29), we have
(30)V̇f=V̇1+e2ė2.Substituting Equation (23) in (30) results to
(31)V̇f=−K1e12−e1e2C1+e2ė2=−K1e12+e2ė2−e1C1.To ensure that Equation (31) is negative definite, we define
(32)ė2−e1C1=−K2e2.Under this setting, Equation (21) combined with Equation (32) results to
(33)V̇f=−K1e12−K2e22.Under this new setting, the Lyapunov law has been verified. The nonlinear control law is then deduced from Equations (4), (28), and (32).
(34)−K2e2=x1L−x3L1−u−C1−K12e1−K1e2C1−kK1p+i̇̇PVC1+C1x¨1ref−ke1−e1C1.Further simplification of Equation (34) results to
(35)u=1−Lx3K2e2+x1L+K12C1e1+K1e2+kK1C1p−i̇̇PV+C1x¨1ref−C1ke1−e1C1.The presence of derivative terms in the derived control law renders its practical implementation unfeasible due to the presence of noise during numerical differentiation. In this light, a higher-order sliding mode differentiator (HOSMD) designed in [49] is used to obtain in this situation a noiseless and robust nonlinear controller suitable for the boost converter control. The stability analysis and global convergence of the closed-loop system with the various subsystems, namely, the INC-IBSC and HOSMDs, are based on the separation principle theorem [49, 50] which permits the application of differentiation in nonlinear control. According to this principle, real-time robust exact differentiation is conceivable up to order L, provided that the next (L+1)th derivative is bounded by a known constant [51]. The finite-time convergence of the INC-IBSC and HOSMDs allows for the separate design of the control subblocks. Therefore, the separation principle is verified [51]. The main limitation of the practical implementation of this nonlinear controller is the satisfaction of the assumption on the boundedness of the state space variables in the operating domain. This assumption is fulfilled, given the convergence of INC-IBSC and HOSMDs and the nonlinear control law which warrants the global stability of the system.


## 6. Results and Discussions

Numerous investigations using numerical simulations, implemented in the MATLAB/Simulink environment, are performed to validate the performance of the proposed hybrid incremental conductance-integral backstepping controller under fast-changing operating conditions. The objective is to force the MSX-60-watt solar module [20] to operate at its maximum power point. Since most MPPT algorithms fail under fast-changing operating conditions, we investigate the robustness of our systems under the most difficult operating scenarios involving fast-changing temperature, fast-changing irradiance, and abrupt variations in the load for robustness. The numerical realization of the proposed hybrid system is presented in Figure 4. At the simplest level, it shows that the incremental conductance algorithm (INC) is fed with measurements of the PV current and voltage. It outputs a reference voltage that is exploited by the nonlinear integral backstepping controller which is governed by Equations (16) and (35). The controller drives the PV system via a corresponding control law. The system is made up of a variable load and a boost converter with parameters input and output capacitor C1=C2=37μF, inductor L=0.3mH, and switching frequency of 250kHz as shown on Table 1. The parameters of the controller as shown in Table 1, k,K1,K2, were obtained via the empirical method [43]. Resistances in the range of 15Ω−30Ω are considered on the load side. The rest of this section is further subdivided as follows: in Subsection 6.1, the proposed hybrid INC-IBSC is evaluated at standard test conditions (STC). In Subsection 6.2, the results of the extended comparison with the classical incremental conductance (INC) algorithm are presented and discussed. The two systems are further evaluated and compared under changing load conditions at STC in Subsection 6.3. In Subsection 6.4, the systems are evaluated and compared under fast-changing irradiance and temperature scenarios in the presence of load variations. A comparison of the proposed hybrid INC-IBSC is provided in Subsection 6.5. Finally, validation of the proposed system with experimental data is provided in Subsection 6.6.Figure 4
Numerical realization of the proposed hybrid INC-IBSC system.Table 1
Parameters of the proposed system.
ParametersValuesIBSC nonlinear controllerk,K1,K2[47.1853, 13750, 10000]Boost converter [L]0.3 mHBoostC1,C2[37μF,37μF]Load resistance [R][15Ω,25Ω,30Ω]Boost converter switching frequencyFs250 kHz


### 6.1. Evaluation of the Proposed Hybrid System at STC

The proposed hybrid system is first investigated at STC corresponding to the temperature of 25°C and irradiance of1000W/m2. According to the characteristics of the considered PV module, a maximum power of 60 W and a maximum voltage of 16.70 V are expected under these conditions [21]. To this end, when the systems are subjected to the said conditions, the tracking of the MPP power and voltage is presented in Figures 5 and 6, respectively. The good MPP tracking potential of the proposed hybrid system can be appreciated in the power graph. A capture of the transient region reveals that the proposed hybrid system tracks the MPP at 0.77 ms. It is worth mentioning that the INC-IBSC controller exhibits overshoot as in Figure 6, the voltage tracking curve. This overshoot was measured at 11.51% from the reference MPP voltage. This explains the drop in power from 60 W to approximately 53.22 W during the transient as seen from the detailed view of the transient in Figure 5. It is worth appreciating that the controller rapidly restores the PV operating power and the voltage to the MPP in a time measured at t1=0.47ms, hence depicting the very good robustness of the controller. The overshoot occurrence is generally attributable to the integral action in the integral backstepping controller. The detailed transient view in Figure 5 tells the effect of the overshoot in a controller and its consequence on the power of the system. Though the integral action introduces some overshoot in the transient that is asymptotically decayed, the good steady-state response of the controller is preserved with the steady-state error of 0.12 V from the reference voltage. Hence, the efficiency of the controller was measured at 99.9434% according to the method described in [52].Figure 5
Tracking of the maximum power point (MPP) power with the hybrid INC-IBSC at STC.Figure 6
Tracking of the maximum power point (MPP) voltage with the hybrid INC-IBSC at STC.


### 6.2. Comparison of the Proposed Hybrid System with the INC at STC

In the vein of demonstrating the good performance of the proposed controller, it is compared with the conventional incremental conductance (INC) algorithm under the same operating conditions corresponding to STC. Their response to the said condition is presented in Figures7 and 8 for the tracking of the MPP power and voltage, respectively. It can be seen from these figures that the INC lags behind the proposed hybrid system in terms of tracking the MPP by a time td. Consequently, the INC tracked the MPP at 6.31 ms compared to the 0.77 ms of the hybrid system. Arithmetically, the proposed system is approximately 12 times faster than the INC MPPT. The relatively low steady-state performance of the INC can be confirmed in Figures 8. The INC continually oscillates at a steady state and will never gain stability. To this end, the efficiency of the INC was measured at 98.943 according to the method of [52]. Therefore, we conclude that the proposed hybrid system outperforms the conventional INC MPPT at STC.Figure 7
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC at STC.Figure 8
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at STC.


### 6.3. Evaluation and Comparison of the Proposed Hybrid System with the INC at STC under Load Variation

The previous operating conditions are not sufficient to comment on the robustness of the proposed hybrid system. Therefore, to appreciate the robustness of the hybrid system, it is evaluated under abrupt variation in the load, and a comparison with the INC is also provided. Load variations are perceived in the system as disturbances because a change in load turns to alter the operating point of the PV. In this light, a robust MPPT controller must reject the effect of these changes by enforcing the operation at the MPP. To this end, the load variation scenarios considered in this investigation are presented in Figure9.Figure 9
Load variation scenario.The load variation scenario as depicted in Figure9 involves three instances of abrupt load variations, labeled on the figure as A, B, and C at the time instant 0.1 s, 0.2 s, and 0.3 s, respectively. The systems are subjected to a constant load of value 15Ω, up to the time instant at A (0.1 s) when the load instantaneously steps up to30Ω. From this instant, it is held constant up to 0.2 s (B), when the load abruptly steps down to25Ω, and kept constant up to 0.3 s (C). At 0.3 s, the load is restored to its initial value of 15Ω and held constant till 0.4 s. Under this sequence of load variations, the power and voltage tracked by the INC and the proposed hybrid INC-IBSC are presented in Figures 10 and 11, respectively. At the different instances of load variation, it can be seen that the INC-IBSC ensures the stability of the PV operating point at the MPP. From a detailed evaluation point of view, it is measured that when the load changes from 15Ω to 30Ω at 0.1 s, the power under the control of the INC drops by 14.03 W, and the voltage increases by 2.38 V, the operating point is only restored after 4.20 ms of time elapsed, after which the power and voltage continue to oscillate. At this same instant, the voltage and power of the PV under the control of the proposed hybrid INC-IBSC stay at the MPP, as presented in the generalization in Table 2. At B, when the load value drops from 30Ω to 25Ω, it is measured that the PV power drops from the MPP by approximately 2.95 W after which it is restored in 1.8 ms and continues to oscillate. At this same instant, the proposed hybrid system ensures that the PV system operates at MP. The complete comparison of the two systems at the instants A, B, and C is presented in Table 3. Generally, one can note that the proposed system has very good robustness against load disturbances. More to that, it delivers an efficiency of 98.9449% under the aforementioned load variation scenario.Figure 10
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC at STC under load variation scenarios.Figure 11
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at STC under load variation scenarios.Table 2
Characteristics of the MSX-60W [23].
PV parametersValueMaximum powerPmpp60 WVoltage at maximum powerVmpp16.7 VCurrent at maximum powerImpp3.59 AOpen circuit voltageVoc21.1Short-circuit currentIsc3.8 ATemperature coefficient ofVoc-80 mV/°CTemperature coefficient ofIsc-0.065%/°CTable 3
Evaluation and comparison of the proposed INC-IBSC with the INC under changing load at STC.
MPPT system↓Ppv⋂↑tsA W⋂ms↓Ppv⋂↑tsB W⋂ms↓Ppv⋂↑tsC W⋂msΔVpv⋂↑tsAV⋂msΔVpv⋂↑tsB V⋂msΔVpv⋂↑tsC V⋂msEfficiency (%)INC14.03⋂4.202.95⋂~1.86.81⋂3.72.38⋂4.201.58⋂0.52.8⋂3.698.5124INC-IBSC0⋂00⋂00⋂00⋂00⋂00⋂099.9449In Table3, the notations↓Ppv⋂tsand↓Ppvrepresent the drop in power from the MPP at the instant of the load variation (A, B, or C) and ts is the time taken to restore the operating point to the MPP. In the notation, W⋂ms, W (watts) is the unit of power, and ms (millisecond) is the unit of time. In the notation ΔVpv⋂ts, ΔVpv is the deviation of the voltage operating point from the MPP voltage.


### 6.4. Evaluation and Comparison of the Proposed Hybrid System with the INC under Fast-Changing Environmental Conditions and Load Variation Scenarios

The proposed hybrid INC-IBSC system and the INC are now subjected to the worst operating conditions. The objective of this test is to evaluate and appreciate the feasibility of the proposed MPPT system and its better robustness against difficult operating scenarios. In this vein, both systems are subjected to the same load variation profile as was described in Figure9 and to fast-changing irradiance and temperature as depicted in Figures 12 and 13, respectively. It can be seen in Figures 14 and 15 for the voltage and power tracking that the proposed system has a far better dynamic performance than the INC, as it maintains the good stability of the operating point at the MPP.Figure 12
Fast-changing operating irradiance scenario.Figure 13
Fast-changing operating temperature scenario.Figure 14
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC under load variation scenarios and fast-changing environmental conditions.Figure 15
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at under load variation scenarios and fast-changing environmental conditions.


### 6.5. Comparison of the Proposed Hybrid INC-IBSC with Other Controllers and Techniques from the Literature

The proposed MPPT system operates the PV at 99.9434% efficiency under standard test conditions with steady-state ripples measured at 0.294% peak to peak in volts. When compared with 8 recently published MPPT algorithm (controller) methods from the literature, it is appreciated that the proposed MPPT system presents a superior performance. The table of performance comparisons at standard test conditions is presented in Table4.Table 4
Comparison of the proposed MPPT with other MPPT methods in the literature under STC.
RefYearDC-DC converter typeMPPT algorithm (controller)Tracking time (ms)Ripples (peak to peak)Efficiency (%)[53]2021BoostNovel MPPT tactic9.6x99.86[54]2022BoostInnovative MPPT10x99.75[55]2021BoostThermal imaging-based P&O220xx[56]2020BoostExtremum seeking2500xX[57]2022BoostAdvanced MPPT11x99.8[43]2018BuckBackstepping1.50.9x[45]2018Non-Integral backstepping2.20.63xInvertingControllerBuck boost[58]2021BoostBackstepping-sliding mode controller10xxProposed—BoostProposed0.40.17%99.9434%x in the table signifies that the respective parameter was not defined by the author.


### 6.6. Real-Time Validation with Experimental Data

In practice, a PV system is subjected to continually changing environmental conditions. Hence, to validate the proposed MPPT controller, it was subjected to real-time experimental data. The experimental set-up as presented in Figure16 was made of a pyranometer for irradiance measurement, the DHT22 temperature sensor, a sample solar PV panel, an Arduino mega microcontroller, and a computer equipped with data logging applications. The experimental set-up was mounted in the city of Dschang, West Province of Cameroon, on January 5, 2022. Data corresponding to the PV irradiance and temperature were collected and resampled at 15 minutes, corresponding to 8:30 am to 4:30 pm, and represented by 33 data points as shown in Table 5.Figure 16
Experimental set-up for recording real-time solar irradiance and temperature on the PV.Table 5
Experimental data of solar irradiance and temperature.
No.IrrW/m2T (°C)No.IrrW/m2T (°C)No.Irr (W/m2)T (°C)No.Irr (W/m2)T (°C)127031.81052046.119100855.92878533.5234033.51143738.420102547.52950131.633453512116242.92165039.53040533430534.81388640.52244737.83142331.4540037.81450543.823401393234029625936151091452450034.93333928.6745145.216112044.62562634.5836240.617104453.12674034.7941939.318109555.42752533The 33 data points as shown in Table5 were scaled to 40000 data points for a running time of 0.4 seconds in MATLAB/Simulink environment. The maximum experimental irradiance of 1162 W/m2 occurred at noon (Table 5) corresponding to 0.138 seconds on the scaled signal as inferred from Figure 17. On the other hand, the minimum irradiance of 270 W/m2 occurs at the very first sample instant, both for Table 5 and Figure 17. Moreover, it can be seen that the temperature was, respectively, maximum (55.9°C) and minimum (28.6°C) at 1:00 pm and 4:40 pm, inferred from Table 5, and at the 0.225 s and 0.4 s instant, inferred from Figure 17.Figure 17
Experimental irradiance and temperature for 8:30 am to 4:30 pm, scaled up to 0.4 seconds.The proposed hybrid INC-IBSC, with its realization in Figure4, was subjected to the experimental conditions of Figure 17 in a MATLAB/Simulink environment. Therefore, the validation involves assessing the proposed MPPT system in response to the experimental environmental conditions of Figure 17. To comment on the performance of the proposed system in the aforementioned condition, a corresponding comparison was attained with the INC.In response to experimental environmental conditions, the power response of both the proposed INC-IBSC and INC is presented in Figure18. Viewing from the tracking line in Figure 18, image (i), it can be seen that the proposed INC-IBSC tracks the maximum power at 1.7 ms while the INC tracks at 6.9 ms. This conveys that the INC-IBSC is approximately 4 times faster than the INC under experimental conditions. Furthermore, we note that these tracking times are in good agreement with the 6.3 ms and 0.77 ms obtained at STC in nonexperimental environmental conditions, which is in Figure 7 and discussed in Subsection 6.2. However, the respective tracking times increased by 0.6 ms for the INC and by 0.93 ms for the proposed INC-IBSC. This increase is attributable to the fact that conditions are more dynamic under the experimental conditions than those at STC. It is therefore confirmed that the proposed INC-IBSC is faster than the INC.Figure 18
Power response of the proposed INC-IBSC and INC to the experimental conditions.Moreover, it is worth exalting the better dynamic response of the proposed INC-IBSC, relative to INC. Inferences from Figure18, internal images (i) and (ii), show that the INC exhibits consistent power oscillation, a situation which is palliated by the INC-IBSC. Numerically integrating the area under both power curves using the trapezoidal method, it is found that the INC yields 12.51 W/s while the proposed system yields 12.70 W/s of energy. This is a confirmation that from an energy extraction point of view, the proposed INC-IBSC is better than the INC. The lower energy yield of the INC amounts to its slow tracking speed and relatively poor dynamic response.From a voltage response point of view and with inference from Figure19, it is further confirmed that the proposed INC-IBSC has a better dynamic response than the INC. This is obvious given the nonlinear and robust capacities of the INC-IBSC. The faster tracking performance of the proposed INC-IBSC relative to the INC is reconfirmed from internal image (i). The internal image (ii) reveals that the proposed system indeed exhibits voltage oscillations, however lower than the INC, a scenario that is confirmable from internal image (iii) on the same figure.Figure 19
Voltage response of the proposed INC-IBSC and INC to the experimental conditions.


## 6.1. Evaluation of the Proposed Hybrid System at STC

The proposed hybrid system is first investigated at STC corresponding to the temperature of 25°C and irradiance of1000W/m2. According to the characteristics of the considered PV module, a maximum power of 60 W and a maximum voltage of 16.70 V are expected under these conditions [21]. To this end, when the systems are subjected to the said conditions, the tracking of the MPP power and voltage is presented in Figures 5 and 6, respectively. The good MPP tracking potential of the proposed hybrid system can be appreciated in the power graph. A capture of the transient region reveals that the proposed hybrid system tracks the MPP at 0.77 ms. It is worth mentioning that the INC-IBSC controller exhibits overshoot as in Figure 6, the voltage tracking curve. This overshoot was measured at 11.51% from the reference MPP voltage. This explains the drop in power from 60 W to approximately 53.22 W during the transient as seen from the detailed view of the transient in Figure 5. It is worth appreciating that the controller rapidly restores the PV operating power and the voltage to the MPP in a time measured at t1=0.47ms, hence depicting the very good robustness of the controller. The overshoot occurrence is generally attributable to the integral action in the integral backstepping controller. The detailed transient view in Figure 5 tells the effect of the overshoot in a controller and its consequence on the power of the system. Though the integral action introduces some overshoot in the transient that is asymptotically decayed, the good steady-state response of the controller is preserved with the steady-state error of 0.12 V from the reference voltage. Hence, the efficiency of the controller was measured at 99.9434% according to the method described in [52].Figure 5
Tracking of the maximum power point (MPP) power with the hybrid INC-IBSC at STC.Figure 6
Tracking of the maximum power point (MPP) voltage with the hybrid INC-IBSC at STC.


## 6.2. Comparison of the Proposed Hybrid System with the INC at STC

In the vein of demonstrating the good performance of the proposed controller, it is compared with the conventional incremental conductance (INC) algorithm under the same operating conditions corresponding to STC. Their response to the said condition is presented in Figures7 and 8 for the tracking of the MPP power and voltage, respectively. It can be seen from these figures that the INC lags behind the proposed hybrid system in terms of tracking the MPP by a time td. Consequently, the INC tracked the MPP at 6.31 ms compared to the 0.77 ms of the hybrid system. Arithmetically, the proposed system is approximately 12 times faster than the INC MPPT. The relatively low steady-state performance of the INC can be confirmed in Figures 8. The INC continually oscillates at a steady state and will never gain stability. To this end, the efficiency of the INC was measured at 98.943 according to the method of [52]. Therefore, we conclude that the proposed hybrid system outperforms the conventional INC MPPT at STC.Figure 7
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC at STC.Figure 8
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at STC.


## 6.3. Evaluation and Comparison of the Proposed Hybrid System with the INC at STC under Load Variation

The previous operating conditions are not sufficient to comment on the robustness of the proposed hybrid system. Therefore, to appreciate the robustness of the hybrid system, it is evaluated under abrupt variation in the load, and a comparison with the INC is also provided. Load variations are perceived in the system as disturbances because a change in load turns to alter the operating point of the PV. In this light, a robust MPPT controller must reject the effect of these changes by enforcing the operation at the MPP. To this end, the load variation scenarios considered in this investigation are presented in Figure9.Figure 9
Load variation scenario.The load variation scenario as depicted in Figure9 involves three instances of abrupt load variations, labeled on the figure as A, B, and C at the time instant 0.1 s, 0.2 s, and 0.3 s, respectively. The systems are subjected to a constant load of value 15Ω, up to the time instant at A (0.1 s) when the load instantaneously steps up to30Ω. From this instant, it is held constant up to 0.2 s (B), when the load abruptly steps down to25Ω, and kept constant up to 0.3 s (C). At 0.3 s, the load is restored to its initial value of 15Ω and held constant till 0.4 s. Under this sequence of load variations, the power and voltage tracked by the INC and the proposed hybrid INC-IBSC are presented in Figures 10 and 11, respectively. At the different instances of load variation, it can be seen that the INC-IBSC ensures the stability of the PV operating point at the MPP. From a detailed evaluation point of view, it is measured that when the load changes from 15Ω to 30Ω at 0.1 s, the power under the control of the INC drops by 14.03 W, and the voltage increases by 2.38 V, the operating point is only restored after 4.20 ms of time elapsed, after which the power and voltage continue to oscillate. At this same instant, the voltage and power of the PV under the control of the proposed hybrid INC-IBSC stay at the MPP, as presented in the generalization in Table 2. At B, when the load value drops from 30Ω to 25Ω, it is measured that the PV power drops from the MPP by approximately 2.95 W after which it is restored in 1.8 ms and continues to oscillate. At this same instant, the proposed hybrid system ensures that the PV system operates at MP. The complete comparison of the two systems at the instants A, B, and C is presented in Table 3. Generally, one can note that the proposed system has very good robustness against load disturbances. More to that, it delivers an efficiency of 98.9449% under the aforementioned load variation scenario.Figure 10
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC at STC under load variation scenarios.Figure 11
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at STC under load variation scenarios.Table 2
Characteristics of the MSX-60W [23].
PV parametersValueMaximum powerPmpp60 WVoltage at maximum powerVmpp16.7 VCurrent at maximum powerImpp3.59 AOpen circuit voltageVoc21.1Short-circuit currentIsc3.8 ATemperature coefficient ofVoc-80 mV/°CTemperature coefficient ofIsc-0.065%/°CTable 3
Evaluation and comparison of the proposed INC-IBSC with the INC under changing load at STC.
MPPT system↓Ppv⋂↑tsA W⋂ms↓Ppv⋂↑tsB W⋂ms↓Ppv⋂↑tsC W⋂msΔVpv⋂↑tsAV⋂msΔVpv⋂↑tsB V⋂msΔVpv⋂↑tsC V⋂msEfficiency (%)INC14.03⋂4.202.95⋂~1.86.81⋂3.72.38⋂4.201.58⋂0.52.8⋂3.698.5124INC-IBSC0⋂00⋂00⋂00⋂00⋂00⋂099.9449In Table3, the notations↓Ppv⋂tsand↓Ppvrepresent the drop in power from the MPP at the instant of the load variation (A, B, or C) and ts is the time taken to restore the operating point to the MPP. In the notation, W⋂ms, W (watts) is the unit of power, and ms (millisecond) is the unit of time. In the notation ΔVpv⋂ts, ΔVpv is the deviation of the voltage operating point from the MPP voltage.


## 6.4. Evaluation and Comparison of the Proposed Hybrid System with the INC under Fast-Changing Environmental Conditions and Load Variation Scenarios

The proposed hybrid INC-IBSC system and the INC are now subjected to the worst operating conditions. The objective of this test is to evaluate and appreciate the feasibility of the proposed MPPT system and its better robustness against difficult operating scenarios. In this vein, both systems are subjected to the same load variation profile as was described in Figure9 and to fast-changing irradiance and temperature as depicted in Figures 12 and 13, respectively. It can be seen in Figures 14 and 15 for the voltage and power tracking that the proposed system has a far better dynamic performance than the INC, as it maintains the good stability of the operating point at the MPP.Figure 12
Fast-changing operating irradiance scenario.Figure 13
Fast-changing operating temperature scenario.Figure 14
Tracking of the maximum power point (MPP) power with the INC and the hybrid INC-IBSC under load variation scenarios and fast-changing environmental conditions.Figure 15
Tracking of the maximum power point (MPP) voltage with the INC and the hybrid INC-IBSC at under load variation scenarios and fast-changing environmental conditions.


## 6.5. Comparison of the Proposed Hybrid INC-IBSC with Other Controllers and Techniques from the Literature

The proposed MPPT system operates the PV at 99.9434% efficiency under standard test conditions with steady-state ripples measured at 0.294% peak to peak in volts. When compared with 8 recently published MPPT algorithm (controller) methods from the literature, it is appreciated that the proposed MPPT system presents a superior performance. The table of performance comparisons at standard test conditions is presented in Table4.Table 4
Comparison of the proposed MPPT with other MPPT methods in the literature under STC.
RefYearDC-DC converter typeMPPT algorithm (controller)Tracking time (ms)Ripples (peak to peak)Efficiency (%)[53]2021BoostNovel MPPT tactic9.6x99.86[54]2022BoostInnovative MPPT10x99.75[55]2021BoostThermal imaging-based P&O220xx[56]2020BoostExtremum seeking2500xX[57]2022BoostAdvanced MPPT11x99.8[43]2018BuckBackstepping1.50.9x[45]2018Non-Integral backstepping2.20.63xInvertingControllerBuck boost[58]2021BoostBackstepping-sliding mode controller10xxProposed—BoostProposed0.40.17%99.9434%x in the table signifies that the respective parameter was not defined by the author.


## 6.6. Real-Time Validation with Experimental Data

In practice, a PV system is subjected to continually changing environmental conditions. Hence, to validate the proposed MPPT controller, it was subjected to real-time experimental data. The experimental set-up as presented in Figure16 was made of a pyranometer for irradiance measurement, the DHT22 temperature sensor, a sample solar PV panel, an Arduino mega microcontroller, and a computer equipped with data logging applications. The experimental set-up was mounted in the city of Dschang, West Province of Cameroon, on January 5, 2022. Data corresponding to the PV irradiance and temperature were collected and resampled at 15 minutes, corresponding to 8:30 am to 4:30 pm, and represented by 33 data points as shown in Table 5.Figure 16
Experimental set-up for recording real-time solar irradiance and temperature on the PV.Table 5
Experimental data of solar irradiance and temperature.
No.IrrW/m2T (°C)No.IrrW/m2T (°C)No.Irr (W/m2)T (°C)No.Irr (W/m2)T (°C)127031.81052046.119100855.92878533.5234033.51143738.420102547.52950131.633453512116242.92165039.53040533430534.81388640.52244737.83142331.4540037.81450543.823401393234029625936151091452450034.93333928.6745145.216112044.62562634.5836240.617104453.12674034.7941939.318109555.42752533The 33 data points as shown in Table5 were scaled to 40000 data points for a running time of 0.4 seconds in MATLAB/Simulink environment. The maximum experimental irradiance of 1162 W/m2 occurred at noon (Table 5) corresponding to 0.138 seconds on the scaled signal as inferred from Figure 17. On the other hand, the minimum irradiance of 270 W/m2 occurs at the very first sample instant, both for Table 5 and Figure 17. Moreover, it can be seen that the temperature was, respectively, maximum (55.9°C) and minimum (28.6°C) at 1:00 pm and 4:40 pm, inferred from Table 5, and at the 0.225 s and 0.4 s instant, inferred from Figure 17.Figure 17
Experimental irradiance and temperature for 8:30 am to 4:30 pm, scaled up to 0.4 seconds.The proposed hybrid INC-IBSC, with its realization in Figure4, was subjected to the experimental conditions of Figure 17 in a MATLAB/Simulink environment. Therefore, the validation involves assessing the proposed MPPT system in response to the experimental environmental conditions of Figure 17. To comment on the performance of the proposed system in the aforementioned condition, a corresponding comparison was attained with the INC.In response to experimental environmental conditions, the power response of both the proposed INC-IBSC and INC is presented in Figure18. Viewing from the tracking line in Figure 18, image (i), it can be seen that the proposed INC-IBSC tracks the maximum power at 1.7 ms while the INC tracks at 6.9 ms. This conveys that the INC-IBSC is approximately 4 times faster than the INC under experimental conditions. Furthermore, we note that these tracking times are in good agreement with the 6.3 ms and 0.77 ms obtained at STC in nonexperimental environmental conditions, which is in Figure 7 and discussed in Subsection 6.2. However, the respective tracking times increased by 0.6 ms for the INC and by 0.93 ms for the proposed INC-IBSC. This increase is attributable to the fact that conditions are more dynamic under the experimental conditions than those at STC. It is therefore confirmed that the proposed INC-IBSC is faster than the INC.Figure 18
Power response of the proposed INC-IBSC and INC to the experimental conditions.Moreover, it is worth exalting the better dynamic response of the proposed INC-IBSC, relative to INC. Inferences from Figure18, internal images (i) and (ii), show that the INC exhibits consistent power oscillation, a situation which is palliated by the INC-IBSC. Numerically integrating the area under both power curves using the trapezoidal method, it is found that the INC yields 12.51 W/s while the proposed system yields 12.70 W/s of energy. This is a confirmation that from an energy extraction point of view, the proposed INC-IBSC is better than the INC. The lower energy yield of the INC amounts to its slow tracking speed and relatively poor dynamic response.From a voltage response point of view and with inference from Figure19, it is further confirmed that the proposed INC-IBSC has a better dynamic response than the INC. This is obvious given the nonlinear and robust capacities of the INC-IBSC. The faster tracking performance of the proposed INC-IBSC relative to the INC is reconfirmed from internal image (i). The internal image (ii) reveals that the proposed system indeed exhibits voltage oscillations, however lower than the INC, a scenario that is confirmable from internal image (iii) on the same figure.Figure 19
Voltage response of the proposed INC-IBSC and INC to the experimental conditions.


## 7. Conclusion

A hybrid incremental conductance-integral backstepping nonlinear controller was proposed for efficient control and effective maximum energy harvesting from a PV system under fast-changing operating conditions. The proposed hybrid system exploits the benefits of the incremental conductance (INC) algorithm in searching for the maximum power point (MPP) to generate a reference voltage that feeds a Lyapunov-based integral backstepping controller. The hybrid MPPT controller demonstrated exceptional performances and palliated the performance limitations of the isolated INC MPPT algorithm. By evaluation and comparisons, it was found better than the INC and 8 other recently published MPPT methods from the literature. An extended validation under experimental environmental conditions showed that the hybrid system is approximately four times faster than the INC in tracking the maximum power with better energy yield than the latter. The above work has been applied to a PV system under uniform irradiance conditions. It is suggested that future works be directed towards the extension of the proposed hybrid MPPT controller to PV systems under the effects of partial shading.A new hybrid incremental conductance-integral backstepping nonlinear MPPT controller was proposed in this paper.



---

*Source: 9931481-2023-02-15.xml*