Robust optimization and power management of a triple junction photovoltaic electric vehicle with battery storage

Citation: Hamed, S.B.; Hamed, M.B.; Sbita, L.; Bajaj, M.; Blazek, V.; Prokop, L.; Misak, S.; Ghoneim, S.S.M. Robust Optimization and Power Management of a Triple Junction Photovoltaic Electric Vehicle with Battery Storage. Sensors 2022,22, 6123. https://doi.org/10.3390/ s22166123 Academic Editors: Jinghua Guo and Jingyao Wang Received: 4 July 2022 Accepted: 11 August 2022 Published: 16 August 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Article Robust Optimization and Power Management of a Triple Junction Photovoltaic Electric Vehicle with Battery Storage Salah Beni Hamed 1, Mouna Ben Hamed 2, Lassaad Sbita 2, Mohit Bajaj 3,* , Vojtech Blazek 4,* , Lukas Prokop 4, Stanislav Misak 4and Sherif S. M. Ghoneim 5 1Physic Department, High School of Engineers of Tunis, Tunis 1008, Tunisia 2Electrical Department, National Engineering School of Gabes, Gabes 6029, Tunisia 3Department of Electrical Engineering, Graphic Era (Deemed to be University), Dehradun 248002, India 4ENET Centre, VSB—Technical University of Ostrava, 708 00 Ostrava, Czech Republic 5Electrical Engineering Department, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia *Correspondence: [email protected] (M.B.); [email protected] (V.B.) Abstract: This paper highlights a robust optimization and power management algorithm that supervises the energy transfer flow to meet the photovoltaic (PV) electric vehicle demand, even when the traction system is in motion. The power stage of the studied system consists of a triple-junction PV generator as the main energy source, a lithium-ion battery as an auxiliary energy source, and an electric vehicle. The input–output signal adaptation is made by using a stage of energy conversion. A bidirectional DC-DC buck-boost connects the battery to the DC-link. Two unidirectional boost converters interface between the PV generator and the DC link. One is controlled with a maximum power point tracking (MPPT) algorithm to reach the maximum power points. The other is used to control the voltage across the DC-link. The converters are connected to the electric vehicle via a three-phase inverter via the same DC-link. By considering the nonlinear behavior of these elements, dynamic models are developed. A robust nonlinear MPPT algorithm has been developed owing to the nonlinear dynamics of the PV generator, metrological condition variations, and load changes. The high performance of the MPPT algorithm is effectively highlighted over a comparative study with two classical P & O and the fuzzy logic MPPT algorithms. A nonlinear control based on the Lyapunov function has been developed to simultaneously regulate the DC-link voltage and control battery charging and discharging operations. An energy management rule-based strategy is presented to effectively supervise the power flow. The conceived system, energy management, and control algorithms are implemented and verified in the Matlab/Simulink environment. Obtained results are presented and discussed under different operating conditions. Keywords: PV generator; triple junction; first order sliding mode; MPPT; nonlinear control; electric vehicle; DC-DC power converters; energy management 1. Introduction With the fast growth of cars, especially car ownership, the number of vehicles in the world increases day by day [ 1 , 2 ]. This has led to an important rise in oil consumption in the transport sector [ 3 – 5 ]. As more of the vehicle’s energy is obtained by an internal combustion engine, the carbon dioxide (CO 2 ) emissions will increase [ 6 – 8 ]. Nowadays, the CO 2 rate has crossed 400 ppm and will increase. Faced with the energy crisis, climate change, and the need to save the earth and people’s lives, the development of a new vehicle structure is considered by looking for some sustainable technologies that reduce energy consumption or utilize renewable and clean energy sources [ 9 , 10 ]. Other energy sources are the challenge of most proposed solutions. Sensors 2022,22, 6123. https://doi.org/10.3390/s22166123 https://www.mdpi.com/journal/sensors Sensors 2022,22, 6123 2 of 31 To sufficiently reduce both consumption and transportation emissions, electric vehicles are considered an effective transport tool. Two kinds of electric vehicles have been discovered: pure electric vehicles and hybrid vehicles [11–13]. Electric vehicles cover more than a research field. The structure, benefits, and drawbacks of each are thoroughly studied in b. A large number of studies focused on the vehicle’s traction [14–16]. The main objectives in the motor choice for the traction part are to increase the electric vehicle’s performance while minimizing both the vehicle weight and energy consumption. Many research studies are carried out with this aim [ 17 – 19 ]. In Reference [ 17 ], many kinds of electric machines are used for electric vehicles, and the importance of choosing the traction machines is highlighted in this case. Besides, the number of motors and their placement in the vehicle are also discussed [20]. Power management in electrical vehicles was also discussed and research reviews and studies were conducted [ 21 – 23 ]. Among these approaches, we find dynamic programming strategy [ 21 ], all or nothing strategy [ 24 ], rule-based strategy, including deterministic and fuzzy logic rules [25,26], filtration strategy [27], and predictive model strategy [28]. Another interesting research field in electric vehicles concerns vehicle autonomy [ 29 , 30 ]. Suggested solutions for improving electric vehicle autonomy can be divided into two kinds. Some of them concentrate on working on battery technology. The other option is to use rechargeable batteries [ 31 ]. Using PV energy sources is one of the suggested recharge battery solutions [ 32 ]. The PV panels are located on the body of the vehicle. The recharge time is improved by using the PV generator at its maximum power. For this goal, different MPPT algorithms are suggested in the literature: perturb and observe (P & O) [33–35], the incremental inductance (IC) [ 36 – 38 ], fuzzy logic (FL) [ 39 – 41 ], neuronal networks (NN) [ 42 – 44 ], particle swarm optimization (PSO) [ 45 – 47 ], and sliding mode [48,49] etc. High oscillation remains the major weakness of these MPPT approaches. According to a literature review, the used PV generator is designed with mono-junction solar cells. The major drawback of this technology is its low efficiency. This fact led to increasing the PV panel covered area. The aim of this paper is to improve electric vehicle performance. To minimize the overall vehicle weight, a highly efficient PV generator based on multi-junction solar cell technology is conceived. A lithium-ion battery bank storage system is used. A nonlinear robust sliding mode-based MPPT algorithm and a Lyapunov function-based nonlinear control approach for DC-DC converters with an energy management rules-based approach are being investigated as solutions to improve the battery recharge time. This paper is structured as follows. In Section 2, the forward simulation model of the electric vehicle powertrain system is established. Section 3presents the energy management and control approaches. In Section 4, the simulation results are presented and discussed. Conclusions and some suggested prospects are provided in the Section 5. 2. Modeling of the PV Electric Vehicle Powertrain System The structure of the used PV electric vehicle powertrain system is shown in Figure 1. Sensors 2022,22, 6123 3 of 31 Sensors 2022, 11, x FOR PEER REVIEW 3 of 33 2. Modeling of the PV Electric Vehicle Powertrain System The structure of the used PV electric vehicle powertrain system is shown in Figure 1. Figure 1. A description of the studied system. 2.1. Modeling of Triple-Junction Solar Cell InGap/InGaAs/Ge The triple-junction InGap/InGaAs/Ge solar cell includes three sub-cells with different wavelengths in series. Electrical representation, by adapting the decreased energy band-gap from the top to the bottom structure, is given in Figure 2. Based on Figure 1, the solar cell current can be written as follows in Equation (1). ipv p Di Rshi I I I I= − − (1) The index i equals 1 for the top sub-cell. For the medium sub-cell, I = 2, and for the bottom sub-cell, I = 3. The light generated current is given by ( ) pi sccSTCi STC STC G I I T T G   = + −  (2) where TSTC is the temperature solar cell at standard test conditions in °C, T is the temperature solar cell in °C, G and GSTC are the solar radiation and the solar radiation at standard test conditions in w/m2, respectively, sccSTCi I is the short circuit current at standard test conditions, and  is the temperature coefficient of the actual short circuit current in A/°C. The diode current intensity is expressed as in Equation (3). 0exp( ) 1 Di Di i i qU II n BT  =−   (3) Its voltage equation is given in Equation (4). Di pvi si pv U U R I=+ (4) Figure 1. A description of the studied system. 2.1. Modeling of Triple-Junction Solar Cell InGap/InGaAs/Ge The triple-junction InGap/InGaAs/Ge solar cell includes three sub-cells with different wavelengths in series. Electrical representation, by adapting the decreased energy band-gap from the top to the bottom structure, is given in Figure 2. Sensors 2022, 11, x FOR PEER REVIEW 4 of 33 Figure 2. Electrical equivalent circuit of a multi-junctio InGap/InGaA/Ge solar cell. The diode saturation current 0i I is expressed as Equation (5) 03 ( )exp( ) 2i BGi ii i E I K T n BT  +− = (5) where q is the electric charge of an electron, i n is the ideality factor of a diode, BGi E is the band-gap energy, B is the Boltzmann’s constant, and i K and i  are constant. The energy band-gap is given in Equation (6). ² ( ) (0) ( ) i BGi BGi i T E T E T   =+ + (6) With i  is a material energy per Kelvin fitting parameters and i  is a material temperature fitting parameters. By using Equations (3) and (4), the triple-junction solar cell () pv pv IU characteristic is obtained. 1 1 2 2 12 12 33 3 3 ( 1) ( 1) ( 1) p pv Rsh p pv Rsh pv sat sat p pv Rsh s pv sat I I I I I I n BT n BT U Ln Ln q I q I I I I n BT Ln R I qI − − − − = + + + −− + + − (7) Figure 2. Electrical equivalent circuit of a multi-junctio InGap/InGaA/Ge solar cell. Based on Figure 1, the solar cell current can be written as follows in Equation (1). Ipv =Ipi −IDi −IRshi (1) Sensors 2022,22, 6123 4 of 31 The index iequals 1 for the top sub-cell. For the medium sub-cell, I= 2, and for the bottom sub-cell, I= 3. The light generated current is given by Ipi =G GSTC [IsccSTCi +γ(T−TSTC)] (2) where T STC is the temperature solar cell at standard test conditions in ◦ C, Tis the temperature solar cell in ◦ C, Gand G STC are the solar radiation and the solar radiation at standard test conditions in w/m 2 , respectively, IsccSTCi is the short circuit current at standard test conditions, and γ is the temperature coefficient of the actual short circuit current in A/ ◦ C. The diode current intensity is expressed as in Equation (3). IDi =I0iexp(qUDi niBT )−1(3) Its voltage equation is given in Equation (4). UDi =Upvi +Rsi Ipv (4) The diode saturation current I0iis expressed as Equation (5) I0i=KiT(3+δi 2)exp(−EBGi niBT )(5) where q is the electric charge of an electron, ni is the ideality factor of a diode, EBGi is the band-gap energy, Bis the Boltzmann’s constant, and Kiand δiare constant. The energy band-gap is given in Equation (6). EBGi(T) = EBGi(0) + ( αiT2 T+βi )(6) With αi is a material energy per Kelvin fitting parameters and βi is a material temperature fitting parameters. By using Equations (3) and (4), the triple-junction solar cell Ipv(Upv) characteristic is obtained. Upv =n1BT qLn(Ip1−Ipv−IRsh1 Isat1+1) + n2BT qLn(Ip2−Ipv−IRsh2 Isat2+1) +n3BT qLn(Ip3−Ipv−IRsh3 Isat3+1)−RsIpv (7) Rsis the equivalent of serial resistance. It is denoted by Equation (8) Rs= 3 ∑ j=1 Rsi (8) Based on the load demand, a suitable triple junction PV generator is conceived. According to the Matlab/Simulink test of the established model, the electric characteristic curves are obtained, as shown in Figure 3. From these characteristics, we note that for each pair of radiation and temperature, there is one operating point in which the generated power is at its maximum value. Moreover, in general, the meteorological are intermittent. As a result, the power produced may differ from the power demanded. Therefore, an MPPT algorithm seems to be the most suitable solution to extract the maximum power on the one hand. On the other hand, to better manage the energy flow and protect the system, an energy management strategy has to be integrated. Sensors 2022,22, 6123 5 of 31 Sensors 2022, 11, x FOR PEER REVIEW 5 of 33 s R is the equivalent of serial resistance. It is denoted by Equation (8) 3 1 s si j RR = = (8) Based on the load demand, a suitable triple junction PV generator is conceived. According to the Matlab/Simulink test of the established model, the electric characteristic curves are obtained, as shown in Figure 3. From these characteristics, we note that for each pair of radiation and temperature, there is one operating point in which the generated power is at its maximum value. Moreover, in general, the meteorological are intermittent. As a result, the power produced may differ from the power demanded. Therefore, an MPPT algorithm seems to be the most suitable solution to extract the maximum power on the one hand. On the other hand, to better manage the energy flow and protect the system, an energy management strategy has to be integrated. (a) (b) Figure 3. PV generator () pv pv P f V= characteristic: (a) under radiation variation and (b) under temperature variation. 2.2. Modeling of Electric Vehicle Dynamics Due to the multiple performance of the three-phase permanent magnet synchronous motor (PMSM), it is used in the monitoring part of the electric vehicle. The dynamic electrical behavior of the three-phase PMSM can be represented as space vectors by the following nonlinear equations established in the (d, q) frame as [50]. sd sd s sd sd s sq sq sq sq s sq sq s sd sd s a di U R i L L i dt di U R i L L i dt     = + −    = + + +   (9) where sd U , sq U and sd i , sq i direct and quadratic stator voltage and current components, respectively. s R is the stator resistance. sd L and sq L are the direct and reverse self-stator inductance components, s  is the rotor angular speed and a  represents the permanent magnet flux linkage. Its mechanical behavior is as follows. 0100 200 300 400 0 5 10 15 x 104 PV generator current (A) PV generator power (w) 1300 w/m² Maximum power point 1000 w/m² 050 100 150 200 250 300 0 2 4 6 8 10 12 14 x 104 PV generator current (A) PV generator power (w) 40 °C Maximum power point 25 °C Figure 3. PV generator Ppv =f(Vpv) characteristic: ( a ) under radiation variation and ( b ) under temperature variation. 2.2. Modeling of Electric Vehicle Dynamics Due to the multiple performance of the three-phase permanent magnet synchronous motor (PMSM), it is used in the monitoring part of the electric vehicle. The dynamic electrical behavior of the three-phase PMSM can be represented as space vectors by the following nonlinear equations established in the (d, q) frame as [50]. (Usd =Rsisd +Lsd disd dt −ωsLsqisq Usq =Rsisq +Lsq disq dt +ωsLsdisd +ωsφa (9) where Usd , Usq and isd , isq direct and quadratic stator voltage and current components, respectively. Rs is the stator resistance. Lsd and Lsq are the direct and reverse self-stator inductance components, ωs is the rotor angular speed and φa represents the permanent magnet flux linkage. Its mechanical behavior is as follows. Jtdωs dt =np(Tem −Tr)−fv f ωs(10) With np is the pair poles’ number, fv f is the coefficient of the viscous friction, Jt is the total moment of inertia Tem is the electromagnetic torque, and Tris the resistant torque. By neglecting the influence of the vehicle’s lateral and vertical dynamics, Tr is expressed as follows. Tr=FrRt(11) where Fris the total resistive force and Rtis the radius of the vehicle’s tire. The force is given by Fr=Frr +Far +Fsr (12) In which Frr is the rolling resistance, Far is the air resistance, and Fsr is the slope resistance. The forces Frr,Far and Fsr are expressed as Equations (13)–(15), respectively. Frr =Mvg frr (13) Far =1 2ρaAf aCadVv2(14) Sensors 2022,22, 6123 6 of 31 Fsr =Mvgsin(αrs)(15) In which Mv is the total mass of the vehicle, g is the gravity acceleration, frr is the coefficient of the rolling resistance, ρa is air density, Afa is the frontal surface area of the vehicle, Cad is the aerodynamic drag coefficient, Vv is the speed of the vehicle, and αrs is the street inclination angle. 2.3. Modeling of the Lithium-Ion Battery Since the photovoltaic system’s electrical characteristics depend on intermittent weather conditions, its output energy may be insufficient to meet the load demands. Storing energy seems to be necessary. The battery is the most commonly used storage system in a standalone system [ 29 ]. Lithium-ion batteries are chosen as a suitable storage system for electric vehicles due to their power density, high specific energy, and long life expectancy. In the existing literature, various lithium-ion battery models have been developed [ 51 ]. The most commonly used is the one developed with Shepherd [ 52 ]. The extended modified Shepherd model is represented with a controlled voltage source and an internal resistance, as indicated by Equation (16). Usb =Esb −RinIsb (16) With Usb is the battery voltage, Esb is the controlled voltage source, Rin is the internal battery resistance, and Isb is the battery current. For charging mode, we have: Esb(Qa,isb f ) = Esb0−Ksb Qn Qn−Qa isb f −Ksb Qn Qn−Qa Qa+Abexp(−Bbt)(17) In discharge mode, we can write: Esb(Qa,isb f ) = Esb0−Ksb Qn Qn+0.1Qa isb f −Ksb Qn Qn−Qa Qa+Abexp(−Bbt)(18) where Esb is the no-load voltage, Esb0 is the battery constant voltage, Qn and Qa are nominal and available battery capacities, isb f is the low frequency component of the battery current, Kb is the polarization voltage, Ab is the battery exponential zone amplitude, and Bb is the battery exponential zone time constant inverse. The available battery capacity is defined as Qa=Zisbdt (19) Here, isb is the battery current. At any given time, the available charge of the battery is expressed over the battery state of charge (SOC). It is defined as SOC(t) = SOCin −1 Qn τ Z 0 isb(τ)dτ(20) The initial voltage of the battery depends on the state of charge [53]. Esb0(SOCin) = a0+a1Ln(SOCin) + a2Ln(1−SOCin) + a3 SOCin +a4SOCin (21) where a0. . . a4 are parameters to fit the model to a specific battery and SOCin is the initial battery state of charge. The battery power is computed as follows. Psb =Usb isb (22) Sensors 2022,22, 6123 7 of 31 2.4. Modeling of the DC-DC Converters 2.4.1. Modeling of the Unidirectional DC-DC Boost Converter Two cascading DC-DC boost converters are integrated between the main triplejunction PV generator and the electric vehicle. The first is used to track the maximum power points. The other is utilized to adapt the low DC voltage to the desired DC-link inverter voltage. The unidirectional DC-DC boost converter is a suitable configuration in this phase. According to Figure 4, the boost converter is composed of a high frequency coil (Lf), an IGBT transistor (T1), a diode Dar, and Cfas an output voltage filter. Sensors 2022, 11, x FOR PEER REVIEW 8 of 33 Figure 4. Unidirectional boost chopper. In this chopper, there is an on–off switch (T1). The working principle depends on the state of this switch. When the switch T1 is on, the source current follows the inductor and the switch. Only the capacity supplies the load. At this stage, the inductor stores energy, and the capacity discharges energy through the load. When the switch T1 is off, the diode Dar will be ready to conduct. At this phase, the inductor loses the stored energy for charging the capacitor. The switch is controlled by using a pulse width modulation signal a S . A bilinear average switching model is obtained by considering some idealities and taking into account the nature of the switch, 1 1 (1 ) (1 ) in in dc c ff dc in dc cff di U U dt L L dU i i dt C C   = − −    = − −   (23) where in U , dc U , and in i , dc i are, respectively, the input voltage, the DC-link voltage, the input, and the output currents of the boost converter, and 1c u is the averaged value of the pulse width modulation signal u . 2.4.2. Modeling of the Bidirectional DC-DC Buck-Boost Converter The battery is the main storage system in this application. It behaves as a bidirectional system. To manage the energy transfer, the battery is connected to the DC-link by means of two quadrant DC-DC converters. The most commonly used in this stage is the bidirectional DC-DC buck-boost converter. Referring to Figure 5, the buck-boost is composed of a high frequency coil (L2) and two IGBT switches, T2 and T3. Figure 5. Bidirectional buck-boost chopper. Figure 4. Unidirectional boost chopper. In this chopper, there is an on–off switch (T 1 ). The working principle depends on the state of this switch. When the switch T 1 is on, the source current follows the inductor and the switch. Only the capacity supplies the load. At this stage, the inductor stores energy, and the capacity discharges energy through the load. When the switch T 1 is off, the diode D ar will be ready to conduct. At this phase, the inductor loses the stored energy for charging the capacitor. The switch is controlled by using a pulse width modulation signal Sa. A bilinear average switching model is obtained by considering some idealities and taking into account the nature of the switch,    diin dt =Uin Lf−(1−µc1)Udc Lf dUdc dt = (1−µc1)iin Cf−idc Cf (23) where Uin , Udc , and iin , idc are, respectively, the input voltage, the DC-link voltage, the input, and the output currents of the boost converter, and uc1 is the averaged value of the pulse width modulation signal u. 2.4.2. Modeling of the Bidirectional DC-DC Buck-Boost Converter The battery is the main storage system in this application. It behaves as a bidirectional system. To manage the energy transfer, the battery is connected to the DC-link by means of two quadrant DC-DC converters. The most commonly used in this stage is the bidirectional DC-DC buck-boost converter. Referring to Figure 5, the buck-boost is composed of a high frequency coil (L 2 ) and two IGBT switches, T2and T3. When the switch T 2 and the diode D 3 states are on, the battery provides energy to the load. In this case, the bidirectional chopper works in the boost operating mode. Now, let us consider the case when the switch T 3 and the diode D 2 are in conduction. In this case, the battery current is negative and the battery charges. Sensors 2022,22, 6123 8 of 31 Sensors 2022, 11, x FOR PEER REVIEW 8 of 33 Figure 4. Unidirectional boost chopper. In this chopper, there is an on–off switch (T1). The working principle depends on the state of this switch. When the switch T1 is on, the source current follows the inductor and the switch. Only the capacity supplies the load. At this stage, the inductor stores energy, and the capacity discharges energy through the load. When the switch T1 is off, the diode Dar will be ready to conduct. At this phase, the inductor loses the stored energy for charging the capacitor. The switch is controlled by using a pulse width modulation signal a S . A bilinear average switching model is obtained by considering some idealities and taking into account the nature of the switch, 1 1 (1 ) (1 ) in in dc c ff dc in dc cff di U U dt L L dU i i dt C C   = − −    = − −   (23) where in U , dc U , and in i , dc i are, respectively, the input voltage, the DC-link voltage, the input, and the output currents of the boost converter, and 1c u is the averaged value of the pulse width modulation signal u . 2.4.2. Modeling of the Bidirectional DC-DC Buck-Boost Converter The battery is the main storage system in this application. It behaves as a bidirectional system. To manage the energy transfer, the battery is connected to the DC-link by means of two quadrant DC-DC converters. The most commonly used in this stage is the bidirectional DC-DC buck-boost converter. Referring to Figure 5, the buck-boost is composed of a high frequency coil (L2) and two IGBT switches, T2 and T3. Figure 5. Bidirectional buck-boost chopper. Figure 5. Bidirectional buck-boost chopper. To distinguish the operating mode, a binary variable m is defined. Thus, we can write: m=1isbre f >0 boost mode 0isbre f <0 buck mode (24) where isbre f is the target battery current. Hence, the bidirectional buck-boost converter average model is given by the following Equation (25). (disb dt =−[m(1−µc2) + (1−m)µc3]Udc L2+Usb L2 idc2= [m(1−µc2) + (1−m)µc3]isb (25) By combining the two operating modes, a virtual control signal is defined. It is designed with m23. The latter is expressed by m23 =m(1−µc2) + (1−m)µc3(26) The converter model becomes (disb dt =−m23 Udc L2+Usb L2 idc2=m23 isb (27) The DC-link feeding the three-phase inverter is modeled by the DC voltage at the output of the filter capacitor. It is represented by the following Equation (28). Cf dUdc dt = (1−uc1)iin +m23isb −idc (28) 3. Control Approaches and Energy Management Strategy 3.1. MPPT Algorithms 3.1.1. P & O Algorithm Thanks to its ease of implementation and its simplicity, the P & O algorithm is the most commonly used [ 54 ]. As its name suggests, it is based on the disturbing the PV system and then observing the future impact of the added disturbance on the PV generator. In fact, if the reference voltage is disturbed in such a direction, the power of the PV generator increases. This means that disturbing the PV system moves its operating point to the maximum power point (MPP). Therefore, in this case, the P & O algorithm kept going, disturbing the reference voltage in the same direction. However, when the system power decreases, this means that disturbing the reference voltage moves the operating point far away from its optimal one. Then, the P & O reverses the sign of the added perturbation. This working principle is repeated until the MPP is reached. Since this algorithm perturbs the operating point of the PV system, its terminal power will fluctuate around the MPP, although solar radiation and temperature are constant leading to a power loss in the system. The flowchart of the P & O algorithm is given in Figure 6. Sensors 2022,22, 6123 9 of 31 Sensors 2022, 11, x FOR PEER REVIEW 10 of 33 Figure 6. Flowchart of the P & O MPPT algorithm. 3.1.2. Fuzzy Logic Algorithm Fuzzy logic (FL) is a numerical computational approach. This concept was first introduced by Lotfi Zadeh in 1965 [55]. It is based on the fuzzy set theory. As no mathematical model is needed for this approach and a human decision-making concept is used, this strategy may give highly effective results. FL strategy can be a challenge for PV systems as reported in Reference [39]. In the presented work, a mamdani type fuzzy system is used for the FL MPPT approach. The error defined with the PV generator power variation over the PV generator voltage and the change of the error over time are chosen as the FL system inputs. The FL algorithm provides, at its output, the change of the DCDC boost converter duty cycle. The mathematical expressions of the FL system input and output variables are given with Equations (29) and (30), respectively. ( ) ( 1) () ( ) ( 1) pv pv pv pv P k P k ek V k V k −− =−− (29) ( ) ( 1) () s kk ek T  −− = (30) where s T is the sample time. To implement the mamdani FL system, four steps are to be followed, as illustrated in Figure 7. Figure 6. Flowchart of the P & O MPPT algorithm. 3.1.2. Fuzzy Logic Algorithm Fuzzy logic (FL) is a numerical computational approach. This concept was first introduced by Lotfi Zadeh in 1965 [ 55 ]. It is based on the fuzzy set theory. As no mathematical model is needed for this approach and a human decision-making concept is used, this strategy may give highly effective results. FL strategy can be a challenge for PV systems as reported in Reference [ 39 ]. In the presented work, a mamdani type fuzzy system is used for the FL MPPT approach. The error defined with the PV generator power variation over the PV generator voltage and the change of the error over time are chosen as the FL system inputs. The FL algorithm provides, at its output, the change of the DC-DC boost converter duty cycle. The mathematical expressions of the FL system input and output variables are given with Equations (29) and (30), respectively. e(k) = Ppv(k)−Ppv(k−1) Vpv(k)−Vpv(k−1)(29) ∆e(k) = ε(k)−ε(k−1) Ts(30) where Tsis the sample time. To implement the mamdani FL system, four steps are to be followed, as illustrated in Figure 7. Sensors 2022,22, 6123 16 of 31 Sensors 2022, 11, x FOR PEER REVIEW 17 of 33 direct stator current component is either fixed at zero value or to a computed value from the high-speed control strategy depending on the operating mode. Working at a highspeed region is assumed thanks to the field-weakening region algorithms [50]. A strategy based on the maximum torque per ampere (MTPA) is used. Figure 6 denotes the principle of the field-weakening algorithm (Figure 10). Figure 10. High-speed control algorithm. 3.3. Nonlinear Control Combining Equations (23), (27) and (28), the following bilinear switched model of the global system is expressed as Equation (42). 1 23 22 1 23 (1 ) (1 ) in in dc c ff sb dc sb dc in sb dc cf f f di u u u dt L L di u u m dt L L du i i i um dt C C C = − −    = − +   = − + −    (42) At the average model Equation (45) over the switching periods, we get 1 23 22 1 23 (1 ) (1 ) in in dc c ff sb dc sb dc in sb dc cf f f dI U U dt L L dI U U M dt L L dU I I I M dt C C C   = − −    = − +   = − + −    (43) where in I is the average value of in i , dc U is the average value of DC-link voltage dc u , sb U is the average value of the battery voltage sb u , dc I is the average value of the load current, and 1c  and 23 M are the DC-DC converter duty cycles. The obtained model is a multi-input, multi-output system. Moreover, it is highly nonlinear. Therefore, a nonlinear control-based Lyapunov approach, as mentioned in Reference [55], is used. One of the control objectives is to enforce the DC-link voltage udc to track its target reference value Udcref, despite external and internal disturbances. An indirect control strategy is used to cope with this problem. It is based on the control current. Based on the power input equals to power output (PIPO) principle, the desired input current of the DC-DC boost converter at the DC-link Iinref is expressed as Figure 10. High-speed control algorithm. 3.3. Nonlinear Control Combining Equations (23), (27) and (28), the following bilinear switched model of the global system is expressed as Equation (42).        diin dt =uin Lf−(1−uc1)udc Lf disb dt =−m23 udc L2+usb L2 dudc dt = (1−uc1)iin Cf+m23 isb Cf−idc Cf (42) At the average model Equation (45) over the switching periods, we get        dIin dt =Uin Lf−(1−µc1)Udc Lf dIsb dt =−M23 Udc L2+Usb L2 dUdc dt = (1−µc1)Iin Cf+M23 Isb Cf−Idc Cf (43) where Iin is the average value of iin , Udc is the average value of DC-link voltage udc , Usb is the average value of the battery voltage usb , Idc is the average value of the load current, and µc1and M23 are the DC-DC converter duty cycles. The obtained model is a multi-input, multi-output system. Moreover, it is highly nonlinear. Therefore, a nonlinear control-based Lyapunov approach, as mentioned in Reference [ 55 ], is used. One of the control objectives is to enforce the DC-link voltage u dc to track its target reference value U dcref , despite external and internal disturbances. An indirect control strategy is used to cope with this problem. It is based on the control current. Based on the power input equals to power output (PIPO) principle, the desired input current of the DC-DC boost converter at the DC-link Iinref is expressed as Iinre f =fi(Udcre f Idc −Usb Isb Uin )(44) where fi≥1 is an ideality factor representing all losses. Let us design with ε1 and ε2 the DC-DC boost converter input current error, and the DC-link voltage error, respectively. They are expressed as Equation (45). ε1=Iin −Iinre f ε2=Udc −Udcre f (45) Deriving Equation (45), we get    . ε1=Uin Lf−(1−µc1)Udc Lf− . Iinre f . ε2= (1−µc1)Iin Cf+M23 Isb Cf−Idc Cf− . Udcre f (46) Sensors 2022,22, 6123 17 of 31 To enforce that the DC-link voltage regulation is assumed with the current and vice versa, the derivative time of ε1and ε2are forced to a specific equation. . ε1=−d1ε1+ε2 . ε2=−d1ε2−ε1(47) Using Equations (46) and (47), the control law of the DC-link boost converter is obtained in Equation (48). µc1=1−Lf Udc "d1ε1−ε2+Uin Lf − . Iinre f #(48) Let us design with ε3the regulation error of the battery current. ε3=Isb −Isbre f (49) where Isbre f is the desired value of the battery current generated from the proposed energy management algorithm. Its time derivative is defined as Equation (50). . ε3=−M23 Udc L2 +Usb L2 − . Isbre f (50) To ensure the exponential convergence of Isb to its reference value, the forced dynamic behavior of ε3is as follows. . ε3=−d3ε3(51) By combining Equations (50) and (51), the control bidirectional DC-DC converter is obtained. M23 =L2 Udc d3ε3+Usb L2 − . Isbre f (52) 3.4. Energy Management The goal of energy management is to effectively manage the energy transfer flow between the PV generator, batteries, and load. In fact, when the electric vehicle is located in a home garage or a covered area, the solar radiation remains insufficient to supply the needed power for starting the vehicle. The demand power is to be provided by a storage battery. Let us design with switches K1, K2, and K3—the used switches that supervise the energy transfer flow. Switch K1 supervises the transfer of energy for the delivered PV generator energy to the load only. Switch K2 is used to control the transfer of energy between the PV generator and the battery only. Finally, switch K3 is used to supervise the transfer of energy between the battery and the load only. The decision parameters of the energy management are the power delivered by the PV generator, the battery state of charge (SOC), and the demanded power load. The main objectives of the power management algorithm are to extract maximum power from the PV generator, avoid overcharge and deep discharge in the battery, and assume the load energy demands. Depending on demand, the PV generator’s produced energy, and the battery SOC, the system operates in one of the following cases. Taking into account the complexity time Tc , the conceived management algorithm is shown in Algorithm 1. Sensors 2022,22, 6123 18 of 31 Algorithm 1 The power management algorithm Compute starting time Tb Repeat { { if (vehicle will start) The vehicle is totally supplied with the battery else (vehicle is in motion) Extract the maximum power from the PV generator if (weather is sunny) if (produced PV power exceeds the required load power) if (the battery is fully charged) { -Disconnect the battery -The load is supplied with the PV generator } if (the battery ready to charge) { -Charge the battery -Supply the load } if (the battery is ready to discharge) { -Supply the load with the produced PV power -Offset the lack of load energy by the battery stored energy } if (the battery is fully discharged) { -Supply the load -Disconnect the battery } elseif (the battery is fully charged) { { -Supply the load with the produced PV power -Offset the lack of load energy by the battery stored energy } else -Disconnect the battery } } Compute the end time Tend } Until (Tend −Tb≥Tc) 4. Simulation Results and Discussion The performance of the robust optimization and energy management strategy based on nonlinear controllers for electric vehicles is highlighted by means of numerical simulations. 4.1. System Characteristics The specifications of the used electric vehicle in simulation are given in Table 3. Table 3. Electric vehicle parameters. Parameter Value Vehicle mass (kg) 1450 Vehicle frontal area (m2)2.711 Tire radius (m) 0.43 Aerodynamic drag coefficient 0.29 Air density (kg/m3)1.204 rolling resistance coefficient 0.013 The mechanical and electrical characteristics of the used electric three-phase PMSM motor are summarized in Table 4. Sensors 2022,22, 6123 19 of 31 Table 4. 100 kw PMSM parameters. Parameter Value Direct inductance (mH) 0.17 Reverse inductance (mH) 0.29 Flux linkage (wb) 0.071 Stator resistance (Ω) 0.0083 Number of poles 8 Viscous friction (Nm/rad/s) 0.005 Moment of inertia (kg m2)0.089 The used PV generator consists of triple-junction solar panels. It provides 100 kw at standard test conditions of 1000 w/m2and T = 25 ◦C. The parameters of the triple-junction InGap/InGaAs/Ge solar cell are shown in Table 5. Table 5. Triple-junction InGap/InGaAs/Ge solar cell parameters. Top Sub-Cell InGaP Top Sub-Cell InGaAs Top Sub-Cell Ge Band-gap energy (ev) 1.976 1.519 0.744 Short circuit current (A) 6.7522 7.7126 10.094 Diode ideality factor 1.97 1.75 1.96 Ki(A/cm2k4) 1.86 ×10−91.288 ×10−810.5 ×10−6 δi2 2 2 αi7.5 ×10−45.405 ×10−44.774 ×10−4 βi500 204 235 The battery storage bank is obtained by the association of 84 Panasonic Lithium-ion CGR18650E battery cells in series and 40 Panasonic Lithium-ion CGR18650E battery cells in parallel. The following characteristics of the used battery cell are regrouped in Table 6. Table 6. Panasonic Lithium-ion CGR18650E battery cell parameters. Parameter Value Qn(Ah) 2.55 Ksb (v) 0.0152 Ab(v) 0.071 Bb(Ah−1)2.893 Rin(Ω) 0.1138 4.2. Behavior Energy Management and Nonlinear Controllers’ Efficiency In this section, the aim is to verify the performance of the conceived controllers and to testify to the validity of the energy management strategy. Different operating and environmental conditions are considered. 4.2.1. Case of Quick Response In order to validate the performance of the conceived algorithms and the energy management strategy at quick response, specific trajectories for both operating and meteorological conditions are considered. In fact, as shown with Figure 11e, the target vehicle speed is fixed to zero value from t= 1 s to t= 2 s. Since t= 2 s, the speed quickly increase to 40 km/h. The used radiation and temperature trajectories, in this case, are represented in Figure 11a,b, respectively. As it is indicated with these figures, radiation and temperature Sensors 2022,22, 6123 20 of 31 are both fixed to 900 w/m 2 and 60 ◦ C, respectively. Since t= 2 s, they simultaneously increase to 950 w/m 2 and 70 ◦ C, respectively. From t= 4 s to t= 6 s, radiation increases to 1000 w/m 2 and the temperature decreases to 50 ◦ C. Finally, t= 6 s, a 11,000 w/m 2 is associated to radiation and the temperature is fixed to 64 ◦ C. The evolution of the available PV generator and its optimal one is given in Figure 11c. As it is indicated in this figure, the available PV generator precisely and rapidly tracks its optimal value despite simultaneous abrupt alteration in radiation, and temperature and load variation (Figure 11f). Sensors 2022, 11, x FOR PEER REVIEW 22 of 33 (a) (b) (c) (d) 0 1 2 3 4 5 6 7 8 900 950 1000 1050 1100 Time (s) Radiation (w/m²) 0 1 2 3 4 5 6 7 8 50 55 60 65 70 Time (s) Temperature (°C) 0 1 2 3 4 5 6 7 8 0 50 100 150 Time (s) PV generator power (kw) Target power Actual power 0 1 2 3 4 5 6 7 8 0 500 1000 Time (s) DC-link voltage (v) Actual DC-link voltage Target DC-link voltage Figure 11. Cont. Sensors 2022,22, 6123 21 of 31 Sensors 2022, 11, x FOR PEER REVIEW 23 of 33 (e) (f) (g) (h) Figure 11. Simulation results under quick response: (a) radiation trajectory, (b) temperature trajectory, (c) PV generator power, (d) DC-link voltage, (e) vehicle speed, (f) load torque, (g) battery state of charge, and (h) operating mode. 0 1 2 3 4 5 6 7 8 -20 0 20 40 60 80 Time (s) Vehicule speed (km/h) Target vehicule speed Actual vehicule speed 0 1 2 3 4 5 6 7 8 70 75 80 85 Time (s) Tr (Nm) 0 1 2 3 4 5 6 7 8 60.004 60.0045 60.005 60.0055 Time (s) SOC (%) Actual SOC Minimal value of SOC Maximal value of SOC 0 1 2 3 4 5 6 7 8 1 2 3 4 5 Time (s) Modes Figure 11. Simulation results under quick response: ( a ) radiation trajectory, ( b ) temperature trajectory, ( c ) PV generator power, ( d ) DC-link voltage, ( e ) vehicle speed, ( f ) load torque, ( g ) battery state of charge, and (h) operating mode. The validity of the energy management algorithm is also noticed from the obtained results. In fact, the evolution of the DC-link voltage is given in Figure 11d. Despite the changes in radiation, in temperature, and in load torque, DC-link voltage is maintained constant, except some fluctuation appeared at the time disturbances variation. The battery state of charge and the working modes are, respectively, depicted in Figure 11g,h. Suitable Sensors 2022,22, 6123 22 of 31 charging and discharging modes are shown over the evolution of the battery state of charge. This is proved with the system operating modes proving the validity of the used energy management strategy. 4.2.2. Case of Variable Vehicle Speed Response Here, the aim is to verify the tracking behavior of the conceived controllers and to testify to the validity of the energy management strategy under internal and external disturbances for a variable speed operation including both normal and field-weakening operating modes. The adapted temperature and radiation trajectories are shown in Figure 12a. A suitable target speed trajectory for electric vehicle applications, including different operating conditions, is used as reported in Figure 12b. Figure 12c shows that the PV generator rapidly tracks its maximum values despite abrupt meteorological conditions and abrupt load variations (Figure 12a,d). This will significantly reduce the recharge time. As it is indicated in Figure 12e, the DC–link voltage is maintained fixed at its target value, with some fluctuations caused by the load and meteorological variations. Taking into account the vehicle power (Figure 12f) and the battery state of charge (Figure 12g), the states of the switches K 1 , K 2 , and K 3 (Figure 12i, j and k) and the working modes are obtained (Figure 12h). The obtained results show that the energy management approach is working well. Sensors 2022, 11, x FOR PEER REVIEW 24 of 33 The validity of the energy management algorithm is also noticed from the obtained results. In fact, the evolution of the DC-link voltage is given in Figure 11d. Despite the changes in radiation, in temperature, and in load torque, DC-link voltage is maintained constant, except some fluctuation appeared at the time disturbances variation. The battery state of charge and the working modes are, respectively, depicted in Figure 11g,h. Suitable charging and discharging modes are shown over the evolution of the battery state of charge. This is proved with the system operating modes proving the validity of the used energy management strategy. 4.2.2. Case of Variable Vehicle Speed Response Here, the aim is to verify the tracking behavior of the conceived controllers and to testify to the validity of the energy management strategy under internal and external disturbances for a variable speed operation including both normal and field-weakening operating modes. The adapted temperature and radiation trajectories are shown in Figure 12a. A suitable target speed trajectory for electric vehicle applications, including different operating conditions, is used as reported in Figure 12b. Figure 12c shows that the PV generator rapidly tracks its maximum values despite abrupt meteorological conditions and abrupt load variations (Figure 12a,d). This will significantly reduce the recharge time. As it is indicated in Figure 12 e, the DC−link voltage is maintained fixed at its target value, with some fluctuations caused by the load and meteorological variations. Taking into account the vehicle power (Figure 12 f) and the battery state of charge (Figure 12g), the states of the switches K1, K2, and K3 (Figure 12i, j and k) and the working modes are obtained (Figure 12h). The obtained results show that the energy management approach is working well. (a) (b) 0 1 2 3 4 5 6 7 8 9 10 1000 1100 1200 1300 Radiation (w/m²) 0 1 2 3 4 5 6 7 8 9 10 25 30 35 40 Time (s) Temperature (°C) 0 1 2 3 4 5 6 7 8 9 10 -50 0 50 100 150 Time (s) Vehicle speed (Km/h) Target speed Actual speed Figure 12. Cont. Sensors 2022,22, 6123 23 of 31 Sensors 2022, 11, x FOR PEER REVIEW 25 of 33 (c) (d) (e) (f) 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 x 104 Time (s) PV generator powers (w) Actual power Target power 0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 Time (s) Tr (Nm) 0 1 2 3 4 5 6 7 8 9 10 0 100 200 300 400 500 600 700 Time (s) udc (v) Target DC-link voltage Actual DC-link voltage 012345678910 -100 0 100 200 300 400 Time (s) Electric vehicle powers (kw) Actual power Target power Figure 12. Cont. Sensors 2022,22, 6123 24 of 31 Sensors 2022, 11, x FOR PEER REVIEW 26 of 33 (g) (h) (i) (j) 0 1 2 3 4 5 6 7 8 9 10 59.996 59.998 60 60.002 60.004 60.006 Time (s) SOC(%) Minimal value of SOC Maximal value of SOC Actual SOC 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 Time (s) Modes 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 Time (s) K1 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 K2 Time (s) Figure 12. Cont. Sensors 2022,22, 6123 25 of 31 Sensors 2022, 11, x FOR PEER REVIEW 27 of 33 (k) Figure 12. Simulation results under variable target vehicle speed: (a) temperature and radiation trajectories, (b) vehicle speeds, (c) PV generator power, (d) load torque, (e) DC−link voltage, (f) vehicle consumed power, (g) battery state of charge, (h) battery state of charge operating mode, (i) switch K1, (j) switch K2, and (k) switch K3. By using the conceived energy management algorithm, the protection of the battery is assumed. In fact, as it is shown in Figure 12g, the SOC is always maintained between the maximum and minimum value of SOC. 4.2.3. Case of Extra Urban Drive Cycle (EUDC) Response In this simulation test, the EUDC is used for the target vehicle speed (Figure 13g) under simultaneous abrupt radiation and temperature variations. The type of the road in which the electric vehicle moves and the impact of the wind are also considered. Therefore, a simultaneous quick change in radiation and temperature is used. The adopted radiation and temperature trajectories are, respectively, plotted in Figure 13a,b. A vehicle road, including slope inclination, is used as shown in Figure 13c. In fact, the vehicle road is inclined with two slope angles. The first is from t = 2 s to t = 4 s and the second is applied at t = 6 s to t = 8 s. A random trajectory for the wind speed is chosen in simulation test, as illustrated in Figure 13d. Available PV generator power sufficiently tracks its optimal power (Figure 13e) despite atmospheric conditions, the type of road, and load torque variation (Figure 13h). The validity of the power management tacking into account the battery safety is effectively highlighted, as it is shown with the battery state of charge (Figure 13i), the DC-link voltage (Figure 13f), and the operating modes (Figure 13j). (a) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 K3 Time (s) 0246810 12 14 900 950 1000 1050 1100 Radiation (w/m²) Time (s) Figure 12. Simulation results under variable target vehicle speed: ( a ) temperature and radiation trajectories, ( b ) vehicle speeds, ( c ) PV generator power, ( d ) load torque, ( e ) DC–link voltage, ( f ) vehicle consumed power, ( g ) battery state of charge, ( h ) battery state of charge operating mode, ( i ) switch K1, (j) switch K2, and (k) switch K3. By using the conceived energy management algorithm, the protection of the battery is assumed. In fact, as it is shown in Figure 12g, the SOC is always maintained between the maximum and minimum value of SOC. 4.2.3. Case of Extra Urban Drive Cycle (EUDC) Response In this simulation test, the EUDC is used for the target vehicle speed (Figure 13g) under simultaneous abrupt radiation and temperature variations. The type of the road in which the electric vehicle moves and the impact of the wind are also considered. Therefore, a simultaneous quick change in radiation and temperature is used. The adopted radiation and temperature trajectories are, respectively, plotted in Figure 13a,b. A vehicle road, including slope inclination, is used as shown in Figure 13c. In fact, the vehicle road is inclined with two slope angles. The first is from t= 2 s to t= 4 s and the second is applied at t= 6 s to t= 8 s. A random trajectory for the wind speed is chosen in simulation test, as illustrated in Figure 13d. Available PV generator power sufficiently tracks its optimal power (Figure 13e) despite atmospheric conditions, the type of road, and load torque variation (Figure 13h). The validity of the power management tacking into account the battery safety is effectively highlighted, as it is shown with the battery state of charge (Figure 13i), the DC-link voltage (Figure 13f), and the operating modes (Figure 13j). Sensors 2022, 11, x FOR PEER REVIEW 27 of 33 (k) Figure 12. Simulation results under variable target vehicle speed: (a) temperature and radiation trajectories, (b) vehicle speeds, (c) PV generator power, (d) load torque, (e) DC−link voltage, (f) vehicle consumed power, (g) battery state of charge, (h) battery state of charge operating mode, (i) switch K1, (j) switch K2, and (k) switch K3. By using the conceived energy management algorithm, the protection of the battery is assumed. In fact, as it is shown in Figure 12g, the SOC is always maintained between the maximum and minimum value of SOC. 4.2.3. Case of Extra Urban Drive Cycle (EUDC) Response In this simulation test, the EUDC is used for the target vehicle speed (Figure 13g) under simultaneous abrupt radiation and temperature variations. The type of the road in which the electric vehicle moves and the impact of the wind are also considered. Therefore, a simultaneous quick change in radiation and temperature is used. The adopted radiation and temperature trajectories are, respectively, plotted in Figure 13a,b. A vehicle road, including slope inclination, is used as shown in Figure 13c. In fact, the vehicle road is inclined with two slope angles. The first is from t = 2 s to t = 4 s and the second is applied at t = 6 s to t = 8 s. A random trajectory for the wind speed is chosen in simulation test, as illustrated in Figure 13d. Available PV generator power sufficiently tracks its optimal power (Figure 13e) despite atmospheric conditions, the type of road, and load torque variation (Figure 13h). The validity of the power management tacking into account the battery safety is effectively highlighted, as it is shown with the battery state of charge (Figure 13i), the DC-link voltage (Figure 13f), and the operating modes (Figure 13j). (a) 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 K3 Time (s) 0246810 12 14 900 950 1000 1050 1100 Radiation (w/m²) Time (s) Figure 13. Cont.