Open access peer-reviewed chapter

A New Control Strategy for Photovoltaic System Connected to the Grid via Three-Time-Scale Singular Perturbation Technique with Performance Analysis

By Youssef Mchaouar, Abdelmajid Abouloifa, Ibtissam Lachkar and Mohammed Fettach

Submitted: May 6th 2019Reviewed: August 29th 2019Published: April 1st 2020

DOI: 10.5772/intechopen.89434

Downloaded: 50

Abstract

This chapter addresses the problem of controlling single-phase grid-connected photovoltaic system through a full bridge inverter with L-filter. The control objectives are threefold: (i) forcing the voltage in the output of photovoltaic panel to track a reference. This reference has been obtained from the maximum power point tracking strategy; (ii) guaranteeing a tight regulation of the DC-link voltage; and (iii) ensuring a satisfactory power factor correction (PFC) at the grid such as the currents injected must be sinusoidal with the same frequency and the same phase as the grid voltage. The considered control problem entails several difficulties including: (i) the high dimension and strong nonlinearity of the system; (ii) the changes in atmospheric conditions. The problem is dealt with by designing a synthesized nonlinear multi-loop controller using singular perturbation technique, in which a three-time-scale dynamics is artificially induced in the closed-loop system. A formal analysis based on the three-time-scale singular perturbation technique and the averaging theory is developed to proved that all control objectives are asymptotically achieved up to small harmonic errors (ripples). The performance of the proposed approach and its strong robustness with respect to climate changes are evaluated based on the various simulations results carried out under Matlab/Simulink software.

Keywords

  • single-phase grid-connected photovoltaic system
  • nonlinear control
  • three-time-scale singular perturbation technique
  • MPPT
  • power factor correction
  • averaging theory
  • stability analysis

1. Introduction

Due to dramatic increase in energy consumption and thrust to reduce carbon and greenhouse gas emissions from the traditional electric power generation systems, photovoltaic (PV) power generators have gained a great popularity in recent years. Indeed, photovoltaic systems produce electric power without harming the environment, transforming a free inexhaustible source of energy, solar radiation, into electricity. Furthermore, the major advantage of the photovoltaic systems is to meet the basic power requirement of non-electrified remote areas, where grid power has not yet reached. Also, there are other advantages such as the declining cost and prices of solar modules. On the other hand, the importance of PV systems in the solar industry makes these systems more efficient and reliable, especially for utility power in distributed generation (DG) at medium and low voltages power systems [1]. All these considerations assure a promising role for PV generation systems in the near future.

On the other hand, many technical problems, such as untimely failures, could be found on electronic systems related in particular to the transfer and conversion of this energy to the network. Today, most conversion systems often suffer from low yields in real production sites. To meet the requirements of the new international standards on expected performance on associated conversion systems, it is important to make a research effort to solve the many control problems associated with the static power converter and bring this area to a degree of sufficient maturity to make them industrial products in their own right. One of the difficulties caused by the use of a photovoltaic conversion chain is focused on the problem of non-perfect control of the chain between the photovoltaic generator itself and the continuous or alternative type of load.

The efficiency of a PV plant is affected mainly by three factors: the efficiency of the PV panel, the efficiency of the static power converter and its control, and the efficiency of the maximum power point tracking (MPPT) algorithm.

PV grid-connected systems represent the most important field applications of solar energy [2, 3, 4]. In general, the power converter interface, from PV module (the DC source) to the load or to the grid, consists of two-stage converters: The first-stage DC/DC converter is usually used to boost the PV voltage and to implement the maximum power point technique. While the second stage is used to convert this power into high-quality AC voltage, with power factor correction (PFC) respecting to the power supply grid (i.e. sinusoidal and in phase with the AC supply voltage).

Maximum power point tracking (MPPT) is required to match the PV array power to the environmental changes achieving to extract the maximum power output from a solar cell [5]. To this end, different MPPT techniques have been proposed such as incremental conductance [6], perturbation and observation (P&O) [7, 8], the hill-climbing, and some other special methods, such as neural networks, fuzzy logic technique [9]. Among all available techniques, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.

In order to provide a stable controller of DC/DC and DC/AC converters, many linear control methods have been proposed using many methods, such as a fuzzy-proportional integral controlled [10], a simple PR controller [11] where the performances have been illustrated by experimental result. However, in both proposed controllers, the problem of maximizing PV power transfer is not accounted for in the controller design. In contrast to linear control, nonlinear approaches can optimize the dynamic performance of system, such as sliding mode [12], fuzzy-sliding mode [13], feedback linearization [14], singular perturbation technique [15], and many others works [16, 17, 18]. In light of the previous descriptions, no theoretical analysis is made to formally prove that the closed-loop control performances are actually achieved.

In this chapter, a multi-loop nonlinear controller is designed and developed via singular perturbation technique (Chapter 11 in Refs. [19, 20]), as was shown in Refs. [21, 22], where three-time-scale dynamics is artificially induced in the closed-loop system. The control objectives are threefold: (i) achieving the MPPT for the PV array; (ii) ensuring a tight regulation of the DC-link voltage; and (iii) ensuring a grid connection with unity power factor (PF). These objectives must be met despite changes of the climatic variables (temperature and radiation). A theoretical stability analysis, for the closed-loop system, is provided using the three-time-scale singular perturbation technique [23, 24] and averaging technique (Chapter 10 in Refs. [19, 25]. The three-time-scale analysis allows to construct a suitable composite Lyapunov function candidate for the closed-loop photovoltaic system, and the stability properties of the resulting subsystems are analyzed providing mathematical expressions for the upper bounds of the singularly perturbed parameters.

Compared to previous works, the contribution of the new nonlinear controller enjoys several interesting features including the following:

  • Several control objectives are simultaneously taken into account such as: MPPT, DC regulation, and PFC, whereas only some of these objectives have been tackled in previous works [10, 11].

  • A theoretical analysis will prove, using three-time-scale singular perturbation and averaging technique, that the desired multiple objectives are achieved. Such a formal analysis was missing in the previous works [12, 13, 14, 15].

  • The nonlinearity of the controlled system was preserved in the controller design in order to keep all the properties of the studied system, whereas it is partly or totally ignored in previous controllers [14].

  • By including of three-time-scale dynamics in the full-order closed-loop system can ensure to achieve desired properties, such as robust zero steady-state error of the reference input realization, desired output performance specifications (overshoot, settling time), and insensitivity of the output transient behavior with respect to parameter variations and external disturbances.

The content of this chapter is outlined as follows: in Section 2, the grid-connected PV system is described and modeled. Section 3 is devoted to the cascade nonlinear controller design and its performances are formally analyzed in Section 4. The global performance of the closed-loop photovoltaic system will be illustrated by numerical simulation using MATLAB/SIMULINK tool in section 5. A conclusion and a reference list end the chapter.

2. System description

This section describes the modeling of photovoltaic system connected to the grid. The power circuit topology used in the proposed single phase grid connected to the photovoltaic array is shown in Figure 1. It consists of the following components: (i) a photovoltaic array which consists of an arrangement of Ns-series and Np-parallel strings; (ii) an input capacitor Cpvand a DC-DC boost converter used to increase the voltage level and achieve MPPT for photovoltaic array; (iii) a DC link capacitor Cdc; and (iv) a single-phase full-bridge inverter including four power semiconductors with Lfilter that is used to provide the energy to the grid and ensure power factor correction.

Figure 1.

Single phase grid-connected PV system.

Typical (Ip-Vp) characteristics of solar cells arranged in Np-parallel and Ns-series can be found in many places (see, e.g. [26]). The PV array module considered in this paper is of type KC200GT. In this chapter, a simple and effective MPPT of incremental conductance algorithm is applied to attain the maximum power of PV array in different solar irradiance and temperature condition parameters.

By analyzing the circuit and applying the well-known Kirchhoff laws, the system of Figure 1 can be described by the following set of differential equations:

diidt=vpvLi1u1vdcLiriLiiiE1
dvpvdt=1CpvipviiE2
digdt=vgLorgLgig+u2vdcLoE3
dvdcdt=1Cdcu2ig+1u1iiCdcE4

where vpvand ipvare, respectively, the photovoltaic generator voltage and current. vdcand idcare, respectively, DC link voltage and current. iidesignates the input current chopper, Cdcis DC link capacitor, riand rgare, respectively, the equivalent series resistances (ESR) of input inductance Liand the filter inductance Lg. vgand igare, respectively, the voltage and current of the grid. Here, the grid voltage is defined by vg=Egsinωgt, where Egand ωgdenote the constant amplitude and the constant angular frequency. The switching functions µ1and µ2are defined by:

µ1=1ifs0isON0ifs0isOFFE5
µ2=1ifs1s4isONands2s3isOFF1ifs1s4isOFFands2s3isONE6

The instantaneous model (1)(4) cannot be used directly for the development of continuous control laws since it involves, as input variables, the binary signal µ1and µ2. To overcome this inconvenience, the average model is used [27]. Therefore, the state variables ii, vpv, ig, and vdcare replaced by their average values x1, x2, x3, and x4over a cutting period. The control inputs u1and u2denote the average values of μ1and μ2, respectively.

dx1dt=1Lix21Lix4riLix1+1Lix4u1E7
dx2dt=ipvCpv1Cpvx1E8
dx3dt=rgLgx3vgLg+1Lgx4u2E9
dx4dt=1Cdcx3μ21Cdcx1u1+1Cdcx1E10

3. Controller design

3.1 Input inductor current regulation and PFC objectives

3.1.1 Control law design

The first control objective is to enforce the photovoltaic voltage x2to track, as closely as possible, the optimal point VM(called Regulator 2). However, it is well-known that the boost converter has a non-minimum phase feature. Such an issue is generally dealt with by resorting a cascaded loop design strategy that starts with the input current loop (Regulator 1), as it is shown in Figure 2. More specifically, the controller makes the input inductor current x1to track a reference signal x1, the latter is determined from (Regulator 2).

Figure 2.

Schematic diagram of the proposed controller for single phase grid-connected PV system.

In parallel with the input current controller, the network current controller (Regulator 3) will be designed for power factor correction requirement that amounts to forcing the network current x3to match the reference signal of the form x3=βsinωgt. It means that the grid current x3should be sinusoidal and in phase with the AC grid voltage vg, with βis a signal. In fact, the latter is allowed to (and actually will) be time-varying but it must converge to a positive constant value. For both objectives (i.e. the input current x1regulation and PFC objectives), let us consider Eqs. (7), (9) in the following form:

Ẋ1,3=M1U+M2E11

where X1,3=x1x3, M1=x4Lix4Lg, M2=riLix1+x2Lix4LirgLgx3vgLg, and U=u1u2.

Remark 1. Under condition that the DC-link voltage remains all the time positive, it can be showed that 0<M1minM1M1Max<and M2M2Max<are satisfied.

Let us introduce the following current tracking errors:

ei=xixii=13E12

Then, the reference model can be constructed in the following form

dxidt=eiTi=ΔDixixii=13E13

where T1and T3denote the time constants and they are selected based on the desired settling time, respectively, for the currents x1and x3. Based on (13), the realization errors of the desired behaviors of ẋ1and ẋ3, namely Ω1and Ω3, are given by

Ωi=Dixixiẋii=13E14

Therefore, the control problem limteit=0i=13corresponds to the insensitivity condition defined by

Ωi=0i=13E15

Doing so, the behaviors of ẋ1and ẋ3with prescribed dynamics of expression (13) will be fulfilled. Replacing in Eq. (14)ẋ1and ẋ3by their expressions (7), (9), and since the requirement (15), one gets

DM2M1U=0E16

Hence, there is an isolated root called the inverse dynamic solutions for Ugiven by

Uid=u1idu2id=Lix4e1T1riLix1+1Lix21Lix4Lgx4e3T3+x4+vg+rgx3LgE17

The control variable, namely U, has emerged in Eq. (16). At this point, an appropriate control law with the first derivative in feedback has to be found, so that the (e1, e3)-systems are made asymptotically stable. As the objective is to drive the error to zero, it is natural to choose the Lyapunov functions candidate

V3V4=12Ω1u12Ω3u22E18

From expressions (14) and (16), it can be easily checked that the following time-derivatives

ddtV3V4=V3u1du1dtV4u2du2dt=Ω1Ω1u1du1dtΩ3Ω3u2du2dt=Ω1x4Lidu1dtΩ3x4Lgdu2dtE19

are made negative-definite using the following control laws:

ddtu1u2=k1ε1ε2Ω1k3ε1ε2Ω3E20

At this point, k1and k3are any design parameters, ε1and ε2are small positive parameters. In view of M1>0, it is easily seem from expressions (11) and (14) that

ddtV3V4=k1ε1ε2x4Ω12Lik3ε1ε2x4Ω32Lg<0Ωi0E21

for a positive values of ki>0i=13. Therefore, from expressions (14) and (20), the discussed nonlinear control laws are formulated as follows

ε1ε2ddtu1u2=k1e1T1dx1dtk3e3T3dx3dtE22

3.1.2 Singular perturbation system of the inner current loops

Consider the closed-loop system of inner loops composed of the Eqs. (7)(10) and the control laws (22), which can be rewritten in the following form

ε1ε2dZdt=hXZtE23
dXdt=fXZtE24

with Z=z1z2T=u1u2T, X=x1x2x3x4T, hXZt=k1e1T1x4Liz1+riLix1x2Li+x4Lik3e3T3x4Lgz2+vgLg+rgLgx3, fXZt=z1x4LiriLix1+x2Lix4Lix1Cpv+ipvCpvvgLgrgLgx3+x4Lgz2x1Cdcx1Cdcz1x3Cdcz2.

Now, we go to the fast time τ1=t/ε1ε2. Then for ε1=0, the ultra-fast dynamic subsystem (UFDS) is given by

dZdτ1=hXZtE25
dXdτ1=0E26

After the rapid decay of transients in expression (25), the steady state (more precisely, quasi-steady state) tends toward an equilibrium Ze=h˜X. The manifold defined by Zeis called the slow manifold, which is given by

Ze=z1ez2e=Lix4e1T1+x4x2+rix1LiLgx4e3T3+vg+rgx3LgE27

Remark 2.

  1. During the fast transient in expression (25), the variables Xare treated as the frozen parameters.

  2. The equilibrium point given by expression (27) involves a division by the DC link voltage x4, from a practical point of view this division is not a problem because the DC link voltage remains all the time positive for the power converter to work correctly.

By substituting of this equilibrium Zeinto Eq. (24), the slow dynamic subsystem (SDS) of inner loops takes place on the slow manifold, according to the equation

dXdt=e1T1x1Cpv+ipvCpve3T3Lox3Cdcx4e3T3+vg+rgx3LgLix1Cdcx4e1T1+rix1x2LiE28

Proposition 1. Consider the closed-loop system composed of Eqs. (23) and (24). For ε10, the system takes the singular perturbation form where the UFDS is defined by equation (25), while the SDS of inner loop is defined by equation (28). Under the considerations given by Remark 2, one has the following properties

  1. If the design parameters ki(i=1,3) are positives, the UFDS (25) will be exponentially stable, and Zconverge exponentially fast to Ze.

  2. The behaviors of xi(i=1,3) are prescribed by the stable reference equations of the form dxi/dt=xixi/Ti. Then, the requirements limtei=limtxixi=0are maintained.

3.2 MPPT and DC bus voltage regulation objective

The second step consists in completing the inner control loops by outer control loops for PV voltage (Regulator 2) and DC-link voltage (Regulator 4). The aim is now to enforce the photovoltaic voltage x2and the DC-link voltage x4to track, respectively, the optimal point x2=VMand a given reference voltage x4=Vdc, such that tuning laws for the ratio βand x1must be designed. According to the three-time-scale design methodology that is employed in this work, the general formulation of the three-time-scale singular perturbed systems requires the system to possess three different time scales. To this end, the voltage loops will be slow compared to the transients of the current loops. Therefore, the design parameters for voltage loops, in particular (ε2, T2, and T4) must satisfy: 0<ε1ε2ε21, and T1T3<T2T4. In addition, the steady states for the current x1and x3yield, respectively, x1and x3. Therefore, the SDS of inner loop given by equation (28) will be reduced to

dXrdt=ddtx2x4=ipvCpvx1Cpvx2x1vgx3rix12rgx32Cdcx4E29

Now, the first step is to establish the relation between the ratio β(which acts as the control input of the outer loop) and the DC-link voltage x4(representing the output of the outer loop).

3.2.1 Relation between βand x4, and the control law

The relation between the ratio βand the DC-link voltage x4is the subject of the following proposition.

Proposition 2. We consider the second equation of expression (29) and the power factor correction requirement defined by x3=βsinωgt.

  1. The relation between x4and βis described in the following results

    dx4dt=x2x1rix12Cdcx4vg2βCdcx4Egrgvg2β2Cdcx4Eg2E30

  • Therefore, the squared-voltage xr,4=x42varies, in response to the tuning ratio β, according to the following first-order time-varying nonlinear equation:

  • dxr,4dt=2x2x1rix12Cdc2vg2βCdcEg2rgvg2β2CdcEg2E31

    The second step is to establish control laws for the outer loops, in which Y=y1y2T=x1βTrepresent the new control inputs, while Xr=xr,2xr,4T=x2x42Trepresent the new output variables. To this end, introduce the following tracking errors

    ej=xjxr,jj=24E32

    Then, let the desired behavior of Xrbe assigned by

    dxr,jdt=ejTj=ΔDjxjxjj=24E33

    The error of the desired dynamic realization it follows

    Ωj=Djxr,jxr,jẋr,jj=24E34

    Then, the insensitivity condition is given by

    Ωj=0E35

    Similar to the previous subsection and bearing in mind the fact that βand their first derivative must be available, and in order to meet the requirement (35), we should apply the control law given by the following structure

    ε2ddty1y2=k2e2/T2dx2,r/dtk4e4/T4dxr,4/dtE36

    3.2.2 Singular perturbation system of the outer voltage loops

    Combining expressions (31) and (36), and the first equation of expression (29), one obtains

    ε2dYdt=gXrYt=k2e2T2+y1ipvCpvk4e4T4f2E37
    dXrdt=fXrYt=ipvCpv1Cpvy1f2E38

    with f2=2xr,2y1riy12CdcEgvg2y2+rgvg2y22CdcEg2. The fast dynamic subsystem (FDS) is obtained by transforming the slow time-scale tto the fast time-scale τ2=t/ε2, then, by setting ε2=0

    dYdτ2=gXrYtE39
    dXrdτ2=0E40

    Notice that expression (39) has an isolated equilibrium at Ye=y1ey2e, which will be determined (in the mean) in Appendix. As the FDS (39) is nonlinear, the stability properties of its equilibrium can be checked through the analysis of the Jacobian matrix of the linearized version defined as follows

    AF=k2Cpv0k4AFk4AFE41

    with AF=4riy1eCdc2xr,2Cdc, AF=2vg2CdcEg+4rgvg2y2eCdcEg2. Taking into account that y2eis positive, we conclude that all eigenvalues of AFsatisfy Reλ1,2<0for the negative values of k2and k4. Therefore, AFis Hurwitz matrix. By substituting of the equilibrium Yeinto expression (38), the reduced SDS of outer loops takes place on the slow manifold, according to the equation

    dXrdt=e2/T2e4/T4E42

    Proposition 3. Consider the system closed loop composed of expressions (37) and (38). For ε20, this system takes the singular perturbation form, where the FDS is given by expression (39) and the reduced SDS of outer loop is given by expression (42). One has the following properties

    1. If the design parameters k2and k4are negative, the FDS (39), will be exponentially stable and Yconverge exponentially fast to Ye.

    2. The behaviors of xr,j(j=2,4) are prescribed by the stable reference equations of the form dxr,j/dt=xjxr,j/Tj. Then, the requirements limtej=limtxjxr,j=0(j=2,4) are maintained.

    4. Control system analysis

    The objective of the global stability of closed-loop system can be analyzed in the following theorem. It is shown that the control objectives are achieved (in the mean) with an accuracy that depends on the network frequency ωgand the small parameters εii=12.

    Theorem. Consider the overall control system composed of the Pv panel, boost DC-DC converter and DC-AC inverter, described by the model (7)(10), in closed loop with the multi-cascade multi-loop composed of:

    • The inner regulators (23), where (ε1ε2, k1, k3, T1, T3) are the design parameters;

    • The outer regulators (37), where (ε2, k2, k4, T2, T4) are the design parameters.

    Then, one has the following property

    1. The augmented state vector Z=u1u2T=z1z2T, Y=x1βT=y1y2T, and X=x1x2x3x4T=x1x2x3x42Tundergoes the following state equations

      ε1ε2Żt=hXYZtE43
      ε2Ẏt=gXYZtE44
      Ẋt=fXYZtE45

    with

    h=k1e1T1z1x4Li+rix1x2+x4Lik3e3T3x4Lgz2+vg+rgx3Lg, f=riLix1+x2Lix4Li+z1x4Liy1Cpv+ipvCpvx4Loz2vgLgroLgx32x2y1riy12Cdc2vg2y2CdcEg2rgvg2y22CdcEg2, g=k2e2T2+y1CpvipvCpvk4e4T4+2vg2y2CdcEg+2rgvg2y22CdcEg22x2y1riy12Cdc.

  • Let the control design parameters be selected, such that the following inequalities hold T1<T2, T3<T4, k1>0, k2<0, k3>0, k4<0, and 0<ε1ε2ε21.

  • Then, there exist positive constants ρ, εnn=12, such that for and 0<ρ<ρ0<εn<εn1n=12the system (43)(45) has an asymptotically stable 2π/ωgperiodic solution Xtεnρ, Ytεnρ, Ztεnρthat continuously depends on εnn=12and ρ=1/ωg. Furthermore, when (εn=0and ρ=0), one has

    limεn0,ρ0Xtεnρ=X0,limεn0,ρ0Ytεnρ=Y0,limεn0,ρ0Ztεnρ=Z0,E46

    where X0=x1,0x2,0x3,0x4,0T, Y0=y1,0y2,0T, and Z0=z1,0z2,0Twith x2,0=VM, x4,0=Vdc, z1,0=1x2,0x4,0+rix1,0x4,0, z2,0=rox3,0x4,0, y1,0=ipv,0, y2,0=Eg+2Eg/22+2rox2,0ipv,0riipv,022ro.

    See Appendix for the proof.

    5. Simulation and discussion of results

    The experimental setup is described by Figure 2 and the nonlinear controller, developed in Section 3, including the control laws (23) and (37), will now be evaluated by simulation in MATLAB/SIMULINK platform using the electromechanical characteristics of Table 1.

    ParametersSymbolValues
    NetworkE/f2202V/50Hz
    BoostCpv
    Li
    ri
    0.3mF
    6mH
    20
    L-filterLo
    ro
    5mH
    50
    PWM switching frequencyfPWM14kHz
    DC capacitanceCdc7mF

    Table 1.

    PV system and single-phase grid characteristics.

    The numerical values used for the design parameters are chosen as follows: ε1=5.81×102, ε2=7.21×105, T1=2.1×104, T2=9.3×103, T3=9.51×105, T4=1.43×102, k1=7.1×107, k2=2.56×106, k3=2.3×106, K4=8.1×108. These values have proved to be suitable based on several trials respecting the singular perturbation technique. In this simulation, we consider the KC200GT type of PV array module with Ns=54and Np=6.

    The performances of the proposed controller are illustrated by Figures 35. Figure 3 shows that the DC-link voltage x4is well regulated and quickly settles down after each change in the signal reference (stepping from 600Vto 750Vat t=0.4s). The wave frame of the output current x3is showed in Figure 4. The current is sinusoidal and in phase with the network voltage complying with the PFC requirement. This is further demonstrated by Figure 5, which shows that the ratio βtakes a constant value after transient periods following the changes in reference signals.

    Figure 3.

    DC-link voltage.

    Figure 4.

    Power factor correction checking.

    Figure 5.

    Tuning parameter β.

    5.1 Radiation variation effect

    Figure 6ac illustrates the resulting closed-loop control performances in presence of radiation changes. Specifically, the radiation takes a low, medium, and high value (equal to 800, 1000 and step to 600 W/m2 at times 0, 0.4, and 0.7 s, respectively), meanwhile the temperature is kept constant, equal to 298.17K(i.e. 25C). Figure 6a shows that the captured PV voltage varies between 182, 212, and 138 V. These values correspond to the maximum points. Figure 6b shows that the DC-link voltage regulation is recovered after a short transient period following each change of the irradiation. Figure 6c shows that the current amplitude changes whenever the radiation varies. It is seen that the output current x3and the grid voltage vgare actually sinusoidal and in phase. Hence, the converter connection to the supply network is done with a unitary power factor.

    Figure 6.

    Radiation variation effect: (a) Photovoltaic voltage. (b) DC-link voltage. (c) PFC checking.

    5.2 Temperature variation effect

    The perfect MPPT is illustrated by Figure 7a. Here, the temperature steps from 298.15 to 318.15 K, then to 308.15 K while the radiation λ is kept constant equal to 1000W/m2. Figure 7b shows that the DC-link voltage x4is tightly regulated: it quickly settles down after each change in the temperature. Figure 7c illustrates the current amplitude changes whenever the temperature varies. The current remains (almost) sinusoidal and in phase with the network voltage complying with the PFC requirement.

    Figure 7.

    Temperature variation effect: (a) Photovoltaic voltage. (b) DC-link voltage. (c) PFC checking.

    6. Conclusion

    In this chapter, an advanced controller is developed for PV grid-connected system. The latter is described by fourth-order nonlinear averaged model. The multi-loops nonlinear controller has been designed and developed using three-time singular perturbation technique and averaging theory.

    Using the theoretical analysis (via three-time-scale singular perturbation technique and averaging theory) and simulation, it is proved that the controller does meet the performances for which it was designed, namely: (i) Maximum power point tracking of PV array; (ii) tight regulation of the DC bus voltage; (iii) perfect power factor in the grid; and (iv) global asymptotic stability of the all system.

    Several simulation results have been made that illustrate the high performances of the proposed controller in ideal operating conditions (in the presence of meteorological constant) and its robustness against radiation and temperature change.

    Appendix (Proof)

    Part 1: Eqs. (43)(45) are immediately obtained from expressions (37), (38), (23), and the first and the third equation of expression (24).

    Part 2: The stability of the time-varying system (43)(45) will now be performed in two steps using the averaging theory (e.g. Chapter 10 in Refs. [19, 25]) and the singular perturbation theory (e.g., Chapter 11 in Refs. [19, 23, 24]). The next step consists in using the averaging theory. To this end, let us introduce the time-scale change t¯=ωgt. Using this time, one gets ε1ε2Z¯̇t¯=ρh¯X¯Z¯t¯, ε2Y¯̇t¯=ρg¯X¯Y¯t¯, and X¯̇t¯=ρf¯X¯Y¯Z¯t¯with ρ=1/ωg. In view of expressions (43)(45), it is seem that the functions h¯X¯Z¯t¯, g¯X¯Y¯t¯, and f¯X¯Y¯Z¯t¯as functions of t¯are periodic with period–2π, let us introduce the averaged functions:

    ε1ε2Ż0t¯=ρlimt12πh¯X¯Z¯t¯dt¯=ρh0X0Z0E47
    ε2Ẏ0t¯=ρlimρ012πg¯X¯Y¯t¯dt¯=ρg0X0Y0E48
    Ẋ0t¯=ρlimρ012πf¯X¯Y¯Z¯t¯dt¯=ρf0X0Y0Z0E49

    Since the systems here studied present equilibrium different from zero and in order to satisfy this requirement, a change of variables is introduced such that defines the new system in terms of its error dynamics. Therefore, the error dynamics are defined by introducing: Z˜0=z˜1,0z˜2,0T=z1,0z1,0z2,0z2,0T, Y˜0=y˜1,0y˜2,0T=y1,0y1,0y2,0y2,0T, and X˜0=x˜1,0x˜2,0x˜3,0x˜4,0=x1,0x1,0x2,0x2,0x3,0x3,0x4,0x4,0, where the constant x1,0, x2,0, x3,0, x4,0, y1,0, y2,0, z1,0, and z2,0represent the desired average values of the state variables. Then, since expressions (43)(45) and according to expressions (47)(49), the system can be rewritten into its error-dynamics formulation thus defining the closed-loop error dynamics as:

    ε1ε2Z˜̇0=ρh˜0X˜0Z˜0=ρk1x˜1,0T1f˜1,0X˜0Z˜0k3x˜3,0T3f˜3,0X˜0Z˜0E50
    ε2Y˜̇0=ρg˜0X˜0Y˜0=ρk2x˜2,0T2f˜2,0X˜0Y˜0k4x˜4,0T4f˜4,0X˜0Y˜0E51
    X˜̇0=ρf˜0X˜0Y˜0Z˜0=ρf˜1,0X˜0Z˜0f˜2,0X˜0Y˜0f˜3,0X˜0Z˜0f˜4,0X˜0Y˜0E52

    where

    f˜1,0X˜0Z˜0=x˜2,0+x2,0x˜4,0+x4,0Lirix˜1,0+x1,0Li+z˜1,0+z1,0x˜4,0+x4,0LiE53
    f˜2,0X˜0Y˜0=ipv,0Cpvy˜1,0+y1,0CpvE54
    f˜3,0X˜0Z˜0=rgx˜3,0+x3,0Lg+x˜4,0+x4,0Lgz˜2,0+z2,0E55
    f˜4,0X˜0Y˜0=Egy˜2,0+y2,0+rgy˜2,0+y2,02Cdc+2x˜2,0+x2,0y˜1,0+y1,0riy˜1,0+y1,02CdcE56

    In order to get stability results regarding the system of interest (43)(45), it is sufficient (thanks to averaging theory) to analyze the averaged system (50)(52). Now, the asymptotic stability of the resulting three-time-scale photovoltaic single phase grid system (50)(52) is discussed, which is based on the sequential (double) time-scale analysis similar to the one presented in Refs. [23, 24], it is an extension of the two-time-scale analysis presented [19].

    The use of theory of the three-time-scale singular perturbations for the stability analysis is based on the idea that, for 0<ε1ε2ε21, the trajectories in X˜0, Y˜0, and Z˜0of the system (50)(52) can be approximated by three models: the slow dynamic subsystem (SDS) of full system, the fast dynamic subsystem (FDS), and the ultra-fast dynamic subsystem (UFDS). We can thus find Lyapunov functions for each one of the singularly perturbed subsystems.

    For the UFDS, it is necessary to ensure that the dynamic of expression (50) does not to shift from the quasi-steady-state equilibrium Z˜0=h˜0X˜0. Then, the associated Lyapunov function candidate is obtained by introducing a change of variables Ẑ0=Z˜0h˜0X˜0, so that its equilibrium is centered at zero. By letting τ1=t¯/ε1ε2, the UFSD in function of Ẑ0is defined as follows

    dZ˜0dτ1=ρĥ0X˜0Ẑ0=ρk1x˜4,0+x4,0Liẑ1,0k3x˜4,0+x4,0Lgẑ2,0E57

    in which X˜0is treated as a fixed parameter. Thus, the associated Lyapunov function can be defined by

    VU=12ρPU1ẑ1,02+PU2ẑ2,02E58

    In view of ki>0i=13and x˜4,0+x4,0>0, it is clear that the solutions PU1and PU2can be chosen as

    PU1=Li2k1x˜4,0+x4,0qU1E59
    PU2=Lg2k3x˜4,0+x4,0qU2E60

    with qU1and qU2are positive. Similar to the ultra-fast subsystem, it is necessary to ensure that the dynamic of expression (51) does not to shift from the equilibrium Y˜0=g˜0X˜0. Then, by introducing a change of variables Ŷ0=Y˜0g˜0X˜0, so that its equilibrium is centered at zero and by letting τ2=t¯/ε2, the ultra-fast subsystem can be rewritten in the following form

    dY˜0dτ2=ρĝ0X˜0Ŷ0=ρk2Cpvŷ1,0ψX˜0Ŷ0E61

    with ψ=2k4riŷ1,02Cdc+rgŷ2,022Cdc+ψŷ2,0Cdc+2riCpvCdcx˜2,0T2+ipv,0Cpvx˜2,0+x2,0Cdcŷ1,0.

    Since expression (61), it is seem that the fast subsystem is nonlinear. According to the proposition 4 (part i), it is shown that this subsystem can be made asymptotically stable by letting kj>0(for j=2,4). Moreover, we can find a Lyapunov function, which takes the following quadratic form

    VF=12ρpF1ŷ1,02+pF2ŷ2,02+2pF3ŷ1,0ŷ2,0E62

    with the solutions to the associated Lyapunov given as

    pF1=CpvqF12k22riCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,022qF2k2ψψCpv+Cdck22k4Cpv2E63
    pF2=Cdc4k4ψqF2E64
    pF3=CdcriCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,02k4ψ2ψ+Cdck2k4CpvqF2E65

    in which X˜0is treated as a fixed parameter. Finally, the new slow dynamic subsystem is obtained by substituting of the ultra-fast subsystem equilibrium Z˜0=h˜0X˜0and the fast subsystem equilibrium Y˜0=g˜0X˜0into expression (52)

    X˜̇0=x˜̇1,0x˜̇2,0x˜̇3,0x˜̇4,0T=ρx˜1,0/T1x˜2,0/T2x˜3,0/T3x˜4,0/T4TE66

    It is easy to define the associated Lyapunov function for SDS as follows

    VSX˜0=12ρX˜0TPsX˜0=12ρpS1x1,02+pS2x2,02+pS3x3,02+pS4x4,02E67

    where pS1=T1qS1/2, pS2=T2qS2/2, pS3=T3qS3/2, pS4=T4qS4/2, and qS1, qS2, qS3, qS4are the positive constants.

    Based on these Lyapunov function candidates, the double application of the standard two-time-scale stability analysis is divided in two stages: in the first stage, the stability analysis focusses on proving the stability properties of the degenerated SF-subsystem (slow-fast subsystem). The results obtained will be used in the second stage in order to prove the stability properties for the full SFU-system (slow-fast-ultra-fast subsystem).

    SFStability analysis

    In the first stage, the standard method for two-time-scale systems is applied in which the previously derived Lyapunov functions for the slow and fast subsystems, that is, VSand VF, respectively, must satisfy certain inequalities.

    • Isolated equilibrium of the origin for theSF-subsystem

    The origin X˜0=0Y˜0=0is an isolated equilibrium of the SF-subsystem

    0=ĝ000E68
    0=f˜000h˜0X˜0E69

    moreover, Y˜0=g˜0X˜0is the root of 0=ĝ0X˜0Y˜0which vanishes at X˜0=0, and g˜0X˜0<ϑ1X˜0where ϑ1is a κfunction.

    • Reduced system condition for theSF-subsystem

    Using the Lyapunov function (67) and substituting ρf˜0X˜0Y˜0h˜0X˜0yields

    VSX˜0X˜0Tρf˜0X˜0Y˜0h˜0X˜0α1ϕ12X˜0E70

    where α11and ϕ1X˜0=qS12x˜1,02+qS22x˜2,02+qS32x˜3,02+qS42x˜4,02.

    • Boundary-layer system condition for theSF-subsystem

    Since expression (62)–(65), it is seem that

    VUFX˜0Ƶ˜0Ƶ˜0TρĤX˜0Ƶ˜0QF2ŷ2,02QF1ŷ1,02E71

    where QF1and QF2are defined as follows

    QF1=qF12riqF22ψ4riCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,022ψ+Cdck2k4Cpvŷ1,0+ŷ2,0E72
    QF2=qF22rgqF24ψ4riCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,022ψ+Cdck2k4Cpvŷ1,0+ŷ2,0E73

    For physical point of view and domain of working principle, it is supposed that all physical state variables are bounded in domain of interest, where X˜0minX˜0X˜0Max, Y˜0minY˜0Y˜0Max, and Z˜0minZ˜0Z˜0Max. Therefore, QF1and QF2can take positives minimum possible values QF1,min>0and QF2,min>0. This can be done by ensuring the appropriate selection of qF1, qF2, k2, and k4as follows

    k2k4>CpvCdc4riCpvx˜2,0MaxT2+ipv,0Cpvx˜2,0Max+x2,022ψminrgŷ2,0Maxŷ1,0Max2ψminqF1qF2>riψmin4riCpvx˜2,0MaxT2+ipv,0Cpvx˜2,0Max+x2,022ψmin+Cdck2k4Cpvŷ1,0Max+ŷ2,0MaxE74

    Therefore, the boundary-layer system condition for the SF-subsystem is defined as follows

    VFY˜0Tρĝ0X˜0Y˜0QF2minŷ2,02+QF1minŷ1,022α2ϕ22Ŷ0E75

    where α21and ϕ2Ŷ0=QF2minŷ2,02+QF1minŷ1,02.

    Now, we consider the composite Lyapunov function candidate of the SF-subsystem given as follow

    VSF=1η1VSX˜0+η1VFŶ0E76

    with 0<η1<1. The derivative of VSFpresents new terms which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions

    VsX˜0X˜0Tρf˜0X˜0Y˜0h˜0X˜0f˜0X˜0g˜0X˜0h˜0X˜0θ1ϕ2Ŷ0ϕ1X˜0E77
    VFŶ0X˜0Tρf˜0X˜0g˜0X˜0h˜0X˜0θ2ϕ2Ŷ0ϕ1X˜0E78
    VFŶ0X˜0Tρf˜0X˜0Y˜0h˜0X˜0ρf˜0X˜0g˜0X˜0h˜0X˜0γ1ϕ22Ŷ0E79

    where the constants θ1, θ2, and γ1are non-negative.

    Therefore, from the singular perturbation theory (e.g. Theorem 11.3 in Ref. [19]), it follows that the derivative of VSF is negative-definite for

    ε2<ε2forη1<η1E80
    withε2=α1α2α1γ1+θ1θ2andη1=θ1θ1+θ2.E81

    SFUStability Analysis:

    The stability of full system is analyzed now. In this step, the results obtained in above section will be used in order to prove the asymptotic stability properties of the full SFU-system, which, for convenience, is first rewritten as

    χ˜̇0=F˜0χ˜0Z˜0E82
    ε1ε2Z˜̇0=ĥ0χ˜0Z˜0E83

    where χ˜0=x˜1,0x˜2,0x˜3,0x˜4,0y˜1,0y˜2,0T, similarly to the SFgeneral asymptotic stability analysis presented in the above section, the SFU-system is treated like a two-time-scale singularly perturbed system, where the SF-subsystem is treated as the new slow augmented reduced order. The stability of the full system implicates that the previously derived Lyapunov functions for the new slow and new fast subsystems, that is, VSFχ˜0and VUẐ0, respectively, must satisfy certain inequalities.

    • Isolated equilibrium of the origin for theSFU-subsystem

    The origin χ˜0=0Z˜0=0is an isolated equilibrium of the SFU-subsystem

    0=F˜000E84
    0=ĥ000E85

    Moreover, Z˜0=h˜0χ˜0is the unique root of 0=ĥ0χ˜0Z˜0, which vanishes at χ˜0=0, and h˜0χ˜0<ϑ2χ˜0where ϑ2is a κfunction.

    • Reduced system condition for theSFU-subsystem

    Using Lyapunov function candidate VSFχ˜0given by expression (76), it is easily shown that

    VSFχ˜0χ˜0T=VSFχ˜0x˜1,0VSFχ˜0x˜2,0VSFχ˜0x˜3,0VSFχ˜0x˜4,0VSFχ˜0y˜1,0VSFχ˜0y˜2,0TE86

    Recalling that Ŷ0=Y˜0g˜0X˜0, and since (61)(65), one gets

    pF1x˜2,0=rgCpv2riCpvx˜T2+ipvCpvx˜2,0+x2,022CpvT21riCpvT2x˜2+Cpvx2,0T2+ipvCpv2riipvT2k2ψ3ψ+Cdck2/2k4Cpv2×2ψ+Cdck2/2k4Cpv2Cpv2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,02riCpvT21k2ψψ+Cdck2/2k4CpvqF2=A1qF2
    pF1x˜4,0=rgCdcCpv2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,022ψ+Cdck2/2k4Cpv2k2T4ψ3ψ+Cdck2/2k4Cpv2qF2=A2qF2
    pF2x˜2,0=CdcCpvrg2T21riCpvT2x˜2,0+x2,0T2+ipvCpv2riipvT24k4ψ3qF2=A3qF2,
    pF2x˜4,0=rgCdc24T4k4ψ3qF2=A4qF2,

    pF3x˜4,0=Cdc2rg2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,02ψ+Cdck2/2k4Cpv4k4T4ψ3ψ+Cdck2/2k4Cpv2qF2=A6qF2,
    pF3x˜2,0=Cdcrg2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,02CpvT21riCpvT2x˜2,0+Cpvx2,0T2+ipvCpv2riipvT24k4ψ3ψ+Cdck2/2k4Cpv×2ψ+Cdck2/2k4Cpv+2riCpvT21Cdc4k4ψψ+Cdck2/2k4CpvqF2=A5qF2
    pF1ŷ1,0x˜2,0=Cpv22k2T2qF1+Cpv22riCpvx˜2,0T2+ipvCpvx˜2,0+x2,022k2T2ψψ+Cdck2/2k4CpvqF2=A13qF1+A7qF2
    pF2ŷ2,0x˜2,0=Cdc2CpvT21riCpvT2x˜2,0+Cpvx2,0T2+ipvCpv2riipvT24k4ψ2qF2=A8qF2

    pF3ŷ1,0x˜2,0=CpvCdc2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,04T2k4ψψ+Cdck2/2k4CpvqF2=A9qF2, pF3ŷ1,0x˜4,0=0

    pF3ŷ2,0x˜2,0=Cdc2riCpvx˜2,0T2+ipvCpvx˜2,0+x2,02CpvT21riCpvT2x˜2,0+Cpvx2,0T2+ipvCpv2riipvT24k4ψ2ψ+Cdck2/2k4CpvqF2=A10qF2

    pF1ŷ1,0x˜4,0=0, pF3ŷ2,0x˜4,0=Cdc22riCpvx˜2,0T2+ipvCpvx˜2,0+x2,04T4k4ψ2ψ+Cdck2/2k4CpvqF2=A12qF2, pF2ŷ2,0x˜4,0=Cdc24k4T4ψ2qF2=A11qF2.

    Substituting the expressions of ρF˜0χ˜0h˜0χ˜0and by using absolute value version of Young’s inequality aba2+b2/2, it shows that

    V1χ˜0χ˜0TρF0χ˜0h˜0χ˜0Μ1x˜1,02Μ2x˜2,02Μ3x˜3,02Μ4x˜4,02Μ5ŷ1,02Μ6ŷ2,02E87

    where Μ1=1η1qS12, Μ2=1η1qS221η12T22qS22η121T22, Μ3=1η1qS32, Μ4=1η1qS421η12T42qS42η121T42, M5=qF12M51η12Cpv2, and M6=qF22M6η122ψCdcrgCdcŷ2,0Max2

    M5=η1A13qF1+A12ŷ1,0+A5ŷ2,0+A7+A10qF2CpvMax+η1qF2A32ŷ2,0+A8+A92CpvMax+η1qF222x˜2,0+x2,0Cdc+2riCdcŷ1,0Max2+η12A12ŷ1,0+A5ŷ2,0+A7+A10qF2+A13qF1Max2+η1qF2A22ŷ1,0+A6ŷ2,0+A122x˜2,0+x2,0Cdc+2riCdcŷ1,0Max
    +η1qF22A22ŷ1,0+A6ŷ2,0+A122ψCdcrgCdcŷ2,0Max+η1qF2×A42ŷ2,0+A112x˜2,0+x2,0Cdc+2riCdcŷ1,0Max+η1A22ŷ1,0+A6ŷ2,0+A12qF2Max2+η1qF2ri2ψŷ2,0+2riCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,0ri2ψψ+Cdck2/2k4Cpvŷ1,0Max
    M6=η1qF22A32ŷ2,0+A8+A92CpvMax+η12A32ŷ2,0+A8+A9qF2Max2+η12+A42ŷ2,0+A11qF2Max2+η1qF2A42ŷ2,0+A112ψCdcrgCdcŷ2,0Max+η1qF22A22ŷ1,0+A6ŷ2,0+A122ψCdcrgCdcŷ2,0Max+η1qF22+A42ŷ2,0+A112x˜2,0+x2,0Cdc+2riCdcŷ1,0Max+η1qF2rg4ψŷ2,0+rg2riCpvx˜2,0T2+ipv,0Cpvx˜2,0+x2,04ψψ+Cdck2/2k4Cpvŷ1,0Max

    Now, it needs to ensure that Μ1x˜1,02Μ2x˜2,02Μ3x˜3,02Μ4x˜4,02Μ5ŷ1,02Μ6ŷ2,02<0. This can be done by ensuring the appropriate selection of qS1, qS2, qS3, qS4, qF1, and qF2, such that qS1and qS3have sufficient large positive values, qF1and qF2have sufficient small positive values, and qS2and qS4are limited as follows 11η11η1T222<qS2<1+1η11η1T222, 11η11η1T422<qS4<1+1η11η1T422(for η1<13). Therefore, the reduced system condition for the SFUsystem is defined as follows

    V1χ˜0χ˜0TρF0χ˜0h˜0χ˜0α3ϕ32χ˜0E88

    where α31and ϕ3χ˜0=Μ1x˜1,02+Μ2x˜2,02+Μ3x˜3,02+Μ4x˜4,02+Μ5ŷ1,02+Μ6ŷ2,02.

    • Boundary-layer system condition for the SFU-system

    Using the Lyapunov function given by expression (58)–(60), and substituting ρĥ0X˜0Ẑ0of expression (57), one obtains

    Vuχ˜0Z˜0Z˜0ρĥ0χ˜0Z˜0=qU12ẑ1,02+qU22ẑ2,02α4ϕ42Ẑ0E89

    where α41and ϕ4Ẑ0=qU12ẑ1,02+qU22ẑ2,02.

    Now, we consider the composite Lyapunov function candidate of the full system given as follow

    VSFUχ˜0ẑ0=1η2VSFχ˜0+η2VUẑ0E90

    with 0<η2<1. The derivative of VSFUχ˜0ẑ0presents new terms, which represents the effect of the interconnection between the slow and fast dynamics. These interconnections are assumed to satisfy the following conditions:

    VSFχ˜0χ˜0TρF˜0χ˜0Z˜0F˜0χ˜0h˜0χ˜0θ3ϕ4Ẑ0ϕ3χ˜0E91
    VuẐ0χ˜0TρF˜0χ˜0h˜0χ˜0θ4ϕ4Ẑ0ϕ3χ˜0E92
    VuẐ0χ˜0TρF˜0χ˜0Z˜0F˜0χ˜0h˜0χ˜0γ2ϕ42Ẑ0E93

    the constants θ3, θ4, and γ2are non-negative. Therefore, from the singular perturbation theory (e.g. Theorem 11.3 in Refs. [19, 23, 24]), it follows that the derivative of VSFU is negative-definite when

    ε2<ε2forη1<η1E94

    with ε2=α1α2α1γ1+θ1θ2and η1=θ1θ1+θ2.

    Therefore, it can be inferred that the equilibrium X0=x1,0x2,0x3,0x4,0T, Y0=y1,0y2,0T, and Z0=z1,0z2,0Tof full system, is asymptotically stable for all εi<εii=12. Therefore, Part 2 immediately follows from the averaging theory (e.g. Theorem 10.4 in Ref. [19]). Theorem is established.

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    Youssef Mchaouar, Abdelmajid Abouloifa, Ibtissam Lachkar and Mohammed Fettach (April 1st 2020). A New Control Strategy for Photovoltaic System Connected to the Grid via Three-Time-Scale Singular Perturbation Technique with Performance Analysis, Advanced Statistical Modeling, Forecasting, and Fault Detection in Renewable Energy Systems, Fouzi Harrou and Ying Sun, IntechOpen, DOI: 10.5772/intechopen.89434. Available from:

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