Robust Feedback Linearizing Control With Sliding Mode ... - IEEE Xplore

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Robust Feedback Linearizing Control With Sliding Mode Compensation for a Grid-Connected Photovoltaic Inverter System Under Unbalanced Grid Voltages Adel Merabet, Member, IEEE, Labib Labib, Member, IEEE, Amer M. Y. M. Ghias, Member, IEEE, Chaouki Ghenai, and Tareq Salameh

Abstract—A robust feedback linearizing control strategy, based on sliding mode compensation, is proposed for the operation of a grid-connected photovoltaic inverter system under grid faults, characterized by unbalanced voltages, to meet low-voltage ride through requirements. Under normal grid condition, the control system is developed for maximum power transfer from the photovoltaic source to the grid by maximum power point tracking operation of the dc–dc converter, and regulation of the dc-link voltage and the current at the inverter-grid side. Under grid fault, which is unbalanced grid voltage due to voltage dips, the active power is regulated to reduce the current excess and the reactive power is injected to avoid the inverter damage or disconnection, while the dc-link voltage is controlled via the dc–dc converter. A sliding mode compensator is injected into the control system to enhance its robustness to uncertainties. The feedback linearizing control schemes are developed from the grid model at the inverter side and the dc-link model at the dc–dc converter side. The proposed control strategies are experimentally validated on a three-phase grid-connected photovoltaic inverter system and experimental results show that the control system is effective in terms of voltage and power control with smooth transitions between the modes. Index Terms—Compensation, feedback control, grid fault, photovoltaic (PV) systems, sliding mode control, uncertainties.

I. INTRODUCTION

D

UE to the rapid increase of photovoltaic (PV) systems integration into the electric grid, the PV control systems

Manuscript received November 27, 2016; revised January 14, 2017; accepted January 30, 2017. This work was supported in part by the Canada Foundation for Innovation project 30527 and in part by the University of Sharjah under Grant 1602040256-P. A. Merabet is with the Division of Engineering, Saint Mary’s University, Halifax, NS B3H 3C3, Canada, and also with the Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah 27272, United Arab Emirates (e-mail: [email protected]). L. Labib is with the Division of Engineering, Saint Mary’s University, Halifax, NS B3H 3C3, Canada (e-mail: [email protected]). A. M. Y. M. Ghias is with the Department of Electrical and Computer Engineering, University of Sharjah, Sharjah 27272, United Arab Emirates (e-mail: [email protected]). C. Ghenai and T. Salameh are with the Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah 27272, United Arab Emirates (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JPHOTOV.2017.2667724

encounter greater challenges to maintain the grid stability and reliability because of the high intermittency of the input source and the voltage dips occurring in the grid [1]. In order to ensure a proper integration, the utilities have issued voltage grid codes to the PV systems imposing on these systems to contribute to the grid support during grid faults. Therefore, the PV systems are required to provide a low-voltage ride-through (LVRT) capability and remain in the grid-connected mode in the presence of grid faults. In that case, as the PV systems operate in different modes, the control system should be efficient to maintain the power quality under normal operation with a maximum power point tracking (MPPT) and to meet the LVRT requirements in the grid fault mode [2]–[4]. Feedback linearizing control (FLC) has been considered one of the efficient control approaches to deal with nonlinearities and uncertainties in the system [5], [6]. It has been widely applied in power converters for grid-connected PV energy systems due to the availability of adequate mathematical models and the accessibility to the states by measurement, where the main control objective is to regulate the dc-link voltage and the power injected to the grid. Several FLC algorithms were developed for a PV system under balanced voltage grid. In [7], a feedback control was developed for the PV system with three disturbance estimators for the states (d-q currents and dc-link voltage), which complicates the analysis as the three disturbances are estimated and they may not be directly linked to the controlled outputs. In [8], a feedforward decoupling control method was used to regulate separately the d-q currents and the dc-link voltage was regulated to follow a reference generated by an MPPT algorithm and the PV current is required in the control law to deal with variable irradiation. In [9], an FLC was designed without taking into consideration the uncertainties, and the method was upgraded in [10] by including the uncertainties, defined using the PV current and the upper bounds for the parametric variation and solar irradiance, in the control design, which makes it relying on the exact values of the upper bounds. Also, this system is partially linearizable with no discussion about the stability analysis of the internal dynamics. In [11], a feedback linearizing method was applied to control active and reactive power, without including uncertainties, from the MPPT

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Fig. 1.

IEEE JOURNAL OF PHOTOVOLTAICS

Three-phase grid-connected PV inverter system.

algorithm, and the dc-link voltage is maintained constant through the dc–dc converter without details about its duty cycle. In all these works, faults in the grid were not considered and the objective was extracting the maximum power from the PV source, which is not always the case when dealing with unbalanced grid voltages. Sliding mode control is considered an efficient control strategy in ensuring zero tracking error and uncertainties rejection [12], [13]. It can be integrated with the FLC method to deal with uncertainties, which will be investigated in this work. Furthermore, only uncertainties directly linked to the controlled outputs will be compensated, which reduces the amount of computation compared to previous works [7], [9], [10]. The FLC will be designed for normal conditions to regulate the dc-link voltage and the q-current of the grid for maximum power transfer from the PV source, via the dc–dc converter, to the inverter-grid. In case of unbalanced grid voltages, the active and reactive power will be controlled to limit the current excess and inject the reactive current to meet LVRT requirements. Furthermore, a constant dc-link voltage will be maintained through the regulation of the dc–dc converter. The advantage of the proposed control system is that the disturbance compensator will deal only with the disturbances directly acting on the controlled outputs, which decreases the control law complexity and computation. A robust FLC strategy with sliding mode compensation is proposed for simultaneous control of the dc-link voltage and the power at the PV converter and the grid inverter sides under normal grid conditions (mode I) and grid faults characterized by unbalanced voltages (mode II). The paper is organized as follows. The modeling of the three-phase grid connected PV system and the voltage dip characterization are presented in Section II. The FLC scheme, for mode I, is presented in Section III. Power calculation and model under unbalanced voltage with the appropriate control scheme are detailed in Section IV. Stability analysis and control design, for both modes, are discussed in Section V. In Section VI, experimental results are provided and discussed to show the effectiveness of the proposed control strategy. Finally, Section VII provides the conclusion. II. GRID-CONNECTED PV INVERTER SYSTEM A. System Modeling The grid-connected PV energy system, shown in Fig. 1, includes a PV module, a boost dc–dc converter, a capacitive

dc-link, a three-phase two level inverter, and a three-phase grid. Voltage dynamics of the three-phase inverter is governed, in the rotating reference frame (d, q), by ⎧ R 1 ed 1 did ⎪ ⎪ = − id + ωiq − + vd + ηd ⎨ dt L L L L (1) ⎪ R 1 di e ⎪ ⎩ q = − iq − ωid − q + vq + 1 ηq dt L L L L where ed , eq are the (d, q) component of the grid voltages, id , iq are the (d, q) components of grid currents, vd , vq are the (d, q) components of the inverter output voltages, ω is the grid frequency, and R and L are the resistance and the inductance of the filter, respectively. The uncertainties ηd and ηq are given by ⎧ did ⎪ ⎪ + ζd ⎨ ηd = ΔRid − ωΔLiq + Δed + ΔL dt (2) ⎪ ⎪ ⎩ ηq = ΔRiq + ωΔLid + Δeq + ΔL diq + ζq dt where ΔR and ΔL are the parametric variations, Δid and Δiq are the uncertainties in d-q currents due to the parametric variations and measurements, Δed and Δeq are the uncertainties due to the measurements of the grid voltage, and ζd and ζd are any other external disturbances acting on the system such as pulse width modulation (PWM) offset, the angular frequency variation, and unmodeled quantities. It is assumed that all uncertainties are bounded with known bounds. The voltage–current relationship at the dc-link between the dc–dc converter and the inverter is expressed as dvdc = i0 − idc (3) dt where vdc is the dc-link voltage, idc is the current at the input of the inverter, i0 is the current at the output of the dc–dc converter, and C is the dc-link capacitance. The power balance relationship between the dc input and the ac output of the inverter, with power losses in the inverter considered negligible, is given by C

3 (ed id + eq iq ). (4) 2 The dc-link voltage dynamics, at the grid inverter side, is expressed, from (3) and (4), by vdc idc =

3 dvdc 1 =− (ed id + eq iq ) + i0 . dt 2Cvdc C

(5)

The dynamics (5) can be rewritten, by considering system’s uncertainties, as 3 1 dvdc =− (ed id + eq iq ) + ηv dt 2Cvdc C

(6)

where ηv is the uncertainty variable and represents the uncertainties in the system such as ηv = −

3 (ed Δid + Δed (id + Δid ) + eq Δiq 2vdc

+ Δeq (iq + Δiq )) + Δv + i0 + ΔC v˙ dc + ζv

(7)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. MERABET et al.: ROBUST FEEDBACK LINEARIZING CONTROL WITH SLIDING MODE COMPENSATION

where Δv are the uncertainties due to the dc-link voltage measurements, i0 is PV current, considered as an unknown external disturbance, and ζv is any other external disturbance. In the grid-connected PV system, the PV current and power vary slowly compared to the dynamics of the system and by considering the slow changes in model parameters, it can be assumed that the uncertainty variables {ηd , ηq , ηv } are bounded and their time derivatives can be assumed to be around zero, as time tends to infinity. Therefore, with the objective to simplify the control design and the stability analysis, it is assumed that [7]

3

The injected reactive current Ir∗ , to support the grid, is expressed as [14] ⎧ Vdip ≤ 0.1 ⎪ ⎨ 0%, ∗ (11) Ir = 200 Vdip %, 0.1 < Vdip ≤ 0.5 ⎪ ⎩ 100%, Vdip > 0.5.

(8)

In order to meet the LVRT requirements and not exceeding the maximum current limit Im ax of the inverter, to avoid the inverter damage or disconnection, the references of the active power and the injected reactive power follow the expressions P0∗ = |S| 1 − I ∗r 2 (12) Q∗0 = |S| Ir∗

Finally, the grid-connected PV inverter system, from (1) and (6), is expressed under the following state model:

where |S| is the maximum apparent power of the grid and is obtained by

η˙ d = 0, η˙ q = 0, and η˙ v = 0.

x(t) ˙ = f (x, t) + g1 (x)v(t) + g2 (x)η(t)

|S| = (|ea |rm s + |eb |rm s + |ec |rm s ) Im ax . (9)

where x = [id iq vdc ]T is the state vector, v = [vd vq ]T is the input vector, and η = [ηd ηq ηv ]T is the disturbance vector. The vectors f, g1 , and g2 are defined by ⎤ R ed ⎢ − L id + ωiq − L ⎥ ⎥ ⎢ ⎥ ⎢ eq R ⎥ g1 (x) = [ g11 ⎢ i − ωi − − d f (x, t) = ⎢ Lq q L ⎥ ⎥ ⎢ ⎦ ⎣ 3 − (ed id + eq iq ) 2Cvdc ⎡ ⎤ 1 0 ⎢L ⎥ ⎢ ⎥ ⎢ ⎥ 1 =⎢0 ⎥ ⎢ ⎥ L⎦ ⎣ 0 0 ⎡ ⎤ 1 0 0 ⎢L ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎥. g2 (x) = [ g21 g22 g23 ] = ⎢ 0 0 ⎢ ⎥ L ⎢ ⎥ ⎣ ⎦ 1 0 0 C ⎡

Grid fault, due to voltage unbalance, is characterized by a reduction in the voltage magnitudes of the grid phases (one or more) [8]–[10]. In this work, the grid voltages are available by measurement and the voltage dip Vdip is detected through the expression

min (|ea |rm s , |eb |rm s , |ec |rm s ) 1− p.u. Ebase

The apparent power has been considered, instead of the rated power, in power reference (12) in order to take into consideration the voltage sag due to the voltage unbalance in one or more phases with asymmetrical voltage drops compared to conventional scenarios of symmetrical voltage drops in the three phases. III. CONTROL SYSTEM UNDER NORMAL GRID CONDITIONS

g12 ]

B. Voltage Dip Characterization During Grid Fault

Vdip =

(13)

(10)

where Ebase is the base value of the rated grid voltage, and |ea |rm s , |eb |rm s , and |ec |rm s are the rms values of the threephase voltages ea , eb , and ec , respectively (see Appendix C).

During normal grid conditions (mode I), where the grid voltage is balanced, the dc–dc converter is controlled via the conventional MPPT controller, based on the PV voltage and current, operation to harvest the maximum power from the PV module, while the dc-link voltage and the current are regulated through the three-phase inverter to deliver the entire power produced by the PV source to the grid. At the three-phase inverter, the outputs to be controlled are the q-component of the grid current and the dc-link voltage h1 = iq . (14) y(t) = h(x) = h2 = vdc The objective of the FLC is to find a control law that minimizes the tracking error between the system output and a trajectory reference [5]. This control process is developed through differentiating the outputs’ system until the appearance of the control input in the output equation, where the number of differentiation represents the relative degree for each output to the control input. From the system model (9) and the outputs (14), and taking into consideration the assumption (8) with Lie derivatives (see Appendix A), the outputs’ differentiations are given by y˙ 1 = Lf h1 + Lg 11 h1 vd + Lg 12 h1 vq + Lg 22 h1 ηq (15) y˙ 2 = Lf h2 + Lg 11 h2 vd + Lg 12 h2 vq + Lg 23 h2 ηv y¨2 = L2f h2 + Lg 11 Lf h2 vd + Lg 12 Lf h2 vq + Lg 23 Lf h2 ηv + Lg 23 h2 η˙ v = 0.

(16)

It can be noticed from (15) and (16) that the input relative degrees are r1 = 1; r2 = 2 for the q-current and the dc-link



voltage, respectively. Furthermore, the relative degree of the output (iq ) to the disturbance (ηq ) is r1 = 1 and the relative degree of the output (vdc ) to the disturbance (ηv ) is r2 = 1. Under matrix form, the output dynamics representation is given by Lf h1 0 Lg 12 h1 vd y˙ 1 = + 2 Lf h2 y¨2 Lg 11 Lf h2 Lg 12 Lf h2 vq ⎡ ⎤ ηd 0 0 Lg 22 h1 ⎢ ⎥ + ⎣ ηq ⎦ 0 0 Lg 23 Lf h2 ηv = F(x) + G(x)v + B(x)η.

(17)

Since, the total relative degree of the system (r1 + r2 = 3) is equal to the system’s order, there is no zero dynamics and the nonlinear system (9) is fully input–output feedback linearizable. Also, the states are available by measurement and the matrix G(x) is nonsingular and invertible if vdc and ed are different than zero, which is practically valid for the threephase grid connected PV system. Therefore, linear dynamics, for the controlled outputs, can be achieved through the following feedback control: v = G(x)−1 [u − F(x)]

(18)

where u = [u1 u2 ] is the new control input to be identified such as the disturbance will be compensated. The standard pole placement choice for the control u, to ensure stable tracking error dynamics, cannot reject the disturbances ηq and ηv as their relative degrees are less or equal than the input relative degrees [5]. Concerning the disturbance ηd , its compensation is not required as it is not directly affecting the controlled output (iq , vdc ) in (17). In order to achieve the disturbance rejection, a sliding mode compensator is injected into the control law u as follows: u1 = −keq + i˙ ∗q − βq αq sgn(sq ) (19) ∗ − βv αv sgn(sv ) u2 = −k1 ev − k2 e˙ v + v¨dc T

damage of the inverter due to this overcurrent surpassing the inverter maximum rated current. Therefore, a safety mechanism is activated to remove the excess of active power production and inject the reactive power to meet LVRT requirements. In this mode, the power is regulated to track reference trajectories (12) through controlling the grid currents from the three-phase inverter and the dc–dc converter is controlled to regulate the dc-link voltage instead of the MPPT operation. A. Power Calculation Under Unbalanced Grid Voltage The instantaneous active and reactive powers, under unbalanced grid voltage, are expressed as [3] P = P0 + Pc cos(2ωt) + Ps sin(2ωt) (21) Q = Q0 + Qc cos(2ωt) + Qs sin(2ωt) where P0 and Q0 are the average values of the instantaneous active and reactive powers, respectively, and Pc , Ps , Qc , and Qs are the oscillating terms, of second-order, in these instantaneous powers. The power components (P0 Pc , Ps , Q0 , Qc , Qs ) are expressed with respect of positive- and negative-sequences of the voltages and currents, defined in synchronous reference frames ((d, q)+ and (d, q)− ), such as ⎧ 3 + + − − + + − − ⎪ ⎪ ⎪ P0 = 2 (ed id + eq iq + ed id + eq iq ) ⎪ ⎪ ⎪ ⎨ 3 + + − − + + − (22) Pc = (e− d id + eq iq + ed id + eq iq ) ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Ps = 3 (e− i+ − e− i+ − e+ i− + e+ i− ) q d d q d q 2 q d ⎧ 3 + + + − − − − ⎪ ⎪ Q0 = (e+ q id − ed iq + eq id − ed iq ) ⎪ ⎪ 2 ⎪ ⎪ ⎨ 3 + + − − + + − (23) Qc = (e− q id − ed iq + eq id − ed iq ) ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Qs = 3 (−e− i+ − e− i+ + e+ i− + e+ i− ) q q q q d d d d 2

IV. CONTROL SYSTEM FOR GRID FAULTS

− + − where e+ d , eq , ed , eq are the positive- and negative sequences − − + of the grid (d, q) voltage components, and i+ d , iq , id , iq are the positive- and negative sequences of the (d, q) current components. In this work, the objective is to control the active and reactive powers through controlling the current components − − + (i+ d , iq , id , iq ). Therefore, only four power components from (22) and (23) can be controlled such as [3] ⎡ + ⎤⎡ + ⎤ ⎡ ⎤ ed e+ e− e− id P0 q q d ⎥⎢ + ⎥ + − − + ⎢P ⎥ 3 ⎢ ⎢ ed eq ed eq ⎥ ⎢ iq ⎥ ⎢ c⎥ ⎥⎢ ⎥. (24) ⎢ ⎥= ⎢ ⎥⎢ − ⎥ − − + ⎣ Ps ⎦ 2 ⎢ e+ ⎣ eq −ed −eq d ⎦ ⎣ id ⎦ Q0 i− e+ −e+ e− −e− q q q d d

During grid faults (mode II), where one or more grid voltages are reduced, the injected current will increase to maintain the same amount of delivered power previous to the fault. However, this current increase may lead to the disconnection or the

The power components (24) have been selected in order to allow specified active and reactive powers (P0 , Q0 ) to be transferred by the inverter while eliminating the active power oscillations terms (Pc , Ps ) through the following

∗ where i∗q is the q-current reference, vdc is the dc-link volt∗ age reference, eq = iq − iq is the q-current tracking error, ∗ − vdc is the dc-link voltage tracking error, k, k1 , and ev = vdc k2 are positive control gains, αq and αv are the positive sliding gains and βq = Lg 22 h1 and βv = Lg 23 Lf h2 . In this work, i∗q = 0 in order to operate the system under a unity power factor. The sliding surfaces sq and sv are defined, based on dynamics (15) and (16), respectively, as t sq (t) = eq (t) + k 0 eq (τ )dτ (20) t sv (t) = e˙ v (t) + k2 ev (t) + k1 0 ev (τ )dτ .


⎤

⎡

transformation: ⎡ + ⎤ ed i+∗ d ⎢ i+∗ ⎥ 2 ⎢ ⎢ e− ⎢ q ⎥ d ⎢ −∗ ⎥ = ⎢ − ⎣ id ⎦ 3 ⎢ e ⎣ q i−∗ e+ q q ⎡

e+ q e− q −e− d −e+ d

e− d e+ d −e+ q e− q

⎤−1 e− q ⎥ e+ q ⎥ ⎥ ⎥ e+ d ⎦ −e− d

⎡

P0∗ ⎢ ∗ ⎢ Pc = ⎢ ⎢P∗ = ⎣ s Q∗0

1 ⎢L ⎢ ⎢ ⎡ ±⎤ ⎢1 ⎢L g21 ⎢ ⎢ ±⎥ ⎢ ± g2 (x) = ⎣ g22 ⎦ = ⎢ 0 ⎢ ⎢ g23 ⎢ ⎢0 ⎢ ⎣ 0

⎤ ⎥ 0⎥ ⎥. 0⎥ ⎦

(25) The grid current dynamics, using the positive and negativesequences of voltages and currents, is expressed by ⎧ ± di R ⎪ ⎪ d = − i± ± ωi± ⎨ q − dt L d ± ⎪ ⎪ ⎩ diq = − R i± ∓ ωi± − d dt L q

e± 1 d + vd± + L L e± 1 q + vq± + L L

(26)

1 ± η L q

3 dvdc 1 + + + − − − − =− ηv (e+ d id + eq iq + ed id + eq iq ) + dt 2Cvdc C (27) where the second-order oscillating terms of the active power are considered disturbance and included in the variable ηv . The system, of (26) and (27), is expressed under state representation as x˙ ± (t) = f ± (x± , t) + g1± (x± )v± (t) + g2± (x)η ± (t)

(28)

where x± = [id ± iq ± ]T is the state vector, v± = [vd ± vq ± ]T is the input vector, and η ± = [ηd ± ηq ± ηv ]T is the disturbance vector. The vectors f ± , g1 ± , and g2 ± are defined by ⎡

e± R ± d − i± ± ωi − q L d L −

e± R ± q iq ∓ ωi± − d L L

3 + − − − − (e+ i+ + e+ q iq + ed id + eq iq ) 2Cvdc d d ⎤ ⎡1 0 0 0 ⎥ ⎢L ⎥ ⎢ ⎥ ⎡ ±⎤ ⎢1 ⎥ ⎢ g11 0 0 0 ⎥ ⎢L ⎥ ⎢ ⎥ ⎢ ± g1± (x) = ⎣ g12 ⎦ = ⎢ 0 1 0 0⎥ ⎥ ⎢ L ⎥ ⎢ g13 ⎥ ⎢ ⎥ ⎢ 1 ⎣0 0 0⎦ L 0 0 0 0 −

0

0 0

0

0 1 L 1 L 0

0 0

0

0 0

0

0 0

0 1 C

0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

B. Power Control 1 ± η L d

+ − T + − T ± + − T ± where i± d = [ id id ] , iq = [ iq iq ] , vd = [ vd vd ] , and vq± = [ vq+ vq− ]T . Furthermore, the dc-link voltage dynamics (6) becomes

⎢ ⎢ ⎢ ⎢ f ± (x, t) = ⎢ ⎢ ⎢ ⎢ ⎣

5

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

The regulation of the active power P and the reactive power ± Q is conducted through controlling the d-q currents (i± d , iq ) to track the current profile (25). Using the nonlinear system (28), the controlled outputs are ± h1 = i± d ± ± y (t) = h (x) = . (29) ± h± 2 = iq Similar analysis, as in the previous control development, is used to obtain the outputs’ derivatives as follows: ⎧ ± ± ± ± ± ±h v ±h v y˙ 1 = Lf ± h± ⎪ 1 + Lg 11 1 d + Lg 11 1 q ⎪ ⎪ ⎪ ± ± ± ± ⎨ +Lg ± h η + Lg ± h ηq 21

± ⎪ ⎪ ⎪ y˙ 2

⎪ ⎩

=

Lf ± h± 2

1

d

+ Lg 11± h2 vd± ± +Lg 22± h± 2 ηq .

22

1

+ Lg 12± h2 vq± + Lg 21± h2 ηd±

(30)

Details of the derivatives are provided in Appendix B. The input relative degrees are r1± = [1 1]T ; r2± = [1 1]T for ± the i± d and iq outputs, respectively. In this case, the total relative degree of the system (r1 + r2 = 4) is less than the order of the system (28), which is n = 5. Therefore, the nonlinear system (28) is partially input–output feedback linearizable. The dc-link voltage dynamics can be considered as the additional dynamics, which will be controlled separately in the next section. Furthermore, the disturbance relative degrees r1± = [1 · 1]T ; ± r2± = [1 · 1]T for the disturbances n± d and nq are equal to the input relative degrees; therefore, they are required to be compensated by the control law. Under the matrix form, the system (30) is given by ± ± Lg 12± h± Lg 11± h± vd Lf ± h± y˙ 1 1 1 1 = + ± ± ± ± ± ± ± L h L h vq± y˙ 2 Lf h1 g 11 2 g 12 2 ⎡ ⎤ ηd± ± ±h Lg 21± h± L 0 2×1 ⎢ g 22 1 1 ⎥ + ⎣ ηq± ⎦ ± ± ± ± Lg 21 h2 Lg 22 h2 02×1 ηv = F± (x± , t) + G± (x± )v± + B± (x± )η ± .

(31)

G± (x) is diagonal and invertible (see Appendix B). The feedback control law has the following structure: v± = G± (x)−1 [u± − F± (x± , t)].

(32)



± T The control law u± = [u± 1 u2 ] is defined as ± ± ± ˙ ±∗ u1 = −kd e± d + id − αd βd sgn(sd ) ± ± ± ˙ ±∗ u± 2 = −kq eq + iq − αq βq sgn(sq )

(33)

+∗ −∗ T +∗ −∗ T ±∗ where i±∗ d = [ id id ] and iq = [ iq iq ] are the d-q current references carried out from the power reference in (25), and ±∗ ± ± ±∗ ± e± d = id − id and ed = id − id are their respective tracking errors, kd and kq are positive control gains, αd and αq are posi± ± ±h . tive sliding gains, and βd± = Lg 21± h± 1 , βq = Lg 22 2 ± ± The sliding surfaces sd and sq are defined, based on the dynamics (30), as ± t ± sd (t) = e± d (t) + kd 0 ed (τ )dτ (34) t ± ± s± q (t) = eq (t) + kq 0 eq (τ )dτ .

Finally, the voltage command is given by vd = vd+ + vd + v =v +v ⇒ vq = vq+ + vq .

(35)

C. Voltage Control The dc–dc converter is controlled to achieve the dc-link voltage regulation, of dynamics (5), which is reorganized as dvdc 1 1 = i0 + ηv dt C C

(36)

where i0 is the input control, linked to the PV current, and ηv is the new disturbance variable at the inverter-grid side. The disturbance ηv is expressed by 3 ηv = − P +f 2vdc

(37)

where P is the active power under unbalanced grid voltage and f represents uncertainties in the system due to parametric variations, unmodeled quantities, and any external disturbances. The proposed feedback control law has the following form: αv ∗ sgn(sv ) − (38) i∗0 = C −kv ev + v˙ dc C where sv is the sliding surface for the dc-link voltage control. The sliding surface sv is defined, based on the dc-link voltage dynamics (36), as t ev (τ )dτ . (39) sv (t) = ev (t) + kv 0

In the control law (38), the sliding mode compensator has been integrated to deal with uncertainties in the system. The proposed control strategy is illustrated in Fig. 2. V. STABILITY ANALYSIS A. Closed Loop System Under Balanced Grid Voltage In mode I, the closed loop system includes the PV gridconnected PV system (17) and the feedback control law (18)

Fig. 2. system.

Proposed control scheme for the three-phase grid-connected PV

and (19). It has the following error dynamics: e˙ q (t) + keq (t) = βq (ηq (t) − αq sgn(sq (t))) e¨v (t) + k2 e˙ v (t) + k1 ev (t) = βv (ηv (t) − αv sgn(sv (t))). (40) Let us define the following Lyapunov function candidate: V (t) = 12 s2q (t) + 12 s2v (t).

(41)

The time derivative of (41) is carried out, using (40), by V˙ (t) = sq (t)s˙ q (t) + sv (t)s˙ v (t) = − sq {e˙ q (t) + keq (t)} + sv {¨ ev (t) + k2 e˙ v (t) + k1 ev (t)} = βq sq {ηq − αq sgn(sq )} + βv sv {ηv − αv sgn(sv )} ≤ − βq {αq − |ηq |} |sq | − βv {αv − |ηv |} |sv | ≤ 0.

(42)

The condition V˙ (t) ≤ 0 ensures the asymptotic stability of the closed loop system under balanced grid voltage. B. Closed Loop System Under Unbalanced Grid Voltage In mode II, the closed loop system includes the PV gridconnected PV system, (31) and (32), the feedback control law (33) for the three-phase inverter, the dc-voltage dynamics (26) and the feedback control law (38) for the dc–dc converter. It has the following tracking error dynamics: ⎧ ± ± ± ± e˙ d (t) + kd e± ⎪ d (t) = βd (ηd (t) − αd sgn(sd (t))) ⎪ ⎪ ⎪ ⎨ ± ± ± ± e˙ q (t) + kq e± q (t) = βq (ηq (t) − αq sgn(sq (t))) (43) ⎪ ⎪ ⎪ ⎪ ⎩ e˙ (t) + k e (t) = 1 (η (t) − α sgn(s (t))). v v v v v v C


Let us definite the following Lyapunov function candidate: V (t) =

1 1 1 ±2 s (t) + s±2 (t) + s2v (t). 2 d 2 q 2

TABLE I PARAMETERS OF THE PV PANEL (CS6P-260P)

(44) Symbol

The time derivative of (44) is carried out, using (43), by ± ± ± V˙ (t) = s± d (t)s˙ d (t) + sq (t)s˙ q (t) + sv (t)s˙ v (t) ± ± ± ± ± = s± d (t) e˙ d (t) + kd ed (t) + sq (t) e˙ q (t) + kq eq (t)

Pm a x Vo p t Io p t

± ± ± ± ± ± ± =± d sd {ηd − αd sgn(sd )} + βq sq {ηq − αq sgn(sq )}

1 sv {ηv − αv sgn(sv )} C ± ± 1 ± s ≤ − βd αd − ηd d 1 ± ± 1 1 ± sq − {αv − |ηv |} |sv | − ηq − βq αq C 1 +

(45)

The condition V˙ (t) ≤ 0 ensures the asymptotic stability of the closed loop system in mode II. C. Closed Design In the control design, the choice of the parameters kd , kq , and kv is based on the pole placement of the sliding surface dynamics to follow appropriate dynamics. The sliding mode dynamics, carried out from (20), (34), and (39), are the following. 1) Balanced Grid Voltage: s˙ q (t) = e˙ q (t) + keq (t) (46) s˙ v (t) = e¨v (t) + k2 e˙ v (t) + k1 ev (t). 2) Unbalanced Grid Voltage: ⎧ ± ± ± ⎪ ⎨ s˙ d (t) = e˙ d (t) + kd ed (t) ± ± s˙ ± q (t) = e˙ q (t) + kq eq (t) ⎪ ⎩ s˙ v (t) = e˙ v (t) + k1 ev (t).

(47)

Depending on the dynamics order, the control gains can be selected so that the dynamics (46) and (47) are imitating the behavior of the following first- and second-order dynamics: ⎧ ⎨S + 1 α (48) ⎩ 2 S + 2ζωn S + ωn2 where S is Laplace variable, α is the time constant for the firstorder dynamics, ζ is the damping ratio, and ωn is the undamped natural frequency of the second-order dynamics. Pole placement consists on choosing the control parameters based on a predefined selection of α, ζ, and ωn . Table II provides the values of the control parameters.

Quantity

Value

Nominal maximum power Optimum operating voltage Optimum operating current

260 W 30.4 V 8.56 A

TABLE II PARAMETERS OF THE EMULATED GRID

+ sv (t) (e˙ v (t) + kv ev (t))

≤ 0.

7

Symbol

Quantity

Value

Grid (power supply) Vp ea k f L C Vd c

Nominal voltage Frequency Filter

45 V 60 Hz

Inductance Inverter

25 mH

DC-link capacitance Nominal dc-link voltage

2.72 mF 65 V

VI. EXPERIMENTAL RESULTS The experimental three-phase grid connected PV system is depicted in Fig. 2. It consists of PV module, dc–dc converter, inverter, resistive-inductive filter, and a three-phase power supply to emulate the grid. The properties of these elements are provided in Tables I and II. The voltages and the currents were measured by the data acquisition interface (OP8660). The proposed control system was built in Simulink and run in the OPALRT real-time simulator (OP5600) through the RT-LAB software [15], [16]. Conventional sinusoidal PWM, with a carrier wave of frequency 5 kHz, was used to provide switch signals to the converters as shown in Fig. 2. The sampling times for the executed model and the control system are 50 μs and 50 ms, respectively. Initial experimental test was conducted to check the efficiency of the system and the power transfer from the PV panel to the emulated grid under normal conditions, which can be considered appropriate as shown in Fig. 3 for the power responses. The grid fault was accomplished through the fault mechanism, shown in Fig. 2, where variable resistors are connected in parallel with switches. The unfaulty grid (balanced grid voltages) occurs when all switches are selected ON, and the faulty grid (unbalanced grid voltages) occurs when one or more switches are selected OFF to drop the voltage in one or more phases. The amount of the voltage drop is related to the resistance value of the variable resistor to allow creating asymmetric voltage drops. Through that mechanism different faults can be experimentally created as shown in Fig. 4. The control gains were chosen by pole placement method and kept constant in all experimental cases. The values are provided in Table III. The irradiance is manipulated through four industrial lamps, where full irradiance means that all lamps are ON. A. Operation Under Different Grid Faults The controlled grid-connected PV system is tested under full irradiance, constant dc-link voltage reference, and under the



Fig. 5. Operation under different faults. (a) Voltage dip, (b) dc-link voltage regulation, and (c) active and reactive power.

Fig. 3. Powers in the system. (a) PV power versus grid power and (b) efficiency.

Fig. 6. DC-link voltage transient response. (a) Transition from mode I to mode II. (b) Transition from mode II to mode I.

Fig. 4. Grid voltage under different faults. (a) Severe single-phase fault, (b) small single-phase fault, (c) two-phase fault, and (d) three-phase fault.

TABLE III CONTROL GAINS Symbol

Quantity

Value Mode I

k, k 1 , k 2 , α q , α v

Grid side control gains

kv , αv kd , kq , αd , αq

DC-link control gains Grid side control gains

1.5 × 103 , 4.2 × 106 , 25 × 103 , 10, 10

following faults (10–20 s: severe single-phase fault; 25–35 s: small single-phase fault; 40–50 s: two-phase fault; 55–65 s: three-phase fault). It can be observed that the fault is adequately detected by the voltage dip (Vdip ) shown in Fig. 5(a). The dc-link voltage is successfully regulated to follow a constant profile, as shown in Fig. 5(b), despite the transition between normal and fault modes. The power responses, illustrated in Fig. 5(c), show that the produced instantaneous active power decreases and the injected reactive power increases with respect of the voltage dip and the power-current reference (25). Furthermore, the oscillatory effects, from the power calculation (22), are successfully attenuated by the proposed control strategy. These results demonstrate the capabilities of the proposed controller to track a constant voltage reference variable power with a good transition between the modes and under different faults as shown in Fig. 6, where the voltage in mode II presents higher oscillation compared to mode I due to the positive and negative sequences under grid faults. These sequences affect the grid currents as shown in Fig. 7.

Model II 100, 1 2 × 104 , 2.3 × 104 , 1, 1

B. Variable Voltage Tracking In order to verify the voltage tracking performance, the proposed control strategy is tested to track a variable voltage


Fig. 7.

Grid current under fault.

9

Fig. 9. Operation under parametric variations. (a) Voltage dip, (b) dc-link voltage regulation, and (c) active and reactive power.

Fig. 8. Operation under variable voltage reference. (a) Voltage dip, (b) dc-link voltage regulation, and (c) active and reactive power.

reference in both modes as shown in Fig. 8(a) and (b). The tracking occurs very fast with a good transition. It can be observed that the voltage dip is affected by the dc-voltage as the gird voltage is influenced by these changes. Despite the change in the dc-link voltage, the power, active and reactive, is well regulated by the control system as shown in Fig. 8(c) C. Robustness to Parametric Uncertainties In this case, the parameters of R and L are increased, in the control laws (18) and (31), by 50%. From the results, shown in Fig. 9, it can be observed that the regulated voltage is not

Fig. 10. Operation under variable irradiance. (a) PV voltage, (b) grid voltage, (c) dc-link voltage regulation, and (d) active and reactive power.

affected as this variation will be compensated by the sliding mode compensator integrated into the feedback control system. D. Robustness to Unknown External Disturbance (Irradiance) Now, the proposed control system is tested under variable irradiance with different scenarios (0–15 s: two lamps ON, 15– 40 s: all lamps ON, 40–55 s: three lamps ON, 55–65: all lamps ON) as shown in the PV voltage response Fig. 10(a). The normal and fault modes are illustrated by the voltage dip in Fig. 10(b).



system has been experimentally validated under different scenarios and proved to be efficient in operating the grid-connected PV system with a smooth transition between the two modes. APPENDIX A. Derivative Elements Under Balanced Grid Voltage

Lf h1 = −

Fig. 11. Comparison between the proposed controller and PI controller. (a) Proposed controller and (b) PI controller.

Despite the PV voltage change, due to the irradiance and the voltage dip, the dc-link voltage remains constant as shown in Fig. 10(c). The power behaves adequately in the two modes as shown in Fig. 10(d). These results confirm the robustness of the proposed sliding mode compensator to deal with unknown uncertainties. E. Comparison With Conventional PI Control Scheme Finally, the FLC scheme with sliding mode compensation is compared to the conventional PI control strategy to check its robustness to uncertainties (grid fault, parametric and irradiance variations). The grid fault was applied in the time intervals (20– 30 s and 100–155 s), the resistance R has been increased by 50% in (70–75 s and from 148 s) and the irradiance was changed as follows (60–65 s: two lamps ON, 65–135 s: all lamps ON, and 135–145 s: all lamps ON). The dc-link voltage regulation is verified using the two control schemes and the results are shown in Fig. 11. It can be observed, from the dc-link voltage tracking, that the conventional PI control method is highly affected by the uncertainties compared to the robust FLC. In the proposed control scheme, the uncertainties were compensated by the sliding mode compensators without need of changing the control gains, while the PI control scheme is required to tune its gains at each occurrence of the uncertainties in order ensure its robustness. VII. CONCLUSION FLC strategies, with sliding mode compensation for robustness enhancement, have been proposed to operate a grid-connected PV inverter system under normal (mode I) and fault (mode II) conditions. In mode I, the control system has been developed for controlling the dc-link voltage and the currents through the inverter for a maximum power transfer to the grid, while maintaining a constant dc-link voltage. In mode II, the active power is regulated to limit the current excess and the reactive power is injected to support the grid via the inverter through current control and power-current transformation, while the regulation of the dc-link voltage is done through the dc–dc converter. The sliding mode compensation enhances the robustness of the control system to uncertainties, such as parametric variations, unmodeled quantities, and external disturbances, directly linked to the controlled outputs. The proposed control

R eq iq + ωid − Lq L

Lg 11 h1 = 0; Lg 12 h1 = Lf h2 = −

1 1 ; Lg 21 h1 = 0; Lg 22 h1 = ; Lg 23 h1 = 0 L L

3 (ed id + eq iq ); Lg 11 h2 = 0; Lg 12 h2 = 0 2Cvdc

1 Lg 21 h2 = 0; Lg 22 h2 = 0; Lg 23 h2 = C 3 3 3 2 Lf h2 = − ed − eq (ed id + eq iq ) 2 2Cvdc 2Cvdc 2Cvdc ⎡ ⎤ R ed − id + ωiq − ⎢ L L ⎥ ⎢ ⎥ ⎢ eq ⎥ R ⎢ ⎥ × ⎢ − iq + ωid − ⎥ ⎢ Lq L ⎥ ⎢ ⎥ ⎣ ⎦ 3 − (ed id + eq iq ) 2Cvdc Lg 11 Lf h2 = − Lg 23 Lf h2 =

3 3 ed ; Lg 12 Lf h2 = − eq 2CLvdc 2CLvdc

3 (ed id + eq iq ). 2 2C 2 vdc

B. Derivative Elements Under Unbalanced Grid Voltage ± ± ± ±h y ˙ y ˙ L f 1 2 1 , y˙ 2± − Lf ± h± , y˙ 1± = 1 = y˙ 1− y˙ 2 Lf ± h− 1 Lf ± h± 2 ± Lf ± h2 = Lf ± h− 2 Lg + h+ Lg + h+ 0 0 1 1 ± ± 11 12 Lg 11± h1 = , Lg 12± h1 = 0 Lg 11− h− 0 Lg 12− h− 1 1 + + Lg + h2 Lg + h2 0 0 ± ± 11 12 Lg 11± h2 = , Lg 12± h2 = 0 Lg 11− h− 0 Lg 12− h− 2 2 + + Lg + h1 Lg + h1 0 0 ± ± 21 22 Lg 21± h1 = , Lg 22± h1 = 0 Lg 21− h− 0 Lg 22− h− 1 1 + + Lg + h2 Lg + h2 0 0 ± ± 21 22 Lg 21± h2 = , Lg 22± h2 = 0 Lg 21− h− 0 Lg 22− h− 2 2

Lf ± h+ 1 = −

e+ e− R + R − − − d d ±h =− id + ωi+ ; L i − − ωi − f q q 1 L L L d L


e+ e− R + R − q q − − ± iq − ωi+ ; L i − h = − + ωi − f 2 d d L L L q L 1 1 + − −h + h = ; Lg 11− h− 1 = ; Lg 12 1 =0 1 = 0; Lg 12 L L 1 1 + + h ; Lg 12− h− = 0; Lg 11− h− 2 = 0; Lg 12 2 = 2 = L L 1 1 ; L + h+ = 0; Lg 22− h− = ; Lg 21− h− 1 = 1 =0 L L g 22 1 1 1 + + h ; L − h− = . = 0; Lg 21− h− 2 = 0; Lg 22 2 = L g 22 2 L

Lf ± h+ 2 = − Lg + h+ 1 11

Lg + h+ 2 11

Lg + h+ 1 21

Lg + h+ 2 21

C. RMS Calculation The method, described in [17], has been used to calculate 2 the rms value such as Vrm s = 12 {v(t)2 + ( dvdt(t) ) }, v(t) is the grid voltage. REFERENCES [1] R. Teodorescu, M. Liserre, P. Rodriguez, and F. Blaabjerg, Grid Converters for Photovoltaic and Wind Power Systems—Control of Grid Converters Under Grid Faults. Chichester, U.K.: Wiley, 2011. [2] J. L. Sosa, M. Castilla, J. Miret, J. Matas, and Y. A. Al-Turki, “Control strategy to maximize the power capability of PV three-phase inverters during voltage sags,” IEEE Trans. Power Electron., vol. 31, no. 4, pp. 3314–3323, Apr. 2016. [3] M. Mirhosseini, J. Pou, B. Karanayil, and V. G. Agelidis, “Resonant versus conventional controllers in grid-connected photovoltaic power plants under unbalanced grid voltages,” IEEE Trans. Sustain. Energy, vol. 7, no. 3, pp. 1124–1132, Jul. 2016. [4] F. Yang, L. Yang, and X. Ma, “An advanced control strategy of PV system for low-voltage ride-through capability enhancement,” Sol. Energy, vol. 109, pp. 24–35, 2014. [5] A. J. Krener, “Feedback linearization,” in Mathematical Control Theory, J. Baillieul and J. C. Willems, Eds. New York, NY, USA: Springer, 1999, ch. 3, pp. 66–98. [6] C. I. Pop and E. H. Dulf, “Robust feedback linearization control for reference tracking and disturbance rejection in nonlinear systems,” in Recent Advances in Robust Control-Novel Approaches and Design Methods, A. Mueller, Ed. Winchester, U.K.: InTech, 2011, ch. 12, pp. 273–290. [7] R. Errouissi, A. Al-Durra, and S. M. Muyeen, “Offset-free feedback linearisation control of a three-phase grid-connected photovoltaic system,” IET Power Electron., vol. 9, no. 9, pp. 1933–1942, Apr. 2016. [8] R. Kadri, J.-P. Gaubert, and G. Champenois, “An improved maximum power point tracking for photovoltaic grid-connected inverter based on voltage-oriented control,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 66–75, Jan. 2011. [9] M. A. Mahmud, H. R. Pota, and M. J. Hossain, “Dynamic stability of three-phase grid-connected photovoltaic system using zero dynamic design approach,” IEEE J. Photovolt., vol. 2, no. 4, pp. 564–571, Oct. 2012. [10] M. A. Mahmud, H. R. Pota, M. J. Hossain, and N. K. Roy, “Robust Nonlinear controller design for three-phase grid-connected photovoltaic systems under structured uncertainties,” IEEE Trans. Power Del., vol. 29, no. 3, pp. 1221–1230, Jun. 2014. [11] F. Delfino, G. B. Denegri, M. Invernizzi, and R. Procopio, “Feedback linearisation oriented approach to Q–V control of grid connected photovoltaic units,” IET Renew. Power Gener., vol. 6, no. 5, pp. 324–339, 2012. [12] N. Kumar, T. K. Saha, and J. Dey, “Sliding-mode control of PWM dual inverter-based grid-connected PV system: Modeling and performance analysis,” IEEE J. Emerging Sel. Topics Power Electron., vol. 4, no. 2, pp. 435–444, Jun. 2016. [13] A. Merabet, K. T. Ahmed, H. Ibrahim, and R. Beguenane, “Implementation of sliding mode control system for generator and grid sides control of wind energy conversion system,” IEEE Trans. Sustain. Energy, vol. 7, no. 3, pp. 1327–1335, Jul. 2016. [14] F.-J. Lin, K.-C. Lu, T.-H. Ke, B.-H. Yang, and Y.-R. Chang, “Reactive power control of three-phase grid-connected PV system during grid faults using Takagi-Sugeno-Kang probabilistic fuzzy neural network control,” IEEE Trans. Ind. Electron., vol. 62, no. 9, pp. 5516–5528, Sep. 2015.

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[15] A. Merabet, K. Ahmed, H. Ibrahim, R. Beguenane, and A. Ghias, “Energy management and control system for laboratory scale microgrid based wind-PV-battery,” IEEE Trans. Sustain. Energy, vol. 8, no. 1, pp. 145– 154, Jan. 2017. [16] Real-Time HIL/RCP Laboratory, OPAL-RT Technologies, 2014. [Online]. Available: http://www.opal-rt.com/new-product/real-time-hilrcplaboratory [17] Y. Nakata, K. Fujiwara, M. Yoshida, J. Itoh, and Y. Okazaki, “Output voltage control for PWM inverter with electric double layer capacitor as dc power supply,” in Proc. Int. Power Electron. Conf., 2010, pp. 3099– 3104.

Adel Merabet (M’10) received the Ph.D. degree in engineering from the Universit´e du Qu´ebec a` Chicoutimi, Chicoutimi, QC, Canada, in 2007. He is an Associate Professor in the Division of Engineering, Saint Mary’s University, Halifax, NS, Canada. Currently, he is a Visiting Academic in the Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah, UAE. His research interests include renewable (wind-solar) energy conversion systems, energy management, advanced control systems, and smart grid.

Labib Labib (M’16) received the B.S. degree in electrical and electronic engineering from the Khulna University of Engineering and Technology, Khulna, Bangladesh, in 2013. He is currently working toward the M.Sc. degree in applied science at Saint Mary’s University, Halifax, NS, Canada. He held the position of a Lecturer in the Department of the Electrical and Electronic Engineering, Leading University, Sylhet, Bangladesh. His research interests include renewable power systems, control systems, power electronics, and smart grid. Amer M. Y. M. Ghias (M’14) received the Ph.D. degree in electrical engineering from the University of New South Wales, Australia, in 2014. He is an Assistant Professor in the Department of Electrical and Computer Engineering, University of Sharjah, Sharjah, UAE. His research interests include model predictive control of power electronics converter, hybrid energy storage, fault-tolerant converter, modulations, and voltage balancing techniques for a multilevel converter.

Chaouki Ghenai received the Ph.D. degree in mechanical engineering from the Orleans University, Orleans, France, in 1995. He is an Assistant Professor in the Department of Sustainable and Renewable Energy Engineering, and a Coordinator of the Sustainable Energy Development Research Group, Research Institute of Science and Engineering, University of Sharjah, Sharjah, UAE. His research interests include renewable energy, energy efficiency, sustainability, modeling, and simulation of microgird power systems.

Tareq Salameh received the Ph.D. in energy science from Lund University, Lund, Sweden, in 2012. He is an Assistant Professor in the Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah, UAE. His research interests include energy systems, solar collector, numerical simulation, and modeling.