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The aim of this paper is to determine the power losses recorded by a PV generator operating under partial shading conditions. These losses are evaluated through two distinct methods. The first method is based on mathematical modeling, while the second is based on Simulink’s physical model. The losses recorded are considerable and increase as a function of the increase in the percentage of shading up to a limit value where they become constant in the case where an ideal by-pass diode is connected in parallel with the modules. This limit value is non-existent in the case where the bypass diode is not ideal, which in fact corresponds to the real model. However, it emerges that the power losses are minimized in a PV system comprising bypass diodes, in particular in the case where the partial shading is considerable.

Photovoltaic solar cell is a generator able to produce electrical energy when subjected to solar radiation. The characteristics of the photovoltaic generator or module (such as current or voltage) given by the manufacturer are determined under the so-called standard test conditions (Tc = 25˚C, G = 1 kW/m², AM = 1.5). However, in practice several factors can influence these data and negatively impact the electric power produced by the solar generator. Among these factors it is the phenomenon of partial shading which can not only reduce the power of the generator but also can decrease its lifespan due to the hot spot. Among the causes inducing partial shading it is the trees, the clouds, the constructions near that are installed in the photovoltaic generators [

There are several configurations of a photovoltaic solar field. In order to meet the electricity requirement in any application, the solar modules are arranged in series, in parallel, in bridged and in full cross connections to form a photovoltaic array to match the power requirements in terms of voltage and current [

Several theoretical studies have been carried out in order to determine the maximum power produced by a PV generator subjected to the partial shading conditions, thus permitting to evaluate the losses generated by this phenomenon. The determination of the maximum power requests to determine the parameters of the module. The number of these parameters varies according to the solar cell model studied. One distinguishes the photovoltaic solar cell model with one diode (with five parameters) and the photovoltaic solar cell model with two diodes (with seven parameters) [

The aim of this paper is to evaluate the power losses generated by the phenomenon of partial shading in a PV generator. The study is based on the one diode electrical model of the PV solar cell. The mathematical methods used are analytic and iterative. The solar module (made up of several cells) is subdivided into two subgroups or submodules with an equal number of cells. A diode by pass is connected in parallel to each subgroup of the module in order to dampen the influence of partial shading [

This part is devoted to the determination of the parameters of the PV generator subjected to partial shading conditions. The electrical equivalent circuit of a one diode photovoltaic solar cell is presented in _{ph}, I_{o}, n, R_{s} and R_{sh}.

Applying Kirchhoff’s law to this model permits to obtain the characteristic equation current-voltage of the solar cell defined by [

I = I p h – I o ( exp ( q ( V + R s I ) n N s k T c ) − 1 ) − V + R s I R s h (1)

Considering the fact that R_{sh}>>>, thus V + R s I R s h → 0 . Equation (1) turns to [

I = I p h – I o ( exp ( q ( V + R s I ) n N s k T c ) − 1 ) (2)

The configuration of the PV module subjected to partial shading conditions is presented in _{s1} number of cells) while the other is subjected to partial shading (called SG-2 with N_{s}_{2} number of cells).

The total voltage across the module is given by:

{ V T = V 1 + V D if V 2 < V D V T = V 1 + V 2 if V 2 < V D (3)

where:

V 1 = n N s 1 k T c q ln ( I p h – I + I o I o ) (4)

V 2 = n N s 2 k T c q ln ( β I p h – I + I o I o ) (5)

Equation (6) gives the photovoltaic current as [

I p h = ( I s c + Δ T ∗ K i ) G G r (6)

The saturation current is evaluated through Equation (7) [

I o = I O , r e f ( T c T r ) 3 exp ( q E g ( 1 T r – 1 T r ) n N s k ) (7)

For I = 0, V = V_{oc}, I_{ph} = I_{sc}, thus Equation (1) turns to:

0 = I s c − I o , r e f ( exp ( q V o c n N s k T c ) − 1 ) (8)

Then:

I o , r e f = I s c ( exp ( q V o c n N s k T c ) − 1 ) (9)

Replacing Equation (9) into Equation (7) permits to define the value of the saturation current as:

I o = I p h ( exp ( q V o c n N s k T c ) − 1 ) ( T c T r ) 3 exp ( q E g ( 1 T r − 1 T r ) n N s k ) (10)

At the maximum power point (where I = I_{mp} and V = V_{mp}), one has:

I m p = I p h , r e f − I o , r e f ( exp ( q ( V m p + R s I m p ) n N s K T r ) − 1 ) (11)

In Equation (9) and Equation (11), the exponential term is much greater than the first term (by considering the fact that the value of the reverse saturation current for any diode is in the order of 10^{−5} and 10^{−6} A [_{ph,ref} ≈ I_{sc}, Equation (9) and Equation (11) turn to:

I o , r e f = I s c exp ( − q V o c n N s k T c ) (12)

I m p = I s c − I o , r e f ⋅ exp ( q ( V m p + R s I m p ) n N s k T r ) (13)

Combining Equation (12) and Equation (13) leads to:

n = q ( V m p + R s I m p − V o c ) N s k T r ln ( I s c − I m p I s c ) (14)

From Equation (2), one gets the partial derivative of V with respect to I as:

∂ V ∂ I = − 1 + q R s I o n N s k T c ⋅ exp ( q ( V + R s I ) n N s k T c ) q I o n N s k T c ⋅ exp ( q ( V + R s I ) n N s k T c ) (15)

Equation (2) permits to deduce the series resistance at the open circuit point such as:

R s = − ∂ V ∂ I − n N s k T r ⋅ exp ( − q V o c n N s k T r ) q I o , r e f (16)

One approximates the value of the derivative of V with respect to I by fixing two points (V_{oc}, 0) and (V_{mp}, I_{mp}) such as:

∂ V ∂ I = − V o c − V m p I m p (17)

Replacing Equation (12) and Equation (17) into Equation (16) permits to get the series resistance of each solar cell as:

R s = V o c − V m p N s I m p − n k T r q I s c (18)

Replacing Equation (18) into Equation (14) leads to:

= q ( V m p − V o c + V o c N s − V m p N s ) k T r ( N s ln ( I s c − I m p I s c ) + I m p I s c ) (19)

The value of the parameter R_{sh} is calculated by solving the Equation V + R s I R s h = 0 using Newton Raphson method (iterative method).

The power of the PV module is given as follows:

P = V ⋅ I (20)

Using the power-voltage characteristics at the maximum power point implies:

∂ P / ∂ V = 0 (21)

where

∂ P ∂ V = I + V ⋅ ∂ I ∂ V (22)

From Equation (2), the partial derivative of I with respect to V is:

∂ I ∂ V = − q I o n k T ( I p h − I + I o I o ) 1 + q I o n k T ( 2 R s + n k T q ( I p h 1 − I + I o ) ) ( I p h − I + I o I o ) (23)

Let us set

∂ P ∂ V = f ( I max ) (24)

Replacing Equation (23) and Equation (24) into Equation (22) leads to:

f ( I max ) = I max − I o [ log ( ( I p h − I max 1 + I o ) I o ) − q R s I max n k T ] ( I p h − I max + I o I o ) 1 + q R s I o n k T ( I p h − I max + I o I o ) (25)

Equation (25) is solved by the Newton Raphson method.

Principe of Newton-Raphson MethodThe Newton-Raphson method finds the tangent line of the function at the current point and uses the zero of tangent line as the next reference point. The process is repeated until the root is found. The function f ( x ) = 0 can be expanded in the neighborhood of the root x 0 through the Taylor formula:

f ( x ) = f ( x 0 ) + ( x − x 0 ) f ′ ( x 0 ) + ( x − x 0 ) 2 2 ! f ″ ( ξ ( x 0 ) ) = 0

where x can be seen as a trial value for the root. At the n^{th} step of the approximate value of the next step; x n + 1 can be derived from:

f ( x n + 1 ) = f ( x n ) + ( x n + 1 − x n ) f ′ ( x n ) = 0

x n + 1 = x n − f ( x n ) f ′ ( x n ) , n = 0 , 1 , ⋯ is called the Newton-Raphson method.

The flowchart of

Combining Equation (3), Equation (4) and Equation (5) leads to the total maximum voltage of the PV module given by:

V max = { N s 1 ( n k T q log ( ( I p h − I max + I o ) I o ) − R s I max ) + V D if V 2 < V D N s 1 ( n k T q log ( ( I p h − I max + I o ) I o ) − R s I max ) + N s 2 ( n k T q log ( ( β I p h − I max + I o ) I o ) − R s I max ) if V 2 > V D (26)

The Simulink models of the used panels are designed from the solar cell of Simulink Library of Matlab. By double-clicking the component of the _{sc} are modified in accordance to the manufacturer’s datasheet of the PV panel to simulate. As for the temperature, it is preferable to insert the letter T in the relevant field to facilitate change during the simulations. The irradiance (Ir) is simulated from the constant block (

The simulation results are based on the flow chart presented in