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 Figure 1. The five parameters of this model include I_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 [1] [9] [10] :

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 [1] [9] [10] :

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 Figure 2. The module is subdivided into two subgroups of 18 cells each. One of the subgroups is not shaded (called SG-1 with N_s1 number of cells) while the other is subjected to partial shading (called SG-2 with N_s2 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 [9] [11] :

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

The saturation current is evaluated through Equation (7) [9] [10] :

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 [9] ). Thus by assuming that I_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.

The 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 Figure 3 details the method of resolution of Newton Rapson.

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)

Figure 4 presents the Flowchart describing the process for determining the maximum power of a PV module under partial shading conditions based on mathematical modeling.