General modeling procedure for photovoltaic arrays

Electric Power Systems Research 155 (2018) 67–79

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General modeling procedure for photovoltaic arrays Juan David Bastidas-Rodríguez a,∗ , Luz Adriana Trejos-Grisales b , Daniel González-Montoya c , Carlos Andrés Ramos-Paja d , Giovanni Petrone e , Giovanni Spagnuolo e a

Escuela de Ingenierías Eléctrica, Electrónica y de Telecomunicaciones, Universidad Industrial de Santander, Bucaramanga, Colombia Departamento de Electromecánica y Mecatrónica, Instituto Tecnológico Metropolitano, Medellín, Colombia Departamento de Electrónica y Telecomunicaciones, Instituto Tecnológico Metropolitano, Medellín, Colombia d Facultad de Minas, Universidad Nacional de Colombia, Medellín, Colombia e Department of Information and Electrical Eng. and Applied Mathematics, University of Salerno, Salerno, Italy b c

a r t i c l e

i n f o

Article history: Received 2 June 2017 Received in revised form 1 September 2017 Accepted 22 September 2017 Keywords: Modeling PV array Irregular configuration Mismatching Partial shading

a b s t r a c t Photovoltaic (PV) arrays are usually formed by strings of panels connected in parallel creating the seriesparallel (SP) configuration. Additional connections can be implemented among the strings of the SP configuration to mitigate a particular mismatching profile. Those connections may follow regular or irregular patterns to form regular configurations, e.g. Total Cross-Tied (TCT), Honey-Comb (HC) or BridgeLinked (BL), or irregular configurations, respectively. In literature there are proposed models for particular array configurations; however, there has not been reported yet a general model capable to describe the behavior of any regular or irregular configuration. Therefore, testing multiple configurations require to parameterize or design multiple models. This paper introduces a procedure for reproducing the electrical behavior of any PV array operating under uniform or mismatched conditions, which eliminates the requirement of using multiple models to evaluate different configurations. The procedure is based in dividing the array into sub-arrays to construct a system of non-linear equations for each sub-array, which is solved by using the Trust-Region Dogleg method. The proposed approach is validated by comparing the results provided by the proposed model with the circuital implementation of the same PV array in a commercial software and with the data produced with an experimental test bed. Such comparisons put into evidence the satisfactory performance of the general modeling procedure in the reproduction of the electrical behavior of both regular and irregular arrays. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The advantages of photovoltaic (PV) systems (free energy source, zero emissions, modularity, etc.) as well as policies and subsidies to incentive the grow of PV installations have produced an important increase in the installed PV power all over the world. During 2016 a total of 75 GW were installed, 50% more compared to 2015, increasing the installed global PV capacity to 300 GW [1] approximately. The basic unit that form a PV array is the PV module, which is a set of series connected cells with a protection diode (bypass diode)

∗ Corresponding author. E-mail addresses: [email protected] (J.D. Bastidas-Rodríguez), [email protected] (L.A. Trejos-Grisales), [email protected] (D. González-Montoya), [email protected] (C.A. Ramos-Paja), [email protected] (G. Petrone), [email protected] (G. Spagnuolo). http://dx.doi.org/10.1016/j.epsr.2017.09.023 0378-7796/© 2017 Elsevier B.V. All rights reserved.

connected in anti-parallel. Usually two or more PV modules connected in series into a single mechanical structure is denominated a PV panel, which is the commercially available unit to construct a PV array [2]. In a typical PV system, the PV panels are connected in series to form strings, and multiple strings are connected in parallel to form the array. Such a configuration is denominated series-parallel (SP) and it is the most widely adopted configuration [3]. Additional ties can be implemented between two strings of a SP array to mitigate the effects of a particular shading profile. Such additional ties can be implemented following a regular or irregular patterns to form, what in this paper are denominated as, regular or irregular configurations, respectively. Total Cross-Tied (TCT), the Bridge-Linked (BL) and the HoneyComb (HC) [4] are three widely adopted regular configurations. The strict definition of TCT, BL and HC assume an array formed by modules instead of commercial panels [5,6]. On the other side, if a PV array configuration does not match with a regular configuration, i.e.

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SP, TCT, BL, HC or similar, then such a configuration is considered as irregular in this paper. For a given operating condition each configuration, either regular or irregular, provide a particular power vs. voltage (P–V) curve, which may contain different number of local maximum power points (LMPPs) and a different global maximum power point (GMPP). Therefore, for a given operating condition, there is at least one configuration that provides the largest GMPP or, in other words, that best mitigates the effects of a particular mismatching condition [7]. In literature different modeling procedures for regular array configurations like SP [8–11], TCT [11,12], BL [11,13] or HC [4,11,14,15] have been reported. Picault et al. [16] mention the effects of the ties between the strings in the construction of a nonlinear equations system that describe an array, while Liu et al. [17] propose a model for a SP array with modules formed by different number of cells and/or modules without bypass diodes. However, in all these papers, the authors have not found a general modeling procedure able to reproduce the electrical behavior of any regular or irregular PV array configuration. This is evidenced in some papers aimed at comparing the performance of different regular configurations operating under mismatching conditions, since those works use circuital implementations of the arrays in electrical simulators, as in [11,15], or interpolations to calculate the arrays’ power vs. voltage (P–V) and current vs. voltage (I–V) curves, this introducing numerical errors. This paper introduces a novel methodology for modeling and calculating the current of any regular or irregular PV array configuration, with N rows and M columns (N × M) of panels, operating under uniform or mismatched conditions. In comparison with other methods presented in literature, the proposed procedure allows to simulate, in an efficient way, irregular topologies, which can be more effective than regular ones when some shading patterns occur. From such basic calculation it is possible to obtain the I–V and P–V curves or to perform dynamic simulations. In the proposed model the connections between consecutive strings are represented by an (N − 1) × (M − 1) matrix named connection matrix, which is used to divide the array into smaller arrays, named subarrays, connected in parallel. A system of nonlinear equations is constructed for each sub-array using nodal analysis to calculate the sub-array currents from the node voltages. Finally, the array current is calculated by adding the sub-arrays’ currents. This modeling procedure is suitable to be implemented in standard programming languages such as Matlab script, C or C++, which makes it easy to use in different applications like the evaluation of MPPT techniques, the design of PV plants based on the estimation of the energy production or the implementation of reconfiguration algorithms without using costly and computing demanding circuital simulators. The paper is organized as follows: Section 2 describes the proposed modeling approach, Section 3 illustrates the use of the proposed model for a particular PV array configuration. Section 4 validates the performance of the proposed approach through different simulation and experimental tests. Finally, the conclusions close the work. 2. Proposed model This section presents the main elements of the proposed modeling procedure, which is supported with a general flow chart and the pseudo-codes required for its implementation. 2.1. PV module representation The operation of a PV device can be modeled through electrical [9,18,19] or thermal models [20,21] considering the dynamic

weather conditions. In this paper, the widely adopted single-diode (SD) equivalent circuit, shown in Fig. 1, is adopted to represent a PV module due to its accuracy [18,22]. The associated bypass diode [8,22,23] has been also modeled. In the PV module, the current source (Iph ) represents the photovoltaic current, the diode includes the P–N junction nonlinear behavior, and the resistances Rh and Rs represent the leakage currents and ohmic losses, respectively. Moreover, ea and eb are the array node voltages at the positive and negative terminals of the PV module with respect to the array reference voltage. The relationship between the current (I) and voltage (Ve = ea − eb ) of the PV module can be obtained using the Kirchhoff voltage (KVL) and current (KCL) laws [24,25]. However, such relationship is not explicit and strongly non-linear; therefore, it is necessary to use the Lambert-W function, denoted by Wo , to express the module current as function of ea and eb as shown in (1). Moreover, the derivatives of (1) with respect to ea and eb are presented in (2) and (3), respectively. In (1)–(3) Ns is the number of series-connected cells into the module, Isat is the inverse saturation current of the PV module diode, and Vt = n · k · T/q in which n is the ideality factor, k is the Boltzmann constant, q is the electron charge, and T is the module temperature in Kelvin. Isat,db is the inverse saturation current of the bypass diode and Vt,db = ndb · k · T/q in which ndb is the ideality factor of the bypass diode.

 

I=− +

Ns · Vt · W0 

 Rs





+ Isat,db · exp



− (ea − eb ) Vt,db



− Isat,db

Rh · Iph + Isat − (ea − eb ) Rh + Rs

 R ·R I s sat h = · · R h + Rs N s · Vt



+

−Rh Rs · (Rh + Rs )

Isat,db Vt,db



∂I = ∂eb





Ns · Vt · (Rh + Rs )

∂I = ∂ea





(Rh + Rs ) · Iph + Isat + Rh · (ea − eb )

exp

−

(1)



 · exp

Rh Rs · (Rh + Rs )

Isat,db Vt,db



 · exp

·

     −

W0 

1 + W0 

eb − ea Vt,db







·



 (2)

     +

W0 

1 + W0 

eb − ea Vt,db

1 Rh + Rs

1 Rh + Rs



 (3)

The values of Iph , Isat , n, Rs , Rh , Isat,db and ndb can be calculated by means of systematic procedures as the ones introduced in [18,19,26], which use the datasheet information of a PV panel and the bypass diode as well as the weather conditions. The parameters n, Rs , Rh , Isat,db and ndb can be considered constant while Iph and Isat depend on the irradiance S and temperature T as shown in (4) and (5). Isat =

exp





Isc,STC + KIsc · (T − TSTC )





Voc,STC + KVoc · (T − TSTC ) /Vt − 1

Iph = S/SSTC



Iph,STC + KIsc · (T − TSTC )



(4) (5)

In (4) and (5), Isc,STC and Iph,STC are the short-circuit current and photovoltaic current in standard test conditions (STC), respectively, Voc,STC is the open-circuit voltage in STC, KIsc and KVoc are the temperature coefficients of the short-circuit current and open-circuit

J.D. Bastidas-Rodríguez et al. / Electric Power Systems Research 155 (2018) 67–79

69

Fig. 1. PV module single diode model equivalent circuit including the bypass diode Ddb .

voltage, respectively, TSTC is the module temperature in STC in Kelvin degrees and SSTC is the STC irradiance.



2.2. Identification of independent sub-arrays and their parameters

Cz = 0

The ties between consecutive strings of an N × M array are defined by an (N − 1) × (M − 1) matrix named connection matrix (Mconn ), which is filled with ones and zeros to describe the presence or absence of ties, respectively. The column j of Mconn describes the connections between the strings j and j + 1 of the array. A one/zero in the row i and column j of Mconn (i.e. Mconn (i, j)) indicates a connection/no connection between the negative terminals of the modules (i, j) and (i, j + 1) of the array. In this paper a sub-array is defined as a string or set of strings that has no ties with other strings to the left and to the right; hence, the sub-arrays can be identified by using Mconn . If the column j of Mconn is a column of zeros, then there is no connection between strings j and j + 1 of the array; therefore, examining Mconn from left to right, each column of zeros indicates the end of a sub-array and the start of the next one. From this analysis it is possible to define the number of sub-arrays in a PV array (Nsa ) as show in (6), where Nz is the number of columns of zeros in Mconn . Nsa = Nz + 1

Msa,i = SAlc,i − SAfc,i + 1. Finally, if a sub-array has only one column the connection matrix is a column of N − 1 zeros, because there is not any connection with other strings.

(6)

In the proposed model each sub-array is analyzed independently, hence it is necessary to define the SD model parameters and the connection matrix of each sub-array. Initially the seven parameters required for each PV module (Iph , Isat , Vt , Rs , Rh , Isat,bd and Vt,bd ) are introduced in seven N × M matrices: MIph , MIsat , MVt , MRs , MRh , MIsatbd , MVtbd . Then, from such matrices it is necessary to extract the parameters matrices for each sub-array. All the sub-arrays have the same number of rows, therefore, to identify the parameters and connection matrices of sub-array i it is only necessary to identify its first (SAfc,i ) and the last (SAlc,i ) columns. Such an identification uses the vector defined in (7), where the first and last elements are 0 and M, respectively, while the rest of the elements correspond to the index of each column of zeros in the connection matrix. Then, SAfc,i and SAlc,i are defined as shown in (8). sa , M sa , M sa , M sa , M sa , M sa Finally, the parameters (MIph,i , Isat,i Vt,i Rs,i Rh,i Isatbd,i sa sa MVtbd,i ) and connection (Mconn,i ) matrices of the sub-array i are introduced in (9) and (10), respectively. Note that in (9) and (10) the superscript sa indicates that the matrix corresponds to a subarray, moreover the dimensions of parameters and connection matrices are N × Msa,i and (N − 1) × (Msa,i − 1), respectively, with

Zc,1

···

Zc,i

···

Zc,Nsa

M



SAfc,i = Cz (i) + 1, SAlc,i = Cz (i + 1)

(7) (8)

sa sa sa = M (r, c), MIph,i = MIph (r, c), MIsat,i = MIsat (r, c), MVt,i Vt sa = M (r, c), M sa = M (r, c), M sa MRs,i = MIsatbd (r, c), Rs Rh Rh,i Isatbd,i sa MVtbd,i

(9)

= MVtbd , ∀ r [1· · ·N], c [SAfc,i · · ·SAlc,i ]

sa Mconn,i = Mconn (rc , cc ), ∀ rc [1· · ·(N − 1)],

cc [SAfc,i · · ·(SAlc,i − 1)]

(10)

2.3. Calculation of the sub-array current All the sub-arrays in the N × M array are connected in parallel, which implies they are subjected to the same array voltage (Varray ). This section shows the proposed procedure to calculate the output current of a sub-array i (Isa,i ) for a given Varray . The basic idea is to use nodal analysis to obtain the node voltages in the sub-array, then the sub-array currents of the modules in the first row are calculated, and, in the end, those currents are added to obtain Isa,i . Considering that each sub-array is independent from the others, the sub-array i can be analyzed as an N × Msa,i array without internal sub-arrays, where Msa,i = SAlc,i − SAfc,i + 1, according to the explanation given in Section 2.2. The number of nodes of sub-array i (Nn,i ) is calculated as shown in (11), where Nt,i is the sum of all the sa as described in (12). Therefore, in the sub-array elements of Mconn,i i there are Nn,i unknown node voltages: ej,i for j  [1 · · · Nn,i ], which form the sub-array node voltage vector Vn,i . The numbering of the node voltages in Vn,i is performed from top to bottom and from left sa . to right examining Mconn,i





Nn,i = (N − 1) · Msa,i − Nt,i



(11)

Msa,i N

Nt,i =

sa Mconn,i (k1 , k2 )

(12)

k2 =1k1 =1

To apply KCL to each node in the sub-array, the current of each module need to be calculated using (1). Hence, it is necessary to

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identify the sub-array node voltage where the positive (ea (r, c)) and negative (eb (r, c)) terminals of the module in row r and column c (i.e. module (r, c)) are connected to. Such an identification is performed sa using Mconn,i and knowing that the top and bottom nodes voltages are Varray and zero, respectively. To associate each terminal voltage to its corresponding node voltage, the identification is performed sa ); each column of the M sa through a (N − 1) × Msa,i matrix (Mnv,i nv,i matrix contains the number of the nodes of each column of the sub-array from top to bottom and from left to right. In this way the sa (r, c) contain the number of the node voltage of e (r, element Mnv,i b sa (r, c) = a means e (r, c) = V (a) = e . On the other hand, c), i.e. Mnv,i n,i a,i b sa (r, c) contain the number of the node voltage of the element Mnv,i sa (r, c) = a means e (r + 1, c) = V (a) = e . ea (r + 1, c), i.e. Mnv,i a n,i a,i The system of Nn,i nonlinear equations of the sub-array i (Fi (Vn,i )) is obtained by applying KCL to each node of the sub-array, where the unknown variables are the node voltages in Vn,i vector. To solve Fi (Vn,i ) by means of the Trust-Region Dogleg or a similar method, it is necessary to define the Jacobian matrix Ji (Vn,i ), which has a dimension Nn,i × Nn,i . Both, Fi (Vn,i ) and Ji (Vn,i ), depend on the consa figuration of the sub-array specified in Mconn,i ; therefore, it is not possible to define a general structure for Fi (Vn,i ) and Ji (Vn,i ). Instead, this paper provides a general procedure for calculating Fi (Vn,i ) and Ji (Vn,i ), which is described in Algorithm 1. In such an algorithm if , nn and jc are auxiliary variables introduced to simplify the writing of the pseudocode. The for loops in lines 2 and 3 go over the nodes of the sub-array from top to bottom and from left to right. Line 4 identifies if a node has been analyzed or not. Then, SAfc,i and SAlc,i for a given node need to be identified. Lines 6–10 calculate the current of the module above the analyzed node (Ir,c ), calculating also its partial derivatives (∂Ir,c /∂ea (r, c), and ∂Ir,c /∂eb (r, c)) for the module above the analyzed node. The same process is performed for the module below the analyzed node in lines 11–15. The process is repeated for the other modules connected to the analyzed node (for loop between lines 5 and 16). Finally, the process described above is repeated for each sub-array node by means of the for loop between lines 2 and 20. Algorithm 1.

Calculation of Fi (Vn,i ) and Ji (Vn,i ).

sa sa INPUT: Varray , Vn,i , Nn,i , Mnv,i , Mconn,i , sub-array parameters’ matrices OUTPUT: Ji (Vn,i ) 1: Set if = 1 2: for node j = 1 to Msa,i do 3: for node i = 1 to N − 1 do sa sa (i, j − 1) = 0 then: Identify SAfc,i and SAlc,i from Mconn,i 4: if j = 1 OR Mconn,i 5: for column jc = SAfc,i to SAlc,i do sa (i, jc ), eb (i, jc ) = Vn,i (nn ) 6: Set nn = Mnv,i 7: if i = 1 then: ea (i, jc ) = Varray sa 8: else: Set nn = Mnv,i (i − 1, jc ), and ea (i, jc ) = Vn,i (nn ). Calculate ∂Ii,jc /∂ea (i, jc )

with (2). Set Ji (if , nn ) = Ji (if , nn ) + ∂Ii,jc /∂ea (i, jc ) 9: end if sa (i, jc ), 10: Calculate ∂Ii,jc /∂eb (i, jc ) with (3), Ii,jc with (1). Set nn = Mnv,i Ji (if , nn ) = Ji (if , nn ) + ∂Ii,jc /∂eb (i, jc ), Fi (if ) = Fi (if ) + Ii,jc sa 11: Set nn = Mnv,i (i, jc ), ea (i + 1, jc ) = Vn,i (nn ) 12: if i = N − 1 then: eb (i + 1, jc ) = 0 [V] sa (i + 1, jc ), eb (i + 1, jc ) = Vn,i (nn ). Calculate 13: else: Set nn = Mnv,i

∂Ii+1,jc /∂eb (i + 1, jc ) with (3). Set Ji (if , nn ) = Ji (if , nn ) − ∂Ii+1,jc /∂eb (i + 1, jc ) 14: end if 15: Calculate ∂Ii+1,jc /∂ea (i + 1, jc ) with (2), Ii+1,jc with (1). Set sa (i, jc ), Ji (if , nn ) = Ji (if , nn ) − ∂Ii+1,jc /∂ea (i + 1, jc ) and nn = Mnv,i Fi (if ) = Fi (if ) − Ii+1,jc 16: end for 17: Set if = if + 1 18: end if 19: end for 20: end for 21: Return Fi (Vn,i ) and Ji (Vn,i )

The node voltages of the sub-array are obtained solving Fi (Vn,i ) with a numerical method that iteratively evaluates Fi (Vn,i ) and Ji (Vn,i ). The solution of the systems of nonlinear equations obtained

with the proposed method require an algorithm able to guarantee global convergence such as the Line-Search or Trust-Region methods [27]. Then, in this paper the Trust-Region Dogleg method is used; such a method is not presented in this paper, since it is deeply described in [27]. Once the node voltages of the sub-array are known, it is simple to calculate the current Isa,i by using (13), where ea (1, c) and eb (1, c) are the ea and eb voltages of the module in the first row of column c in the sub-array.



Msa,i

Isa,i =

I(ea (1, c), eb (1, c))

(13)

c=1

2.4. Calculation of the array current The array current (Iarray ) for a given Varray is obtained by adding the sub-arrays currents as shown in (14), where Nsa is the number of sub-arrays defined in (6). At this point the model of a PV array with any configuration is complete and it can be used for different applications. For example, it can be used for the reconstruction of the I–V or P–V curves, estimation of the array energy production, dynamic simulations to evaluate MPPT techniques, among others. Finally, the flow chart presented in Fig. 2 summarizes the proposed modeling procedure to calculate the array current for a given array voltage. Iarray =

Nsa 

(14)

Isa,i

i=1

3. Application of the proposed model In this section the irregular 3 × 3 PV array, shown in Fig. 3, is modeled with the procedure described in Section 2 to illustrate the application of the proposed solution. By using this pilot example, with the PV array configuration shown in Fig. 3, the general procedure is applied to a practical case. In such an array N = 3 and M = 3, hence, the seven parameters of the SD model are stored 3 × 3 matrices and Mconn is a 2 × 2 matrix as shown in (15). The array voltage can be settled according to the operating condition to be analyzed. In (15) it can be observed that Mconn has one column of zeros (column 2); therefore, Nsa = 2 according to (6), and Cz is written as shown in (16) according to (7). The array in this example has two sub-arrays named SA1 and SA2 . The first and last columns of SA1 and SA2 are calculated by using (8) with: SAfc,1 = 1, SAlc,1 = 2 (17) and SAfc,2 = 3 and SAlc,2 = 3 (18), respectively. Fig. 4 shows each subarray with their respective nodes voltages, and modules terminal voltages as well as currents.

 Mconn =

1

0

1

0

 (15)

 

M sa

conn,1



M sa

Cz = 0 2 3

conn,2



(16)

SAfc,1 = 1 SAlc,1 = 2

(17)

SAfc,2 = 3 SAlc,2 = 3

(18)

The parameters matrices and connection matrix for each subarray are obtained using (9) and (10), respectively. To illustrate this process, expression (19) shows MIph and the underbraces indicate sa sa the part of the matrix corresponding to MIph,1 and MIph,2 . Following the same procedure, the rest of parameters matrices can be defined sa sa and Mconn,2 are illustrated for each sub-array. The matrices Mconn,1

J.D. Bastidas-Rodríguez et al. / Electric Power Systems Research 155 (2018) 67–79

Fig. 2. Flow chart of the proposed model.

with underbraces in (15). Then, Nn,1 = 2 and Nn,2 = 2 are calculated sa sa by applying (11) and (12) to Mconn,1 and Mconn,2 , respectively.

⎡

Iph

1,1

Iph

1,2

Iph

1,3

MIph = ⎣ Iph

2,1

Iph

2,2

Iph

2,3

Iph

3,1

Iph

3,2

Iph

3,3

⎢ 



M sa

⎤ ⎥ ⎦

(19)

  

Iph,1

M sa

Iph,2

In SA1 and SA2 the unknown voltages to solve are Vn,1 = [e1,1 e2,1 ] and Vn,2 = [e1,2 e2,2 ], respectively. Before applying the Trust-Region sa Dogleg method to solve Vn,1 and Vn,2 , it is necessary to define Mnv,1 sa and Mnv,1 . Following the description introduced in Section 2.3 those matrices are obtained and reported in (20). At this point, all the data required by Algorithm 1 to calculate F1 (Vn,1 ), J1 (Vn,1 ), F2 (Vn,2 ), and J2 (Vn,2 ) have been constructed.



sa Mnv,1 =

1 1 2 2



 

sa Mnv,2 =

1 2

(20)

To illustrate the application of the procedure described in Algorithm 1, the structures of F1 (Vn,1 ) and J1 (Vn,1 ) are presented in (21) and (22) respectively, while (23) and (24) present the structures of F2 (Vn,2 ), and J2 (Vn,2 ).

Fig. 3. 3 × 3 PV array used as example.

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F 1 Vn,1



=

⎡



I1,1 Varray , e1,1



I 2,1 e1,1 , e2,1







− I2,1 e1,1 , e2,1









+ I1,2 Varray , e1,1



− I3,1 e2,1 , 0 + I2,2 e1,1 , e2,1

∂I1,1 ∂I2,1 ∂I1,2 ∂I2,2 ⎢ ∂eb(1,1) − ∂ea(2,1) + ∂eb(1,2) − ∂ea(2,2) J1 (Vn,1 ) = ⎢ ⎣ ∂I2,1 ∂I2,2 + ∂ea(2,1) ∂ea(2,2)



F2 (Vn,2 ) =

⎡

I1,3 (Varray , e1,2 ) − I2,3 (e1,2 , e2,2 ) I2,3 (e1,2 , e2,2 ) − I3,3 (e2,2 , 0)

∂I1,3 ∂I2,3 ⎢ ∂eb(1,3) − ∂ea(2,3) J2 (Vn,2 ) = ⎢ ⎣ ∂I2,3 ∂ea(2,3)







− I2,2 e1,1 , e2,1





− I3,2 e2,1 , 0

 (21)

⎤

∂I2,1 ∂I2,2 − ⎥ ∂eb(2,1) ∂eb(2,2) ⎥(22) ∂I3,1 ∂I2,2 ∂I3,2 ⎦ − + − ∂ea(3,1) ∂eb(2,2) ∂ea(3,2) −

∂I2,1 ∂eb(2,1)



(23)

⎤

∂I2,3 ⎥ ∂eb(2,3) ⎥ ∂I2,3 ∂I3,3 ⎦ − ∂eb(2,3) ∂ea(3,3) −

(24)

Applying the Trust-Region Dogleg method to F1 (Vn,1 ) and F2 (Vn,2 ) the node voltages of each sub-array (Vn,1 and Vn,2 ) are solved, then Isa,1 and Isa,2 are obtained using (13). Finally, Iarray is calculated from (14), which enables to reproduce the electrical behavior of the PV array at the voltage Varray . 4. Validation of the proposed model The modeling procedure presented in Section 2 was implemented in C language using the GNU Scientific Library (GSL) [28] to calculate the Lambert-W function and to perform operations with vectors and matrices. In this paper the Trust-Region Dogleg method [27] was implemented in C. This section presents simulation and experimental results validating the proposed model for SP, TCT and BL arrays formed with PV panels of two modules each, as well as a practical application of the model in reconfiguration. 4.1. Simulation and experimental results To evaluate the performance of the proposed procedure, simulations and experimental tests were made with a 3 × 3 array connected in SP, TCT, and BL configurations. Such configurations were formed using nine JS65 Yingli Solar PV panels, whose main electrical characteristics were obtained from the manufacturer

datasheet [29]. The required parameters were calculated using the procedure introduced in [18,19] obtaining the following data at standard test conditions (STC): Iph = 3.557 A, Isat = 25.183 ␮A, Vt = 0.842 V, Rs = 0.347 , Rh = 202.237 . The bypass diode parameters are Isat,bd = 48.804 ␮A and Vt,bd = 0.141 V. In addition, different irradiance profiles were introduced by modifying the Iph value of each module, this taking advantage that Iph is directly proportional to the irradiance [18,19]. Each JS65 PV panel has two modules (two bypass diodes), each one with 18 cells connected in series and a bypass diode connected in anti-parallel. Since the PV array connections are made at panel level the TCT and BL configurations are, in fact, irregular configurations in terms of modules. Indeed the connection points between two consecutive modules inside a PV panel are not accessible, so the connection becomes irregular when the consecutive modules in a panel work in mismatched conditions. Instead, the SP configuration can be considered a regular structure at both panel and module levels. The modeled SP, TCT and BL configurations are illustrated in Fig. 5, where the PV panels are represented by dashed rectangles. In addition, the experimental platform also includes an electronic load BK8510 and a switching matrix as it is shown in Fig. 6; the switching matrix enables to change the connection between the PV panels to settle the SP, TCT and BL structures previously described. Two irradiance profiles were considered: one profile to obtain two LMPPs in the P–V curve of SP, TCT, and BL arrays; and another profile to obtain three LMPPs in the P–V curve of SP configuration. The irradiance profiles are described in Table 1, where the values represent the percentage of Iph under STC for each module in the array. Fig. 7 present the I–V and P–V curves obtained with the proposed model, the experimental platform, and the circuital implementation of the arrays in Simulink/SimElectronics toolbox,

Fig. 4. PV sub-arrays. (a) Sub-array 1. (b) Sub-array 2.

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Fig. 5. 3 × 3 PV arrays implemented with JS65 Yingli Solar PV panels (a) SP, (b) TCT, (c) BL.

Fig. 6. Experimental platform.

Table 1 Percentage values of Iph for experimental tests.

Table 2 Mean absolute percentage errors and execution times for experimental tests.

Mismatching 1 [%] 100 100 100 100 4.5 4.5

100 100 100 100 4.5 4.5

Mismatching 1

Mismatching 2 [%] 100 100 100 100 100 100

100 100 30 30 55 55

100 100 100 100 70 70

100 100 100 100 10 10

for the two mismatching profiles. The results presented in Fig. 7 put into evidence the usefulness and correctness of the proposed model in the reproduction of the I–V and P–V curves of irregular and regular PV array configurations. The appendix of the paper reports the system of equations and Jacobian matrix, evaluated for the SP configuration, to illustrate the calculation of the nodes voltages of a sub-array. Similar systems of equations are obtained for the other configurations (TCT and BL). The mean absolute percentage error (MAPE) has been evaluated in order to compare the difference in the prediction of the current (Ei ), power (EP ) and GMPP (EGMPP ) of the proposed model

Ei [%] EP [%] EGMP [%] tmod [s] tsim [s]

Mismatching 2

SP

TCT

BL

SP2

1.52 1.51 1.14 1.22 × 10−2 4.47

1.52 1.50 1.17 1.17 × 10−2 4.85

1.46 1.45 1.17 1.30 × 10−2 3.53

1.24 1.22 1.23 1.35 × 10−2 4.16

with respect to the experimental results. The MAPE was calculated by applying (25), where Aj is the reference value, Pj is the predicted value and k is the number of elements of each vector. Table 2 shows Ei , EP and EGMPP for Mismatching 1 (SP, TCT and BL configurations) and Mismatching 2 (SP configuration), as well as the processing time required to process both the proposed model (tmod ) and circuital implementation in Simulink/SimElectronics toolbox (tsim ). Those results put into evidence the satisfactory performance of the proposed model in the reproduction of regular and irregular PV arrays operating under mismatching conditions. The results also

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Fig. 7. Experimental I–V and P–V curves (a) SP with mismatching profile 1, (b) TCT with mismatching profile 1, (c) BL with mismatching profile1, and (d) SP with mismatching profile 2.

show a reduction in two orders of magnitude in the processing time with respect to the circuital implementation of a PV array in a commercial software.

⎛





⎞

1   Aj − Pj  ⎠ MAPE = ⎝  Aj  × 100 k k

(25)

j=1

4.2. Application example: evaluation of PV array configurations In actual power plants SP and TCT configurations are widely used and under uniform conditions both configurations provide nearly the same power [3,30]. The authors of [31,32] show that TCT configurations exhibit better performance under some particular partial shading conditions; however, the authors of [4] and [14] use differ-

ent shading profiles and array sizes to conclude that HC and BL are better options to mitigate the effects of partial shading on PV arrays. Hence, it is difficult to generalize that a single configuration is better for any shading pattern or array size. A better option is to simulate the particular PV system accounting for different possible configurations and shading profiles to select the most efficient one for that particular case. This subsection uses the proposed mathematical model to compare the performance of SP, TCT and BL configurations (using the PV array of Section 4.1), under different partial-shading patterns presented in Table 3. This confirms the usefulness of the model proposed in this paper. The P–V curves of the three configurations (SP, TCT, and BL) for each shading profile (shading profile 1, 2 and 3), obtained with the proposed model, are presented in Fig. 8. For shading profile 1 (Fig. 8a) SP configuration provides the highest GMPP power, while

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75

Fig. 8. P–V curves for the three configurations under different shadowing profiles: (a) shading profile 1, (b) shading profile 2, and (c) shading profile 3.

for shading profiles 2 (Fig. 8b) and 3 (Fig. 8c) the highest GMPP power is provided by BL and TCT configurations, respectively. This confirms that there is not a single configuration providing the best operation conditions for all the shading patterns. In order to maximize the produced power, SP configuration should be set when shading profile 1 is present, and BL and TCT configurations should be set when shading profiles 2 and 3 appear, respectively. To experimentally test the model predictions, the prototype designed for this particular example enables to settle SP, TCT and BL configurations by means of a switching matrix. It receives a digital signal from a digital acquisition system (DAQ) interfacing a processing device (e.g. a PC with Matlab) as presented in Fig. 9. The control of the switching matrix is performed in Matlab so that the SP, TCT and BL configuration are selected. The experimental results are presented in Fig. 10, where the array voltage, current and power are measured at the output of the switching matrix. At the beginning of Fig. 10 the shading profile 1 is imposed, therefore, the reconfiguration system sets the best configuration (SP) and the array power (162 W) and voltage (45 V) correspond to the GMPP of the SP configuration shown in Fig. 8a. From the time 0.17 s to 0.7 s, shading profile 2 is imposed and the best configuration is BL, hence the array power (132 W) and voltage (35 V) correspond to the GMPP highlighted in Fig. 8b. For the last part of the experiment the best configuration is TCT because the shading profile 3 is imposed, hence the array power (70 W) and voltage (42 V) are in agreement with Fig. 8c.

Table 3 Mismatching profiles for the evaluation of PV configurations. Profile 1 [W/m2 ] 500 470 780 220 480 880

560 830 720 570 240 650

Profile 2 [W/m2 ] 80 600 640 700 870 960

620 930 230 660 280 660

680 60 250 220 650 830

60 310 760 680 120 130

90 8 410 640 700 520

This experiment confirms the model predictions, hence the best connection depends on the particular operating conditions of the PV array. Moreover, those results put into evidence the applicability of the proposed model for evaluating PV configurations in real-time, so that new reconfiguration algorithms can be designed to reduce the impact of partial-shading. 5. Conclusion

Profile 3 [W/m2 ] 330 760 660 7 590 380

Fig. 9. Basic structure of the system for evaluating PV configurations.

100 620 120 130 96 140

A modeling procedure for PV arrays connected in any configuration, including the classical SP, TCT, BL or HC, and operating under uniform or mismatched conditions, has been presented. The array configuration is represented by a connection matrix, which indicates the connections among the strings of the array. Analyzing the

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Fig. 10. Experimental results of the pre-calculated PV array reconfiguration process.

connection matrix it is possible to divide the array into sub-arrays that are solved independently. Each sub-array is analyzed by using nodal equations to obtain the node voltages, which in turn are used to calculate the sub-array current. Finally, the currents of all subarrays are added to obtain the array current. The proposed model also introduces an algorithm to evaluate the system of nonlinear equations and the Jacobian matrix associated to each sub-array, which are used by the Trust-Region Dogleg method to obtain the node voltages. The proposed approach has the flexibility to represent a different configuration by only modifying the connection matrix. Hence it makes simple to evaluate different PV array configurations without the need to change the mathematical model. Moreover, the proposed procedure avoid the time consuming task of represent each module by its circuital equivalent as it is needed in several commercial softwares, e.g. electrical simulators such as PSIM or Simulink/SimElectronics. Finally, the model is suitable to be implemented in standard programming languages; in particular this paper reports the model implementation in C language, which demonstrates that it can be implemented on embedded systems. The proposed method has been validated with simulation and experimental tests. The I–V curves of different configurations (SP, TCT and BL configurations) generated with the proposed model were compared with the circuital implementation of those PV arrays in Simulink/SimElectronics toolbox and with experimental measurements obtaining current prediction errors of 1.52%, 1.52% and 1.46% for SP, TCT and BL configurations, respectively. Moreover, the processing times achieved by the proposed model were significantly lower in comparison with the Simulink/SimElectronics toolbox implementation. Those results put into evidence the satisfactory reproduction of I–V and P–V curves of regular and irregular PV arrays under mismatching conditions. Moreover, the errors and processing times are better compared with other reported procedures such as [16] or [8,33,25], which in fact are valid only for SP arrays, hence the model proposed in this paper is general. Finally, the applicability of the proposed model was illustrated with a simple example evaluating PV configurations under different shading profiles. The main limitation of the proposed model concerns the requirement of having the same number of modules in each col-

umn of the array. In any case, this symmetric structure is usually adopted in commercial PV arrays exhibiting SP, TCT, BL or HC configurations. Finally, the development of a new modeling procedure to overcome this limitation is under investigation.

Acknowledgements This work was supported by the Universidad Industrial de Santander under the project 1896, the Universidad Nacional de Colombia, the Instituto Tecnológico Metropolitano, the University of Salerno and Colciencias (Fondo Nacional de Financiamiento para la Ciencia la Tecnología y la Innovación Francisco José de Caldas) under the project MicroRENIZ – 25439 (Code 1118-669-46197) and with the Colciencias Doctoral Scholarships34065242 and 5672012.

Appendix A. This appendix presents the numerical matrices used to calculate the nodes voltages of one of the sub-arrays of the SP array simulated in Section 4.1 and illustrated in Fig. 5a; those matrices were parameterized with the values given in Section 4.1. The SP array has three independent sub-arrays calculated by using (6), and each sub-array has five nodes as it is obtained by using (11). The unknowns of the first sub-array SA1 (first string from left to right of array shown in Fig. 5(a)) are the nodes voltages Vn,1 = [e(1,1) , e(2,1) , e(3,1) , e(4,1) , e(5,1) ], which are defined in terms of the terminal voltages given in (26)–(32). In this way, by applying (1)–(3) and (7)–(10), the elements of the functions matrix [F1 ]5,1 , constructed using the Algorithm 1, are shown in (33)–(37). Similarly, elements of matrix [J1 ]5,5 , also constructed using the Algorithm 1, are shown in (38)–(50), where elements J1(1,3) , J1(1,4) , J1(1,5) , J1(2,4) , J1(2,5) , J1(3,1) , J1(3,5) , J1(4,1) , J1(4,2) , J1(5,1) , J1(5,2) , J1(5,3) are equal to zero. Finally, an initial guess solution based on the open-circuit voltage is defined, and the system is solved by applying the Trust-Region Dogleg method described in [27] to obtain the current Isa,1 . The process is repeated for SA2 and SA3 (second and third strings from left to right of array shown in Fig. 5(a), respectively) to calculate Isa,2 and Isa,3 , respectively, and (14) is used to calculate Iarray . In this way,

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77

different values of Varray are evaluated through the same process going from zero to 65 V to obtain the array I–V and P–V curves. e(1,1) = eb(1,1) = ea(2,1)

(26)

e(2,1) = eb(2,1) = ea(3,1)

(27)

e(3,1) = eb(3,1) = ea(4,1)

(28)

e(4,1) = eb(4,1) = ea(5,1)

(29)

e(5,1) = eb(5,1) = ea(6,1)

(30)

Varray = ea(1,1)

(31)

eb(6,1) = 0

(32)



 



FN1 = −43.7W0 1,1 + 48.8 × 10−6 exp



 



+43.7W0 2,1 − 48.8 × 10−6 exp



 

 



 



 

+43.7W0 4,1 − 48.8 × 10



 −6

 

exp



 

+43.7W0 5,1 − 48.8 × 10



 

exp





 



J1,1 = 2.88





e3





 −V

e4





 −V

e5





 

 





 

− 48.8 × 10−6 + 3.55 − 5 × 10−3 Ve5

0.141

0.141



+ 48.8 × 10−6 − 3.55 − 5 × 10−3 Ve5(36)

0.141

e6

 

− 48.8 × 10−6 + 3.55 − 5 × 10−3 Ve4

0.141

 −V

 

+ 48.8 × 10−6 − 3.55 − 5 × 10−3 Ve4(35)

0.141

e5



− 48.8 × 10−6 + 3.55 − 5 × 10−3 Ve3

0.141

 −V

 



+ 48.8 × 10−6 − 3.55 − 5 × 10−3 Ve3(34)

 −V e4

 

− 48.8 × 10−6 + 3.55 − 5 × 10−3 Ve2



 





+ 48.8 × 10−6 − 3.55 − 5 × 10−3 Ve6(37)

    −V  −V W0 2,1 e1 e2   + 2.88   3.46 × 10−4 exp + 3.46 × 10−4 exp + 9.8 × 10−3 (38)

1 + W0 1,1

J1,2 = −2.88

J2,1 = −2.88

0.141

0.141

     −V  −V W0 2,1 W0 3,1 e2 e3   + 2.88   − 3.46 × 10−4 exp + 3.46 × 10−4 exp + 9.8 × 10−3 (41)

1 + W0 2,1

J2,3 = −2.88



0.141

1 + W0 3,1

  −V e3   − 4.9 × 10−3 − 3.46 × 10−4 exp (42)

W0 3,1

1 + W0 3,1



0.141

   −V W0 2,1 e2   − 4.9 × 10−3 − 3.46 × 10−4 exp (40)

1 + W0 2,1



0.141

1 + W0 2,1

   −V W0 2,1 e2   − 4.9 × 10−3 − 3.46 × 10−4 exp (39)

1 + W0 2,1



J3,2 = −2.88



W0 1,1



J2,2 = 2.88



0.141

 −V

FN5 = −43.7W0 5,1 + 48.8 × 10−6 exp +43.7W0 6,1 − 48.8 × 10−6 exp

e2

0.141



 −6

 −V e3

FN4 = −43.7W0 4,1 + 48.8 × 10−6 exp

 



+ 48.8 × 10−6 − 3.55 − 5 × 10−3 Ve2 (33)

 −V



 

− 48.8 × 10−6 + 3.55 − 5 × 10−3 Ve1



0.141

FN3 = −43.7W0 3,1 + 48.8 × 10−6 exp



0.141

e2



+43.7W0 3,1 − 48.8 × 10−6 exp



e1

 −V

FN2 = −43.7W0 2,1 + 48.8 × 10−6 exp



 −V



0.141

  −V e3   − 4.9 × 10−3 − 3.46 × 10−4 exp (43)

W0 3,1

1 + W0 3,1

0.141

0.141

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J3,3 = 2.88



    −V  −V W0 4,1 e3 e4   + 2.88   − 3.46 × 10−4 exp + 3.46 × 10−4 exp + 9.8 × 10−3 (44)

W0 3,1

1 + W0 3,1

J3,4 = −2.88



J4,3 = −2.88

  −V e4   − 4.9 × 10−3 − 3.46 × 10−4 exp (45) 0.141



  −V e4   − 4.9 × 10−3 − 3.46 × 10−4 exp (46)

W0 4,1

0.141

1 + W0 4,1

J4,4 = 2.88



    −V  −V W0 5,1 e5 e4   + 2.88   − 3.46 × 10−4 exp + 3.46 × 10−4 exp + 9.8 × 10−3 (47)

W0 4,1

1 + W0 4,1

J4,5 = −2.88



0.141

1 + W0 5,1



0.141

  −V e5   − 4.9 × 10−3 − 3.46 × 10−4 exp (48)

W0 5,1

0.141

1 + W0 5,1

J5,4 = −2.88

0.141

W0 4,1

1 + W0 4,1



J5,5 = 2.88

0.141

1 + W0 4,1

   −V W0 5,1 e5   − 4.9 × 10−3 − 3.46 × 10−4 exp (49) 0.141

1 + W0 5,1



    −V  −V W0 6,1 e5 e6   + 2.88   − 3.46 × 10−4 exp + 3.46 × 10−4 exp + 9.8 × 10−3 (50)

W0 5,1

1 + W0 5,1

0.141

1 + W0 6,1

0.141

Where: 1,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve1 ) 2,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve2 ) 3,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve3 ) 4,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve4 )

(51)

5,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve5 ) 6,1 = (5.8 × 10−7 ) exp(0.23 + 0.07Ve6 ) Ve1 = Varray − eb(1,1) Ve2 = ea(2,1) − eb(2,1) Ve3 = ea(3,1) − eb(3,1) Ve4 = ea(4,1) − eb(4,1)

(52)

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