Composite Structures 119 (2015) 477–486

Contents lists available at ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Model order reduction applied to heat conduction in photovoltaicmodules

http://dx.doi.org/10.1016/j.compstruct.2014.09.0080263-8223/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +39 0583 4326 604; fax: +39 0583 4326 565.E-mail address: marco.paggi@imtlucca.it (M. Paggi).

S.O. Ojo a, S. Grivet-Talocia b, M. Paggi c,⇑a Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italyb Department of Electronics and Telecommunications, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italyc IMT Institute for Advanced Studies Lucca, Piazza San Francesco 19, 55100 Lucca, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Available online 16 September 2014

Keywords:PhotovoltaicsThermal systemModel order reductionNumerical methods

Modeling of physical systems may be a challenging task when it requires solving large sets of numericalequations. This is the case of photovoltaic (PV) systems which contain many PV modules, each modulecontaining several silicon cells. The determination of the temperature field in the modules leads to largescale systems, which may be computationally expensive to solve. In this context, Model Order Reduction(MOR) techniques can be used to approximate the full system dynamics with a compact model, that ismuch faster to solve. Among the several available MOR approaches, in this work we consider the DiscreteEmpirical Interpolation Method (DEIM), which we apply with a suitably modified formulation that is spe-cifically designed for handling the nonlinear terms that are present in the equations governing the ther-mal behavior of PV modules. The results show that the proposed DEIM technique is able to reducesignificantly the system size, by retaining a full control on the accuracy of the solution.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

A photovoltaic (PV) system usually consists of an array of PVmodules (e.g. 10), and each module contains several solar siliconcells (e.g. 60 or 70 in commercial modules). Each module is a lay-ered composite such that the silicon cells are sandwiched betweenthe different layers (see Fig. 1 for a schematic representation of amodule cross-section). The computation of thermal stresses andthermo-elastic displacements is of paramount importance bothduring the production process, and during the operating conditionsof the module. In the former case, compressive residual thermo-elastic stresses may arise and are beneficial in case of cracks withinsilicon due to the activation of partial crack closure [1]. In the lat-ter, cyclic thermoelastic stresses are responsible for crack growthin silicon cells and a power-loss of the PV system in time. More-over, thermoelastic deformation may induce failure of the busbarsconnecting solar cells, due to an increase in the gap between cells,as experimentally and numerically studied in [2,3]. In all of thesecases, it is important to accurately compute the temperature distri-bution in the plane of the solar cells [4], but also the temperaturein the various layers [5] for the study of fully coupled thermome-chanical problems.

So far, semi-analytical and numerical solutions [2,3] for theassessment of the change in the gap between solar cells have beenproposed by assuming a uniform temperature field across themodule, which is an assumption holding for stationary conditions.In reality, temperature contour plots obtained from finite elementthermal analysis [6] show that there is a temperature gradientacross each layer, with the regions near the frame being signifi-cantly cooler, while the temperature distribution across the cellsin the center of the module is found to be quite uniform. In addi-tion, existence of cracks in the silicon cells may induce a localizedtemperature increment (hot spots) in the region near the cracksdue to a localized electric resistance [7]. Moreover, transientregimes, such as those taking place in accelerated environmentaltests within climate chambers, or under operating conditions, haveonly marginally been investigated due to the inherent complexityrelated to the very different thicknesses of the layers composinga PV module. In such cases, accurate predictions require the solu-tion of large systems of equations resulting from the finite elementor finite difference approximation of the (nonlinear) partial differ-ential equation governing the problem of heat conduction. Suitabletechniques for reducing the computational requirements for suchsimulations are therefore highly desirable.

The purpose of this study is to formulate a thermal model of aPV system, and to identify an appropriate class of Model OrderReduction (MOR) techniques, capable of approximating the full

478 S.O. Ojo et al. / Composite Structures 119 (2015) 477–486

system by a compact dynamical model characterized by a muchsmaller number of degrees of freedom (state variables), and whosenumerical solution requires a significantly reduced time withrespect to the full system.

Choosing a particular MOR technique that is suitable for a givenproblem depends on several factors, such as type of system (linearor nonlinear), number and structure of equations to be solved forand, in case of nonlinear systems, the degree of nonlinearity. Modelorder reduction is a mature field when applied to linear systems,and several excellent books on the subject are available [8–11].Many techniques are available for linear order reduction, includingKrylov subspace projection based on orthogonal Arnoldi [12,13] orbiorthogonal Lanczos [14] processes, principal components analysisand balanced truncation [15], Hankel norm approximation [16], andsingular value decomposition (SVD) based methods, which includeProper Orthogonal Decomposition (POD) in its many variants [17].Many extensions to nonlinear system are also available, see e.g.[18–22], which combine system projection or truncation with suit-able approximations of the nonlinear terms.

In this work, we propose a POD projection based approach forthe reduction of the model dynamic order, combined with a Dis-crete Empirical Interpolation Method (DEIM) [22] for handlingthe nonlinear terms. We show that this approach is ideally suitedto the inherent structure of the PV discretized heat transfer equa-tions obtained through a finite difference scheme, allowing formassive order reduction at almost no loss of accuracy. A systematicvalidation study is conducted on a PV template system, showingthe influence of the various parameters governing the reductionscheme. Numerical results confirm the excellent scalability of theproposed technique.

2. Formulation of the differential problem and finite differenceapproximation

A 2D thermal model of a PV module is proposed here based onthe work by Jones [23]. We consider a PV module containing 12 sil-icon cells embedded in a composite made of glass, EVA, Silicon,EVA, backsheet and tedlar layers with the properties described inTable 1 [24]. Although solar cells are separated from each otherby a small amount of EVA in their plane, in this work we slightlysimplify the structure and we consider all layers as uniform inthe x and y directions, see Fig. 1. Further, we consider the y direc-tion as infinite.

Under the above assumptions, the following general 2D heatequation for the composite PV panel holds:

C@T@t¼ kx

@2T@x2 þ kz

@2T@z2 þ H � G ð1Þ

where T(x,z, t) represents the space and time dependent tempera-ture profile of the module. C(x,z) is an equivalent volumetric heat

Glass layer (h=4mm)

125 mm

1580 mm

Fig. 1. A sketch of a cross-section of a PV module, not in scale. For

capacity (J/m3 K), which is equal to an equivalent mass densitytimes the equivalent specific heat capacity (C = q � cp) taking intoaccount the composite structure of the laminate, see next section.The function H(x,z, t) represents the heat losses by radiation andconvection taking place at any point within the PV module, andG(x,z, t) is the electrical energy generated by the cell layer. The coef-ficients kx(x,z) and kz(x,z) are the thermal conductivities in the x andz directions respectively. According to Fourier’s law of heat conduc-tion, the heat flows in the x and z direction are related to thesethermal conductivities by

qz ¼ �kz@T@z; qx ¼ �kx

@T@x

ð2Þ

Substituting (2) into (1), we have:

C@T@t¼ � @qx

@x� @qz

@zþ H � G ð3Þ

Using now a finite difference (FD) discretization scheme defined bygrid spacing Dxi and Dzj in the x and z-direction, respectively, withassociated discretization indices i for 1 6 i 6 l and j for 1 6 j 6 s (seeFig. 2), we can rephrase (3) as

Ci; jdTi; j

dt¼

qi�12; j� qiþ1

2; j

Dxiþ

qi; j�12� qi; jþ1

2

Dzjþ Hi; j � Gi; j ð4Þ

where qi�12; j

and qiþ12; j

are the heat flows through the left and rightboundaries of element (i, j), and qi; j�1

2and qi; jþ1

2are the heat flows

through the upper and lower boundaries of the element in its plane.Multiplying (4) by the area Ai,j = DxiDzj of each grid element in theFD discretization leads to

Ci; jAi; j@Ti; j

@t¼ Q i�1

2; j� Q iþ1

2; jþ Q i; j�1

2� Q i; jþ1

2þ Ai; jHi; j � Ai; jGi; j ð5Þ

where Q represents, consistently with energy conservation princi-ples, the heat flow between adjacent cells, which can be furtherexpressed as Q = KDT [24], where DT is the temperature changebetween the two cells, and K is the corresponding thermal conduc-tance. The latter is a function of the equivalent thermal conductiv-ities of the two cells and the width and length of the elements, i.e.Dzj and Dxi.

2.1. Thermal conductances and heat flows

The discretized thermal conductances Ki,j (W/m K) provideinformation on the thermal coupling between the elements inthe discretization of the PV module. Assuming perfect bonding atthe various interfaces between the layers, the thermal conductanceper unit length in the x-direction between cells (i � 1, j) and (i, j) isgiven by [25]

Ki�12; j¼ Dzj

Dxi�1=2kx i�1; j þ Dxi=2kx i; j þ Ri�12; j

ð6Þ

EVA (h=0.5mm)

Si cell (h=0.166mm)

Backsheet(h=0.1mm)

Tedlar (h=0.1mm) 2mmx-axis

z-axis

the actual value of the thicknesses, see the labels in the figure.

Table 1Material properties of the layers of the PV module.

Layer Thickness (mm) Thermal conductivity (W/m K) Density (kg/m3) Specific heat capacity (J/kg K)

Glass 4 1.8 3000 500EVA 0.5 0.35 960 2090PV Cells 0.166 148 2330 677EVA 0.5 0.35 960 2090Back contact 0.1 237 2700 900Tedlar 0.1 0.2 1200 1250

z-direction

x-direction

Fig. 2. Finite difference discretization of the PV module in its plane.

S.O. Ojo et al. / Composite Structures 119 (2015) 477–486 479

where R is the thermal resistance at the interface between the ele-ments. Since in the present approximation, the PV layers are uni-form in the x-direction, the thermal conductivity kx does notchange in the x direction and we have Ri�1

2; j¼ 0 and we can simplify

notation as kx i�1, j = kx i, j = kj. This assumption is still reasonable forthe silicon cell layer, since the cells are separated from each otherby a small amount of EVA (2 mm), much smaller than the lateralsize of each silicon cell (125 mm). Grid spacing Dx is also consid-ered to be constant in the x direction. Therefore (6) becomes

Ki�12;j¼ Dzj

Dx=2kj þ Dx=2kj¼ kj

Dzj

Dxð7Þ

and Ki�12; j¼ Kiþ1

2; j¼ Kj due to material homogeneity in the x-

direction.In the z-direction, we have:

Kj�12¼ Dx

Dzj�1=2kz j�1 þ Dzj=2kz j þ Rj�12

ð8Þ

Kjþ12¼ Dx

Dzjþ1=2kz jþ1 þ Dzj=2kz j þ Rjþ12

At the top and bottom boundary elements of the PV module, (i,1)and (i,s), we have

K12¼ Dx

Dz1=2kz 1 þ R12

ð9Þ

Ks�12¼ Dx

Dzs=2kz s þ Rs�12

ð10Þ

where R12

and Rs�12

are the thermal resistances between the top andbottom elements and the free surfaces.

From (5), the heat flows through the left and the right bound-aries of the element (i, j) are thus defined as

Q i�12; j¼ KjðTi�1;j � Ti; jÞ ð11Þ

Q iþ12; j¼ KjðTi;j � Tiþ1; jÞ ð12Þ

whereas the heat flows through the lower and upper boundary ofthe element (i, j) are

Q i; j�12¼ Kj�1

2ðTi; j�1 � Ti; jÞ ð13Þ

Q i; jþ12¼ Kjþ1

2ðTi; j � Ti; jþ1Þ ð14Þ

2.2. Boundary conditions

In this study, a constant (Dirichlet) temperature is applied tothe right and left boundary of the module. Thus, the heat flow atthe left and right boundary elements of the PV module (1, j) and(l, j) are

Q 12; j¼ KjðTb1 � T1; jÞ ð15Þ

Ql�12; j¼ KjðTl; j � Tb2Þ ð16Þ

where Tb1 and Tb2 are the fixed temperatures imposed at the rightand left of the module. In all subsequent simulations, we will setthem equal to 343 K and 313 K in order to simulate a distinct differ-ential temperature profile from one end of the PV module to theother.

The heat flow at the top and bottom boundary elements of thePV module (i,1) and (i,s) are instead

Qi; 12¼ K1

2ðTsky � Ti;1Þ ð17Þ

Qi; s�12¼ Ks�1

2ðTi; s � TskyÞ ð18Þ

where Tsky is the temperature of the sky.

2.3. Heat loss

The heat loss, which varies through the layer thickness of themodule, is given by the sum of the following contributions [23]

Hðx; z; tÞ ¼ qlwðx; z; tÞ þ qswðx; z; tÞ þ qconvðx; z; tÞ ð19Þ

where the long wave, short wave and convection heat transfers aredenoted by qlw, qsw and qconv respectively.

The short wave radiation heat transfer of a body of area A isgiven by

qsw ¼ AaU ð20Þ

where a and U are the absorptivity of the material and the totalincident irradiance input to the module surface, respectively. Thelong wave radiation heat transfer is given by the Stefan Boltzmannlaw

qlw ¼ reT4 ð21Þ

where r is the Boltzmann’s constant (5.607 � 10�8 J s�1 m�2 K�4)and e is the emissivity of the body. It is assumed that the net longwave exchange is negligible for the rear of the module. Thus, it isonly necessary to calculate the long wave exchange from the sur-face of the module. The net long wave radiation exchange betweentwo surfaces x and y is given by [23]

qlwxy¼ AxFxyðLx � LyÞ ¼ AyFxyðLy � LxÞ ð22Þ

Here Lx and Ly are long wave irradiance per unit area for surface xand y respectively which are given by

Lx ¼ rexT4x and Ly ¼ reyT4

y

where Fxy is the view factor, a fraction of the radiation leaving sur-face x that reaches surface y.

0 5 10 15 20 25 300

100

200

300

400

500

600

700

800

Time (mins)

Irra

dian

ce (

W/s

q.m

)

Experimental dataSimulated data

Fig. 3. Experimental and simulated irradiance input (from 09:52–10:22, 11/01/2011).

480 S.O. Ojo et al. / Composite Structures 119 (2015) 477–486

A tilted module surface not overlooked by adjacent buildings atan angle h from the horizontal has a view factor of ð1þcos hÞ

2 for thesky and ð1�cos hÞ

2 for the horizontal ground [23]. Thus, inserting theview factor coefficient for sky and ground into Lx we have

Lx ¼ r ð1þ cos hÞ2

eskyT4sky þ r ð1� cos hÞ

2egroundT4

ground ð23Þ

Ly ¼ remodT4 ð24Þ

where emod is the module emissivity.Substituting (23) and (24) into (22), we have

qlw ¼ Ar ð1þ cos hÞ2

eskyT4sky þ

ð1� cos hÞ2

egroundT4ground � emodT4

� �ð25Þ

Further, Tsky = Tambient � dT for clear sky condition in which dT = 20 Kand Tsky = Tambient for overcast condition.

The convection heat transfer is related to the temperature gapbetween the upper part of the solar panel and the ambient [23]

qconv ¼ �Aðhc;forced þ hc;freeÞðT � TambientÞ ð26Þ

where hc,forced and hc,free in W/m2 K are the forced and free convec-tion heat transfer coefficients which depend on the wind speed.

Collecting all the heat loss contributions together, we finallyobtain

Hi; j ¼Ai; j ai; jUþr ð1þcoshÞ2

eskyT4skyþ

ð1�coshÞ2

egroundT4ground�ei; jT

4i; j

� ��� hc; forcedþhc; free

�Ti; j�Tambient

� �� �ð27Þ

where ai, j and ei, j denote the absorptivity and emissivity coeffi-cients of the discretized elements in the different layers.

2.4. Power generated by the PV Cell

The power generated by the PV cell at location (i, j) can be esti-mated as [23]

Gi; j ¼ CFFE lnðcEÞ

Ti; jð28Þ

where CFF is the fill factor model constant (1.22 K m2) and E(t) in W/m2 is the incident irradiance input through the thickness of the PVmodule. The constant c is equal to 106 m2/W. It should be notedthat the power generated by the discretized PV cells in (28) isnon-zero only for the silicon cell layer.

The incident irradiance input into the system is obtained fromexperimental data [23]. To obtain a validated result of the reducedorder model to be derived in Section 3, a minute by minute irradi-ance input obtained from the solar resource and meteorologicalassessment project website (http://www.nrel.gov/midc/kalael-oa_oahu/) will be used. The plots for irradiance for a period of30 min are shown below in Fig. 3.

2.5. System of nonlinear ODEs for the PV module

Considering all the relations established in Section 2, the dis-cretized thermal Eq. (5) can be rewritten after substituting the cor-responding expressions for Q, H and G as

Ci; jAi; j@Ti; j

@t� KjTiþ1; j � KjTi�1; j þ Ti; j 2Kj þ Ki; jþ1

2þ Ki; j�1

2

� �� Kjþ1

2Ti; jþ1 � Kj�1

2Ti; j�1 � Ai; j ai; jUþ r ð1þ cos hÞ

2eskyT4

sky

��þ ð1� cos hÞ

2egroundT4

ground � ei; jT4i; j

�� hc; forced þ hc; freeÞðTi; j � TambientÞ� �

¼ �Ai; jCFFE lnðcEÞ

Ti; jð29Þ

At the left and right boundaries of the PV module, Qi�12; j

and Qiþ12; j

from (5) are replaced by Q 12; j

and Ql�12; j

respectively from (15) and(16), while at the top and bottom boundaries of the PV module,Qi; j�1

2and Qi; jþ1

2are replaced by Qi; 1

2and Qi; s�1

2respectively from

(17) and (18).The discretized thermal Eq. (29) can finally be written in a com-

pact matrix form

CAdTðtÞ

dt¼ KTðtÞ þ AIðTðtÞ; EðtÞÞ ð30Þ

or, in explicit form, as

dTðtÞdt¼ KTðtÞ|fflffl{zfflffl}

linear term

þ FðTðtÞ; EðtÞÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Non-linear term

ð31Þ

where K ¼ ðCAÞ�1 � ðKÞ and F = C�1I. The independent variablet 2 [0,h] denotes time, and TðtÞ ¼ ½T1ðtÞ; . . . ; TnðtÞ�T 2 Rn is theunknown temperature vector for all the elements in the FD discret-ization, where we use a single subscript with n denoting the totalnumber of nodes. E(t) is the time-varying irradiance input to thesystem, the matrix K 2 Rn�n contains constants and F(T(t),E(t)) isa nonlinear function evaluated at T(t) component-wise i.e.

F ¼ ½F1; . . . ; Fn�T ð32Þ

2.6. Reference solution for the complete system

Before applying our proposed model order reduction approach,we derive a reference solution for (31) by direct time discretiza-tion. A backward finite difference scheme is selected to solve thethermal problem to avoid any convergence issues associated withexplicit methods in terms of choice of time step. The numericalmethod is implemented in Matlab. A uniform discretization ofthe module in the x-direction is adopted with l = 361 grid points,while there are s = 6 strips in the z-direction with different thick-nesses so that n = s � l = 2166. The solution of this problem is per-formed for ns = 186 time steps, each step representing 10 s ofphysical time. Fig. 4(a) shows the temperature profile for node741 in the silicon layer for all the 186 time steps. The temperaturealong the silicon layer vs. position at the last time step is shown inFig. 4(b). As it can be seen, the transient regime is quite evident andthe temperature in the silicon cell layers is significantly differentfrom cell to cell.

0 50 100 150 200290

300

310

320

330

340

Time step

Tem

pera

ture

(K

)Temperature for node 741

0 0.5 1 1.5305

310

315

320

325

330

335

340

345

Length of the module(m)

Cel

l Tem

pera

ture

(K)

last time step

(a) (b)Fig. 4. Temperature profile of the full system for (a) a node within the silicon layer vs. time step and (b) along the silicon layer at the last time step of the simulation.

S.O. Ojo et al. / Composite Structures 119 (2015) 477–486 481

3. Model order reduction

The direct numerical simulation of (31) may be quite demand-ing in terms of computing resources, especially in view of its exten-sion to a full 3D geometry. For this reason, we investigate in thiswork a MOR technique, with the objective of approximating thelarge-scale system (31) with a lower order compact dynamicalmodel, that is able however to preserve accuracy in its input/out-put transient response. The two key aspects of proposed MORapproach are: (i) a massive reduction in the degrees of freedom(states), and (ii) an accurate representation of the nonlinear termsthat influence the heat exchange of the structure. These twoaspects are analysed in detail.

3.1. System projection

The reduction in the degrees of freedom is here performedthrough a standard projection approach. The vector T collectingall n cell temperatures is approximated as a linear superpositionof a small number k of ‘‘basis vectors’’, which span a reduced ordersubspace. More precisely, we consider the representation T � Vk

eT ,where eT 2 Rk is a reduced temperature vector collecting the coef-ficients of T into a reduced basis, defined by the columns of matrixVk 2 Rn�k. We consider an orthonormal basis, so thatVT

k Vk ¼ I 2 Rk�kðk� nÞ with I an identity matrix. Introducing theabove reduced expression for T into (31), we have

VkdeT ðtÞ

dt� KVk

eT ðtÞ þ FðVkeT ðtÞ; EðtÞÞ ð33Þ

Projecting now these equations along the subspace generated by Vk

leads to

deT ðtÞdt� VT

k KVk|fflfflfflffl{zfflfflfflffl} eT ðtÞ þ VTk FðVk

eT ðtÞ; EðtÞÞwith VT

k KVk ¼ eK and where eK 2 Rk�k

The reduced form of the thermal Eq. (31) reads

deT ðtÞdt¼ eK eT ðtÞ þ VT

k FðVkeT ðtÞ; EðtÞÞ ð34Þ

The above system represents a reduced order model, since its mainvariables are the coefficients of a reduced basis. In order to deter-mine Vk, we use a Proper Orthogonal Decomposition (POD), whichextracts the basis vectors from the actual transient solution of thefull system by means of a truncated singular value decomposition.

In particular, we collect the ns snapshots T(th) obtained from the fullsolution of the system at discrete time steps of size h in the follow-ing snapshot matrix

S ¼ ½Tðt1Þ; . . . ;TðtnsÞ� ð35Þ

and we apply the POD algorithm below

INPUT: S ¼ ½Tðt1Þ; . . . ; TðtnsÞ� � Rn�ns

OUTPUT: Vk ¼ ½v1; . . . ;vk� 2 Rn�k

1. Form the snapshot matrix S = {T(t1), . . . ,T(tns)}2. Perform the singular value decomposition T = VRWT to

produce orthogonal matrices V ¼ ½v1; . . . ;v r � 2 Rn�r andW ¼ ½w1; . . . ;wr � 2 Rns�r and diagonal matrixR ¼ diagðr1; . . . ;rrÞ 2 Rr�r where r is the rank of S.

3. Set a threshold to pick the k highest modes from the diag-onal matrix R

4. Pick the columns in matrix V which correspond to themodes selected in 3 to generate the POD basisVk ¼ ½v1; . . . ;vk� 2 Rn�k

Note that in the present case we choose ns = 186, as the totalnumber of time steps in the full solution. The choice of ns shouldbe carefully considered since it can strongly influence the accuracyof the approximation and the computational cost, as shown later.

3.2. Discrete Empirical Interpolation Method (DEIM)

System (34) is a reduced order model, but the evaluation of thenonlinear term still requires the mapping Vk to the full-size space.The DEIM approach is used here to further approximate the nonlin-ear terms, thus reducing the computational cost associated withthe simulation of the reduced model. According to [10], we writethe nonlinear term of (34) as

NðeT Þ ¼ VTk|{z}

k�n

FðVkeT ðtÞ; EðtÞÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

n�1

ð36Þ

and we define

f ðtÞ ¼ FðVkeT ðtÞ; EðtÞÞ ð37Þ

The basic idea is to approximate f(t) by projecting it onto the sub-space spanned by a suitable set of m� n basis vectors u1, . . . ,um via

f ðtÞ � UcðtÞ ð38Þ

482 S.O. Ojo et al. / Composite Structures 119 (2015) 477–486

where U ¼ ½u1; . . . ;um� 2 Rn�m. The corresponding coefficient vectorc(t) is determined by selecting m significant rows from the overde-termined system (38). This can be achieved by considering the map-ping matrix

P ¼ ½e.1; . . . ; e.m

� 2 Rn�m ð39Þ

where e.i¼ ½0; . . . ; 0;1; 0; . . . ;0�T 2 Rn is the .ith column of the iden-

tity matrix I 2 Rn�n for i = 1, . . . ,m. The coefficient c(t) can thus bedetermined by inverting system

PTf ðtÞ ¼ ðPTUÞcðtÞ ð40Þ

Provided PTU is non-singular, the final approximation of (38) is

f ðtÞ � UcðtÞ ¼ UðPTUÞ�1PTf ðtÞ ð41Þ

Since f ðtÞ ¼ FðVkeT ðtÞ; EðtÞÞ, (41) can thus be written as

FðVkeT ðtÞ; EðtÞÞ � UðPTUÞ�1

PTFðVkeT ðtÞ; EðtÞÞ ð42Þ

Eq. (42) ensures that the nonlinear function F is evaluated for thefull system and then interpolated by matrix P, an operation whichstill shows the dependence of the reduced system on the completesystem size. To avoid this dependence, DEIM interpolates the inputvector of the nonlinear function F and then evaluates F component-wise at its interpolated input vector. Based on this, (42) can be writ-ten as

FðVkeT ðtÞEðtÞÞ � UðPTUÞ�1eF ðPTVk

eT ðtÞ; EðtÞÞ ð43Þ

where eF denotes the selected components of F. This approximationis particularly effective when the full nonlinear function F is evalu-ated independently for each component of its vector argument, asin present FD formulation. The nonlinear term in (36) can now berepresented as

NðeT ðtÞÞ ¼ VTkUðPTUÞ�1|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

k�m

eF ðPTVkeT ðtÞ; EðtÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

m�1

ð44Þ

Now, to evaluate NðeT Þ in (44), we must specify the projectionbasis [u1, . . . ,um] and the interpolation indices [.1, . . . ,.m]. Wecan obtain the basis [u1, . . . ,um] by applying the above describedPOD scheme to the matrix collecting the nonlinear snapshotsF = {F(T(t1)), . . . ,F(T(tns))} resulting from a direct evaluation of thenonlinear function of the full system at different time steps, andthen using the DEIM algorithm described in [22]. The followingimplementation is used to iteratively construct the basis vectorsand the set of interpolation indices.

INPUT: fuigmi¼1 � Rn linearly independent

OUTPUT: .!¼ ½.1; . . . ;.m� 2 Rm

1. [jqj,.1] = max{ju1j}2. U ¼ ju1j; P ¼ ½e.1

�; .!¼ ½.1�

3. for i = 2 to m do4. Solve (PTU)c = PTui for c5. r = ui � Uc6. [jqj,.i] = max{jrj}7. U ½U ui�; P ¼ ½P e.i

�; .!¼ .

!

.i

" #end for

3.3. Modification of DEIM formulation

To control the accuracy of the reduced system more efficiently,we notice that: (i) there are two nonlinear terms with differentcharacteristics in the thermal formulation of the PV module, andthat (ii) these two terms influence different layers of the PV mod-ule. In fact, since we assumed that the net long wave exchange forthe rear of the module is negligible (see Section 2.3), the heat loss

term in the thermal system formulation has most impact on thesurface of the PV module. On the other hand, the power output isgenerated only by the silicon cell (third layer). On this note, theDEIM operation is here performed separately for the two nonlinearterms using two different sets of snapshots. Accordingly, we haveEq. (43) respectively expressed for the two nonlinear terms inthe reduced system as

F1ðVkeT ðtÞ; EðtÞÞ � U1 PT

1U1

� ��1eF 1 PT1Vk

eT ðtÞ; EðtÞ� �ð45Þ

F2ðVkeT ðtÞ; EðtÞÞ � U2 PT

2U2

� ��1eF 2 PT2Vk

eT ðtÞ; EðtÞ� �ð46Þ

Finally, the nonlinear term in (36) can now be represented as

NðeT Þ ¼ VTkU1 PT

1U1

� ��1

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}k�m1

eF 1 PT1Vk

eT ðtÞEðtÞ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

m1�1

þ VTkU2 PT

2U2

� ��1

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}k�m2

eF 2 PT2Vk

eT ðtÞ; EðtÞ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

m2�1

ð47Þ

With this modification, the interpolation of the nonlinear terms canbe handled independently, enabling a finer control on reduced sys-tem complexity and efficiency.

4. Numerical results

In this section, we assess the accuracy of the proposed reducedmodeling scheme by comparing the responses of the compactmodel and the original system. In particular, we investigate theconvergence of the reduced system as a function of the threeparameters that measure its complexity, namely the size k of thereduced basis used in the state-space (linear) projection, and thetwo orders m1, m2 of the nonlinear interpolation.

As shown in Fig. 5, the cell temperatures at the last time step ofthe iterative solution of the full system and of the reduced systemare in fair good agreement by increasing the order k of the reducedsystem. As the order of the reduced system increases from k = 1 tok = 7, the approximation of the reduced system approaches theexact value of the complete system. At k = 7, the reduced systemapproximation fits well the complete system such that furtherincreasing the order of the reduced system does not change theresult significantly.

It is noticed that the number of snapshots ns used to constructthe compact model also affects its convergence to the full solution.Convergence is achieved more efficiently using a high number ofsamples in the snapshots matrix S than using a small number ofsamples. The plots in Fig. 6 illustrate the convergence of thereduced solution using 185, 100, 70 and 40 snapshots while fixingthe order of k = 7 and interpolation order m1 = m2 = 5.

4.1. Validation of the reduced model using simulated irradiance data

In this section, we investigate the sensitivity of the reducedmodel to the input irradiance signal. To this end, the reducedmodel is first constructed based on snapshots derived from exper-imental irradiance data, see Fig. 3. Then, simulated irradiance datais used to excite the model, and the corresponding response iscompared to the full system response computed by direct time dis-cretization. The irradiance data for this validation was carefullyselected to have different environmental characteristics with theidentification experimental data. On this basis, a day is chosen inautumn of November, 2011 with average air temperature of22 �C and average wind speed of 3 m/s. It is verified that thereduced model approximates the full system to a reasonabledegree of accuracy also under this different excitation, as shownin Fig. 7.

0 50 100 150 200290

300

310

320

330

340

350

Time step

Tem

pera

ture

(K

)

Temperature for node number 741

Completek=1k=3k=5k=7

0 0.5 1 1.5300

310

320

330

340

350

360

Length of the module(m)

Cel

l Tem

pera

ture

(k)

last time step

Completek1=1k1=3k1=5k1=7

(a) (b)Fig. 5. Convergence of reduced solution to the full solution with increasing order of k and fixed interpolation points m1 = m2 = 3 for (a) a specific point in the silicon layer, and(b) for the entire silicon layer.

0 50 100 150 200292

294

296

298

300

302

Time step

Tem

pera

ture

(K

)

Temperature for node 250

Complete185 Snapshots100 Snapshots70 Snapshots40 Snapshots

Fig. 6. Convergence of the reduced solution by increasing the number of snapshotsfor order of basis k = 7 and interpolation points with m1 = m2 = 5.

S.O. Ojo et al. / Composite Structures 119 (2015) 477–486 483

4.2. Error analysis

To demonstrate the efficiency and accuracy of the nonlinearorder reduction, an error plot is deemed necessary to observe theconvergence of the responses by varying the interpolation pointsm1,m2 and the dynamical order k. To do this, we compute anormalized error e(k,m1,m2) as

e ¼ kT �eTk2

kTk2ð48Þ

where the norm is defined either in time domain by fixing the celllocation, or in the space domain by fixing time step.

In order to observe the rate of convergence of the reduced sys-tem as we increase its dynamical order k with a fixed number ofinterpolation points m1 = m2 = m, we compute the error at specifictime steps representative of the beginning, middle and end of sim-ulation. A time variation error plot is also obtained for a selectedelement in the PV module for all the time steps. The results areshown in Fig. 8.

A clear lower error bound can be observed from the error plots,which is an indication that the reduced system response converges,as the order of the system is increased, only to the extent allowedby the representation of the nonlinear terms. It should be generally

noted that the error obtained by using only one interpolation point(i.e. an equivalent of a linear system) is in the range of 10�5 to 10�6,an indication that the nonlinearity in the system is not at all strong.Increasing the number of interpolation points further shifts theerror bound from 10�5 to less than 10�8 which confirms the excel-lent suitability of the DEIM algorithm for this thermal modelingtask.

In order to verify the improved efficiency that can be achievedby using different interpolation points for the two nonlinear terms,as against using the same interpolation order for the two nonlinearterms, we perform various experiments by independently varyingm1 and m2. The results are shown in Fig. 9.

The plots shown in Fig. 9 confirm that by independently varyingm1 and m2, we can achieve better control of the accuracy of thereduced model. In Fig. 9(a) where the same interpolation order isused for the two nonlinear terms, the reduced model becomes effi-cient as order k is increased above 50 when the lower bound errorbecomes more stable. By varying m1 and m2 independently, thestability of the lower error bound is attained with a smaller orderk. It can be clearly observed in Fig. 9(b)–(d) that there is a reductionin the lower bound error from less than 10�6 to less than 10�8 asm1 is increased from 1 to 7 while m2 is varied for each fixed m1.Furthermore with independent variation of m1 and m2, a lowerorder k (<50) of the linear subspace projection is required to attaina stable error bound. This observation proves that, a better approx-imation of the full system can be achieved by independent varia-tion of the interpolation of the nonlinear terms.

4.3. Runtime

Finally, we emphasize the advantages of proposed MOR tech-nique by reporting the runtime required for the various simula-tions on the same commodity laptop. A transient analysis of thefull system requires 61.40 s. Based on available snapshots, the con-struction of the reduced model via the proposed POD/DEIMrequires as few as 0.86 s, whereas the transient simulation of thereduced model (k = 7 and m1 = m2 = 3) takes only 1.40 s. Excludingmodel setup and construction, the overall speedup is almost 44X.Considering that the proposed model is a quite simplified and 2Dstructure, more dramatic speedup is expected when applying thisprocess to a full 3D geometry.

0 50 100 150 200290

300

310

320

330

340

350

Time step

Tem

pera

ture

(K

)Temperature for node number 741

Completek=1k=3k=5k=7

0 0.5 1 1.5300

310

320

330

340

350

360

Length of the module(m)

Cel

l Tem

pera

ture

(k)

last time step

Completek1=1k1=3k1=5k1=7

(a) (b)Fig. 7. Convergence of reduced solution to the full solution with increasing order of k and fixed interpolation points m1 = m2 = 3 for (a) a specific point in the silicon layer, and(b) for the entire silicon layer.

0 50 100 150 20010

-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 50th time step

m=1m=2m=4m=5m=7

0 50 100 150 20010-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 150th time step

0 50 100 150 20010-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 186th time step

0 50 100 150 20010

-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 150th node

(a) (b)

(c) (d)

m=1m=2m=4m=5m=7

m=1m=2m=4m=5m=7

m=1m=2m=4m=5m=7

Fig. 8. Error plot for the topmost layer for fixed m1 = m2 = m at (a) 50th time step (b) 150th time step (c) 186th time step, and for (d) node number 150 in the discretizedmodule for all 186 time steps.

484 S.O. Ojo et al. / Composite Structures 119 (2015) 477–486

0 50 100 150 20010-10

10-8

10-6

10-4

10-2

100

k

Err

orError plot for 186th time step

m=1m=2m=4m=5m=7

0 50 100 150 20010-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 186th time step

m2=1m2=2m2=4m2=5m2=7

0 50 100 150 20010-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 186th time step

m2=1m2=2m2=4m2=5m2=7

0 50 100 150 20010-10

10-8

10-6

10-4

10-2

100

k

Err

or

Error plot for 186th time step

m2=1m2=2m2=4m2=5m2=7

(a) (b)

(c) (d)Fig. 9. Error plot for the third layer at the 186th time step for (a) m1 = m2 = m (b) m1 = 1 (c) m1 = 5 (d) m1 = 7.

S.O. Ojo et al. / Composite Structures 119 (2015) 477–486 485

5. Conclusion

The partial differential equation governing the heat conductionwithin a 2D photovoltaic system has been derived and a numericalsolution based on a finite difference scheme has been proposed. Inorder to limit the computational burden, we have identified theDEIM as a suitable technique to reduce the system through a combi-nation of linear subspace projection and interpolation of nonlinearterms. In the numerical examples, we show that the heat conductionof a PV system discretized into 2166 nodes along the module span issuccessfully reduced to a compact model with dynamical orderk = 7, based on interpolation with only m1 = m2 = 3 points. For thespecific system under investigation, we see that further incrementin k does not result into a significant accuracy, and that the efficiencyin the numerical solution of the reduced model, which depends onthe number of interpolation points, can be fine- tuned by carefullyselecting a different interpolation order for individual nonlinearterms. From the validation results, it is also concluded that thereduced solution is not very sensitive to the input function as itapproximates well the simulated irradiance data much the sameway as the experimental irradiance data used in the constructionof the compact model. As a future development of the presentresearch, the generalization of the present formulation for coupledthermo-mechanical problems is envisaged. In this case, the pres-ence of cohesive crack bi-material interfaces between the layers,

to model temperature-induced debonding of the back-sheet, shouldbe exploited in order to assess the durability of PV modules. Also,extension to a full 3D geometry is under way.

Acknowledgments

MP would like to thank the European Research Council for theERC Starting Grant ‘‘Multi-field and multi-scale ComputationalApproach to Design and Durability of PhotoVoltaic Modules’’(ERC Grant Agreement n. 306622 – CA2PVM) granted under theEuropean Union’s Seventh Framework Programme (FP/2007-2013). SOO would like to acknowledge the support of the ItalianMinistry of Education, University and Research to the project FIRB2010 Future in Research ‘‘Structural mechanics models for renew-able energy applications’’ (RBFR107AKG).

References

[1] Paggi M, Berardone I, Infuso A, Corrado M. Fatigue degradation and electricrecovery in Silicon solar cells embedded in photovoltaic modules. Sci Rep2014;4:4506. http://dx.doi.org/10.1038/srep04506.

[2] Eitner U, Köntges M, Brendel R. Use of digital image correlation technique todetermine thermomechanical deformations in photovoltaic laminates:measurements and accuracy. Sol Energy Mater Sol Cells 2010;94:1346–51.

[3] Köntges M, Kunze I, Kajari-Schröder S, Breitenmoser X, Bjrneklett B. The risk ofpower loss in crystalline silicon based photovoltaic modules due tomicrocracks. Sol Energy Mater Sol Cells 2011;95:1131–7.

486 S.O. Ojo et al. / Composite Structures 119 (2015) 477–486

[4] Sapora A, Paggi M. A coupled cohesive zone model for transient analysis ofthermoelastic interface debonding. Comput Mec 2014;53:845–57.

[5] Paggi M, Corrado M, Rodriguez MA. A multi-physics and multi-scale numericalapproach to microcracking and power-loss in photovoltaic modules. ComposStruct 2013;95:630–8.

[6] Leea Y, Andrew A, Taya B. Finite element thermal analysis of a solarphotovoltaic module. Energy Procedia 2012;15:413–20.

[7] Paggi M, Sapora A. Numerical modelling of microcracking in PV modulesinduced by thermo-mechanical loads. Energy Procedia 2013;38:506–15.

[8] Antoulas AC. Approximation of large-scale dynamical systems. Philadelphia,PA, USA: Society for Industrial and Applied Mathematics; 2005.

[9] Benner Peter, Hinze Michael, Ter Maten E Jan W. Model reduction for circuitsimulation. Lecture notes in electrical engineering, vol. 74. Springer; 2010.

[10] Tan SXD, He L. Advanced model order reduction techniques in VLSIdesign. New York: Cambridge University Press; 2007.

[11] Schilders WHA, Van Der Vorst HA, Rommes J. Model order reduction: theory,research aspects and applications. Springer Verlag; 2008.

[12] Odabasioglu A, Celik M, Pileggi LT. PRIMA: passive reduced-order interconnectmacromodeling algorithm. IEEE Trans Comput Aided Des Integr Circuits Syst1998;17(8):645–54.

[13] Arnoldi WE. The principle of minimized iterations in the solution of the matrixeigenvalue problem. Q Appl Math 1951;9(1):17–29.

[14] Lanczos C. An iteration method for the solution of the eigenvalue problem oflinear differential and integral operators1. J Res Natl Bureau Stand 1950;45(4).

[15] Moore B. Principal component analysis in linear systems: controllability,observability, and model reduction. IEEE Trans Autom Control1981;26(1):17–32.

[16] Glover K. All optimal Hankel-norm approximations of linear multivariablesystems and their l – inf error bounds. Int J Control 1984;39(6):1115–93.

[17] Willcox K, Peraire J. Balanced model reduction via the proper orthogonaldecomposition. AIAA J 2002;40(11):2323–30.

[18] Bond B, Mahmood Z, Li Y, Sredojevic R, Megretski A, Stojanovic V, et al.Compact modeling of nonlinear analog circuits using system identification viasemidefinite programming and incremental stability certification. IEEE TransCAD Integr Circuits Syst 2010;29(8):1149–62.

[19] Phillips JR. Projection-based approaches for model reduction of weaklynonlinear, time-varying systems. IEEE Trans Comput Aided Des IntegrCircuits Syst 2003;22(2):171–87.

[20] Rewienski M, White J. A trajectory piecewise-linear approach to model orderreduction and fast simulation of nonlinear circuits and micro-machineddevices. IEEE Trans Comput Aided Des Integr Circuits Syst 2003;22(2):155–70.

[21] Astolfi A. Model reduction by moment matching for linear and nonlinearsystems. IEEE Trans Autom Control 2010;55(10):2321–36.

[22] Chaturantabut S, Sorensen D. Nonlinear model reduction via discrete empiricalinterpolation. SIAM J Sci Comput 2010;32(5):2737–64.

[23] Jones A, Underwood C. A thermal model for photovoltaic systems. Sol Energy2001;70(4):349–59.

[24] Armstrong S, Hurley W. A thermal model for photovoltaic panels undervarying atmospheric conditions. Appl Therm Eng 2010;30:1488–95.

[25] Blomberg T. Heat conduction in two and three dimensions: computermodelling of building physics applications. Report TVBH-1008, Departmentof Building Physics, Lund University, Sweden; 1996. p. 5–13.