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CHEMICAL ENGINEERING TRANSACTIONS

VOL. 45, 2015

A publication of

The Italian Association

of Chemical Engineering

www.aidic.it/cet
Guest Editors: Petar Sabev Varbanov, Jiří Jaromír Klemeš, Sharifah Rafidah Wan Alwi, Jun Yow Yong, Xia Liu

ISBN 978-88-95608-36-5; ISSN 2283-9216 DOI: 10.3303/CET1545063

Please cite this article as: Thakare M.S., Ghosh P.C., Bandyopadhyay S., 2015, Optimal sizing of a photovoltaic-thermal

hybrid system using energy ratio, Chemical Engineering Transactions, 45, 373-378 DOI:10.3303/CET1545063

373

Optimal Sizing of a Photovoltaic-Thermal Hybrid System

using Energy Ratio

Mangesh S. Thakare, Prakash Chandra Ghosh, Santanu Bandyopadhyay*

Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India

santanub@iitb.ac.in

A photovoltaic-thermal hybrid system (PVT) is a renewable energy based cogeneration system as it

simultaneously converts solar radiation to electricity and low temperature heat. For applications with limited

space, such as building integrated systems, use of a PVT system may be better than installing separate

collectors to produce electrical and thermal energy. Reduction of total space requirement for an integrated

PVT system depends on the electrical and thermal energy requirements. The concept of energy ratio is

introduced in this paper for selection of appropriate solar device to satisfy the thermal and electrical energy

demands. A mathematical optimization problem is formulated to identify the range of energy ratio for which

installation of PVT system is optimal.

1. Introduction

A photovoltaic-thermal hybrid system (PVT) is a renewable energy based cogeneration technology. It

produces both electricity and heat by means of one integrated component, in which cells are applied on the

thermal absorber. Dupeyrat et al. (2014) performed TRNSYS simulation to show that the integration of PV

and thermal energy capture on a limited roof area provides higher electrical and exergy output. Many efforts

are directed towards increasing thermal and electrical performances of PVT system: channel type with

parabolic reflector (Garg and Agarwal, 1995), sheet and tube type (Zondag et al., 2002); booster reflectors

(Tripanagnostopoulos et al., 2002), two absorber type (Bakker et al., 2002), polymer channel absorber

(Sandnes and Rekstad, 2002), single-crystalline silicon with flat reflectors (Kostić et al., 2010), single glazed

flat plate PVT with heat exchanger (Dupeyrat et al., 2011), unglazed box type prototype collector (Thakare et

al., 2013), prototype PVT collector with square channel (Evola and Marletta, 2014). An integration of
renewable energy in the form of photovoltaic power for running a heat and water system also presented by

Martínez-Patiño et al. (2012) The performance improvement includes efforts to improve system performance

by maintaining an optimum water flow rate (Kalogirou, 2001; Fudholi et al., 2014). The optimum flow rate

result in increase the thermal and electrical performance of the PVT system (Chow, 2010). Aste et al. (2012)

presented a methodology to calculate the best value of solar fraction ‘f’ for hybrid PVT systems under

energetic end economic point of views.

The concept of energy ratio is introduced in this paper for selection of appropriate solar device to satisfy the

thermal and electrical demands. A mathematical model is proposed to determine energy ratio (α) for choice of

side by side PV and Thermal collector systems vis-a-vis PVT system. A simple methodology is developed to

take decision about the solar device with the help of two dimensionless parameters i.e. area ratio (ar) and α.

The performance of PVT system over side by side PV and Thermal system is assessed for given solar

insolation, ambient temperature and load profiles.

2. Mathematical model

A typical PVT based cogeneration system contains a PVT collector which is made up from a PV laminate. In

this system, thermal energy is recovered which is otherwise lost from the PV laminate. It decreases the

operational temperature of the PV laminate and increases the electrical efficiency. The thermal energy

generated by the system is stored in storage tank and the thermal load is served at specific load temperature

374

where as electrical energy is stored in battery bank. The electrical load is supplied with the help of charge

controller and inverter. As a consequence, in a limited space application, in comparison with separate PV and

thermal system, an appropriate ‘α’ of PVT based cogeneration system can serve the both thermal and

electrical demand efficiently. A PVT panel model proposed by Zondag et al. (2002) is adopted in this paper to

explain the concept of ‘α’ of the PVT based renewable cogeneration system. A mathematical problem is

formulated with the objective of minimizing the panel area with given thermal and electrical energy

requirements. The optimization model formulation is mixed integer non linear programming (MINLP) problem

for unity solar fraction PVT system with immediate water replenishment. The model divides in two parts as

follows.

2.1 Thermal subsystem formulation
For PVT system, useful hourly thermal energy generation Eq(1) and electrical efficiency Eq(2) are the basic

equations. In these, Apv is the area of collector and ηel the electrical efficiency of the cells. Fr is the heat

removal factor and Tsti and Ta are the hourly inlet and ambient temperatures respectively. The average solar

radiation, incident on the PVT collector, in an hour and loss coefficient are given by IT and UL, respectively. τα

and τ are the transmission absorption factor and transmission factor of glass. The mean plate temperature

and reference temperature of cell are given by Tpm and Tref. The symbol ηref indicate reference cell efficiency

at 25 °C and β is the cell efficiency temperature coefficient.

)))()(()()(()( tTtTUtIFAtQ astiLTelrpvu   (1)

  )))((1( refpmrefel TtTt   (2)

Selection of useful thermal output is carried out through binary variables x1(t) and x2(t). The binary variable

x1(t) is used to separate out positive useful thermal gain. The same is then used to calculate hourly mean

plate temperature (Tpm) of the collector using Eq(3). The Tpm is a function of useful heat gain, heat transfer

coefficient across cell and absorber (hca), Apv and mass flow rate ( m ). The hourly useful thermal output and
electrical efficiency of the PVT collector are determined as the Tpm converged to constant value. Getting the

value of electrical efficiency, the electrical output is determined by multiplying it with collector area and solar

insolation.

   
    













 















Tel
a

uu
stipm

tIt* -
+ ( t)( Tx2( t)

Apvhca

( t)Q

Cp* m*2

( t)Q
+ ( t)( Tx1( t)tT




)(

(3)

The demand met by solar energy (qls(t)) to meet the load at a mass flow rate m is assumed to be constant
over a time step t and is calculated based on the initial temperature of the storage tank (Tsti).The thermal

energy supplied to load is calculated as,

))()(()( tTtTCmtq astipls   (4)

The surface area of the cylindrical storage tank is assumed to be related to the storage volume (Vst) by

following relation with equal height to diameter ratio (Kulkarni et al., 2009).

3/2
)(54.5 stst VA  (5)

The storage heat losses in the storage tank are estimated as:

))()(()( tTtTAUtQ astistststl  (6)

The model predicts the hourly storage tank temperature using Eq(7).

 



















VstC

qtqtQ
tTtT

stllsu
stistf



))()(
)()( (7)

In order to supply the thermal load at specific load temperature, load temperature constraint; and to avoid

phase change of working fluid in a collector, saturation temperature constraint are imposed on the storage

tank temperature respectively. The hourly storage temperature is obtained iteratively for a day assuming

thermal steady state operation of storage tank.

2.2 Electrical subsystem formulation
The electrical energy generated by the PVT collector in an hour is calculated by Eq(8). The change in storage

energy level can be represented as the difference between the electrical energy generated by the PVT

375

collector (P(t)), the energy required by the load (D(t)) and the excess energy (Pdu(t)) which is to be dumped.

The energy balance on the electrical subsystem is accounted using Eq(9).

     tIApvttP Tel  (8)

       tPtDtPtPS du (9)

The binary variables y1(t), y2(t) are used to select amount of energy transfer during charging or discharging

respectively. The battery energy level for an hour is given by Eq(10).

        CD tPSy2( t) CD tPSy1( t)1-tQtQ BB 21  (10)

where CD1 and CD2 are the charging and discharging efficiencies of the battery. During operation some of

the excess energy needs to be dumped so that the battery energy level is kept to a minimum while meeting

the load at all times. The daily excess energy dumped is considered as a non negative quantity.

The battery bank has to store larger energy than the hourly battery energy level. The constraint of maximum

amount of energy that battery can store in a day is accounted by the variable BBC. The battery size (Br) is

calculated using Eq(11) considering the depth of discharge of battery (DOD).

BBC/DOD Br  (11)

2.3 Objective function
In order to satisfy given demands for a given time period, area of PVT collector has load temperature and

storage related constraints. The solar useful thermal output, mean plate temperature and electrical output of a

collector is decided by the area of PVT collector. The objective function described in Eqs(1), (3), (8) and (12)

is to minimize area of PVT collector.

A Minimize pv (12)

The optimum solution is obtained using General Algebraic Modeling System (GAMS, 2007) software with

BARON solver for the above MINLP formulation.

2.4 Computation of Energy ratio
Being solar renewable cogeneration, PVT generates both thermal and electrical energy. Hence, the daily

thermal energy (QTL) and electrical energy (QEL) require to supply the thermal and electrical demands are

need to be considered to compute the energy ratio. The energy ratio, α is then defined as the ratio of the daily

total thermal energy require to supply the thermal demand to the sum of total thermal and electrical energy

require to meet the both thermal and electrical demands. The value of α is of then calculated by obtaining the

optimal area of PVT collector for the particular thermal and electrical demands with the mentioned thermal
and electrical constraints by Eq(13).

)/( ELTLTL QQQ  (13)

The value of α lies between 0 and 1 for combine thermal and electrical demands where as it is 0 for only

electrical demand and 1 for only thermal demand. In the analysis of the proposed energy ratio, a numerical

optimization is carried out to determine minimum area of PVT collector and side by side PV and thermal

system to meet the various thermal and electrical demands.

To determine the area require for individual PV and thermal system, the above explained model is adapted by

considering the thermal and electrical subsystem formulation separately. The MINLP formulation constitutes

by Eqs(8-11, 12) of electrical subsystem are used to determine area of PV system. The performance of the

PV module depends on the PV cell temperature. The factors affecting cell temperature includes mainly

ambient temperature, solar irradiance and wind speed etc. To account effect of cell temperature, mean plate

temperature (Tpm) in electrical efficiency Eq(2) is modified with the cell temperature (Tcell) for PV system.

Typically, over a range of environmental conditions, the nominal operating cell temperature (NOCT) is used to

predict the cell temperature. According to the definition, NOCT is the temperature of the cells at a solar

irradiance (GNOCT) of 800 W/m
2
, an ambient temperature (Ta,NOCT) of 20 °C, and a wind speed of 1 m/s. As

polycrystalline cell is considered for PVT and PV panel, the NOCT value of polycrystalline cell is taken as 44

°C (Davis et al., 2001). The performance of PV system is analyzed with the following cell temperature

equation (Markvart, 2000).

)(*)( tI
G

T - NOCT
( t)TtT T

NOCT

NOCTa,
acell 















(14)

376

It is to be noted the behaviour of thermal system constitute non linear programming (NLP) mathematical

formulation for separate thermal system. As only thermal output is generated by thermal system, the useful

thermal output Eq(1) of PVT system is changed to Eq(15) to analyze the thermal system performance. The

area require to meet the thermal demand by separate thermal system is determined using Eqs (3-7, 12 and

15).

)))()(()()(()( t
a

Tt
sti

T
L

U
r

Ft
T

I
r

F
pv

At
u

Q  
(15)

The optimal area obtained in individual PV and thermal subsystem formulation is then added to get total area

of side by side PV and thermal system.

2.5 Computation of Area ratio

The concept of non dimensional area, area ratio ar, is introduced here to analyze area requirement for PVT

system in comparison with separate PV and thermal system. The area ratio, ar is defined as the ratio of the

area of PVT to supply the both thermal and electrical demands to the sum of total area require by individual

PV (Aphotovoltaic) and Thermal (AThermal) system to meet the thermal and electrical demand separately.

)/( ThermalicPhotovoltaPVr AAAa  (16)

The value of ar more than 1 infers demands can be met by separate PV and thermal system where as for ar

less than 1, PVT requires less area to serve the both demands.

3. Sizing of solar devices using energy ratio

Influence of energy ratio on sizing of solar devices is analyzed in this section. Two cases of electrical load

dominant and thermal load dominant systems are presented to show the comparison of space requirement

for PVT over side by side PV and thermal system. Keeping the same time-varying nature of load profile,

energy ratio is obtained by varying total energy requirement. For the given insolation and ambient

temperature, the energy ratio obtained in the range of 0 to 1. The size of the each solar device to satisfy

thermal and electrical demands is obtained using optimization model proposed in earlier section. The options

available to satisfy thermal and electrical demands include combine PVT system and individual PV panel and

thermal collector. The methodology is explained with the help of flowchart in Figure 1. The parameter values

used for the calculation of thermal system are referred from Zondag et al. (2002).The reference cell efficiency

is considered as 9.7 % at 25 °C where as battery charging and discharging efficiencies are assumed to be 86

%.The DOD of battery is assumed to be 50 %. The results obtained with both kind of electrical and thermal

load dominant system are presented below.

3.1 Electrical load dominant system
The effect of change in energy ratio on sizing of PVT and separate PV and Thermal system is analyzed in this

section. The solar insolation and ambient temperature data for March 15th for a rural location near Mumbai

(19º17’N, 72º48’E) has been used for calculations (Krishna Priya et al., 2013). The electrical load pattern is

adapted from Arun et al. (2008) and the thermal load pattern from Pillai and Banerjee (2007). The thermal

load is 180 litres per day (LPD) at 60 °C. The Figure 2 shows sizing curve for the electrical load dominant

system. It shows nature of ‘α’ verses area ratio ‘ar’ for solar devices. It indicates for ‘ar’ more than 1, individual

PV panel or solar thermal collector requires less space as compared to integrated PVT system. The choice of

integrated PVT panel can be beneficial for ‘ar’ less than 1. From the Figure 2, it can be seen that the value of

‘ar’ is more than 1 for ‘α’ nearer to 0 or 1. It means for the ‘α’ approaches to 0 electrical energy requirement is

dominant where as ‘α’ closed to 1 thermal energy requirement is predominant. The value of ‘α’ closed to zero

suggests PV panel require less space as compared to PVT panel where as the ‘α’ closed to 1 indicate solar

thermal collector require less space over integrated PVT panel. It is due to the fact that the cogeneration of

electrical energy and thermal energy decide the area of PVT system. It indicates, if one kind of the load is not

present then that energy generated is not being used, which results in increase in area as compare to side by

side PV and Thermal system. It is observed for electrical dominant system, for ‘α’ in the range of 0.1 to 0.93,

integrated PVT panel has ‘ar’ less than 1. The area required to satisfy both thermal and electrical demands is

less for PVT system during this ‘α’ range. It is due to the combined efficiency of integrated PVT panel is more

than the individual PV panel and solar thermal collector (Zondag et al., 2003). It can be seen that the point ‘x’

in the Figure 2 corresponds to ‘α’ of 0.67 and ‘ar’ of 0.6 represent optimal value of α to fetch maximum benefit

of space reduction. The point ‘x’ indicates balance kind of thermal and energy requirement for PVT panel. The

appropriate value of ‘α’ can result in less space requirement in building integrated solar applications. It also

helps to use of solar energy optimally to satisfy both thermal and electrical demands.

377

Figure 1: Methodology for sizing of solar devices

3.2. Thermal load dominant system
To analyze effect of ‘α’ on sizing of PVT and individual PV and Thermal system, the electrical load pattern is

adapted from Arun et al. (2008) and the thermal load pattern from Kulkarni et al. (2009). The domestic

thermal load is considered as 4,500 LPD at 60 °C. For this case, monthly mean values of hourly solar

radiation (Mani, 1981) on April 15 are used. The sizing curve for the thermal load dominant system is shown

in Figure 3. It shows usefulness of ‘α’ to decide option available to serve thermal and electrical demands. The

area requires to supply both the demands is less for a PVT system for ‘α’ in the range of 0.08 to 0.95. The ‘α’

lying outside the range (0.08 - 0.95), suggests less area require for individual PV and thermal system to

supply electrical and thermal demands respectively. The point ‘y’ in Figure 3 corresponds to ‘α’ of 0.82 and ‘ar’

of 0.55 indicates the optimal balance of thermal and electrical demand for the thermal load dominant system.

It gives idea of balanced thermal and electrical load for PVT system to serve at overall optimum. Such

balanced load can results into significant reduced area for PVT system over side by side PV and thermal

system in limited space application.

Figure 2: Sizing curve for electrical load dominant Figure 3: Sizing curve for thermal load dominant

system system

The ‘α’ infers the proportion of both kind of load to be supplied by PVT system and individual PV and thermal
system. The thermal energy is predominant after the point ‘y’ i.e. above ‘α’ of 0.82, hence the curve is having
positive slope. The higher the area required to meet the prominent thermal load for both PVT and solar
thermal collector result in increase in value of ‘ar’.

4. Conclusions

A mathematical model, based on MINLP formulation, is presented to select appropriate solar device to meet
simultaneously thermal and electrical energy demands. A simple methodology is proposed to take decision
about appropriate solar device in view of limited space availability. The concept of energy ratio and area ratio
is proposed to decide for sizing either of integrated PVT and side by side PV and thermal system. The energy
ratio infers significant reduction in area for PVT system over individual PV and Thermal system for balance
kind of thermal and electrical demands. It is seen that there exists a range of energy ratio (0.1 to 0.93 for two
examples considered in this paper), within which it is optimal to use PVT system. The future direction of
research is to evaluate the effect of energy ratio for other configuration of PVT over side by side PV and

End

Given : Demands, insolation data, ambient temperature

Determine the minimum area for PVT, individual PV and

Thermal system

Calculate energy ratio and area ratio

Start

Plot the energy ratio verses area ratio of PVT, and side by side

PV and T

0.2

0.4

0.6

0.8

1.2

0 0.2 0.4 0.6 0.8 1

A
r
e
a
r
a
ti
o
,
a

Energy ratio, α

PV + T

PV T

P
V

+
T

P
V

+
T

0.2

0.4

0.6

0.8

1.2

0 0.2 0.4 0.6 0.8 1

A
re

a
r
a
ti
o
a

Energy ratio, α

P
V
+

PV + T

PV T

378

Thermal system. The model formulated is so for unity solar fraction. The further study is directed towards the
development of model for variation of solar fractions. The effect of variation in packing factor of PVT on
energy ratio is required to be carried out.

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