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Multi-objective optimization of hybrid photovoltaic–thermal collectors integrated in a DHW heating system
Accepted Manuscript Title: Multi-objective optimization of hybrid photovoltaic-thermal collectors integrated in a dhw heating system Author: Jos´e Tamayo Vera Timo Laukkanen Kai Sir´en PII: DOI: Reference:
S0378-7788(14)00049-8 http://dx.doi.org/doi:10.1016/j.enbuild.2014.01.011 ENB 4765
To appear in:
ENB
Received date: Revised date: Accepted date:
23-4-2013 22-8-2013 8-1-2014
Please cite this article as: J.T. Vera, T. Laukkanen, K. Sir´en, Multi-objective optimization of hybrid photovoltaic-thermal collectors integrated in a dhw heating system, Energy and Buildings (2014), http://dx.doi.org/10.1016/j.enbuild.2014.01.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
MULTI-OBJECTIVE OPTIMIZATION OF HYBRID PHOTOVOLTAIC-THERMAL COLLECTORS INTEGRATED IN A DHW HEATING SYSTEM
a*
b
c
José Tamayo Vera , Timo Laukkanen , Kai Sirén
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
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a*
00076 Aalto FINLAND
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
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b
00076 Aalto FINLAND
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
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c
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00076 Aalto FINLAND
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Abstract
A mathematical model for making quantitative and qualitative predictions regarding the performance of
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water-cooled photovoltaic/thermal collectors integrated with a building domestic hot water preparation system has been developed. A genetic algorithm has been applied to the model in order to simultaneously find optimal design parameters affecting photovoltaic/thermal collectors‟ feasibility. For all formulated
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problems, Pareto optimal sets of conflicting solutions are obtained giving the designer information on the trade-off relationships between solutions.
Keywords: solar energy, photovoltaic thermal collector,
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optimization, NSGA-II, genetic algorithm
*Correspondent author.
E-mail address: [email protected], Tel: +358 40 4516077, Fax. +358 9 4355 2555
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1
Solar cells act as good heat collectors in hybrid photovoltaic -thermal (PV/T) systems and can be integrated and optimized with buildings for the simultaneous use of renewable DC power and heat. For the last forty years, many innovative PV/T systems and products have been documented. The academic and
are reviewed by Charalambous P. G. et al. [1] and later by Zondag [2].
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professional evaluation of these innovations, a range of theoretical models and validated experimental data
PV/T systems‟ practical implementations are still limited at present mostly to experimental applications. A
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PV/T collector is only a part of a given energy application. If a Domestic Hot Water (DHW) preparation
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system is chosen, the systems include thermal storage, piping, pumping unit and control equipment. Building–integrated applications (BiPVT) have been reviewed by Bazilian and Prasad [3]. A detailed review
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of thermosyphon applications, PV/T integrated heat pumps and concentrator-type PV/T has been carried out by Chaw [4].
Even though the presented research is restricted to water-cooled PV/T technology, it nonetheless
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remains a broad area of research and a number of technical details and applications have been omitted to be able to give focused overview.
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As a summary, the following are the contribution of the present paper:
A procedure for simultaneously analyzing performance parameters of a PV/T system connected
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with a DHW heating system using Metaheuristics. The literature for the last 40 years does not provide for a procedure that simultaneously finds optimal performance parameters of PV/T models and systems.
Multi-objective optimization of PV/T with a DHW heating system is performed with solar cells
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Introduction
effectiveness as one objective vs. various conflicting objectives like maximization of thermal efficiency, minimization of backup hot utility, minimization of pressure drop in the tubes and minimization of initial investment on the system. These problems have not been reported in literature.
Fluid mass-flow rate, solar cells aspect ratio, model length, air gap, storage tank volume and number of PV/T collectors are chosen as design variables. These design parameters have not been selected as a group to be optimized in previous research in the subject.
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Two PV/T models are studied, one with a front glass cover and one without a glass cover. The authors first apply a thermal procedure which yields performance parameters separately. Secondly, the thermal model is adapted in order to apply the elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II). The NSGA-II is described in details by its author K. Deb in Ref. [5]. The predicted results from applying multiobjective optimization on PV/T models is in good agreement with the first thermal model. These results are
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obtained in one single run of the model. Nomenclature
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Nt Nu Pr Q R Re Rg rc (Vc) SF T t V Vs W z
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ṁ
surface area, m specific heat, J/kg/K diameter of water pipe, m DC power, W Control parameter 2 heat transfer coefficient W/m /K 2 solar radiation density, W/m initial investment, € thermal conductivity, W/m/K length, m mass, kg; number of objectives mass-flow rate, kg/s number of water tubes, number of individuals Nusselt number Prandlt number heat flux, W thermal resistance, K/W Reynolds number refractive index of glass cover ratio of solar cell area to aperture area lumped heat capacity, J/K solar factor temperature, °C time, s volume, l number of generations width, m thickness, cm
cr
2
A c D E F h G I k L m M
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Greek α β ε
η ρ τ
() ψ ξ
absorptance temperature coefficient, 1/K emissivity extinction coefficient of glass cover, 1/m efficiency, distribution index density, reflectance transmittance Angle of incidence, angle of refraction of direct beam, Absolute temperature, K effective absorptance 2 4 Stefan–Boltzmann constant, W/m /K pipe friction coefficient hydraulic resistance coefficient
o
3
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2.1
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Analytical models for PV/T collectors’ design Temperature influence on PV module efficiency
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2
air adhesive layer solar cell, collector, crossover electrical, environment glass cover hydraulic insulation material mutation outer absorber plate radiation, reference stratification section system tube, tank utility water inlet incident beam outlet, refracted beam
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Subscripts a ad c e g h i m o p r s sys t ut w 0 1 2
c = ηr [1-β( Tc – Tr)]+γLog G; where:
(1)
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[6], with the following experimental equation:
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The best known model for photovoltaic module efficiency as a function of temperature is given by Evans
o
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ηr is the reference module efficiency at a PV cell temperature Tr of 25 C and at a reference solar irradiance G on the module. Tc is the PV cell temperature. γ and β are, respectively, the solar irradiance and temperature coefficients for the PV module. γ and β depend on the material used for the PV cells. Evans [6] suggested for silicon β=0.00488 C
-1
and γ =0.12. In this research the
assumption is that the term (γLog G) =0. This assumption is in line with all other researchers [1-4], [6]
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and [7], who studied similar PV temperatures.
2.2
The thermal model
One of the most common ways of using solar energy is for DHW preparation. Among all the interacting parts of a DHW system; two main components can be highlighted as the most important regarding system efficiency: the solar collectors and the storage tank (Fig. 1). In order to investigate the feasibility of PV/T
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technology connected with a DHW system, a thermal model of the system has been developed and an optimization procedure has been applied to the model. Thermal models for stand alone PV/T collectors‟ performance evaluation have been presented in literature [7 - 10]. The models explain the essential energy flows by conduction, convection and radiation throughout a serial assembly of many one-dimensional elementary layers.
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The thermal model we present searches for optimal daily PV efficiency, thermal efficiency and energy gain for the PV/T collectors connected with a DHW heating system to find optimal designs (optimal flow rate,
PV/T collector model with glass cover
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2.2.1
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aspect ratio, optimal collector‟s length, optimal air gap, hot water tank‟s capacity and number of collectors).
Thermal behaviour of the entire collector can be well defined from the heat transfer analysis in the vicinity
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of a single water tube. Fig. 2 is the schematic, presented in the resistance network‟s analogy, of the collectors‟ cross section along the Z-axis perpendicular to each water tube. The thermal analysis is derived
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based on this model.
;
(2)
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Transparent front cover
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Photovoltaic cell
;
(3)
;
(4)
is the PV layer effective absorptance.
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The generated electrical power Ec varies with the temperature-dependent solar cell operating efficiency
c. If rc is the ratio of cell area to aperture area, then: ;
(5)
c is calculated from equation (1). Absorber plate ;
(6) 5
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Metallic bond and tube (7)
Insulation layer ;
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Working fluid (water)
(8)
(9)
cr
;
Vc is the lumped thermal capacitance of the corresponding layer. T is each layer‟s average temperature.
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The various heat transfer coefficients h correspond to the heat transfer coefficients between corresponding layers. The convective surface heat transfer coefficient for flat plates exposed to outside winds is hag, and the
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heat transfer coefficient with the environment heg, are calculated as proposed by Duffie and Beckman [11]. The heat transfer parameter in the air gap hgc, is a combination of the thermal radiation between limiting
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surfaces and convective heat transfer for 45° inclined parallel plates as described by Duffie and Beckman [11].
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The convective heat transfer hw, for fully developed turbulent flow can be obtained from the Petukhov equation. If Reynolts number indicates laminar flow, Nusselt‟s number for laminar flow in short tubes is calculated as [11].
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Heat transfer coefficients are calculated dynamically as function of temperature and feed to the corresponding equation of the thermal model on the run.
2.2.2
(10)
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PV/T collector model without a cover glass
In the case of the second model, PV/T without cover glass, the first layer‟s differential equation around the PV node is as follows:
(11)
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Values for the different material‟s optical parameters, density, and heat capacity accepted in various literature sources [8 - 10] are gathered in table 1.
2.2.3
Clarification of the radiation part of the models
At the front glass, there are two interfaces to cause reflection losses. The solutions of its transmittance g,
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reflectance ρg and absorptance αg, allowing for both reflection and absorption can be obtained via ray-tracing techniques [11]. The whole procedure describing the radiation phenomena on the models are exposed below
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as in Duffie & Beckman [11].
If the solar irradiation falling on the glass surface is at a rate of G, the radiation energy absorbed by the
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glass cover is:
(12)
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;
a is the transmittance of the glass considering only absorption loss given in [11]: (13)
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;
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is the extinction coefficient of glass cover which is a property of the material. Applying the Bouguer‟s law with the angles of incidence and refraction of the direct beam represented by θ1 and θ2, the following
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equation is obtained:
;
(14)
The energy accumulated in glass is a result of solar radiation absorbed, Qg, and the net heat exchange with:
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the ambient air at temperature Ta for convective heat exchange,
o
the background equivalent (sky, ground and surroundings) environment at
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temperature Te for long-wave radiation heat exchange,
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the PV plate at temperature Tc through the air gap for combined convection plus radiation heat exchange.
The convective surface heat transfer coefficient for flat plates exposed to outside winds is reported in Reff [12] as: ;
(15)
where va is the outside wind velocity. 7
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The heat transfer coefficient with the environment is calculated as proposed by Duffie and Beckman [11]: ;
(16)
where σ is the Stefan–Boltzmann‟s constant, Θ and are respectively the absolute temperature and the emissivity of the corresponding layer.
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Swinbank [13] related environment temperature Θe to the local ambient temperature:
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(17)
The heat transfer parameter in the air gap hgr, is a combination of the thermal radiation between limiting
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surfaces with coefficient hr, and convective heat transfer by mean of the coefficient hc for 45° inclined parallel
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plates, [11]:
(18)
(19)
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;
;
ed
(20)
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where k, and Nu are respectively the thermal conductivity, the thickness and the Nusselt correlation parameter of the corresponding layer. c and g are the hemispherical thermal emissivity of the limiting surfaces.
The procedure explained above for the covered model is applied for the model without the cover glass
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except from the terms in the equations that consider combined convection plus radiation heat exchange through the air gap.
2.2.4
The solar thermal water accumulator
The efficiency of a solar heating system can be improved by providing stratified storage for hot water. Thermal stratification in hot water accumulators has been studied widely and some analytical and
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experimental work performed by a number of researchers [14] show that thermal stratification can effectively improve the performance of solar energy systems. In order to simulate the heating system in Fig. 1, the case of a passive house located in Helsinki is considered (i. e. no air supply or space heating is needed for the nearly 6 month period of functioning of the solar system). Assumed house inhabitants‟ number is 4 with a DHW load profile given by Perlman and Mills
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in Fig. 3 [15]. If the incoming water flow to the accumulator is slow and occupies its own density related level, an
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acummulator with two stratification sections s1 and s2 can be modelled mathematically as follows:
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For the upper section,
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an
For the lower section,
(22)
and temperatures T, in
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Note that terms of equations (21 ‒ 22) are included depending on mass-flow
(21)
the system. The following conditions have been considered:
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Control parameter
The conductance of the hot water accumulator UA varies depending on the volume.
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The numerical computing for both thermal models is carried out in Matlab following the flowchart presented in Appendix A. The differential equations developed above in the thermal model are solved based th
th
on Runge-Kutta 4 and 5 order formulas.
3
Multi-objective optimization procedure for system connected PV/T
The authors propose the use of multiple objectives in order to effectively investigate the conflicting features during performance evaluation of PV/T systems. Conflicting means that a single solution that is optimal with respect to every objective function does not exist. Evolutionary algorithms are used to efficiently 9
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solve multi-objective optimization problems (MOOP) by obtaining diverse and near-optimal solution sets. Multiple evolutionary techniques have been proposed for MOOPs. One such approach is NSGA-II. A version of NSGA-II lies behind MATLAB optimization toolbox. The mentioned toolbox has been used in several studies dealing with MOOPs used in building, [16]. Simulation based optimization has been used increasingly during the last 10 years at Aalto University
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Department of Energy Technology, especially in such important research on heat exchanger networks (HEN) by Laukkanen T. documented in his doctoral dissertation, [17]. Important contributions on optimization of
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energy in buildings are found in [18 -19]. The major attraction in my opinion of such studies is that they allow
3.1
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for evaluating a predesigned energy system without the use of expensive real life experiments.
Multi-Objective optimization using NSGA-II
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The design of a PV/T system can be described mathematically as a multi-objective optimization problem with two or more conflicting parameters which are to be minimized or maximized subject to certain
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constraints [5]:
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subject to
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Minimize/maximize
A solution x is a vector of n decision variables:
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The different steps of the NSGAII procedure are described in [5] are the follows:
1. A random population is initialized for all design variables. 2. Objective functions for all objectives are evaluated. 3. The front ranking of the population is performed based on the dominance criteria. Crowding distance is calculated. 4. Selection is performed using crowded tournament selection operator. 5. Crossover and mutation operators are applied to generate an offspring population. 10
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6. Parent and offspring populations are combined and a non-dominated sorting is done. 7. The parent population is replaced by the best members of the combined population.
In Step 3, each solution is assigned a non-domination rank (a smaller rank to a better non-dominated front). In Step 4, for each i-th solution of a particular front, the density of solutions in its surroundings is estimated by taking the average distance of two solutions on either side for each objective [5]. This average
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distance is called the crowding distance.
Selection is done based on the front rank of an individual solution and for solutions having the same front
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rank, they are selected base on their crowding distances (larger, the better). To create new offspring, two
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operators are used: simulated binary crossover (SBX) and polynomial mutation [20, 21]. In Step 8, initially solutions of better fronts replace the parent population.
The multi-objective optimization problems
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3.2
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Four multi-optimization problems (MOP) have been formulated. The problems on glass covered PV/T type carry subindex a, and problem on coverless PV/T type subindex b. The heating system has been simulated for a 24 hour period, using the test year for Helsinki TRY2012 [22].
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MOP1
This 2–dimensional problem maximizes daily electrical efficiency of the PV layer and daily system thermal
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efficiency simultaneously. The design variables are: fluid‟s mass-flow rate, collector‟s length, the aspect ratio and the air gap thickness for the covered PV/T. Minimize:
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MOP2 This 2–dimensional problem maximizes daily electrical efficiency and minimizes the system‟s hot utility consumption Qut, which supports the solar system. The focusing set of design variables are: fluid‟s mass-flow rate, collector‟s length, the aspect ratio and the air gap thickness for the covered PV/T.
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Minimize:
;
is the DHW load.
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were;
MOP3
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The following approach optimizes the PV layer efficiency, the system thermal solar fraction and the
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resistance to water flow through the piping network:
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Minimize
f2(x) evaluates the thermal solar fraction SF, of the system. The thermal solar fraction is the amount of
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thermal energy provided by the solar technology divided by the total thermal energy required by the system. , and
SF is to be maximized subject to PV/T design variables
choosen in previous problems and additionally the thermal storage tank size and the number of PV/T collectors.
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f3(x) evaluates the Bernoulli equations additional parameter which counts for pressure losess in the system ∆P as Hausen [23]. It is important to minimize these losses in order to minimize operational cost. Here pipe friction coefficient ψ computed as a function of Reynolds number (Re) and ξ is the sum of local hydraulic resistances. Solar main pipe‟s hydraulic diameter is computed dynamically. MOP4 In order to obtain the trade-off between PV layer‟s efficiency, the system solar share and the amount of initial investment on the system, MOP4 was formulated. Function I, which considers the initial investments, have been constructed using the prices for solar system components from Table 2. 12
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Minimize
The fixed value for the investment cost is the sum of the cost on the pump unit, the control unit and one
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set of fixed pipes. Vs1 is the first stratification section volume and Vs2 is the lower stratification section
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volume.
Table 3 presents the decision variables to be analysed in all MOPs. The decision variables are
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Results and discussion Thermal model
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4.1
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constrained by the lower and upper bounds derived from the simulations on the thermal models.
The thermal model deals with thermal evaluation of various performance parameters for two PV/Ts,
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covered and coverless, coupled with a DHW heating system. The c-Si PV cells are considered pasted with highly conductive material directly to a conventional aluminium solar absorber with copper tubes, Di =10 mm.
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A 19 cm air gap between the PV layer and the cover glass was assumed. Below the collector plate 15 cm mineral wool insulation was assumed. The simulation was carried out for a DHW tank with two stratification sections. When water‟s temperature at the collectors‟ outlet is lower than the lower sections of the tank, the corresponding solar energy gain is zero.
The DC power and heat gains resulting from a hybrid glass covered PV/T water-cooled collector coupled
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with a 4 persons DHW consumption profile is shown in Fig. 4. The results correspond to a typical summer day in Helsinki Finland.
Fig. 5 explicitly shows the poor performances of a PV/T system connected with a medium-high temperature heating system for the chosen parameters. The effect of the presence of the DHW storage that shifts the benefit of thermal solar energy to the hour of the day of higher DHW consumption is also appreciable. This figure also shows the actual variation of the system thermal efficiency each hour for various PV/T cooling water mass-flow rates. Based on this result, it is concluded preliminarily that in a PV/T 13
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set-up with a DHW system, the collector cooling-fluid mass-flow has a lower impact on the system thermal efficiency than the daily DHW load profile. By doubling the DHW load from 48l/day/person to 96l/day/person, a level of consumption that may, for example, be observed in a four family dwelling that uses a spa pool, the PV/T thermal performance is significantly increased. It has been observed in previous research that optimal water-cooled stand alone PV/T designs with
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respect to electrical and thermal efficiency are short in length and have a water mass-flow rate closer to 0.01kg/s per tube, [7] and [24]. The present thermal model, where the design is coupled with a DHW heating
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system, shows that larger cooling water mass-flow rates do not result in better thermal solar fraction for the given building application. Obviously, looking for the optimal set of parameters that maximize both electrical
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and thermal performances should include the building application and has to be handled as a multi-objective
4.2
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optimization problem.
Optimization problems
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MOP1a is started with 120 individuals and run for 240 generations. Each front is recorded in a spreadsheet and plotted in order to illustrate the evolution process, Fig 6a. The final Pareto optimal solutions
with Windows 7 (AMD Turion
TM
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were obtained with 28800 function evaluations needing 122,7h to reach the final population on a computer II Dual-Core Mobile M520 2,30 GHz processor; 4,00 GB RAM). This first
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problem indicated that the last 120 fronts are substantially the same as the preceeding front, generation one hundredth i.e., convergence is equal to 1 already at 100 generation. Based on this finding the population number is reduces to 40 individuals for MOP1b and the iteration is done only for 100 generations. MOP1b made 3838 function evaluations needing only 17,5h running time. The evolution process for MOP1b is illustrated in Fig. 6b.
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Both problems resulted in very dense final populations and the final fronts are very similar to each other. Clear trade-offs between the chosen objectives can be appreciated. After applying the non-dominated sorting and the crowded distance tournament selection, both problems converged to a very small number of Pareto optimal solutions as follows: MOP1a
-
All final solutions are designs with fluid mass-flow equal to the upper bound.
-
Collectors of minimum length strongly dominate the other designs. 14
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-
All final solutions have an air gap thickness of 9 cm.
-
Final front contains designs with Pareto optimal solar cells covering ratio ranging from 20 % to 60%. The lower covering ratio results in lower electrical efficiency and higher thermal efficiency. Higher covering ratio increases the electrical efficiency and decreases the thermal efficiency.
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MOP1b
All final solutions are designs with fluid mass-flow equal to the upper bound.
-
Collectors of minimum length strongly dominate the other designs.
-
Final front contains designs with Pareto optimal solar cells covering ratio ranging from 20 % to
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-
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90%. The lower covering ratio results in lower electrical efficiency and higher thermal efficiency.
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Higher covering ratio increases the electrical efficiency and decreases the thermal efficiency.
It is not possible to check for convergence on the run with the version of NSGA-II used. For this reason
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the following parameter settings are commonly used for all the rest of the formulated problems: 1. For M=2 objective problems: population size (N) =40; number of generations (Vs) =200; probability of crossover (Pc) =0.9; probability of mutation (Pm) =0.033; distribution index for crossover (ηc)=20;
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distribution index for mutation (ηm) =20.
2. For M=3 objective problems: population size (N) =60; number of generations (Vs) =300; probability
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of crossover (Pc) =0.9; probability of mutation (Pm) =0.033; distribution index for crossover (ηc) =5; distribution index for mutation (ηm) =15.
MOP2 allowed analyzing which designs achieve less hot utility consumption in comparison to daily DC
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power efficiency. The final Pareto optimal set was obtained after 7626 function evaluations in the case of MOP2a and 7614 function evaluations in the case of MOP2b. This problem has broadened the final set of Pareto optimal solutions as seen in Fig. 7. Problem MOP2 allows choosing among PV/T collectors which combine various design parameters that fit a given application when hot utility consumption and electric efficiency are important. These design parameters are presented in Table 4. Short designs with collector loop water mass-flow rate near the upper bound are optimal with respect to electric efficiency. These designs are accompanied with high hot utility consumption.
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An interesting outcome of problem MOP2 is that the Pareto optimal subsets „A‟ and „II‟ for both PV/T types are overlapping in objective function space; this means that in order to achieve similar performance for both models, the coverless model needs a larger area for the thermal absorber. The PV/T model without the glass cover in connection with a medium-high temperature DHW heating system will not achieve significant hot utility savings with high thermal efficiency as can be seen from the
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results of problems MOP1 and MOP2. The 3-dimensional problem MOP3, allows a more holistic picture of the conflicting features in objective
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function space as well as the trade-off among the various designs to be visualised (Fig. 8 and 9). The
resulting Pareto optimal set was obtained after 17151 function evaluations. The numerical values for the
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various chosen decision variables are presented in Appendix B. Parameters aiming at improving the system‟s thermal solar fraction are detrimental to the system‟s DC power efficiency and vice versa.
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Solutions marked with ‟A‟ in Fig. 9 are 1 m 1.5 m long, they share a cooling water mass-flow rate which is near the lower bound and have a large storage size. This set of design variables leads to low electrical
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efficiency; however high thermal efficiency is achieved. Solutions marked with ‟B‟ in Fig. 9 are short in length, with a cooling water mass-flow rate near the upper bound and have a small storage size. The electric
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efficiency is kept near the reference for these designs. Solutions in the „C‟ region correspond to designs that are 1 1.5 m in length, cooling water mass-flow rate is greater than 0.007 kg/s and the storage size lies between 0.6 and 1 m . The DC power efficiency lies in the region of 0.11 0.116.
ce pt
3
The thermal solar fraction is directly and strongly proportional to the number of collectors and the storage tank volume. The number of collectors nevertheless, seems to be high, which can result in a high number of standby hours and also in a high initial investment on the system. The final Pareto optimal front for MOP4 was obtained with 17128 function evaluations. Appendix C is a
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spread sheet with the numerical values obtained for each variable and objective function exposing the tradeoff between the three objectives. A system aiming at higher SF, would work at lower PV efficiency and need larger system sizing. Knowing this obvious correlation is not sufficient when designing an optimal PV/T system. MOP4 allows presenting the set of Pareto optimal solutions, from which the designer can choose one for a given application. This set of Pareto optimal solutions are obtained with one single run of the optimization model.
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4.3
There was a trend towards improving both electrical and thermal efficiencies for all designs which reached the final Pareto optimal fronts. The optimal performance parameters in the final fronts vary depending on the correlation between the chosen design-variables. This resulted multiplicity in solutions is possible due to the applied genetic algorithm NSGA-II.
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The introduction of a third dimension in objective function space f3 (x), investigated in MOP3, resulted in better trade off in the decision variable space. Problem MOP3 leads to the obvious decision making
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situation when low fluid velocities and therefore low fluid pumping expenses are to be chosen for the design.
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Optimum mass-flow rate found from MOP3 match the observations made in the thermal model, although with the difference that NSGA-2 allows many solutions to be obtained in one run. The lower values found for f3(x)
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represent lower electrical efficiencies, the higher values represent higher cost on design and operation. The pressure drop bound given by a particular design can be derived in an optimum design procedure with the presented problem. The pumping power as a consequence of pressure drop variations is obtained
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and the operational cost can be estimated. Results from analyzing problem MOP3 showed that the obtained pressure drop were in the range of 0.025 - 48 kPa in the formulated unconstrained problem.
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The two 3-dimensional problems studied allowed to conclude that depending on how the optimization problem is formulated; electrical efficiency vs thermal efficiency is to be optimized, the useful electrical and
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thermal energies are optimized or whether cost on investment is included in the problem, the correlation of optimal design variables varies; i.e. we obtain a different Pareto optimal set for each problem, demonstrating that PV/T systems cannot be optimized using thermal computations alone, but need a holistic multi-objective optimization approach.
A more realistic indicator of electric energy to thermal energy ratio is the primary energy cost, although
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Additional observations into the Pareto optimal sets
there is not a simple answer in determining on which standard the cost for primary energy output should be based, future studies on PV/T systems have to go beyond the optimization of system‟s performance parameters and have to consider primary energy cost.
17
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4.4
In order to validate the models we use two approaches. At this stage due to the complexity of the used mathematical procedures, a “naïve” approach is used that establishes a series of intermediate results from:
thermal models, eg. Fig 4 and Fig 5;
o
subroutines that calculate key system parameters on the run, printing them to the screen for
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o
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check up;
at the optimization stage we follow the evolution and shape of each generation Pareto front,
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Fig 6a and Fig 6b.
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In the near future the models are validated by the experiments that are under design currently. At the present stage the models are validated with the "naive" approach and with the fact that the results obtained with the models are in-line with existing systems. Suggestions from the obtained Pareto
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optimal solutions provide experimental meaning to the next step, which consist on designing and developing new hybrid PV/T collectors‟ prototypes using these optimal solutions to measure their
Conclusions
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5
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thermal and electrical performances under the climate of Helsinki.
PV/T technology has received increasing attention in recent years and it has been clearly demonstrated using numerical modeling of single performance parameters that this method is tedious, and in the authors‟ opinion does not present a holistic view of the problems investigated. The use of PV/T technology in building energy systems currently presents a challenge to decision makers when choosing among many design
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Validity of the models
variables which are partially conflicting and incommensurable, for example the electrical efficiency of the PV layer and the hot utility consumption, as well as the thermal solar fraction and investment cost of the system. The potential benefits of multi-objective optimization using evolutionary algorithms in PV/T systems are demonstrated on a model of a potential real world application in a DHW preparation system. This research presents a detailed multi-objective optimization procedure that finds the set of Pareto optimal solutions in one run for each formulated problem using the elitist multi-objective evolutionary algorithm NSGA-II. The decision variables chosen for the procedure are fluid mass-flow rate, collector‟s length, aspect ratio, air gap, 18
Page 18 of 37
storage tank capacity and number of collectors. A set of Pareto optimal solutions was obtained for each problem formulated. Clear trade-offs between solutions can be observed in the final Pareto optimal sets for the studied problems. This provides decision makers more detailed understanding of trade-off relationships when designing a PV/T system which would meet various needs in practice. PV/T technology in connection with a DHW preheating in a medium-high temperature system is a very
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attractive variant which can achieve very high thermal solar fraction keeping the PV layer temperature near reference in Helsinki climate. This technology could compete with other renewable energy applications.
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The continuation of this work will include extending the modeling and optimization procedures for the covered PV/T collector to avoid reduction of electrical output of the PV layer due to optical loses of the
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additional glass, as well as avoiding high temperatures of the PV layer during standby hours which may
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subject the PVs to thermal stress with possible detrimental effect on the PV structure.
References
Charalambous, P. G., Maidment, G., Kalogirou, S. A., Yiakoumetti, K. (2007). Photovoltaic thermal
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[1].
(PV/T) collectors: a review. . Applied Thermal Eng , 27:275–86. Zondag , H. (2008). Flat-plate PV-Thermal collectors and systems: A review. Renewable and
ed
[2].
Sustainable Energy Reviews , 12:891–959. [3].
Bazilian, M. D. and Prasad, D., (2002). Modelling of a photovoltaic heat recovery system and its role
[4].
Chow , T. T. (2010). A review on photovoltaic/thermal hybrid solar technology. Apply Energy , 87:365379.
[5].
ce pt
in a design decision support tool for building professionals. Renew Energy , 27:57-68.
Deb, K. (2001). Multi-objective Optimization Using Evolutionary Algorithms. Chichester: John Wiley
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and Sons Ltd. [6].
Evans, D. L. (1981). Simplified method for predicting photovoltaic array output. Solar Energy , 27(6):555-60.
[7].
Chow, T. T. (2003). Performance analysis of photovoltaic.thermal collector by explicit dynamic model. Solar Energy , 75.143-152.
[8].
Garg, H. P.;& Adhikari, R. S. (1997). Conventional hybrid photovoltaic/thermal (PV/T) air heating collector: steady-state simulation. Renewable Energy , 11(3):363-85.
19
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[9].
70:349-59. [10]. Notton, G., Cristofari, C., Mattei, M., Poggi, P., (2005). Modelling of a double-glass photovoltaic module using finite differences. Applied Thermal Engineering , 25 2854–2877. [11]. Duffie, J. A. and Beckman, W.A. (1991). Solar engineering of thermal processes. New York: Wiley.
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[12]. Watmuff, J.H., Charters, W.W.S., Proctor, D., 1977. Solar and wind induced external coefficients for solar collectors. COMPLES 2, 56
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[13]. Swinbank, S. L., (1963). Long-Wave Radiation from Clear Skies. Quarterly J. Royal Meteorological Soc., 89, 339
us
[14]. Hollands , K.G.T. and Lightstone, M.F., (1989). A review of low-flow, stratified-tank solar water heating systems. Solar Energy, 43(2):97-105.
an
[15]. Perlman M. and Mills B. E., (1985). Development of residential hot water use patterns. ASHRAE Transactions , 91(part2):657–79.
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[16]. Attia Sh., Hamdy M., O'Brien W., Carlucci S. (2013). Assessing gaps and needs for integrating building performance optimization toos in net zero energy buildings design. Energy and Buildings 60 ,
ed
110 - 124.
[17]. Laukkanen, T. (2012). Multiobjective Heat Exchanger Network Synthesis Based on Grouping of Process Streams. Espoo : Aalto University publication.
ce pt
[18]. Hamdy, M., Hasan, A., Siren, K. . (2013). A multi-stage optimization method for cost-optimal and nearly-zero-energy building solutions in line with the EPBD-recast 2010 . Energy and Buildings 56 , 189 - 203.
[19]. Hamdy, M., Hasan, A., Siren, K. . (2011). Impact of adaptive thermal comfort criteria on building
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Jones A. D., Underwood C.P ., (2001). A thermal model for photovoltaic systems. Solar Energy ,
energy use and cooling equipment size using a multi-objective optimization scheme . Energy and Buildings 43 , 2055 - 2067. [20]. Deb, K. and Agrawal, R. B., (1995). Simulated binary crossover for continuous search space. Complex Systems , 9(2):115–148. [21]. Deb, K. and Goyal, M., (1996). A combined genetic adaptive search (GeneAS) for engineering design. Computer Science and Informatics , 26(4):30–45.
20
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[22]. Jylhä, K., Kalamees, T., Tietäväinen H., Ruosteenoja K., Jokisalo, J., Hyvönen, R., Ilomets, S., Saku, S. and Hutila, A. (2011). Buildings energy calculations test year 2012 and reviews on the effects of climate change. Helsinki: Finnish Meteorological Institute. [23]. Hausen, H. (1976). Wärmeübertragung im Gegenstrom, Gleichstrom und Kreuzstrom. Berlin: Springer-Verlag GmbH.
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[24]. Zondag, H. A., De Vries, D. W., Van Helden, W. G. J., Van Zoligen R. J. C. and Van Steenhoven A.
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ed
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an
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A. (2002). The Thermal and Electrical Yield of a PV-Thermal Collector. Solar Energy 72(2):113-128.
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Figure captions.doc
Figure captions Page
Fig. 1 Schematic of the modelled DHW preparation system---------------------------------------------------------5 Fig.2Thermal network schematic for PV/T collector with single cover glass------------------------------------6
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Fig. 3 − Daily DHW draw profile [13]----------------------------------------------------------------------------------------9
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Fig. 4− Daily energy profiles for covered PV/T model (ṁw=0,004kg/s, rc = 0, 6; Vt=500l; L=1m)----------15 Fig. 5 Daily thermal efficiency profiles for various water mass-flow rates (covered model; rc = 0, 6;
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Vt=500l; L=1m)----------------------------------------------------------------------------------------------------------------------16 Fig 6a− Pareto-optimal front in objective space, MOP1a ------------------------------------------------------------17
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Fig 6b− Pareto-optimal front in objective space, MOP1b-------------------------------------------------------------18
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Fig. 7 Final Pareto optimal front, MOP2---------------------------------------------------------------------------------19 Fig 8− Final Pareto optimal front, MOP3----------------------------------------------------------------------------------20
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Fig 9− Scatter-plot matrix with Pareto-optimal solutions, MOP3----------------------------------------------------21
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Appendix A: Thermal model flowchart-------------------------------------------------------------------------------------25
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Tables and table captions.docx
Table 1 List of thermophysical parameters used in the simulations. Parameter
Value
Parameter
Value
αp
0,9
g
2700
kg/m
αc
0,9
c
2330
kg/ m
3
αg
0,04
p
2675
kg/ m
3
ρg
0,04
w
995
kg/ m
3
p
0,9
cg
840
J/kgK
c
0,4
cp
500
J/kgK
cc
500
J/kgK
cw
4180
J/kgK
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Table 2
3
Estimated prices of pieces of solar equipment (prices are based on the authors’ professional practice in
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Finland)
Price
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Equipment
PV/T solar collector fully packed with solar cells
Controll Unit
3
600 € 600 €
One set of pipes, fittings and installation details
535 €
Extra pipes
25 €/m
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Table 3
2
2,95 €/dm
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Energy Starage Tank Pump Unit
720 €/m
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Decision variables for multi-objective optimization problems Decision Variable
Range
Type
0.001≤x(1)≤0.01
Continuous
1≤x(2)≤5
Discrete
PV/T aspect ratio
0.2≤x(3)≤1
Continuous
PV/T air gap, cm
4≤x(4)≤40
Continuous
0.2≤x(5)≤2
Continuous
1≤x(6)≤30
Discrete
2
Fluid flow rate, kg/(sm ) PV/T segments
Storage Volume, m
3
No of PV/T collectors
Page 23 of 37
Table 4
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cr
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Set of optimal parameters in design variable space, MOP2
Appendix B: MOP3 numerical results for the various chosen decision variables and objective functions
0,5 0,7 0,7 0,7 0,3 0,8 0,4 0,4 0,4 0,4 0,4 0,8 0,8 0,8 0,8 0,9 0,8 0,9 0,9 0,8 0,9 0,9 0,8 0,8 0,8 0,4 0,7 1,0
cm 25 21 21 22 24 18 21 21 21 12 13 17 12 15 14 9 15 9 9 7 8 5 27 28 27 15 11 40
l 101 164 161 166 271 370 267 281 265 289 291 373 442 459 454 500 448 495 483 500 490 478 351 349 370 360 496 500
N
c
an
V
M
zag
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m 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2,0 2,0 2,0 1,0 1,5 2,5
rc
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kg/s 0,0099 0,0093 0,0095 0,0098 0,0093 0,0092 0,0078 0,0079 0,0078 0,0068 0,0071 0,0091 0,0085 0,0084 0,0082 0,0069 0,0086 0,0067 0,0062 0,0060 0,0061 0,0050 0,0040 0,0036 0,0034 0,0010 0,0014 0,0013
L
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ṁw
16 15 14 14 25 29 21 21 21 26 26 29 29 28 28 28 29 28 28 28 28 29 26 27 27 25 28 24
-0,1203 -0,1192 -0,1185 -0,1168 -0,1156 -0,1153 -0,1148 -0,1146 -0,1136 -0,1130 -0,1128 -0,1123 -0,1118 -0,1110 -0,1107 -0,1100 -0,1093 -0,1081 -0,1070 -0,1057 -0,1053 -0,1042 -0,1032 -0,1027 -0,1017 -0,0999 -0,0978 -0,0954
SF
P
-0,7771 -0,7892 -0,7954 -0,8079 -0,8224 -0,8253 -0,8272 -0,8338 -0,8372 -0,8445 -0,8474 -0,8542 -0,8596 -0,8660 -0,8713 -0,8751 -0,8818 -0,8807 -0,8861 -0,8998 -0,9007 -0,8987 -0,9062 -0,9087 -0,9174 -0,9110 -0,9371 -0,9444
Pa 48040 41847 43291 46385 41193 40672 27241 28520 27360 20299 22043 39079 32950 32242 31174 20774 34784 19297 16109 14701 15554 9627 5736 4516 1006 19 27 25 2
Page 24 of 37
Appendix C: MOP4 numerical results for the various chosen decision variables and objective functions
c
SF
1 2 21 21 25 19 17 2 30 26 24 24 23 23 25 24 24 3 6 5 23 25 19 20 4 24 24
-0.1206 -0.1197 -0.1192 -0.1182 -0.1174 -0.1170 -0.1158 -0.1150 -0.1143 -0.1138 -0.1133 -0.1129 -0.1108 -0.1107 -0.1094 -0.1085 -0.1076 -0.1038 -0.1024 -0.1013 -0.1002 -0.0992 -0.0990 -0.0982 -0.0971 -0.0964 -0.0944
-0.7752 -0.7842 -0.7910 -0.7977 -0.8072 -0.8117 -0.8208 -0.8303 -0.8337 -0.8410 -0.8382 -0.8439 -0.8626 -0.8660 -0.8834 -0.8855 -0.9006 -0.8762 -0.9041 -0.9137 -0.9153 -0.9255 -0.9018 -0.9280 -0.9167 -0.9312 -0.9531
I € 3403 3547 3775 3844 4050 4482 4563 4647 4898 4926 3778 3854 3906 4742 5412 4669 5598 3320 3929 4099 4388 5275 3716 4192 3954 4319 5610
us
cr
ip t
N
an
Vs1 l 116 140 179 190 225 299 312 327 369 374 179 192 201 343 456 330 488 102 205 234 283 433 169 250 209 271 490
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zag cm 37 37 33 33 34 39 39 35 20 30 28 27 28 31 25 30 25 38 40 40 27 39 40 27 40 25 23
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L m 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.0 1.5 2.0 2.0 2.0 1.5 2.0 2.0 2.5 1.0 2.0 2.5 1.5 2.0 1.0 1.5 2.5 2.0 3.0
Ac
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ṁw kg/s 0.0095 0.0100 0.0100 0.0097 0.0100 0.0087 0.0085 0.0100 0.0083 0.0092 0.0100 0.0098 0.0098 0.0078 0.0100 0.0081 0.0100 0.0011 0.0033 0.0031 0.0017 0.0020 0.0010 0.0010 0.0010 0.0010 0.0013
3
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*Highlights (for review)
Highlights A procedure for simultaneously analyzing the performance of a PV/T is presented
Multi-objective optimization of PV/T with a DHW heating system is performed
The studied objectives functions have not been reported in the literature
Chosen design parameters have not been selected to be optimized in previously
Clear trade-offs between solutions can be observed in the final Pareto optimal sets
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S0378-7788(14)00049-8 http://dx.doi.org/doi:10.1016/j.enbuild.2014.01.011 ENB 4765
To appear in:
ENB
Received date: Revised date: Accepted date:
23-4-2013 22-8-2013 8-1-2014
Please cite this article as: J.T. Vera, T. Laukkanen, K. Sir´en, Multi-objective optimization of hybrid photovoltaic-thermal collectors integrated in a dhw heating system, Energy and Buildings (2014), http://dx.doi.org/10.1016/j.enbuild.2014.01.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript
MULTI-OBJECTIVE OPTIMIZATION OF HYBRID PHOTOVOLTAIC-THERMAL COLLECTORS INTEGRATED IN A DHW HEATING SYSTEM
a*
b
c
José Tamayo Vera , Timo Laukkanen , Kai Sirén
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
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a*
00076 Aalto FINLAND
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
cr
b
00076 Aalto FINLAND
Aalto University, School of Engineering, Department of Energy Technology, P.O.Box 14100,
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c
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00076 Aalto FINLAND
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Abstract
A mathematical model for making quantitative and qualitative predictions regarding the performance of
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water-cooled photovoltaic/thermal collectors integrated with a building domestic hot water preparation system has been developed. A genetic algorithm has been applied to the model in order to simultaneously find optimal design parameters affecting photovoltaic/thermal collectors‟ feasibility. For all formulated
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problems, Pareto optimal sets of conflicting solutions are obtained giving the designer information on the trade-off relationships between solutions.
Keywords: solar energy, photovoltaic thermal collector,
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optimization, NSGA-II, genetic algorithm
*Correspondent author.
E-mail address: [email protected], Tel: +358 40 4516077, Fax. +358 9 4355 2555
Page 1 of 37
1
Solar cells act as good heat collectors in hybrid photovoltaic -thermal (PV/T) systems and can be integrated and optimized with buildings for the simultaneous use of renewable DC power and heat. For the last forty years, many innovative PV/T systems and products have been documented. The academic and
are reviewed by Charalambous P. G. et al. [1] and later by Zondag [2].
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professional evaluation of these innovations, a range of theoretical models and validated experimental data
PV/T systems‟ practical implementations are still limited at present mostly to experimental applications. A
cr
PV/T collector is only a part of a given energy application. If a Domestic Hot Water (DHW) preparation
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system is chosen, the systems include thermal storage, piping, pumping unit and control equipment. Building–integrated applications (BiPVT) have been reviewed by Bazilian and Prasad [3]. A detailed review
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of thermosyphon applications, PV/T integrated heat pumps and concentrator-type PV/T has been carried out by Chaw [4].
Even though the presented research is restricted to water-cooled PV/T technology, it nonetheless
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remains a broad area of research and a number of technical details and applications have been omitted to be able to give focused overview.
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As a summary, the following are the contribution of the present paper:
A procedure for simultaneously analyzing performance parameters of a PV/T system connected
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with a DHW heating system using Metaheuristics. The literature for the last 40 years does not provide for a procedure that simultaneously finds optimal performance parameters of PV/T models and systems.
Multi-objective optimization of PV/T with a DHW heating system is performed with solar cells
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Introduction
effectiveness as one objective vs. various conflicting objectives like maximization of thermal efficiency, minimization of backup hot utility, minimization of pressure drop in the tubes and minimization of initial investment on the system. These problems have not been reported in literature.
Fluid mass-flow rate, solar cells aspect ratio, model length, air gap, storage tank volume and number of PV/T collectors are chosen as design variables. These design parameters have not been selected as a group to be optimized in previous research in the subject.
2
Page 2 of 37
Two PV/T models are studied, one with a front glass cover and one without a glass cover. The authors first apply a thermal procedure which yields performance parameters separately. Secondly, the thermal model is adapted in order to apply the elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II). The NSGA-II is described in details by its author K. Deb in Ref. [5]. The predicted results from applying multiobjective optimization on PV/T models is in good agreement with the first thermal model. These results are
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obtained in one single run of the model. Nomenclature
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Nt Nu Pr Q R Re Rg rc (Vc) SF T t V Vs W z
ce pt
ṁ
surface area, m specific heat, J/kg/K diameter of water pipe, m DC power, W Control parameter 2 heat transfer coefficient W/m /K 2 solar radiation density, W/m initial investment, € thermal conductivity, W/m/K length, m mass, kg; number of objectives mass-flow rate, kg/s number of water tubes, number of individuals Nusselt number Prandlt number heat flux, W thermal resistance, K/W Reynolds number refractive index of glass cover ratio of solar cell area to aperture area lumped heat capacity, J/K solar factor temperature, °C time, s volume, l number of generations width, m thickness, cm
cr
2
A c D E F h G I k L m M
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Greek α β ε
η ρ τ
() ψ ξ
absorptance temperature coefficient, 1/K emissivity extinction coefficient of glass cover, 1/m efficiency, distribution index density, reflectance transmittance Angle of incidence, angle of refraction of direct beam, Absolute temperature, K effective absorptance 2 4 Stefan–Boltzmann constant, W/m /K pipe friction coefficient hydraulic resistance coefficient
o
3
Page 3 of 37
2.1
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Analytical models for PV/T collectors’ design Temperature influence on PV module efficiency
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2
air adhesive layer solar cell, collector, crossover electrical, environment glass cover hydraulic insulation material mutation outer absorber plate radiation, reference stratification section system tube, tank utility water inlet incident beam outlet, refracted beam
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Subscripts a ad c e g h i m o p r s sys t ut w 0 1 2
c = ηr [1-β( Tc – Tr)]+γLog G; where:
(1)
ed
[6], with the following experimental equation:
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The best known model for photovoltaic module efficiency as a function of temperature is given by Evans
o
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ηr is the reference module efficiency at a PV cell temperature Tr of 25 C and at a reference solar irradiance G on the module. Tc is the PV cell temperature. γ and β are, respectively, the solar irradiance and temperature coefficients for the PV module. γ and β depend on the material used for the PV cells. Evans [6] suggested for silicon β=0.00488 C
-1
and γ =0.12. In this research the
assumption is that the term (γLog G) =0. This assumption is in line with all other researchers [1-4], [6]
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and [7], who studied similar PV temperatures.
2.2
The thermal model
One of the most common ways of using solar energy is for DHW preparation. Among all the interacting parts of a DHW system; two main components can be highlighted as the most important regarding system efficiency: the solar collectors and the storage tank (Fig. 1). In order to investigate the feasibility of PV/T
4
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technology connected with a DHW system, a thermal model of the system has been developed and an optimization procedure has been applied to the model. Thermal models for stand alone PV/T collectors‟ performance evaluation have been presented in literature [7 - 10]. The models explain the essential energy flows by conduction, convection and radiation throughout a serial assembly of many one-dimensional elementary layers.
ip t
The thermal model we present searches for optimal daily PV efficiency, thermal efficiency and energy gain for the PV/T collectors connected with a DHW heating system to find optimal designs (optimal flow rate,
PV/T collector model with glass cover
us
2.2.1
cr
aspect ratio, optimal collector‟s length, optimal air gap, hot water tank‟s capacity and number of collectors).
Thermal behaviour of the entire collector can be well defined from the heat transfer analysis in the vicinity
an
of a single water tube. Fig. 2 is the schematic, presented in the resistance network‟s analogy, of the collectors‟ cross section along the Z-axis perpendicular to each water tube. The thermal analysis is derived
M
based on this model.
;
(2)
ed
Transparent front cover
ce pt
Photovoltaic cell
;
(3)
;
(4)
is the PV layer effective absorptance.
Ac
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The generated electrical power Ec varies with the temperature-dependent solar cell operating efficiency
c. If rc is the ratio of cell area to aperture area, then: ;
(5)
c is calculated from equation (1). Absorber plate ;
(6) 5
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Metallic bond and tube (7)
Insulation layer ;
ip t
Working fluid (water)
(8)
(9)
cr
;
Vc is the lumped thermal capacitance of the corresponding layer. T is each layer‟s average temperature.
us
The various heat transfer coefficients h correspond to the heat transfer coefficients between corresponding layers. The convective surface heat transfer coefficient for flat plates exposed to outside winds is hag, and the
an
heat transfer coefficient with the environment heg, are calculated as proposed by Duffie and Beckman [11]. The heat transfer parameter in the air gap hgc, is a combination of the thermal radiation between limiting
M
surfaces and convective heat transfer for 45° inclined parallel plates as described by Duffie and Beckman [11].
ed
The convective heat transfer hw, for fully developed turbulent flow can be obtained from the Petukhov equation. If Reynolts number indicates laminar flow, Nusselt‟s number for laminar flow in short tubes is calculated as [11].
ce pt
Heat transfer coefficients are calculated dynamically as function of temperature and feed to the corresponding equation of the thermal model on the run.
2.2.2
(10)
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PV/T collector model without a cover glass
In the case of the second model, PV/T without cover glass, the first layer‟s differential equation around the PV node is as follows:
(11)
6
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Values for the different material‟s optical parameters, density, and heat capacity accepted in various literature sources [8 - 10] are gathered in table 1.
2.2.3
Clarification of the radiation part of the models
At the front glass, there are two interfaces to cause reflection losses. The solutions of its transmittance g,
ip t
reflectance ρg and absorptance αg, allowing for both reflection and absorption can be obtained via ray-tracing techniques [11]. The whole procedure describing the radiation phenomena on the models are exposed below
cr
as in Duffie & Beckman [11].
If the solar irradiation falling on the glass surface is at a rate of G, the radiation energy absorbed by the
us
glass cover is:
(12)
an
;
a is the transmittance of the glass considering only absorption loss given in [11]: (13)
M
;
ed
is the extinction coefficient of glass cover which is a property of the material. Applying the Bouguer‟s law with the angles of incidence and refraction of the direct beam represented by θ1 and θ2, the following
ce pt
equation is obtained:
;
(14)
The energy accumulated in glass is a result of solar radiation absorbed, Qg, and the net heat exchange with:
o
the ambient air at temperature Ta for convective heat exchange,
o
the background equivalent (sky, ground and surroundings) environment at
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temperature Te for long-wave radiation heat exchange,
o
the PV plate at temperature Tc through the air gap for combined convection plus radiation heat exchange.
The convective surface heat transfer coefficient for flat plates exposed to outside winds is reported in Reff [12] as: ;
(15)
where va is the outside wind velocity. 7
Page 7 of 37
The heat transfer coefficient with the environment is calculated as proposed by Duffie and Beckman [11]: ;
(16)
where σ is the Stefan–Boltzmann‟s constant, Θ and are respectively the absolute temperature and the emissivity of the corresponding layer.
ip t
Swinbank [13] related environment temperature Θe to the local ambient temperature:
cr
(17)
The heat transfer parameter in the air gap hgr, is a combination of the thermal radiation between limiting
us
surfaces with coefficient hr, and convective heat transfer by mean of the coefficient hc for 45° inclined parallel
an
plates, [11]:
(18)
(19)
M
;
;
ed
(20)
ce pt
where k, and Nu are respectively the thermal conductivity, the thickness and the Nusselt correlation parameter of the corresponding layer. c and g are the hemispherical thermal emissivity of the limiting surfaces.
The procedure explained above for the covered model is applied for the model without the cover glass
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except from the terms in the equations that consider combined convection plus radiation heat exchange through the air gap.
2.2.4
The solar thermal water accumulator
The efficiency of a solar heating system can be improved by providing stratified storage for hot water. Thermal stratification in hot water accumulators has been studied widely and some analytical and
8
Page 8 of 37
experimental work performed by a number of researchers [14] show that thermal stratification can effectively improve the performance of solar energy systems. In order to simulate the heating system in Fig. 1, the case of a passive house located in Helsinki is considered (i. e. no air supply or space heating is needed for the nearly 6 month period of functioning of the solar system). Assumed house inhabitants‟ number is 4 with a DHW load profile given by Perlman and Mills
ip t
in Fig. 3 [15]. If the incoming water flow to the accumulator is slow and occupies its own density related level, an
cr
acummulator with two stratification sections s1 and s2 can be modelled mathematically as follows:
us
For the upper section,
M
an
For the lower section,
(22)
and temperatures T, in
ed
Note that terms of equations (21 ‒ 22) are included depending on mass-flow
(21)
the system. The following conditions have been considered:
ce pt
Control parameter
The conductance of the hot water accumulator UA varies depending on the volume.
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The numerical computing for both thermal models is carried out in Matlab following the flowchart presented in Appendix A. The differential equations developed above in the thermal model are solved based th
th
on Runge-Kutta 4 and 5 order formulas.
3
Multi-objective optimization procedure for system connected PV/T
The authors propose the use of multiple objectives in order to effectively investigate the conflicting features during performance evaluation of PV/T systems. Conflicting means that a single solution that is optimal with respect to every objective function does not exist. Evolutionary algorithms are used to efficiently 9
Page 9 of 37
solve multi-objective optimization problems (MOOP) by obtaining diverse and near-optimal solution sets. Multiple evolutionary techniques have been proposed for MOOPs. One such approach is NSGA-II. A version of NSGA-II lies behind MATLAB optimization toolbox. The mentioned toolbox has been used in several studies dealing with MOOPs used in building, [16]. Simulation based optimization has been used increasingly during the last 10 years at Aalto University
ip t
Department of Energy Technology, especially in such important research on heat exchanger networks (HEN) by Laukkanen T. documented in his doctoral dissertation, [17]. Important contributions on optimization of
cr
energy in buildings are found in [18 -19]. The major attraction in my opinion of such studies is that they allow
3.1
us
for evaluating a predesigned energy system without the use of expensive real life experiments.
Multi-Objective optimization using NSGA-II
an
The design of a PV/T system can be described mathematically as a multi-objective optimization problem with two or more conflicting parameters which are to be minimized or maximized subject to certain
M
constraints [5]:
ce pt
subject to
ed
Minimize/maximize
A solution x is a vector of n decision variables:
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The different steps of the NSGAII procedure are described in [5] are the follows:
1. A random population is initialized for all design variables. 2. Objective functions for all objectives are evaluated. 3. The front ranking of the population is performed based on the dominance criteria. Crowding distance is calculated. 4. Selection is performed using crowded tournament selection operator. 5. Crossover and mutation operators are applied to generate an offspring population. 10
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6. Parent and offspring populations are combined and a non-dominated sorting is done. 7. The parent population is replaced by the best members of the combined population.
In Step 3, each solution is assigned a non-domination rank (a smaller rank to a better non-dominated front). In Step 4, for each i-th solution of a particular front, the density of solutions in its surroundings is estimated by taking the average distance of two solutions on either side for each objective [5]. This average
ip t
distance is called the crowding distance.
Selection is done based on the front rank of an individual solution and for solutions having the same front
cr
rank, they are selected base on their crowding distances (larger, the better). To create new offspring, two
us
operators are used: simulated binary crossover (SBX) and polynomial mutation [20, 21]. In Step 8, initially solutions of better fronts replace the parent population.
The multi-objective optimization problems
an
3.2
M
Four multi-optimization problems (MOP) have been formulated. The problems on glass covered PV/T type carry subindex a, and problem on coverless PV/T type subindex b. The heating system has been simulated for a 24 hour period, using the test year for Helsinki TRY2012 [22].
ed
MOP1
This 2–dimensional problem maximizes daily electrical efficiency of the PV layer and daily system thermal
ce pt
efficiency simultaneously. The design variables are: fluid‟s mass-flow rate, collector‟s length, the aspect ratio and the air gap thickness for the covered PV/T. Minimize:
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MOP2 This 2–dimensional problem maximizes daily electrical efficiency and minimizes the system‟s hot utility consumption Qut, which supports the solar system. The focusing set of design variables are: fluid‟s mass-flow rate, collector‟s length, the aspect ratio and the air gap thickness for the covered PV/T.
11
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Minimize:
;
is the DHW load.
ip t
were;
MOP3
cr
The following approach optimizes the PV layer efficiency, the system thermal solar fraction and the
us
resistance to water flow through the piping network:
ed
M
an
Minimize
f2(x) evaluates the thermal solar fraction SF, of the system. The thermal solar fraction is the amount of
ce pt
thermal energy provided by the solar technology divided by the total thermal energy required by the system. , and
SF is to be maximized subject to PV/T design variables
choosen in previous problems and additionally the thermal storage tank size and the number of PV/T collectors.
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f3(x) evaluates the Bernoulli equations additional parameter which counts for pressure losess in the system ∆P as Hausen [23]. It is important to minimize these losses in order to minimize operational cost. Here pipe friction coefficient ψ computed as a function of Reynolds number (Re) and ξ is the sum of local hydraulic resistances. Solar main pipe‟s hydraulic diameter is computed dynamically. MOP4 In order to obtain the trade-off between PV layer‟s efficiency, the system solar share and the amount of initial investment on the system, MOP4 was formulated. Function I, which considers the initial investments, have been constructed using the prices for solar system components from Table 2. 12
Page 12 of 37
ip t
Minimize
The fixed value for the investment cost is the sum of the cost on the pump unit, the control unit and one
cr
set of fixed pipes. Vs1 is the first stratification section volume and Vs2 is the lower stratification section
us
volume.
Table 3 presents the decision variables to be analysed in all MOPs. The decision variables are
4
Results and discussion Thermal model
M
4.1
an
constrained by the lower and upper bounds derived from the simulations on the thermal models.
The thermal model deals with thermal evaluation of various performance parameters for two PV/Ts,
ed
covered and coverless, coupled with a DHW heating system. The c-Si PV cells are considered pasted with highly conductive material directly to a conventional aluminium solar absorber with copper tubes, Di =10 mm.
ce pt
A 19 cm air gap between the PV layer and the cover glass was assumed. Below the collector plate 15 cm mineral wool insulation was assumed. The simulation was carried out for a DHW tank with two stratification sections. When water‟s temperature at the collectors‟ outlet is lower than the lower sections of the tank, the corresponding solar energy gain is zero.
The DC power and heat gains resulting from a hybrid glass covered PV/T water-cooled collector coupled
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with a 4 persons DHW consumption profile is shown in Fig. 4. The results correspond to a typical summer day in Helsinki Finland.
Fig. 5 explicitly shows the poor performances of a PV/T system connected with a medium-high temperature heating system for the chosen parameters. The effect of the presence of the DHW storage that shifts the benefit of thermal solar energy to the hour of the day of higher DHW consumption is also appreciable. This figure also shows the actual variation of the system thermal efficiency each hour for various PV/T cooling water mass-flow rates. Based on this result, it is concluded preliminarily that in a PV/T 13
Page 13 of 37
set-up with a DHW system, the collector cooling-fluid mass-flow has a lower impact on the system thermal efficiency than the daily DHW load profile. By doubling the DHW load from 48l/day/person to 96l/day/person, a level of consumption that may, for example, be observed in a four family dwelling that uses a spa pool, the PV/T thermal performance is significantly increased. It has been observed in previous research that optimal water-cooled stand alone PV/T designs with
ip t
respect to electrical and thermal efficiency are short in length and have a water mass-flow rate closer to 0.01kg/s per tube, [7] and [24]. The present thermal model, where the design is coupled with a DHW heating
cr
system, shows that larger cooling water mass-flow rates do not result in better thermal solar fraction for the given building application. Obviously, looking for the optimal set of parameters that maximize both electrical
us
and thermal performances should include the building application and has to be handled as a multi-objective
4.2
an
optimization problem.
Optimization problems
M
MOP1a is started with 120 individuals and run for 240 generations. Each front is recorded in a spreadsheet and plotted in order to illustrate the evolution process, Fig 6a. The final Pareto optimal solutions
with Windows 7 (AMD Turion
TM
ed
were obtained with 28800 function evaluations needing 122,7h to reach the final population on a computer II Dual-Core Mobile M520 2,30 GHz processor; 4,00 GB RAM). This first
ce pt
problem indicated that the last 120 fronts are substantially the same as the preceeding front, generation one hundredth i.e., convergence is equal to 1 already at 100 generation. Based on this finding the population number is reduces to 40 individuals for MOP1b and the iteration is done only for 100 generations. MOP1b made 3838 function evaluations needing only 17,5h running time. The evolution process for MOP1b is illustrated in Fig. 6b.
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Both problems resulted in very dense final populations and the final fronts are very similar to each other. Clear trade-offs between the chosen objectives can be appreciated. After applying the non-dominated sorting and the crowded distance tournament selection, both problems converged to a very small number of Pareto optimal solutions as follows: MOP1a
-
All final solutions are designs with fluid mass-flow equal to the upper bound.
-
Collectors of minimum length strongly dominate the other designs. 14
Page 14 of 37
-
All final solutions have an air gap thickness of 9 cm.
-
Final front contains designs with Pareto optimal solar cells covering ratio ranging from 20 % to 60%. The lower covering ratio results in lower electrical efficiency and higher thermal efficiency. Higher covering ratio increases the electrical efficiency and decreases the thermal efficiency.
ip t
MOP1b
All final solutions are designs with fluid mass-flow equal to the upper bound.
-
Collectors of minimum length strongly dominate the other designs.
-
Final front contains designs with Pareto optimal solar cells covering ratio ranging from 20 % to
cr
-
us
90%. The lower covering ratio results in lower electrical efficiency and higher thermal efficiency.
an
Higher covering ratio increases the electrical efficiency and decreases the thermal efficiency.
It is not possible to check for convergence on the run with the version of NSGA-II used. For this reason
M
the following parameter settings are commonly used for all the rest of the formulated problems: 1. For M=2 objective problems: population size (N) =40; number of generations (Vs) =200; probability of crossover (Pc) =0.9; probability of mutation (Pm) =0.033; distribution index for crossover (ηc)=20;
ed
distribution index for mutation (ηm) =20.
2. For M=3 objective problems: population size (N) =60; number of generations (Vs) =300; probability
ce pt
of crossover (Pc) =0.9; probability of mutation (Pm) =0.033; distribution index for crossover (ηc) =5; distribution index for mutation (ηm) =15.
MOP2 allowed analyzing which designs achieve less hot utility consumption in comparison to daily DC
Ac
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power efficiency. The final Pareto optimal set was obtained after 7626 function evaluations in the case of MOP2a and 7614 function evaluations in the case of MOP2b. This problem has broadened the final set of Pareto optimal solutions as seen in Fig. 7. Problem MOP2 allows choosing among PV/T collectors which combine various design parameters that fit a given application when hot utility consumption and electric efficiency are important. These design parameters are presented in Table 4. Short designs with collector loop water mass-flow rate near the upper bound are optimal with respect to electric efficiency. These designs are accompanied with high hot utility consumption.
15
Page 15 of 37
An interesting outcome of problem MOP2 is that the Pareto optimal subsets „A‟ and „II‟ for both PV/T types are overlapping in objective function space; this means that in order to achieve similar performance for both models, the coverless model needs a larger area for the thermal absorber. The PV/T model without the glass cover in connection with a medium-high temperature DHW heating system will not achieve significant hot utility savings with high thermal efficiency as can be seen from the
ip t
results of problems MOP1 and MOP2. The 3-dimensional problem MOP3, allows a more holistic picture of the conflicting features in objective
cr
function space as well as the trade-off among the various designs to be visualised (Fig. 8 and 9). The
resulting Pareto optimal set was obtained after 17151 function evaluations. The numerical values for the
us
various chosen decision variables are presented in Appendix B. Parameters aiming at improving the system‟s thermal solar fraction are detrimental to the system‟s DC power efficiency and vice versa.
an
Solutions marked with ‟A‟ in Fig. 9 are 1 m 1.5 m long, they share a cooling water mass-flow rate which is near the lower bound and have a large storage size. This set of design variables leads to low electrical
M
efficiency; however high thermal efficiency is achieved. Solutions marked with ‟B‟ in Fig. 9 are short in length, with a cooling water mass-flow rate near the upper bound and have a small storage size. The electric
ed
efficiency is kept near the reference for these designs. Solutions in the „C‟ region correspond to designs that are 1 1.5 m in length, cooling water mass-flow rate is greater than 0.007 kg/s and the storage size lies between 0.6 and 1 m . The DC power efficiency lies in the region of 0.11 0.116.
ce pt
3
The thermal solar fraction is directly and strongly proportional to the number of collectors and the storage tank volume. The number of collectors nevertheless, seems to be high, which can result in a high number of standby hours and also in a high initial investment on the system. The final Pareto optimal front for MOP4 was obtained with 17128 function evaluations. Appendix C is a
Ac
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spread sheet with the numerical values obtained for each variable and objective function exposing the tradeoff between the three objectives. A system aiming at higher SF, would work at lower PV efficiency and need larger system sizing. Knowing this obvious correlation is not sufficient when designing an optimal PV/T system. MOP4 allows presenting the set of Pareto optimal solutions, from which the designer can choose one for a given application. This set of Pareto optimal solutions are obtained with one single run of the optimization model.
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4.3
There was a trend towards improving both electrical and thermal efficiencies for all designs which reached the final Pareto optimal fronts. The optimal performance parameters in the final fronts vary depending on the correlation between the chosen design-variables. This resulted multiplicity in solutions is possible due to the applied genetic algorithm NSGA-II.
ip t
The introduction of a third dimension in objective function space f3 (x), investigated in MOP3, resulted in better trade off in the decision variable space. Problem MOP3 leads to the obvious decision making
cr
situation when low fluid velocities and therefore low fluid pumping expenses are to be chosen for the design.
us
Optimum mass-flow rate found from MOP3 match the observations made in the thermal model, although with the difference that NSGA-2 allows many solutions to be obtained in one run. The lower values found for f3(x)
an
represent lower electrical efficiencies, the higher values represent higher cost on design and operation. The pressure drop bound given by a particular design can be derived in an optimum design procedure with the presented problem. The pumping power as a consequence of pressure drop variations is obtained
M
and the operational cost can be estimated. Results from analyzing problem MOP3 showed that the obtained pressure drop were in the range of 0.025 - 48 kPa in the formulated unconstrained problem.
ed
The two 3-dimensional problems studied allowed to conclude that depending on how the optimization problem is formulated; electrical efficiency vs thermal efficiency is to be optimized, the useful electrical and
ce pt
thermal energies are optimized or whether cost on investment is included in the problem, the correlation of optimal design variables varies; i.e. we obtain a different Pareto optimal set for each problem, demonstrating that PV/T systems cannot be optimized using thermal computations alone, but need a holistic multi-objective optimization approach.
A more realistic indicator of electric energy to thermal energy ratio is the primary energy cost, although
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Additional observations into the Pareto optimal sets
there is not a simple answer in determining on which standard the cost for primary energy output should be based, future studies on PV/T systems have to go beyond the optimization of system‟s performance parameters and have to consider primary energy cost.
17
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4.4
In order to validate the models we use two approaches. At this stage due to the complexity of the used mathematical procedures, a “naïve” approach is used that establishes a series of intermediate results from:
thermal models, eg. Fig 4 and Fig 5;
o
subroutines that calculate key system parameters on the run, printing them to the screen for
ip t
o
o
cr
check up;
at the optimization stage we follow the evolution and shape of each generation Pareto front,
us
Fig 6a and Fig 6b.
an
In the near future the models are validated by the experiments that are under design currently. At the present stage the models are validated with the "naive" approach and with the fact that the results obtained with the models are in-line with existing systems. Suggestions from the obtained Pareto
M
optimal solutions provide experimental meaning to the next step, which consist on designing and developing new hybrid PV/T collectors‟ prototypes using these optimal solutions to measure their
Conclusions
ce pt
5
ed
thermal and electrical performances under the climate of Helsinki.
PV/T technology has received increasing attention in recent years and it has been clearly demonstrated using numerical modeling of single performance parameters that this method is tedious, and in the authors‟ opinion does not present a holistic view of the problems investigated. The use of PV/T technology in building energy systems currently presents a challenge to decision makers when choosing among many design
Ac
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Validity of the models
variables which are partially conflicting and incommensurable, for example the electrical efficiency of the PV layer and the hot utility consumption, as well as the thermal solar fraction and investment cost of the system. The potential benefits of multi-objective optimization using evolutionary algorithms in PV/T systems are demonstrated on a model of a potential real world application in a DHW preparation system. This research presents a detailed multi-objective optimization procedure that finds the set of Pareto optimal solutions in one run for each formulated problem using the elitist multi-objective evolutionary algorithm NSGA-II. The decision variables chosen for the procedure are fluid mass-flow rate, collector‟s length, aspect ratio, air gap, 18
Page 18 of 37
storage tank capacity and number of collectors. A set of Pareto optimal solutions was obtained for each problem formulated. Clear trade-offs between solutions can be observed in the final Pareto optimal sets for the studied problems. This provides decision makers more detailed understanding of trade-off relationships when designing a PV/T system which would meet various needs in practice. PV/T technology in connection with a DHW preheating in a medium-high temperature system is a very
ip t
attractive variant which can achieve very high thermal solar fraction keeping the PV layer temperature near reference in Helsinki climate. This technology could compete with other renewable energy applications.
cr
The continuation of this work will include extending the modeling and optimization procedures for the covered PV/T collector to avoid reduction of electrical output of the PV layer due to optical loses of the
us
additional glass, as well as avoiding high temperatures of the PV layer during standby hours which may
an
subject the PVs to thermal stress with possible detrimental effect on the PV structure.
References
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M
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in a design decision support tool for building professionals. Renew Energy , 27:57-68.
Deb, K. (2001). Multi-objective Optimization Using Evolutionary Algorithms. Chichester: John Wiley
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Garg, H. P.;& Adhikari, R. S. (1997). Conventional hybrid photovoltaic/thermal (PV/T) air heating collector: steady-state simulation. Renewable Energy , 11(3):363-85.
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[9].
70:349-59. [10]. Notton, G., Cristofari, C., Mattei, M., Poggi, P., (2005). Modelling of a double-glass photovoltaic module using finite differences. Applied Thermal Engineering , 25 2854–2877. [11]. Duffie, J. A. and Beckman, W.A. (1991). Solar engineering of thermal processes. New York: Wiley.
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[12]. Watmuff, J.H., Charters, W.W.S., Proctor, D., 1977. Solar and wind induced external coefficients for solar collectors. COMPLES 2, 56
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[13]. Swinbank, S. L., (1963). Long-Wave Radiation from Clear Skies. Quarterly J. Royal Meteorological Soc., 89, 339
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[14]. Hollands , K.G.T. and Lightstone, M.F., (1989). A review of low-flow, stratified-tank solar water heating systems. Solar Energy, 43(2):97-105.
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[15]. Perlman M. and Mills B. E., (1985). Development of residential hot water use patterns. ASHRAE Transactions , 91(part2):657–79.
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[16]. Attia Sh., Hamdy M., O'Brien W., Carlucci S. (2013). Assessing gaps and needs for integrating building performance optimization toos in net zero energy buildings design. Energy and Buildings 60 ,
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[17]. Laukkanen, T. (2012). Multiobjective Heat Exchanger Network Synthesis Based on Grouping of Process Streams. Espoo : Aalto University publication.
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[18]. Hamdy, M., Hasan, A., Siren, K. . (2013). A multi-stage optimization method for cost-optimal and nearly-zero-energy building solutions in line with the EPBD-recast 2010 . Energy and Buildings 56 , 189 - 203.
[19]. Hamdy, M., Hasan, A., Siren, K. . (2011). Impact of adaptive thermal comfort criteria on building
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Jones A. D., Underwood C.P ., (2001). A thermal model for photovoltaic systems. Solar Energy ,
energy use and cooling equipment size using a multi-objective optimization scheme . Energy and Buildings 43 , 2055 - 2067. [20]. Deb, K. and Agrawal, R. B., (1995). Simulated binary crossover for continuous search space. Complex Systems , 9(2):115–148. [21]. Deb, K. and Goyal, M., (1996). A combined genetic adaptive search (GeneAS) for engineering design. Computer Science and Informatics , 26(4):30–45.
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[22]. Jylhä, K., Kalamees, T., Tietäväinen H., Ruosteenoja K., Jokisalo, J., Hyvönen, R., Ilomets, S., Saku, S. and Hutila, A. (2011). Buildings energy calculations test year 2012 and reviews on the effects of climate change. Helsinki: Finnish Meteorological Institute. [23]. Hausen, H. (1976). Wärmeübertragung im Gegenstrom, Gleichstrom und Kreuzstrom. Berlin: Springer-Verlag GmbH.
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[24]. Zondag, H. A., De Vries, D. W., Van Helden, W. G. J., Van Zoligen R. J. C. and Van Steenhoven A.
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A. (2002). The Thermal and Electrical Yield of a PV-Thermal Collector. Solar Energy 72(2):113-128.
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Figure captions.doc
Figure captions Page
Fig. 1 Schematic of the modelled DHW preparation system---------------------------------------------------------5 Fig.2Thermal network schematic for PV/T collector with single cover glass------------------------------------6
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Fig. 3 − Daily DHW draw profile [13]----------------------------------------------------------------------------------------9
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Fig. 4− Daily energy profiles for covered PV/T model (ṁw=0,004kg/s, rc = 0, 6; Vt=500l; L=1m)----------15 Fig. 5 Daily thermal efficiency profiles for various water mass-flow rates (covered model; rc = 0, 6;
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Vt=500l; L=1m)----------------------------------------------------------------------------------------------------------------------16 Fig 6a− Pareto-optimal front in objective space, MOP1a ------------------------------------------------------------17
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Fig 6b− Pareto-optimal front in objective space, MOP1b-------------------------------------------------------------18
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Fig. 7 Final Pareto optimal front, MOP2---------------------------------------------------------------------------------19 Fig 8− Final Pareto optimal front, MOP3----------------------------------------------------------------------------------20
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Fig 9− Scatter-plot matrix with Pareto-optimal solutions, MOP3----------------------------------------------------21
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Appendix A: Thermal model flowchart-------------------------------------------------------------------------------------25
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Tables and table captions.docx
Table 1 List of thermophysical parameters used in the simulations. Parameter
Value
Parameter
Value
αp
0,9
g
2700
kg/m
αc
0,9
c
2330
kg/ m
3
αg
0,04
p
2675
kg/ m
3
ρg
0,04
w
995
kg/ m
3
p
0,9
cg
840
J/kgK
c
0,4
cp
500
J/kgK
cc
500
J/kgK
cw
4180
J/kgK
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Table 2
3
Estimated prices of pieces of solar equipment (prices are based on the authors’ professional practice in
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Finland)
Price
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Equipment
PV/T solar collector fully packed with solar cells
Controll Unit
3
600 € 600 €
One set of pipes, fittings and installation details
535 €
Extra pipes
25 €/m
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Table 3
2
2,95 €/dm
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Energy Starage Tank Pump Unit
720 €/m
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Decision variables for multi-objective optimization problems Decision Variable
Range
Type
0.001≤x(1)≤0.01
Continuous
1≤x(2)≤5
Discrete
PV/T aspect ratio
0.2≤x(3)≤1
Continuous
PV/T air gap, cm
4≤x(4)≤40
Continuous
0.2≤x(5)≤2
Continuous
1≤x(6)≤30
Discrete
2
Fluid flow rate, kg/(sm ) PV/T segments
Storage Volume, m
3
No of PV/T collectors
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Table 4
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cr
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Set of optimal parameters in design variable space, MOP2
Appendix B: MOP3 numerical results for the various chosen decision variables and objective functions
0,5 0,7 0,7 0,7 0,3 0,8 0,4 0,4 0,4 0,4 0,4 0,8 0,8 0,8 0,8 0,9 0,8 0,9 0,9 0,8 0,9 0,9 0,8 0,8 0,8 0,4 0,7 1,0
cm 25 21 21 22 24 18 21 21 21 12 13 17 12 15 14 9 15 9 9 7 8 5 27 28 27 15 11 40
l 101 164 161 166 271 370 267 281 265 289 291 373 442 459 454 500 448 495 483 500 490 478 351 349 370 360 496 500
N
c
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V
M
zag
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m 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,0 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2,0 2,0 2,0 1,0 1,5 2,5
rc
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kg/s 0,0099 0,0093 0,0095 0,0098 0,0093 0,0092 0,0078 0,0079 0,0078 0,0068 0,0071 0,0091 0,0085 0,0084 0,0082 0,0069 0,0086 0,0067 0,0062 0,0060 0,0061 0,0050 0,0040 0,0036 0,0034 0,0010 0,0014 0,0013
L
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ṁw
16 15 14 14 25 29 21 21 21 26 26 29 29 28 28 28 29 28 28 28 28 29 26 27 27 25 28 24
-0,1203 -0,1192 -0,1185 -0,1168 -0,1156 -0,1153 -0,1148 -0,1146 -0,1136 -0,1130 -0,1128 -0,1123 -0,1118 -0,1110 -0,1107 -0,1100 -0,1093 -0,1081 -0,1070 -0,1057 -0,1053 -0,1042 -0,1032 -0,1027 -0,1017 -0,0999 -0,0978 -0,0954
SF
P
-0,7771 -0,7892 -0,7954 -0,8079 -0,8224 -0,8253 -0,8272 -0,8338 -0,8372 -0,8445 -0,8474 -0,8542 -0,8596 -0,8660 -0,8713 -0,8751 -0,8818 -0,8807 -0,8861 -0,8998 -0,9007 -0,8987 -0,9062 -0,9087 -0,9174 -0,9110 -0,9371 -0,9444
Pa 48040 41847 43291 46385 41193 40672 27241 28520 27360 20299 22043 39079 32950 32242 31174 20774 34784 19297 16109 14701 15554 9627 5736 4516 1006 19 27 25 2
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Appendix C: MOP4 numerical results for the various chosen decision variables and objective functions
c
SF
1 2 21 21 25 19 17 2 30 26 24 24 23 23 25 24 24 3 6 5 23 25 19 20 4 24 24
-0.1206 -0.1197 -0.1192 -0.1182 -0.1174 -0.1170 -0.1158 -0.1150 -0.1143 -0.1138 -0.1133 -0.1129 -0.1108 -0.1107 -0.1094 -0.1085 -0.1076 -0.1038 -0.1024 -0.1013 -0.1002 -0.0992 -0.0990 -0.0982 -0.0971 -0.0964 -0.0944
-0.7752 -0.7842 -0.7910 -0.7977 -0.8072 -0.8117 -0.8208 -0.8303 -0.8337 -0.8410 -0.8382 -0.8439 -0.8626 -0.8660 -0.8834 -0.8855 -0.9006 -0.8762 -0.9041 -0.9137 -0.9153 -0.9255 -0.9018 -0.9280 -0.9167 -0.9312 -0.9531
I € 3403 3547 3775 3844 4050 4482 4563 4647 4898 4926 3778 3854 3906 4742 5412 4669 5598 3320 3929 4099 4388 5275 3716 4192 3954 4319 5610
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cr
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N
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Vs1 l 116 140 179 190 225 299 312 327 369 374 179 192 201 343 456 330 488 102 205 234 283 433 169 250 209 271 490
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zag cm 37 37 33 33 34 39 39 35 20 30 28 27 28 31 25 30 25 38 40 40 27 39 40 27 40 25 23
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L m 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.0 1.5 2.0 2.0 2.0 1.5 2.0 2.0 2.5 1.0 2.0 2.5 1.5 2.0 1.0 1.5 2.5 2.0 3.0
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ṁw kg/s 0.0095 0.0100 0.0100 0.0097 0.0100 0.0087 0.0085 0.0100 0.0083 0.0092 0.0100 0.0098 0.0098 0.0078 0.0100 0.0081 0.0100 0.0011 0.0033 0.0031 0.0017 0.0020 0.0010 0.0010 0.0010 0.0010 0.0013
3
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*Highlights (for review)
Highlights A procedure for simultaneously analyzing the performance of a PV/T is presented
Multi-objective optimization of PV/T with a DHW heating system is performed
The studied objectives functions have not been reported in the literature
Chosen design parameters have not been selected to be optimized in previously
Clear trade-offs between solutions can be observed in the final Pareto optimal sets
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