Nayab Bushra, Timo Hartmann
Design optimization method for roof-integrated
TSSCs
Open Access via institutional repository of Technische Universität Berlin
Document type
Preprint
Date of this version
11th March 2022
This version is available at
https://doi.org/10.14279/depositonce-15328
Citation details
Bushra, Nayab; Hartmann, Timo (2022). Design optimization method for roof-integrated TSSCs. Technische
Universität Berlin, Preprint, http://dx.doi.org/10.14279/depositonce-15328.
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cb This work is licensed under a Creative Commons Attribution 4.0 International license:
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1
Design optimization method for roof-integrated TSSCs
1
Nayab Bushra a, *, Timo Hartmann a
2
a Civil Systems Engineering, Technical University of Berlin, Gustav-Meyer-Allee 25, 13355 Berlin, Germany 3
Email addresses: [email protected] (Nayab Bushra); timo.hartmann@tu-berlin.de (Timo Hartmann) 4
* Corresponding author: Email address: [email protected] 5
Abstract 6
This paper proposes a two-step design optimization method for roof-integrated two-stage solar 7
concentrators (TSSCs) as energy supply systems. The integration process of these systems with 8
buildings is complex as several conflicting and multi-disciplinary concerns need to be addressed. Thus, 9
the proposed approach is intended to be adaptable to informed decision-making processes in early 10
design stages, and yet to be collaborative where several key stakeholders are involved. The method 11
is an extension to our previously developed approach where the performance of roof-integrated TSSCs 12
in several design scenarios is accessed along with multiple performance indicators by developing a 13
parametric model and controlling a set of design inputs. In the current study, the proposed method is 14
combined with a multi-objective design optimization method, aiming to optimize, building and TSSCs 15
geometry. The method was validated in an illustrative case study of a single-family house (California) 16
for a number of conflicting objectives e.g., maximization of direct normal solar irradiance (DNI) and 17
annual average load match index (av.LMI), and minimization of covered roof area. The validation of 18
the method shows a number of interesting results. The method enables the generation of performance-19
driven designs and searches for the most appropriate solutions, that can help to meaningfully support 20
the decision-making process. 21
Keywords: two-stage solar concentrator; roof-integrated; parametric; multi-objective; design 22
optimization; decision-making. 23
24
25
26
27
2
1 Introduction 28
The push for a less carbon-intensive built environment has led to several questions about how 29
self-sufficient buildings should be designed. Sustainability-related issues can be addressed by working 30
on building envelopes to maximize solar gains [1,2,3], and integrating advanced solar technologies 31
[1,4,5,6,7]. In recent years, building-integrated photovoltaics (PVs) have gained significant interest in 32
building energy research [2,3,8,9,10,11,12,13]. However, PVs are still behind the solar concentrators 33
(using mirrors and a receiver to convert sunlight into usable energy) in many aspects. For instance, a 34
PV requires two times more space than a concentrator to produce the same amount of energy (~ 550 35
kW), where space can be a critical factor, especially in urban areas [14]. Further, PV efficiency is very 36
low (16–22%) [15] compared with concentrator efficiency (40%) [16]. Concentrators with high 37
concentration ratios have high efficiency and energy yield and need a small-sized receiver 38
[17,18,19,20,21]. Among several designs, two-stage solar concentrators (TSSCs) show 50% to 200% 39
[18,19,20] more concentration ratio and require lesser (i.e., 77%) solar cells [21] compared with 40
traditional concentrators. TSSCs are prominent for high energy yield, efficient power delivery, and 41
deployment modularity [22,23,24]. In TSSCs, light is reflected from a primary mirror to a secondary 42
mirror, which is focused on the receiver [22]. Despite growing interest in building-integrated 43
concentrators [25], there exists less research [24,26,27,28,29,30,31,32,33] on integrating TSSCs with 44
buildings. Further, the integration of TSSCs with buildings reflects a complex decision-making process, 45
involving stakeholders from different domains e.g., building architects, civil engineers, and energy 46
specialists having multiple and conflicting objectives e.g., energy demand vs. energy yield vs. energy 47
cost [2,34]. To address this, design optimization can help to find trade-offs and support the quick 48
decision-making process. This facilitates the setup of design parameters (decision variables) and 49
fitness functions (design objectives) for generating, evaluating, and optimizing multiple designs. Design 50
optimization can be achieved by applying optimal combinations of different design strategies and 51
ranking design options according to a set of objectives [35]. Nevertheless, design optimization for 52
building integrated TSSCs requires optimization at the building level and system design level. 53
On the design level, TSSCs have several limitations e.g., complex architecture, the requirement 54
of efficient trackers, poor performance in case of misaligned mirrors, and the requirement of high-55
3
manufacturing skills [22]. Some studies [26,27,29] optimized TSSCs for minimum size and cost and 56
maximum yield by considering decision variables e.g., the number of modules and arrangements, and 57
receiver properties. However, there is still more research needed to include several other decision 58
variables in the optimization process that include but are not limited to geometric concentration ratio 59
[24], mirrors’ size [31,33], the distance between mirrors [24,33], or the mirrors’ shape [30,36] to 60
generate optimal TSSC solutions. Additionally, the research on optimization of building-integrated 61
TSSCs is scant [26,27,29]. Unlike PVs, TSSCs have the leverage of design flexibility (e.g., by varying 62
mirror shapes, system dimensions, etc.) since the technology is still far from maturity [22]. Because 63
concentrators only work under direct normal irradiance (DNI) unlike PVs. Thus, successful integration 64
of TSSCs with buildings requires exploration of building surfaces that receive most of DNI, to ensure 65
optimal performance of these systems. However, in existing research [26,27,29], building-integrated 66
TSSCs are optimized as stand-alone designs before installation on buildings and are limited to existing 67
buildings. Thus, building-related parameters are ignored for optimization of TSSCs while buildings 68
control a significant proportion of incoming sunlight [2,3,8,9,10,11,12,13]. Hence, optimization of 69
building surfaces, especially roofs that are more optimal locations in urban areas, is still missing for 70
maximization of DNI, before installation of TSSCs. 71
This research envisions that design optimization of both; TSSCs and building can enable 72
informed decision making, by generating several solutions and evaluating across different objectives. 73
In this sense, the building roof can be optimized to maximize DNI, and TSSC geometry can be 74
optimized for optimal performance. One possibility is to develop parametric models by mimicking 75
design parameters [37] and applying multi-objective optimization [26,38,39] where genetic algorithm 76
(GA) based methods are widely adopted in the buildings and energy research [26,39]. In the current 77
research, there exist several parametric models combined with multi-objective optimization to 78
maximize solar gains on building envelopes, ultimately energy yield [2,3,8,9,10,11,12,13]. To the 79
authors’ best knowledge, no study proposed any model to maximize DNI by applying parametric 80
modeling and multi-objective optimization approaches. Existing models [2,3,8,9,10,11,12,13] enabled 81
building design optimization, by mimicking building-related decision variables e.g., roof design and 82
slope, or orientation. However, these models are limited to integrating PVs with buildings. Further, 83
4
these models are limited to using fixed, and commercially available PVs, and do not include PV-related 84
decision variables. 85
To the authors’ best knowledge, there exists no parametric modeling approach combined with 86
multi-objective optimization for building-integrated TSSs considering building- and TSSCs-related 87
decision variables, thus a collaborative design optimization of both is still missing. This motivates to 88
development of an integrated design optimization approach in subsequent steps: optimization of roof 89
design for maximizing DNI followed by optimization of TSSC design for improved performance. To 90
begin to address this knowledge gap, this paper introduces a two-step design optimization method that 91
allows for automatically integrating TSSCs with building roofs. The proposed method is based on our 92
previously developed performance assessment method [40] of roof-integrated TSSCs, where several 93
design alternatives were developed by applying a parametric modeling approach. Previously [40], we 94
accessed the performance of roof-integrated TSSCs by manipulation of the building- and TSSCs-95
related design parameters i.e., roof shape, roof slope, building orientation, TSSC type, geometric ratio, 96
and separation distance between mirrors. However, we did not optimize these designs through 97
optimization algorithm(s), and manually filtered designs according to performance criteria. The method 98
presented in this study enables a two-step optimization: (1) optimizing the roof design to maximize DNI 99
(single-objective), and (2) optimizing TSSC configuration for two performance objectives (multi-100
objective): maximize energy reflected by annual average load match index (av.LMI) [41,42] and 101
minimize covered roof area by TSSC modules. The proposed method applies NSGA-II (GA algorithm) 102
due to its wide applications in the design optimization of buildings and energy systems [43,44,45,46]. 103
In the proposed method, we consider decision variables that are related to both, building design (i.e., 104
roof shape, slope, and orientation) and TSSC design (type, solar cell size, number of modules, 105
geometric ratio, and separation distance between mirrors). Thus, our method helps us to find the best 106
roof design achieving maximum DNI, and best TSSC designs that can be integrated well with buildings 107
achieving maximum energy gain and requiring minimum installation space on the building’s roof. The 108
main hypothesis is that the performance of building-integrated TSSCs can be improved by applying 109
parametric modeling and multi-objective optimization approaches to building scale, and TSSC scale 110
5
designs. The method presented in this research allows exploring trade-offs among performance 111
objectives, enables performance-driven design, and serves to guide decision-makers. 112
The paper is structured as follows. Section 2 provides an overview of modeling and optimization 113
considerations for installing TSSCs with buildings. We then describe the proposed method, highlighting 114
the principal components of our approach in section 3. Section 4 represents the implementation of the 115
proposed method in a case study. In section 5, we present our optimization results followed by the 116
discussion and limitations of the proposed method in section 6. Finally, section 7 concludes the paper. 117
2 Background 118
Design problems related to building-integrated solar technologies are complex where conflicting 119
goals are often required at the same time. This requires a holistic and integrated design approach 120
where multiple teams can work together. Even in integrative and collaborative teamwork, finding a 121
meeting point that allows the optimal solution for all necessities becomes challenging. Surely, suitable 122
multi-objective optimization methods, aimed at tailored and reliable evaluations of the performance, 123
are a possible answer where a range of solutions are sought that span the trade-off between each 124
design objective [38,47]. In the field of building energy research, researchers usually define only two 125
objectives, such as energy cost and CO2 emissions [43,48], energy use and cost [44], or energy use 126
and daylight [35]. In a few cases, a few studies proposed three objectives such as [35], energy use, 127
energy generation, and daylight [11], energy use, energy generation, and visual discomfort time [3], 128
energy use, cost, and energy generation [2], or energy use, cost, and thermal comfort [10,49]. 129
Nevertheless, a complex problem solving through multi-objective optimization usually covers two 130
major aspects that includes: (1) optimization method, and (2) design optimization scope. Concerning 131
the optimization method, evolutionary algorithms appear to resolve multi-objective optimization 132
problems by mimicking the systems and techniques encountered in evolutionary biology [47]. In this 133
context, concepts such as inheritance, mutation, natural selection, and crossover assist in the search 134
for an optimal set of solutions to a given problem [47]. Several types of evolutionary algorithms have 135
been identified in building energy research such as multi-objective genetic algorithm (MOGA) [50,51], 136
micro-GA [26,27], fast non-dominated sorting genetic algorithm (NSGA-II) [43,44,45,46], multi-137
6
objective evolutionary algorithms (MOE) [52], generalized pattern search optimization algorithms 138
(GPSOA) [53], hybrid-GPSOA [54], and trust-region-reflective least-squares algorithms [55]. GAs are 139
the population-based methods and have a good diffusion in the buildings and energy research 140
community, producing a sub-optimal solution in a reasonable time where each individual of the 141
population (decision variables) represents a solution for the target problem. The population of solutions 142
evolves during several generations, where at each generation, all the individuals are evaluated by a 143
fitness function that measures how good the solution represented by the individual is for the target 144
problem [39]. Typically, the outcomes of multi-objective optimization are divided into feasible and 145
Pareto solutions. Feasible solutions are found by the optimization algorithm in searching for optimal 146
solutions, that satisfy all defined constraints. While a solution is Pareto optimal if there are no other 147
feasible solutions that are better with respect to one objective without being worse with respect to at 148
least one other objective [37]. In Pareto fronts, objectives can be mapped in single- or multi-149
dimensional representations, while the problems with more than three objectives are more challenging 150
due to complexity of data to display [56]. The parallel coordinate plots are scalable alternative to plot 151
optimization results. In general, number of generation and population size varies based on model 152
complexity, where most studies reported generation size between 10 and 200, and population of 20 to 153
150 [10,11,35,44,45,48]. 154
Regarding the design scope, during the multiple design stages of projects regarding the 155
installation of solar technologies with buildings, several specialists need to interact concerning building 156
and solar technology design to be integrated with building, to predict the overall performance across 157
several disciplines. Typically, to achieve a sustainable building design, two main concerns are, how to 158
design buildings to enhance solar energy yield [1,2,3], and how to design high-performing solar energy 159
systems [6,7]. Although the optimization methods described above are undoubtfully promising, due to 160
their inherent complexity, these methods are not commonly used in the design practice yet and 161
currently are limited to a few academic research studies [35]. The complexity comes from the large 162
number of multi-disciplinary interrelated parameters involved in optimizing building and system 163
performance. Because of the high complexity in setting up a model for multi-objective optimization, 164
there is a great demand for utilizing and integrating the advanced modeling and simulation 165
7
technologies, such as parametric models and optimization algorithms with energy simulations [35]. In 166
general, design optimization covers two aspects, choice of decision variables and selection of solar 167
technology. Typically, existing models take decision variables that are either related to building [1,2,3] 168
or energy systems [6,7]. However, there is a lack in considering the decision variables associated with 169
both, buildings, and solar technologies into a single framework. Moreover, in the process of integrating 170
solar technologies with building envelopes, one of the commonly developed approaches is the 171
parametric simulation method. This approach enables us to set design parameters within a proper 172
range and to see their effects on some objective functions, while other variables are constant [37]. 173
Several studies proposed parametric models that were limited to optimization of building-integrated PV 174
systems [2,8,9,10,11,12,13]. Additionally, most of the fundamental work in this field is limited to 175
investigating the decision variables associated primarily with building geometries [2,8,9,10,11,12,13] 176
by manipulating roof forms, orientations, and roof slopes [2,3,57] aiming to increase the solar irradiance 177
and ultimately the energy yield from building-integrated solar energy systems. Regarding the 178
technology choice, PV systems are widely applied, however, there is little work towards practical 179
implementation of solar concentrators, in particular TSSCs, that are far better than PV in terms of their 180
performance. 181
In the current practice, for solar concentrators, design optimization is performed only for existing 182
buildings, not considering the optimization of building designs [26,27,45,46,53,55]. For instance, 183
studies aimed to optimize concentrator designs for maximizing energy yield or energy efficiency and, 184
minimizing the system cost, by investigating system-related design parameters such as mirrors’ 185
diameter as well as operational parameters [45,53,58]. Considering other design approaches, TSSCs 186
have the advantage of higher yield by utilizing smaller-sized solar cells, over conventional single-stage 187
concentrators as highlighted in our previous study [40]. Despite their applications in buildings 188
[24,26,27,28,29,30,31,32,33], only a few studies reported design optimization models of TSSCs 189
integrated with buildings [26,27,29]. For example, Burhan et al. [26,27] optimized a TSSC design using 190
a micro-GA and aimed to find the optimum system configuration and dimension with zero failure time 191
and minimum cost. They [26] considered the module number and the initial storage as decision 192
variables. Later, they [27] incorporated other aspects e.g., solar cell (receiver) properties, and the 193
8
optical parameters and arrangement, and aimed to optimize the system size at minimum cost. Another 194
study [29] used a trust-region optimization algorithm to maximize electrical yield from a TSSC by 195
varying solar cell properties. 196
One of the major concerns in their successful implementation in the built environment is 197
optimizing TSSCs for several design-related and performance-related issues. Future cost-effective and 198
energy-efficient solutions can be developed by increasing energy yield and reducing the size of system 199
size [31,33] and solar cell receiver [21]. Further small or lesser modules can lead to reduction in energy 200
generation costs, especially in urban areas with significant space values [14]. Therefore, a significant 201
amount of effort is needed to design novel solutions by exploring several decision variables related to 202
design e.g., geometric concentration ratio [24], mirrors’ size [31,33], the distance between mirrors 203
[24,33], or mirrors’ shape [30,36]. However, an analysis of TSSCs for optimal performance by 204
considering all these parameters as decision variables is still missing in the current literature. 205
Considering building-integrated TSSCs, energy performance of TSSCs is not only influenced by 206
system design, but also building design achieving a certain DNI level. Therefore, the integration of 207
TSSCs with building also requires optimal building designs, ensuring a maximum level of DNI [59]. 208
Building Roofs are logical places for efficiently harvesting solar energy [60,61,62,63], therefore roof 209
designs should be optimized to maximize DNI. However, there is still missing literature pointing towards 210
the analysis of DNI of various roof designs before the installation of TSSCs. This requires taking several 211
building-related decision variables including roof types, orientation, and slopes to investigate several 212
optimal building roof designs. Therefore, to assess the actual potential of TSSCs in buildings, their 213
integration with the building envelopes needs to be accurately addressed. This requires developing a 214
parametric model and optimizing building and TSSC designs. Employing such an approach can help 215
to investigate performances of several optimal designs and allow evaluating the trade-offs between 216
each design objective e.g., maximum energy balance, minimum system size, and cost. Therefore, to 217
enable integral designs of building-integrated TSSCs and evaluate several optimal solutions, decision 218
variables related to buildings and TSSCs should be considered in a single design optimization method. 219
Because previous studies either aimed to optimize building roofs to maximize solar gains 220
[2,3,8,9,10,11,12,13,57] or to optimize TSSC configurations [26,27] for improved performance, 221
9
concurrent design optimization of both is still missing. This requires an integrated design optimization 222
approach in subsequent phases of the design process that is optimization of roof design followed by 223
optimization of TSSC design across a number of conflicting performance-related objectives. In building 224
energy research, a load match index (LMI) reflecting a match between yield and demand is one of 225
important energy performance indicators [41]. Inspired by previous studies on parametric modeling 226
methods for building-integrated PV [41,42], we intend to increase an annual average LMI (av.LMI) for 227
TSSCs by varying building and system-related design variables. Further, this research envisions that 228
reduced roof space occupied by TSSC modules can lead to cost effectiveness of system where space 229
value is high in urban sector [14]. 230
To this end, we propose a design optimization method by employing a parametric modeling 231
approach by taking several decision variables of both building and TSSCs. Due to their popularity in 232
both, buildings, and solar energy systems, we use NSGA-II as an optimization technique [43,44,45,46]. 233
The proposed design optimization method is based on our previously developed performance 234
assessment method of roof-integrated TSSCs, where several design alternatives are developed by 235
applying parametric modeling approaches [40]. In our previous study, we assessed the performance 236
of several roof-integrated TSSCs by manipulation of the building- and TSSCs-related design 237
parameters i.e., roof shape, roof slope, building orientation, TSSC type, geometric ratio, and separation 238
distance between mirrors. The method presented in this study performs a two-step optimization, first 239
optimizing the building’s roof to maximize DNI (single-objective), and then optimizing TSSCs for a 240
number of performance objectives (multi-objective). In multi-objective optimization, the method 241
evaluates av.LMI, and covered roof area. The two-step optimization helps finding optimal roof shape, 242
slope, and orientation as well as optimal TSSC type, solar cell size, number of TSSC modules, 243
geometric ratio, and separation distance between mirrors of TSSCs for good performance. Thus, the 244
method serves to guide decision-makers in the design and operation of TSSCs as part of building 245
geometry as discussed in the following section. 246
247
248
10
3 The proposed design optimization method 249
The proposed method is structured in four successive steps (Fig. 1). The first step is the 250
geometric generation step that requires the parametric model creation for roof and TSSC designs 251
considering several decision variables and constraints. 252
253
254
255
256
257
258
259
260
261
The second step is the design optimization of the building roof created in step 1, using a single-262
objective optimization that aims to maximize the solar irradiance in terms of DNI on the roof. The third 263
step is the design optimization of TSSCs created in step 1 using multi-objective optimization aims to 264
find optimal solutions in terms of two key objectives that are maximizing the av.LMI and minimizing the 265
covered roof area by integrating optimal TSSC designs with optimal roof design created at step 2. The 266
fourth step is the evaluation of generated results to investigate several optimal solutions in terms of 267
av.LMI and covered roof area in the design space. The following sections describe each step in detail. 268
3.1 Parametric model 269
In the first step, we assume several design-related (i.e., building dimension) and environmental 270
constraints (i.e., energy demand and weather data), as well as the continuous and discrete type of 271
Fig. 1 The proposed design optimization method, where DNI represents the direct normal irradiance,
av.LMI represents the annual average load match index, and cov.roof represents the percentage of
the available roof area covered by TSSC modules.
11
decision variables based on the variation domains we set (Table 1, Fig. 2, Fig. 3). 272
273
Variable(s)
Lower limit
Upper limit
Type
Roof slopes (o)
5
30
Continuous
Building orientation (o)
0
315
Continuous
Geometric concentration ratio
26
122
Continuous
Number of modules
10
100
Continuous
Solar cell width (mm)
7
10
Continuous
Separation distance (m)
0.20
0.71
Continuous
Variable(s)
Min. index
Max. index
Type
Roof type
0
2
Discrete
TSSC type
0
3
Discrete
274
275
276
Table 1 Decision variables, their acceptable range, and types.
Fig. 2 Continuous and discrete decision variables of roof designs investigated in the method.
12
277
278
279
280
281
282
283
284
285
286
287
288
289
290
In our previous study [40], we considered varying roof type, slope, and orientation, as well as 291
TSSC type, geometric ratio, and separation distance. However, in this study, we further include two 292
design variables related to TSSCs as decision variables that are solar cell size and number of modules 293
[26,27]. The method begins with parametrically generating roof and TSSCs geometries. The roof 294
design is generated based on roof slope and orientation as continuous, and roof type as discrete 295
decision variables (Fig. 2). The scope of this study is limited to three roof types including shed, gable, 296
and saltbox roof. To enable the parametric changes, we assign each roof type with an index as 0 for 297
shed, 1 for gable, and 2 for saltbox roof design. Additionally, TSSC geometries are generated based 298
Fig. 3 Continuous and discrete decision variables of TSSC designs investigated in the method,
where AM1 represents the primary mirror area, AM2 represents he secondary mirror area, and
AR represents the receiver area.
13
on geometric concentration ratio, a number of modules, the width of the solar cell, and the separation 299
distance between mirrors as continuous, and TSSC type as discrete decision variables, with the 300
receiver properties and dimension (i.e., solar cells) as main design constraints (Fug. 3). There are 301
several TSSC design types based on mirror shapes. However, cassegrain design has attracted more 302
attention in solar energy applications as identified by Bushra and Hartmann [22]. In this study, we focus 303
on four cassegrain types as cassegrain employing parabolic primary and secondary mirrors (‘cass-I’), 304
cassegrain employing a parabolic primary and a hyperbolic secondary mirror (‘cass-II’), cassegrain 305
employing a parabolic primary, and a wide elliptical secondary mirror (cass-III), and cassegrain 306
employing a parabolic primary and a long elliptical secondary mirror (cass-IV). We assign each TSSC 307
type with an index as 0 for ‘cass-I’, 1 for ‘cass-II’, 2 for ‘cass-III’, and 3 for ‘cass-IV. After defining 308
parametric relations and creating roof and TSSC designs, the next step is to perform roof design 309
optimization to obtain maximum solar irradiance in terms of DNI as discussed in the following section. 310
3.2 Roof design optimization 311
Considering the weather conditions as environmental constraint, this step allows optimizing the 312
roof design by maximizing incoming sunlight on the roof. We use weather data (.WEA) file containing 313
hourly data of the cumulative radiation of a particular geographic location over a certain period. The 314
optimization process through solar simulations uses this data to determine the amount of radiation in 315
particular DNI on a selected roof with respect to roof type, orientation, slope, time range, and shading. 316
We use a genetic algorithm, in particular, NSGA-II for a single-objective optimization that is maximizing 317
the solar energy gain in terms of DNI as given below: 318
𝑀𝑎𝑥 𝑍1=𝐷𝑁𝐼 (𝑋)
(1)
Subject to: 𝑋={𝑥𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛, 𝑥𝑆𝑙𝑜𝑝𝑒, 𝑥𝑅𝑜𝑜𝑓𝑦𝑝𝑒 } 319
Our method allows estimating hourly, daily, to annually-averaged DNIs, we only focus on annual 320
DNI values because we aim to design a system fulfilling an annual average energy demand. At this 321
step, we define a fitness function for solar assessment analysis on the roof to estimate the DNI. We 322
assign this fitness function to the initial population list with a defined size and apply the optimization 323
algorithm. Then the results transfer to the generation loop to improve the values in each generation. 324
14
The generation loop runs the generation and sorting processes until the run iteration counter reaches 325
the limit we set. With different roof design options, we aim to test the maximum solar potential on the 326
roof surface generated at step 1 before integrating TSSCs on the roof. 327
3.3 Energy system design optimization 328
Based on the maximum DIN value achieved at the first-step optimization (at step 2), we perform 329
TSSC design optimization at this step. Our primary goal is to design a TSSC as an energy system that 330
can be used for electrical and thermal energy applications in buildings. This step requires information 331
regarding the maximum DNI for optimal roof solutions for step 2. We define fitness functions for two 332
objectives, av.LMI and covered roof area (CRA) as follows: 333
𝑀𝑎𝑥 𝑍2=𝑎𝑣.𝐿𝑀𝐼 (𝑋)
(2)
𝑀𝑖𝑛 𝑍3=𝐶𝑅𝐴 (𝑋)
(3)
Subject to: 𝑋={𝑥𝐷𝑁𝐼, 𝑥𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑟𝑎𝑡𝑖𝑜, 𝑥𝑆𝑒𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒,334
𝑥𝑇𝑆𝑆𝐶𝑇𝑦𝑝𝑒,𝑥𝑆𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙 𝑠𝑖𝑧𝑒,𝑥𝑁𝑢𝑚𝑏𝑒𝑟 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 } 335
The fitness functions measure how good the solution represented by the individual are for the 336
target problem that is to maximize the av.LMI and minimize the covered roof area. The first fitness 337
function is based on electrical and thermal energy calculations. To begin with this, we perform an 338
optical simulation of a TSSC design and estimate the concentration ratio as well as optical efficiency. 339
Then the electrical energy from a TSSC module is estimated (using a solar cell and thermal receiver) 340
as follows [64]: 341
𝑃𝑚𝑜𝑑,𝑟 =((𝑘𝑡.𝜂𝑐.𝜏.[𝜌+ 1
𝐶𝑅.(1− 𝜌
0.98)].𝑓.𝜂𝑚𝑜𝑑)−𝑝𝑝𝑎𝑟).𝜂𝑖𝑛𝑣.𝐶𝑅.𝐷𝑁𝐼.𝐴𝑠𝑐.𝑛𝑐
(4)
Where 𝑘𝑡 is the power thermal coefficient, 𝜂𝑐 is the solar cell efficiency, 𝜂𝑜𝑝 is the optical 342
efficiency, 𝜂𝑚𝑜𝑑 is the module efficiency, 𝜂𝑖𝑛𝑣 is the inverter efficiency, 𝑓 is the tracking factor, 𝑛𝑐 is the 343
number of cells per module, 𝑝𝑝𝑎𝑟 is a loss factor (0.023), 𝐴𝑠𝑐 is the solar cell area, 𝐶𝑅 is the 344
concentration ratio, 𝜏 is the transmittivity of mirrors, 𝜌 is the reflectivity of mirrors, and 𝐷𝑁𝐼 is the direct 345
normal irradiance. We then calculate the thermal energy yield as [64]: 346
15
𝑄𝑡ℎ,𝑟 = ([(1−𝜂𝑐.𝜂𝑚𝑜𝑑.𝑘𝑡).𝜂𝑜𝑝.𝐶𝑅.𝐷𝑁𝐼.𝑓] − [ℎ𝑐
.(𝑇𝑐−𝑇𝑜)+𝜖𝑐.𝜎.(𝑇𝑐4−𝑇𝑜4)]).𝐴𝑠𝑐.𝑛𝑐
(5)
where 𝜖𝑐 is the cell emissivity and ℎ𝑐
is the system working hours, 𝑇𝑐 is the cell temperature, 𝑇𝑜 347
is the ambient temperature (25 °C), and 𝜎 is the Stefan Boltzmann constant (5.670373*10-8 W/m2.k4). 348
Finally, this allows estimating overall av.LMI as temporal demand coverage ratio as calculated below 349
[41,42,65]: 350
𝑎𝑣.𝐿𝑀𝐼=1
𝑁.∑𝑚𝑖𝑛 [1,𝑁𝑚𝑜𝑑.𝑔𝑖(𝑡)
𝑙𝑖(𝑡) ]
𝑦𝑒𝑎𝑟
(6)
Where N𝑚𝑜𝑑 is the number of total modules, g is the electrical and thermal yield from a single 351
module, l is the energy load, i is the energy carrier, t is the time interval, and N is the number of data 352
samples. We also estimate the roof covered area (𝐶𝑅𝐴) as the percent of the roof area occupied by 353
modules relative to the total roof area given as: 354
𝐶𝑅𝐴=(𝐴𝑚𝑜𝑑.𝑁𝑚𝑜𝑑
𝐴𝑟𝑜𝑜𝑓 ).100
(7)
Where A𝑚𝑜𝑑 is the module area, and A𝑟𝑜𝑜𝑑=𝑓 is the available roof area. We assign the list of 355
fitness functions to the initial population list. Then the results are transferred to a generation loop to 356
improve the values in each generation. The generation loop runs the generation and sorting processes 357
until the iteration counter reaches the defined limit. Thus, the methods allow performing a multi-358
objective optimization based on decision variables related to TSSC design and the fitness function. In 359
the following step, we assess multiple designs by evaluating multi-objective optimization results. 360
3.4 Design assessment 361
This step allows the user to assess several sub-optimal solutions in terms of contradictory 362
objectives by updating the building and TSSC decision variables according to a pre-defined range and 363
moving through the above process described by steps 1 to 3. Thus, in this step, the method reports a 364
set of sub-optimal solutions for the key objectives from two-step optimization. As indicated in previous 365
sections, the key objectives we consider are maximum DNI, maximum av.LMI, and minimum covered 366
roof area. By applying a multi-objective optimization algorithm called a genetic algorithm (NGGA-II) in 367
two subsequent steps, we first seek solutions with the highest DNI, and then we seek a range of 368
16
solutions with a trade-off between design objectives that are av.LMI and covered roof area. By doing 369
this, we generate a large design space with several different design solutions. This further automates 370
the process which is essential to be performed on a design from key decision variables as input to the 371
design assessment. The scope of design options is provided by the opportunity to combine different 372
possible values for the decision variables. Hence, our method allows making changes in the 373
optimization settings, and decision variables (i.e., continuous, discrete), and repeating the same 374
process from steps 1 to 3 until we achieve the good or sub-optimal solutions. We validate the method 375
in an illustrative case study as discussed in the following section. 376
4 Validation approach 377
To validate the method, we conducted an optimization study for a single-family detached building 378
(size 186 m2 – 232 m2) - the most common type of residential buildings in California [66]. As inputs to 379
the model, we considered building- and TSSC-related decision variables (Table 1) as well as several 380
environmental [67] and design constraints (Table 2). We used an hourly cumulative radiation dataset 381
(.WEA file) that is available in Dynamo for the chosen location over a year. We implemented the 382
method using Dynamo [68], SolTrace [69], and R [70] (Fig. 4). 383
384
385
386
387
388
389
390
391
392
Fig. 4 Implementation of the proposed method.
17
Table 2 Design and environmental constrains in the method. 393
Environmental
constraints
Value
Design constraints
value
Location:
Latitude
33.61
Building
design:
Width (W) x Length (L)
x Height (H)
16 m x 13 m x
10 m
Longitude
-114.58
Floor area
208 m2
Energy
demand:
Annual
(electrical)
15,545
kWh
TSSC
design:
module efficiency
(𝜂𝑚𝑜𝑑)
90 %
Annual
(thermal)
7,567
kWh
mirrors transmittivity
(𝜏)
90 %
Annual total
23,112
kWh
mirrors reflectivity (𝜌)
94 %
Tracking factor (𝑓)
0.9
cell size ( 𝐴𝑐)
81 mm2
cell emissivity (𝜖𝑐)
85 %
inverter efficiency
(𝜂𝑖𝑛𝑣)
90 %
394
Dynamo is a visual programming application that allows specifying the design spaces quickly, 395
interactively, and accurately. Calculations were performed in Dynamo using a text-scripting interface. 396
We used the Optimo package in Dynamo was used for single- and multi-objective optimization based 397
on the genetic algorithm (i.e., NSGA-II). SolTrace is an optical ray-tracing tool for optical modeling and 398
simulating of TSSCs, where the Python scripting functionality in Dynamo was used for triggering optical 399
simulations, and for data exchange between SolTrace and Dynamo. R is a programming language for 400
statistical computing and graphics. The implementation begins with the development of a parametric 401
model in Dynamo by generating a random population list of decision variables followed by roof design 402
optimization. At this step, a fitness function is defined for estimating DNI on the roof through Dynamo’s 403
solar irradiance analysis package. This analysis is performed over the period of one year (01-01-2020 404
– 31-12-2020) using the climate data of a chosen location in California. The fitness function for DNI is 405
assigned to the initial population list and finally, the NSGA-II algorithm is applied in Dynamo. This 406
reports the maximum DNI values (single-objective) and corresponding decision variables of the 407
building that are building roof type, roof slope, and orientation. In the following step, TSSC designs are 408
18
optimized, where all the calculations are performed in Dynamo. For this, a fitness function is defined 409
that enables the optical simulations of TSSCs in SolTrace. To enable these simulations directly from 410
Dynamo, we developed a python script that triggers optical simulations in SolTrace using the inputs of 411
population generations and optimal DNI. This script also allows exporting all results from SolTrace 412
back to Dynamo. We then calculate the energy yield as well as the required number of modules to 413
meet the energy demand of the building. The fitness function is defined for further estimating the 414
covered roof area by modules that are required to fulfill a specific pre-defined energy demand for a 415
building. The fitness function is assigned to the initial population list and the NSGA-II algorithm is 416
applied in Dynamo. For first- and second-step optimization, the generation loop runs the generation 417
and sorting processes until the iteration counter limit is reached. Finally, the method exports all decision 418
variables and their corresponding results to a comma-separated values (CSV) file. In optimization at 419
steps 2 and 3, the loop could in theory run endlessly unless a defined end criterion is reached. For this 420
study, the end criterion was chosen to be 10 generations with 30 individuals each. We chose the 421
number of generated solutions as a compromise between computational time and having a meaningful 422
number of cases for the optimization algorithm to be able to find sub-optimal solutions. Finally, the 423
results of the optimization study are analyzed graphically in R. Our design problem involves three 424
objectives (i.e., DNI, av.LMI, and covered roof area), thus, we chose to represent our results in Pareto 425
Front. These plots represent the trade-off front between design objectives and allow finding equally 426
optimal solutions as discussed in the following section. 427
5 Results 428
Fig. 5 illustrates the results of our design optimization study in parallel coordinate plots that help 429
to explore design space and steer the optimization procedure where each line represents an alternative 430
solution for multiple objectives. Moreover, we also highlight the Pareto optimal solutions (Fig. 6). The 431
first-step optimization of roof design shows that a maximum of annually-averaged DNI of more than 432
1900 kWh/m2 is achievable. Thus, maximum DNI is achievable at an optimized roof design that reflects 433
the saltbox roof type (index 2) with a slope of 12o and orientation of 45.5o (Fig. 5). This roof design will 434
be considered for the integration of TSSC designs that are optimized using the maximum DNI as an 435
input. During the second-step design optimization, TSSCs are optically simulated under maximum DNI 436
19
and for a whole set of the population of input design parameters for different generations. Fig. 6 also 437
shows the design space of all evaluations based on a set of decision variables and corresponding 438
performance objectives well as of Pareto optimal solutions. Moreover, we also represent the trade-off 439
between two objectives, av.LMI versus covered roof area (Fig.6). Our results show that the optimization 440
process can improve the performance of TSSCs and find the sub-optimal solutions from the design 441
space. 442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
Fig. 5 The design space of all evaluations in three roofs and four TSSC designs, where SD represents
the separation distance, GCR represents the geometric concentration ratio, and highlighted design
spaces represent sub-optimal solutions for maximum direct normal irradiance (DNI), and annual average
load match index (av.LMI), and minimum covered roof area.
20
458
459
460
461
462
463
464
465
466
467
Results of the multi-objective optimization (Fig. 6) procedure show that there is a widely 468
distributed initial randomly generated solution set. For the higher generations, the results are getting 469
more and more clustered toward the optimum (Pareto) solution in terms of av.LMI and roof covered 470
area. Moreover, the solution of the final population is better than the solution of the initial population, 471
where Pareto sub-optimal solutions can even provide a high av.LMI of more than 1 and the covered 472
roof area of below 10%. Our results show that very few alternatives of the initial population set satisfy 473
the high-performance criteria to get the maximum av.LMI and minimum roof covered area. However, 474
the higher population set alternatives satisfy these criteria. The Pareto optimal shows a conflicting 475
relationship between performance objectives (Fug. 6), for instance, higher av.LMI leads to a higher 476
number of required modules and ultimately a large fraction of the roof area is occupied by TSSC 477
modules. In Fig. 6 and Fig. 7, we highlight a few good, Pareto sub-optimal solutions representing high-478
performing TSSC designs that can be integrated with roof design extracted from the first-step 479
optimization process for maximizing the DNI. 480
481
All evaluations
Pareto solutions
d
f
c
b
e
a
Fig. 6 Pareto optimal solutions for optimization annual average load match index (av.LMI) and
covered roof area (%).
21
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
For instance, the first design alternative (Fig. 7b) consists of TSSC type 1 that is cass-II design 499
(consisting of a parabolic primary and a hyperbolic secondary mirror), with a separation distance of 62 500
cm, and a geometric concentration ratio of 3821 using 8.66 mm wide solar cell and 23 modules. With 501
this design configuration, the modules cover only 1.4 % of the roof, which is very low, but the av.LMI 502
Fig. 7 The design space of all evaluations in three roofs and four TSSC designs for maximum direct
normal irradiance (DNI), and annual average load match index (av.LMI), and minimum covered roof
area, and number of solar cells (Num.cells), where GCR represents the geometric concentration ratio,
and SD represents the separation distance between mirrors.
22
becomes too small, that is 0.17. Hence, with this design, only 17% of the building energy is met. The 503
second design (Fig. 7c) combined 95 modules of TSSC type 2 that is cass-III design (consisting of a 504
parabolic primary and a wide elliptical secondary mirror), with a wider solar cell receiver (9.9 mm) and 505
the higher geometric ratio of mirrors (4314) compared with the first solution, but with a lower separation 506
distance between mirrors (41 cm). Modules based on this design configuration cover 5.9 % of the roof 507
and lead to av.LMI of 0.98. The third design (Fig. 7d) consists of the same TSSC type, a separation 508
distance between mirrors as and solar cell with the second design but has a slightly lower geometric 509
concentration ratio (4179) and slightly lesser modules (93) and uses slightly small-sized solar cells (9.4 510
mm). This leads to slightly improved av.LMI of 1.1 but covers roof area of 7.5 % due to wider (primary) 511
mirrors. The fourth (Fig. 7e) design consists of TSSC type 3 that is cass-IV design (consisting of a 512
parabolic primary and a long elliptical secondary mirror) using a slightly wider solar cell (9.9 mm), with 513
a slightly larger separation distance (46 cm) and small geometric ratio (1366) compared with previous 514
three designs. Moreover, with a higher number of modules (99) compared with the other three designs, 515
av.LMI can be improved to 1.17 but with a slightly higher covered roof (8 %) compared with other 516
designs. 517
The fifth (Fig. 7f) design is based on the same TSSC type and uses the same width of the solar 518
cell as the second and third design configurations, but with a separation distance of 43 cm and a 519
geometric ratio of 4101. Moreover, with only 78 modules, av.LMI of 1.29 is achievable where roof 520
covered area is 10%. Finally, the last design (Fig. 7g) is of TSSC type 0 that is cass-I (consisting of 521
parabolic primary and secondary mirrors), with a separation distance of 56.4 cm, and a geometric ratio 522
of 1478. With 87 modules, av.LMI of such design is significantly improved to more than 1.5 but covers 523
roof area of more than 10% that is higher among all design configurations. In general, designs shown 524
in Fig. 7d– Fig. 7f show good performance with av.LMI of more than 1 and cover less than 10 % of the 525
roof. However, designs in Fig. 7b and Fig. 7c are good for achieving lower covered roof area but not 526
so good av.LMI values. Further, the design shown in Fig. 7f is impressive in terms of achieving higher 527
av.LMI but the covered roof area increases to more than 10%. In both single- and multi-objective 528
optimization, our results become more clustered and converged at a higher generation size of more 529
than 7. We observed a slight variation in performance objectives at higher generation compared with 530
23
initial solutions. 531
Regarding the simulation time, in our illustrative example, with a generation size of 10, and 532
population size of 30, the simulation time for each generation was 12 minutes, where the overall 533
optimization process took about two hours. The process of parametric model settings, model 534
regeneration, and exporting results took about 10 minutes of the total time. We spent about an hour 535
on first step optimization (roof design for single-objective) and about one hour on second step 536
optimization (TSSC design for multi-objective). This time could be further reduced by running our 537
simulations on a computer with a high computation power. In general, the time consumption and less 538
computation power were the major issues that restricted us to perform optimization for the limited 539
number of generations and population size. Our method offers a unique opportunity for design 540
optimization of building roof and TSSCs, however, the evaluation of our results is limited to the example 541
building. Moreover, there are a few challenges that should be addressed in future studies, as discussed 542
in the following section. 543
6 Discussion 544
Our proposed method allows performing a two-step design optimization, where building roof and 545
TSSC geometry can be optimized in consecutive optimization steps. The first-step optimization allows 546
accessing roof design in terms of type, slope, and orientation that can achieve maximum DNI. The 547
second-step optimization allows TSSC design configuration with respect to type, separation distance, 548
geometric ratio, size of the solar cell, and a number of modules, to maximize av.LMI and to minimize 549
roof covered area by modules. The exemplary application shows that the proposed method offers an 550
opportunity for integrative, collaborative, and concurrent design of building geometry and the TSSCs 551
as building-integrated energy systems in the early design stage. Thus, the method aims to help 552
designers to perform a broad variety of simulation-based analyses for design optimization of building 553
and energy system and facilitates performance-driven design generation. The method enables 554
parametric variation in the roof and TSSC design combines multi-objective optimization techniques 555
and suggests calculating several building-related and system-related performance indicators as key 556
objectives. In our previous study, we assessed the performance of several design options by manually 557
manipulating several building-related and TSSC-related design parameters [40]. However, the method 558
24
proposed in this study demonstrates the process of design space exploration through parametric 559
modeling combined with multi-criteria optimization to find solutions that are good or sub-optimal. The 560
method is demonstrated using a single-family house in California. To understand how the decision 561
variables related to building and TSSCs are driving variables for certain key objectives, we optimized 562
a large design space featuring various combinations of roof slope, orientation, geometric concentration 563
ratio, solar cell receiver’s width, number of modules, and separation distance between mirrors in TSSC 564
design as continuous, and roof type and TSSC type as discrete decision variables. Our method helps 565
finding the solutions that are good or sub-optimal in terms of maximizing the DNI as solar gain on 566
building roof, maximizing an av.LMI, and minimizing the covered roof area. The method allows to trade-567
off between each objective, which can help the decision-making, for example, to decide on which 568
building design gains high solar irradiance, and what are good solutions in terms of high energy yield 569
(i.e., av.LMI), and small covered roof area. The method can serve as a guiding framework for informed 570
decision-making processes where experts from several different domains, such as building architects 571
and energy specialists are involved, having their own, domain-specific concerns. Unlike traditional 572
approaches for modeling such systems, the method proposed in this study allows multi-disciplinary 573
teams to work together towards automatically developing high-performing, self-sufficient building 574
designs, and high-performing energy system designs. The method helps the seamless integration 575
process of integrating TSSCs with building roofs close to satisfying and meeting several key objectives 576
at the same time. By applying multi-objective optimization, the method can significantly support the 577
quick and informed decision-making process by considering several conflicting design-related and 578
performance-related objectives. The method supports incorporating a broader variety of simulations in 579
different domains into the multi-objective optimization methods and leads to a more comprehensive 580
exploration of the solution space and provides better decision support for the designers. 581
However, there exist several key issues that should be addressed in future research. For 582
example, the proposed method is implemented using several tools that include Dynamo for parametric 583
modeling and optimization, SolTrace for optical simulations of TSSCs, Python for bridging SolTrace 584
and Dynamo, and R Cran for graphical representation of results, however, to develop such methods, 585
compatibility among tools remains a key issue. Moreover, the proposed method is not limited to these 586
25
tools and can be implemented with other relevant tools available to designers and engineers. In this 587
study, we considered energy (i.e., av.LMI), and covered roof area, as key objectives of the system, 588
however, the application of the method can be extended to several other performance-related 589
objectives. For example, the fitness functions in the method can be modified with the inclusion of life-590
cycle related aspects, environmental, social, or economical aspects. Moreover, we only investigated 591
the building design for higher solar gain in terms of DNI, however, a number of building energy 592
performance objectives could be included which are in conflict with the solar gain in particular surface 593
area to volume ratio (SV). A high SV ratio leads to lower solar gains, and this should also be explored 594
in future research by optimizing the building geometry for example to maximize DNI and minimize the 595
SV value. This could yield valuable insights regarding building energy performance and energy 596
efficiency. Further, the optimization problem can be extended by using several other decision variables 597
related to building design, such as building shape factor, surface area, volume, height, or to system 598
design such as material properties of mirrors and receiver and tracker choices. Further, in the discrete 599
decision variable choices, we considered three roofs (indices of 0 to 2) and four TSSC types (indices 600
of 0 to 3), several other types of building roofs, and design configurations of TSSCs can be added in 601
future research. 602
Moreover, we combined parametric simulations with a specific genetic algorithm-based multi-603
objective optimization method (NSGA-II). However, several other evolutionary optimization algorithms 604
can also be used in future research which are widely used in building energy research fields such as 605
MOGA [50,51], micro-GA [26,27], MOE [52], GPSOA [53], hybrid-GPSOA [54], and trust-region-606
reflective least-squares algorithms [55]. Another key issue with our method was that the generation 607
size was limited to 10 and population size to 30, due to limited computation power and large time 608
consumed in each simulation run. This time could be reduced by running the simulations on a high-609
power computer. Further, the method should be tested for other generation size and/or larger 610
population number and the overall simulation time can be reduced by using cloud computing services. 611
Additionally, we used tools to implement the method which are extensively validated; however, we did 612
not verify the optimization results according to real data. In this study, the workflow is exemplified in 613
the climatic context of California and used the definitions of the USA energy codes for the simulation 614
26
parameters. For the purpose of generalization of results, future work should test the proposed method 615
in other climatic contexts and for different decision variables as well as design and environmental 616
constraints. Additionally, the scalability of the proposed method can be extended to multiple buildings 617
and multi-family or multi-story buildings, and to larger districts. This could include the building design 618
optimization with aim of minimizing the shading of roof surfaces as one of design objectives. 619
7 Conclusions 620
This research proposes a two-step design optimization method that facilitates the integration 621
process of TSSC geometries with building roofs as renewable energy supply systems. The first-step 622
optimization allows designing an optimal roof for maximum DNI and a second-step optimization allows 623
designing optimal TSSC configurations for maximum energy yield in terms of av.LMI and minimum 624
space requirements in terms of covered roof area by modules. Thus, the method allows the optimal 625
design of both energy autarkic building and energy supply system collaboratively. The method 626
combines optimization algorithms with parametric models of building roofs and TSSC based on a 627
number of decision variables. In the parametric model, we considered several decision variables 628
related to the building (roof type, slope, orientation) and TSSC (type, geometric ratio, solar cell width, 629
module number, and separation distance between mirrors). Moreover, in the parametric model, we 630
applied the optimization algorithm, NSGA-II with an initial population size of 30, and generation of 10, 631
resulting in 330 runs of performance simulation. The proposed method is validated in an illustrative 632
case study of a single-family house (California). The proposed method allows to find a trade-off 633
between conflicting performance objectives in large design space and helps to identify the set of sub-634
optimal solutions. Our results indicate that the method enables us to assess the performance of 635
different designs and search for the most appropriate design options. Our method helps finding optimal 636
roof and TSSC design configurations in terms of multiple and conflicting objectives. Ultimately, the 637
method can meaningfully support the decision-making process for a low-carbon intensive built 638
environment, where multiple stakeholders can work concurrently and collaboratively. 639
640
641
27
Acknowledgments 642
This research did not receive any specific grant from funding agencies in the public, commercial, 643
or not-for-profit sectors. 644
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