Charline Lasnon

Nicolas Coudrais

Benjamin Houdu

Catherine Nganoa

Thibault Salomon

Blandine Enilorac

Nicolas Aide
email: aide-n@chu-caen.fr

Prof Nicolas Aide

How fast can we scan patients with modern (digital) PET/CT systems?

Keywords: PET, Digital, 18 F-FDG, pain, dyspnea

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Results:

The following reconstructions, which harboured a noise not significantly higher than that of the standard reconstruction, were selected for clinical evaluation:

1iterations/10subsets/20sec (1i10s20sec), 1i10s30sec, and 2i10sPSF60sec.

Only the 60s per bed acquisition displayed similar target-to-background ratios compared to the standard reconstruction, but mean ratios were still higher than 2.0 for the 30s reconstruction. Inter-rater agreement for the number of involved anatomical sites and detected lesion was good or almost perfect (Kappa: 0.64-0.91) for all acquisitions. In particular, kappa for the 30s per bed acquisition was 0.78 and 0.91 for lesion and anatomical sites number, respectively. Intra-rater agreement was also excellent for the 30s

Introduction

Over the last decade, Positron emission tomography (PET) has been marked by technological involvements, including improved reconstruction algorithms and hardware evolutions [START_REF] Van Der Vos | Quantification, improvement, and harmonization of small lesion detection with state-of-the-art PET[END_REF],

the most recent one being digital PET that is now available from the major PET vendors [START_REF] Fuentes-Ocampo | Digital vs. analog PET/CT: intra-subject comparison of the SUVmax in target lesions and reference regions[END_REF][START_REF] Lopez-Mora | Comparison of image quality and lesion detection between digital and analog PET/CT[END_REF][START_REF] Pan | Performance evaluation of the 5-Ring GE Discovery MI PET/CT system using the national electrical manufacturers association NU 2-2012 Standard[END_REF][START_REF] Rausch | Performance Evaluation of the Vereos PET/CT System According to the NEMA NU2-2012 Standard[END_REF][START_REF] Van Sluis | Image quality and semi-quantitative measurements of the Siemens Biograph Vision PET/CT: Initial experiences and comparison with Siemens Biograph mCT PET/CT[END_REF]. Fully digital PET cameras, such as the Vereos system (Philips Heathcare®, Ohio, US), feature small digital silicon photomultipliers instead of much larger photomultiplier tubes, providing true digital photon counting with 1-to-1 crystal coupling. The properties of digital systems, in particular the enhancement in the time-of-flight capabilities, have been shown to improve signal-to-noise ratio and ultimately image contrast and lesion detectability compared to analog PET [START_REF] Lopez-Mora | Comparison of image quality and lesion detection between digital and analog PET/CT[END_REF][START_REF] Aljared | BSREM Reconstruction for Improved Detection of In-Transit Metastases With Digital FDG-PET/CT in Patients With Malignant Melanoma[END_REF][START_REF] Salvadori | Headto-head comparison of image quality between brain (18)F-FDG images recorded with a fully digital versus a last-generation analog PET camera[END_REF]. From a daily routine perspective, these PET advancements are usually used to decrease injected activity and reduce acquisition time, the decrease in the injected activity being made with the aim of reducing radiation exposure but also for obvious economic reasons. However, evaluation of the physics performance of modern PET system with regards to noise equivalent count rate curves (NECR), which account for useful information (true coincidences) and events creating noise (random and scattered coincidences), show that recent PET scanners could handle high activity in the FOV without compromising image quality (i.e noise level).

An issue faced by PET unit teams is the management of bedridden patients experiencing pain or dyspnea. Also challenging is the scan of uncompliant pediatric patients when premedication is not efficient or feasible. Positioning of the patient on the table and PET acquisition being the bottleneck of the total scan time, the fastest PET acquisition would be very useful in all of these situations, provided the diagnostic performances are maintained.

The aim of this study was to seek the minimal duration per bed position being feasible with a recent PET system without compromising image quality and tumour or other target detection in patients for whom fast or even ultra-fast PET imaging may be useful. We focused on cancer patients with dyspnea or in pain, as well as paediatric patients in whom recommendation for injected activity (PEDdose: https://www.eanm.org/publications/dosage-calculator/) lead to activity per unit of weight higher than those used in adult.

The minimal time per bed position was investigated by reconstructing list mode data with decreasing time down to 10 s per bed position. We also evaluated whether modifications had to be made regarding reconstruction parameters when using ultra-fast imaging (i.e in images acquired with less than 60 seconds per bed position), in order to maintain an acceptable noise level. A comprehensive evaluation of the impact of reducing acquisition time was performed on simulated data, including image quality, lesion detectability as well as inter-and intra-observer agreement.

Materials and methods

Patients' selection

From January 2018 to September 2018, patients experiencing pain or dyspnea, in whom a standard 18 F-FDG PET scan duration (around 20 min) was deemed to be infeasible, received a higher injected activity (arbitrary set to 5 MBq/Kg) and underwent a faster scan, as described below in the materials and methods section. Also included in this series were pediatric patients in whom the injected activity, according to the EANM guidelines (using the PETdose recommendations), usually leads to an injected activity around 5-6 Mbq/kg, which is almost twice the injected activity per unit weight in adults at our PET centre. This study was part of the NUMANATEP project, meant to evaluate digital PET systems at our institution, which was approved by the regional Ethics Committee (registration number:

CERES 27669) and authorization to use additional reconstructions for research purpose was sought from every patient. Anonymised data were collected and used as per the European mutually agreed General Data Protection Regulation (GDPR) with the French committee on data privacy registration number CNIL 2204611 v 0.

PET/CT acquisition and reconstruction parameters

The 18 F-FDG usual injected activity was increased from 3 MBq/kg to 5 MBq/kg in adult patients and the acquisition time was decreased from 2 min to 1.5 min. Pediatric patients' activities were computed using the EANM PEDdose application) and these patients were also scanned at 1.5 min per bed position.

Data were acquired in list mode on a Vereos digital PET/CT system. At our centre, the standard reconstruction is as follows: 2 iterations, 20 subsets (2i20s), point spread function modelling and time-of-flight enabled, no post filtering and voxel size of 2mm 3 . In addition, frames of 60s, 30s, 20s and 10s per bed position were reconstructed with the same parameters. Given that an increase in the noise level was expected for reconstructions shorter than 60s per bed position, we also reconstructed 30, 20 and 10s per bed simulated acquisitions with (i) the same number of iterations but with PSF modeling disabled and (ii) a two-fold decrease in the number of iterations (1i10sec) and PSF disabled. Therefore, a total of 15 reconstructions per patient were studied (Figure 1). Within this complete manuscript, the reconstructions are defined and labeled as follows: xixsxSec where xi, xs and xsec stand for the number of iterations, the number of subsets and the time per bed position in seconds.

PSF: point spread function modeling.

Evaluation of noise and tumour/local background when decreasing the acquisition time

Images were analyzed using the MIM software. For all analysis, region of interest (ROI) and volume of interest (VOI) were drawn by one PET reader, using a gradient-based contouring method (PET edge) on images obtained with the standard reconstruction and then copied and pasted to the ten other set of images to avoid any intra-observer variability.

Noise was evaluated in large reference organs and in the vascular background as follows:

-Liver: a 3 cm diameter VOI centred on the right liver lobe -Aorta: a 1.5 cm VOI centre on the descending aorta -Muscles: a 1.5 cm diameter VOI placed on the right ilio-psoas muscle at the level of the 4th lumbar vertebra

In order to assess the potential lesion detectability as a function of lesion/surrounding background visual ratio, a doughnut shaped VOI surrounding the tumours' VOI was used and tumour/background ratios were computed using the SUVmean values within the tumours and doughnut VOIs (supplemental figure 1). As most of included patients had multiple lesions, we focused on the most obvious and less obvious lesions on the standard reconstruction.

Comparison of short and standard acquisitions

Based on the analysis of noise level described below, we selected three reconstructions for which the noise was not statistically different, compared to the standard reconstruction (2i10sPSF90sec): 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, which were randomly assigned to 4

PET readers with at least a 12-month experience in the reading of PET images using the Vereos system. A fifth reader was asked to read all these reconstructions as well as the standard reconstruction, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time. The order of review was (1) 1i10s10sec, (2) 1i10s20sec, (3) 1i10s30sec, (4) 2i10sPSF60sec and finally the standard of reference 2i10sPSF90sec.

PET readers rated the image quality according to a 3-point scale (1=poor; 2=fair; 3= good), recorded the number of malignant foci (0, 1, 2-5, >5) and the number of anatomical sites (0, 1, 2, 3, 4, ≥5). The 5 cutoff value was chosen as this figure is commonly used by the oncology community to define oligometastic disease. We also focused on the number of anatomical sites, which is even more important than the number of lesions by organ in the staging system of many cancer types. Readers were aware of the reasons why the patients had been referred to an 18 F-FDG PET examination, in order to avoid bias in interpretations.

Statistical analysis

Quantitative data are presented as mean ± standard deviation (SD).

Noise (in the liver, aorta and psoas), lesion/background ratios and mean image quality were compared using the non-parametric rank-sum Friedmann test for multiple paired sample.

For that test, the mean rank of each reconstruction was compared to the standard acquisition and reconstruction set (i.e 90s per bed, 2i20s, PSF modeling enabled), which was used as a control reconstruction. The aim was to seek reconstruction settings where the noise level was not statistically significant, compared to the standard reconstruction, which would therefore be useable in clinical practice with an acceptable loss in image quality. The reported P values were corrected for multiple comparison testing using the Dunns' method.

Inter-observer agreement was evaluated using the Cohen's Kappa. Kappa values were reported using the benchmarks of Landis and Koch (0.81-1 almost perfect agreement, 0.61-0.80 substantial agreement, 0.41-0.60 moderate agreement and 0.21-0.40 fair agreement).

Graphs and statistical analysis were performed using Prism 8 (Graphpad, graphpad Software, San Diego, CA, USA) and XLSTAT (addinsoft, Paris France).

Results

Comparison of short and standard acquisitions: noise level, contrast and visual image quality

Over a nine-month period, twenty-eight patients were included. During this period, 873 patients were referred for 18 F-FDG PET. Therefore, the prevalence of patients potentially requiring fast PET imaging was 3.4%. Their characteristics are displayed on Table 1.

Noise level, expressed as the coefficient of variation (CoV) in the blood pool (aorta), liver and muscle was 0.12 (0.04), 0.12 (0.04) and 0.17 (0.08), respectively for the standard reconstruction (2i10sPSF90sec) (Figure 2). If the same reconstruction parameters were maintained (2 iterations, 10 subsets and PSF modeling), only the 60s per bed reconstruction displayed a noise level comparable to the standard reconstruction, with CoV-AORTA, CoV-LIVER and CoV-MUSCLE of 0.14 (0.04), 0.15 (0.04) and 0.20 (0.09), respectively. For reconstruction with a shorter time per bed, maintaining a noise level not significantly higher than that of the standard reconstruction required disabling PSF and decreasing the number of iterations by a two-fold factor in images with less than 60s per bed position. Using these parameters, the 30s (1i10s30sec) and 20s (1i10s20sec) per bed reconstructions displayed CoV-AORTA, CoV-LIVER and CoV-MUSCLE of 0.11 (0.03), 0.12 (0.03) and 0.14 (0.05) and 0.13 (0.04), 0.14 (0.03) and 0.17 (0.07), respectively. Despite adaptation of the reconstruction parameters, the shortest reconstruction used (10s per bed) showed a significantly higher noise, compared to the standard clinical reconstruction (Figure 2).

Based on this analysis of noise level, we selected three reconstructions for comparison of lesion to background ratios and visual image quality: 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec

(Figure 4). Only the 60s per bed reconstruction displayed a mean lesion/background ratio comparable to the standard reconstruction: 3.19 (1.51) versus 3.22 (1.57). For the 30s and 20s time per bed reconstructions, mean lesion/background ratios for the most obvious lesion were still both higher than 2.0 {2.43 (0.93) and 2.43 (0.91)} but represented a decrease of 22% and 23%, respectively compared to the standard reconstruction (Figure 3a).

For the less obvious lesion, mean lesion/background ratios were almost similar to those obtained with the most obvious lesion {1.99 (0.54) and 2.01 (0.56)} and represented a decrease of 21% and 22%, respectively compared to the standard reconstruction (Figure 3b).

Concerning mean quality scores, the only reconstruction displaying no significant difference as compared to the standard reconstruction is the 1i10s30sec reconstruction. It showed a mean quality score equal to 2.6 (0.5) versus 2.7 (0.5) for the standard reconstruction. All other reconstructions showed significantly slightly lower mean quality scores (Figure 4a).

Impact of acquisition time on Lesion detectability

Intra observer agreement

As described in the Material and Methods section, intra rater agreement was evaluated by a PET reader who interpreted all reconstructions, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time. The best intra rater agreements were observed with the 2i10sPSF60s for both the number of detected lesions and the number of detected anatomical sites. Regarding the number of detected lesions, the Cohen kappa coefficients were equal to 0.78, 0.72 and 0.89 for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, respectively.

For the number of detected anatomical sites, the Cohen kappa coefficients were equal to 0.72, 0.72 and 0.95 for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec, respectively. However, all Cohen kappa coefficients were ranged at least good, whatever the reconstruction used.

Regarding the number of detected lesions, 3 (11%) patients had fewer lesions detected using the 1i10s20sec, 1i10s30sec reconstructions as compared to the 2i10sPSF60sec and/or the standard reconstruction (Figure 4b). In line with those findings, 6 (22%) had fewer anatomical sites detected with the 1i10s20sec, 1i10s30sec reconstructions (Figure 4c).

Representative images of one of these patients are displayed on Figure 5. Of note, in line with noise level evaluation (Figure 1), no false positive findings related to noise level were observed on the 1i10s20sec and 1i10s30sec reconstructions.

Inter observer agreement

As described in the Material and Methods section, the three selected reconstructions (1i10s20sec, 1i10s30sec, and 2i10sPSF60sec) were randomly assigned to 3 other PET readers.

Inter-rater agreement for the number of involved anatomical sites and detected lesion was good or almost perfect (Kappa: 0.64-0.91) for all the selected acquisitions (Table 2). In particular, kappa coefficients for the 30s per bed acquisition were 0.79 and 0.91 for lesion and anatomical sites number, respectively. Regarding the number of detected lesions, discordances occurred in 6, 4 and 4 patients for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec reconstructions, respectively. Regarding the number of detected anatomical sites, discordances occurred in 7, 2 and 4 patients for 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec reconstructions, respectively.

Discussion

For a PET centre, being able to perform fast imaging may be a challenge, as shown in Figure .   6 where a lymphoma patient with a bulky mediastinal mass and a superior vena cava syndrome had to be scanned with several pillows to cope with dyspnea. This study was conducted as a non-inferiority study to seek the fastest acquisition time possible with our system, focusing on a specific population likely to benefit from fast or ultra-fast imaging:

patients in pain, experiencing dyspnea and pediatric patients. Claustrophobic patients were not included but could also benefit from fast imaging.

This type of study could be applied to modern PET scanners capable of high-count rate.

Hence, a recent evaluation of the Vereos system showed that its NECR is 513.4 Kcps @ 54.9

KBq/cc [START_REF] Rausch | Performance Evaluation of the Vereos PET/CT System According to the NEMA NU2-2012 Standard[END_REF]. This can be compared to a NECR of 185 Kcps @ 29 KBq/cc and 266.3 Kcps @ 20.8

KBq/cc for the Siemens mCT [START_REF] Rausch | Performance evaluation of the Biograph mCT Flow PET/CT system according to the NEMA NU2-2012 standard[END_REF] and the GE Discovery MI (5 ring version) [START_REF] Pan | Performance evaluation of the 5-Ring GE Discovery MI PET/CT system using the national electrical manufacturers association NU 2-2012 Standard[END_REF], respectively.

The liver, which is the organ of reference in many circumstances (Deauville score, and marker of quality for PERCIST, where absolute and relative variation in liver uptake have to remain below 0.3 units and 20%, respectively [START_REF] Wahl | From RECIST to PERCIST: Evolving Considerations for PET response criteria in solid tumors[END_REF]) was used, together with the vascular background in the aorta and muscle. Concordant results were obtained when comparing the level of noise in these three reference organs amongst simulated short acquisitions, provided that the shortest acquisitions were improved by decreasing the number of iterations and disabling PSF modeling. TOF was enabled for all reconstructions, as it has been shown that the improved TOF capabilities of modern digital PET systems leads to a dramatically enhanced image quality, as compared with analog PET, which may prove particularly useful for PET exams performed with reduced injected activities and/or recording times [START_REF] Salvadori | Head-tohead comparison between digital and analog PET of human and phantom images when optimized for maximizing the signal-to-noise ratio from small lesions[END_REF].

Based on noise in reference organs, we studied intra-and inter-observer analysis amongst the reconstructions deemed to be best suitable for fast imaging, i.e 1i10s20sec, 1i10s30sec, and 2i10sPSF60sec. The 30s per bed position reconstruction appeared to be suitable for clinical practice, with intra observer agreement of 0.79 and 0.90 for lesion and anatomical sites number and a mean image quality comparable to that of the standard reconstruction (2.6 +/-0.5 versus 2.7 +/-0.5, P ns).

Of note, while we focused on adapted reconstructions displaying a noise level comparable to a standard reconstruction, one could decide to perform ultra-fast PET acquisitions despite an increase in noise level. As shown in Figure 1, an acquisition with 10 s per bed position would visually lead to a significant level of noise in the liver but still allows visualization of the tumour. This increase in noise level in an organ frequently used as a reference (for example for PERCIST [START_REF] Pinker | Evaluating tumor response with FDG PET: updates on PERCIST, comparison with EORTC criteria and clues to future developments[END_REF], Deauville scoring [START_REF] Nanni | Report of the 6th International Workshop on PET in lymphoma[END_REF]) and being part of the metastatic spread of many cancer types [START_REF] Viadana | The metastatic spread of cancers of the digestive system in man[END_REF] was 73% on average when comparing the 1i10s10sec and 1i10s30sec reconstructions (Figure 2).

The relatively small number of patients included in this study could be regarded as a limitation of it, but in fact reflects the prevalence of patients potentially requiring fast PET imaging (3.4%) at our institution. This figure is related to the population scanned, in our case 90% of the PET scans performed in our department are oncology-related [START_REF] Lasnon | Patient's weight: a neglected cause of variability in SUV measurements? A survey from an EARL accredited PET centre in 513 patients[END_REF]. Secondly, we arbitrarily decided to increase the injected activity. One might assume that a clinical situation requiring fast imaging may occur after the 18 F-FDG has been injected. Data regarding this situation are provided as supplementary file and show similar results for noise level but tumour to background ratio smaller than 2 for the less obvious lesions, requiring further studies before being applied.

Our data could be useful to other centres facing the situation of patients experiencing pain, dyspnea or any other disease requiring a decreased acquisition time. In our experience, from a practical point of view, fast scanning requires a sufficient number of technologists and physicians to position the patient on the table of the PET scanner and remove him/her promptly and efficiently (with the appropriate equipment in the case of bedridden patients) so that the time spared during acquisition (and the gain in patient' comfort) is not lost during these phases. Median estimated total PET acquisition time for the 1i10s30sec was 4 min, compared to 12 min for the standard reconstruction.

It is noteworthy that PET systems with very long axial field-of-view such as total body PET [START_REF] Cherry | Total-Body PET: Maximizing Sensitivity to Create New Opportunities for Clinical Research and Patient Care[END_REF][START_REF] Badawi | First Human Imaging Studies with the EXPLORER Total-Body PET Scanner[END_REF] would obviously allow ultra-fast imaging but this new technology is unlikely to be widely available given its cost. Perspectives of fast PET imaging using modern PET systems will require adaptation of the reconstruction parameters for a given system, and are likely to be impacted by evolution of reconstruction capabilities, especially implementation of deep learning reconstruction [START_REF] Cui | PET image denoising using unsupervised deep learning[END_REF][START_REF] Lu | An investigation of quantitative accuracy for deep learning based denoising in oncological PET[END_REF][START_REF] Wang | Image Reconstruction is a New Frontier of Machine Learning[END_REF], enabling handling of noise level at low-count statistics.

Conclusion

Acquisition time per bed on the Vereos system can be reduced down to 30s without significant impact on quantitative and visual image quality and with preservation of a good detectability as compared to the standard reconstruction. These fast acquisitions require optimization of reconstruction parameters.

the clinical data; BH collected the clinical data; TS collected the clinical data. All authors read and approved the final version of the manuscript.

Figure 1 :

 1 Figure 1: Illustration of the multiple reconstructions performed to simulate fast acquisition

Figure 2 :

 2 Figure 2: Noise in reference organs (a: aorta, b: liver, c: muscle) expressed as the coefficient

Figure 3 :

 3 Figure 3: Lesion to background ratio using a doughnut VOI for background definition. Fast

Figure 4

 4 Figure 4 Comparison of the number of image quality (a), anatomical sites (b) and number of

Figure 5 :

 5 Figure 5: 63-year-old female patient with severe pain due to a myeloma recurrence. A

Figure 6 :

 6 Figure 6: 62-year-old female lymphoma patient with bulky mediastinal mass and superior

Table 1 : Patient characteristics (n = 28)

 1

	Clinical indication
	Dyspnea (n = 8, 28%)
	Pain (n = 11, 40%)
	Pediatric patients (n = 9,
	32%)
	Gender -(15M/13F)
	Age (years)*:
	Adult patients: 64.8 (12.9)
	Paediatric patients: 7.5 (4.4)
	Injected activity (MBq/kg)* :
	Adult patients: 5 (0.7)
	Paediatric patients: 5.8 (0.4)
	Delay (min)*: 63 (6,67)
	Glycaemia (g/l)*: 1,03 (0,27)
	* Mean (SD)

Table 2 :

 2 Intra-observer agreement evaluated for a PET reader who interpreted all reconstructions, starting with the supposedly worst reconstruction (in terms of noise level) and increasing the time per bed position up to the standard acquisition time.

	< 0.2: poor agreement, 0.2 -0.4: fair agreement, 0.41 -0.6: moderate agreement, 0.61 -0.8:
	good agreement, 0.8 -1: very good

Table 3 :

 3 Inter-observer agreement: interpretation was compared between one reader who read the four reconstructions, and four other observers who interpreted one reconstruction each.

	< 0.2: poor agreement, 0.2 -0.4: fair agreement, 0.41 -0.6: moderate agreement, 0.61 -0.8:
	good agreement, 0.8 -1: very good

Availability of data and material:

The datasets used and/or analysed during the current study are available from the corresponding author upon reasonable request.

Declarations

Ethics approval and consent to participate and consent for publication: All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Approval to collect data for our study was validated by the national committee for data privacy, the National Commission on Information technology and Liberty (CNIL), with registration n° 2204611 v 0 and patients' consent to use anonymised data for research purposes was sought on an opt-out basis.

Consent of publication: Not applicable

Competing interests: The authors declare that they have no competing interests Funding: The authors declare that they had no funding Authors' contribution: NA designed the study and wrote the manuscript; NC performed the PET/CT images analysis CL collected the clinical data, helped to design the study, performed the statistical analysis and wrote the manuscript; CN collected the clinical data; BE collected


Omar Behar

Benjamin Grange

Gilles Flamant
email: gilles.flamant@promes.cnrs.fr

combined cycle solar power plant using the fluidized particle solar receiver technology

Keywords: solar gas turbine, solar combined cycle, particles receiver, thermal energy storage, particle heat exchanger, performance modeling

Introduction

The development of the Solarized Gas Turbine (SGT) and the associated combined cycles has attracted a lot of interest in recent years because it is one of the ways (with supercritical cycles) to improve solar to electric efficiency.

Initially, very small (micro) SGT have been developed as an alternative to Stirling engines integrated to solar dish collector using turbocharger technology [START_REF] Six | Solar Brayton engine/alternator set[END_REF][START_REF] Mazzonib | A simulation tool for concentrated solar power based on micro gas turbine engines[END_REF]. Next, a concept with dedicated GT has been developed within the framework of the OMSOP project for producing an efficient, reliable, and cost-effective system that can generate about 3 -25 kW [START_REF] Aichmayer | Scaling effects of a novel solar receiver for a micro gas-turbine based solar dish system[END_REF][START_REF] Omsop | Optimised Microturbine Solar Power system): Final Report Summary[END_REF]. However, solar dish technologies have not been fully deployed to the market yet [START_REF] Coventry | Dish systems for CSP[END_REF]. This is due to technical issues of scale-up, lack of thermal energy storage solution, and high investment costs associated with solar dish technology. Thus, R&D activities have been reoriented to the improvement of central receiver technology. The latter is easier to scale up and already reached sizes above 100 MW. Besides, it promises to be one of the most viable renewable options to replace fossil fuel power generation technologies in the near future [START_REF] Behar | A review of studies on central receiver solar thermal power plants[END_REF].

Indeed, the integration of the GT cycle to the solar central receiver system was demonstrated at both small and pre-commercial scales. Research projects including SOLGATE [START_REF] Heller | Test and evaluation of a solar powered gas turbine system[END_REF][START_REF] Schwarzbozl | Solar gas turbine systems: Design, cost and perspectives[END_REF][START_REF]SOLGATE solar hybrid gas turbine electric power system[END_REF], SOLHYCO [10], and SOLUGAS [START_REF] Quero | Solugas -Operation experience of the first solar hybrid gas turbine system at MW scale[END_REF][START_REF] Behar | A novel hybrid solar gas turbine[END_REF] have tested and proved the concept of SGT. Heller et al. [START_REF] Heller | Test and evaluation of a solar powered gas turbine system[END_REF][START_REF] Schwarzbozl | Solar gas turbine systems: Design, cost and perspectives[END_REF][START_REF]SOLGATE solar hybrid gas turbine electric power system[END_REF] have reported the results of the SOLGATE project, which aimed to investigate the feasibility of operating SGT at the hundred-kilowatt scale. During this project, a 250 kW gas turbine was powered using three solar air receivers, connected in series, to heat the compressed air up to 800-1000°C. The SOLGATE SGT was operated for 135 h including about 96 h of on-sun tests. SOLHYCO project aimed at the development and demonstration of a SGT-cogeneration system able to operate in both solar and hybrid modes. A 100 kW GT has successfully tested during a period of more than 165 hours. The main objective of SOLUGAS was the qualification and validation of the first SGT at the pre-commercial scale (megawatt-scale). Thus, a 4.6 MW gas turbine has been modified to integrate a pressurized air receiver. The compressed air has been extracted from the compressor to be preheated by a 3.2

MWth solar air receiver up to 800°C [START_REF] Quero | Solugas -Operation experience of the first solar hybrid gas turbine system at MW scale[END_REF][START_REF] Behar | A novel hybrid solar gas turbine[END_REF]. The compressed air is then injected into the combustion chamber to reach the turbine inlet temperature (TIT). The SOLUGAS SGT was operated during one year under a wide range of conditions, including a standard working regime for commercial plants [START_REF] Quero | Solugas -Operation experience of the first solar hybrid gas turbine system at MW scale[END_REF].

The cited projects have successively proved the SGT with pressurized air receiver at both experimental and pre-commercial scales. Following these projects, various studies have further investigated the performance of the SGT. Behar [START_REF] Behar | A novel hybrid solar gas turbine[END_REF] proposed to preheat the compressed air of the SGT using parabolic trough technology. The simulation showed that a solar to electric efficiency of 17% is achievable. Bellos et al. [START_REF] Bellos | Parametric analysis and optimization of a solar assisted gas turbine[END_REF] used a multi-objective approach to optimize the SGT with a parabolic trough technology. The analysis showed that the optimum design leads to 64% fuel savings but with 2.8% penalty on the produced electricity. Spelling [START_REF] Spelling | Hybrid Solar Gas-Turbine Power Plants -A Thermoeconomic Analysis[END_REF] used a multi-objective optimization approach to evaluate the thermoeconomic performance of several configurations of the SGT. The author concluded that the integration of thermal energy storage and the use of combined-cycle configurations are mandatory to overcome the limitations of the simple-cycle SGT. Grange et al. [START_REF] Grange | Impact of thermal energy storage integration on the performance of a hybrid solar gas-turbine power plant[END_REF] investigated the impact of packed bed energy storage on the performance of SGT with pressurized air receiver. Simulations showed that the use of a packed bed heat storage allows increasing the solar fraction by 10-15%. Sánchez-Orgaz et al. [START_REF] Sánchez-Orgaz | Recuperative solar-driven multi-step gas turbine power plants[END_REF] investigated the impact of the recuperator effectiveness in a multi-step SGT. The authors pointed out that for the optimum design the increase in the effectiveness of the recuperator is always associated with an increase in the overall efficiency of the SGT. Santos et al. [START_REF] Santos | Seasonal thermodynamic prediction of the performance of a hybrid solar gas-turbine power plant[END_REF] developed a modular thermodynamic model for predicting the performance of the SGT. Based on the model developed by previous authors, Merchán et al. [START_REF] Merchán | Modeling hybrid solar gas-turbine power plants: Thermodynamic projection of annual performance and emissions[END_REF] have evaluated the annual performance of several configurations and operation modes of a SGT with pressurized air receiver. The authors analyzed the margin for improvement for each component and concluded that the Brayton cycle is the key element to improve the overall efficiency. Petrakopoulou et al. [19] have investigated the integration of a SGT with a high-temperature air volumetric receiver.

The results showed that the significant amount of solar energy required to increase the inlet temperature of the combustion chamber to 800-1000 °C causes a relative decrease in the energy efficiency of the SGT.

The previous scientific literature review highlights the limitations of the use of pressurized air receiver including low thermal efficiency, low allowable solar heat flux, and limited working temperature.. Besides, the use of air as a heat transfer fluid (HTF) is associated with unsolved issue about thermal energy storage, which results in limited capacity factor and solar share and daily startup and shut-down of the SGT. This induces a significant increase in the O&M costs and cost of the electricity produced.

The use of particles as a HTF was introduced to improve the thermal efficiency of the solar receiver, to increase the working temperature (allowing high allowable heat flux), and to integrate efficient and cost-competitive thermal energy storage. Up to now, there are three promising concepts of solar particle receivers. The first one is the so-called falling particle receiver, which is developed by Sandia Laboratory (SNL) [20]. The second concept is the centrifugal receiver, which is developed by the German aerospace center (DLR) [START_REF] Wu | Proof of concept test of a centrifugal particle receiver[END_REF]. The third one, known as particles-in-tube (or dense suspension of particles) solar receiver, has been developed by CNRS within the framework of the CSP2 [START_REF]CSP2: Concentrated Solar Power in Particles[END_REF] and Next-CSP [START_REF]Next-CSP: High Temperature concentrated solar thermal power plant with particle receiver and direct thermal storage[END_REF] European projects. This concept relies on a simple-proven design [START_REF] Flamant | Dense suspension of solid particles as a new heat transfer fluid for concentrated solar thermal plants: On-sun proof of concept[END_REF][START_REF] Gal | Thermal analysis of fluidized particle flows in a finned tube solar receiver[END_REF]. The main advantages are the use of a tube-receiver similar to the standard receiver of solar power tower and small diameter particles that exhibit high wall-to-fluidized bed heat transfer [START_REF] Zhang | Particle circulation loops in solar energy capture and storage: gas-solid flow and heat transfer considerations[END_REF].

The review of the previously published studies showed that there are few published works on the SGT with fluidized particle receiver. Farsi and Dincer [START_REF] Farsi | Thermodynamic assessment of a hybrid particle-based concentrated solar power plant using fluidized bed heat exchanger[END_REF] have conducted a thermodynamic analysis of a solar power plant with a fluidized particle receiver. They concluded that the integration of the particle solar receiver with direct energy storage in the Rankine and Brayton cycles enables to achieve the energy and the exergy efficiencies of 50% and 53.8% respectively. Kang et al. [START_REF] Kang | Energy analysis of a particle suspension solar combined cycle power plant[END_REF] have carried out the energy analysis of a hybrid solar combined cycle with particles-in-tube solar receiver and direct energy storage. The authors indicated that the overall efficiency of the thermal conversion (without the optical part) might achieve 48%. An attempt to evaluate the LCOE (Levelized Cost Of Electricity) of modular solar tower plants using solid particles as Heat Transfer Media (HTM), allowing HTM temperatures up to 1000°C is proposed in [START_REF] Buck | Solar tower system temperature range optimization for reduced LCOE[END_REF]. A LCOE of about 6 c€/kWh is found for high insolation site like in Chile. The previous cited works proposed a simplified analysis of some parts of the SGT and did not account for the actual performance of the SGT at part-load.

Besides, the practical design of the solar receiver and the heliostat field for a given capacity of the GT is not investigated. The design of some comopents such as the multi-stages fluidized bed heat exchanger and the steam cycle were arbitrary selected rather than designed based on practical approach. This study proposed a complete approach that fills the gap between purely academic approaches and detailed engineering developments providing a realistic design tool that integrates comprehensive modeling of each component. Moreover, from the academic point of view, this work establishes a methodology for the modeling of complex solar thermal energy conversion systems that integrate components modeled on the basis of optical, thermal science, and thermodynamic concepts.

Reliable simulation tools are needed to predict the performance of large-scale solarized gas turbine cycles with a particles-in-tube solar receiver and direct storage of particles. Therefore providing a flexible approach necessary to upscale the technology to a commercial level,.

The present work aims at designing and evaluating the performance of a solar gas turbine combined cycle with a fluidized particles-in-tube receiver and direct thermal storage using practical modeling techniques. We propose the design of a representative medium-scale module using an on-the-shelf gas turbine. Section 2 describes the modeling approaches while section 3 provides the design and the validation of each component of the system. Section 4 and 5 present the performance evaluation during typical days and under different operation strategies. The most important conclusions are summarized in the last section.

Modeling the components of the system

Description of the system

Figure 1 shows a typical solar gas turbine combined cycle with a fluidized particles-in-tube receiver and direct thermal storage. It consists of four main sub-systems:

• The heliostat field,

• The solar particle receiver,

• The two-Tank particle storage and fluidized particles/pressurized air heat exchanger,

• The combined-cycle integrating a modified simple cycle GT and a Rankine steam cycle.

During sunny periods, the particles flow from the cold tank toward the receiver where they are heated up in tubes. The particles circulate upward inside the tubes in the fluidized state (fluidization with air). Then they are stored in the hot tank. When power production is needed, they are introduced in a fluidized bed heat exchanger to heat the compressed air of the GT.

The GT can work in two modes: hybrid solar-fuel mode and solar-only mode. In solar-only mode, the compressed air is heated into the FB-HEX and then passes directly in the turbine section where it expands. In the case of hybrid mode, the compressed air is preheated in the FB-HEX and flows in the combustion chamber to be heated up to reach the turbine inlet temperature (TIT). The exhaust gases of the GT passes through a Heat Recovery Steam Generator (HRSG) to produce superheated steam for the Rankine cycle. The complete design started with the solar gas turbine (SGT) since it defines the necessary power to run it. Then, the other parts have been designed upstream. However, to make this section easier for the readers, it is organized into three main sub-sections starting from solar energy collection to power production.

• Modeling of the solar field and the particles-in-tube solar receiver,

• Modeling of the fluidized bed heat exchanger,

• Modeling of the simple gas turbine cycle.

The model of the steam cycle is given in Appendix A.

The main assumptions used in the modeling are as follows:

-Each component is at steady state during each time-step; -The start-up and shut-down losses are ignored; -The temperature in the cold and hot tanks is uniform as well as in each stage of the fluidized bed heat exchanger; -The absorber surface of the solar cavity receiver is flat, tube shape is not taken into account;

-The tower shadow is neglected.

Modeling of the heliostat field and the particle-in-tube solar receiver

The models of the solar field and the receiver are presented together because they are strongly linked. The solar field is designed using SolarPILOT, a software from NREL [START_REF]Solar Power Tower Integrated Layout and Optimization Tool[END_REF], with a design power, depending on the desired particle outlet temperature (750°C or 880°C).

The cavity receiver consists of absorber tubes at the back positioned on a curved line when looking at the top view. The tube number and geometry are detailed in section 3.3. The cavity is made of five refractory panels and the aperture has a 5.5 m height and a 6 m width, with an angle of 20° relative to the vertical plane.

To avoid any deterioration of the receiver due to uneven flux distribution leading to thermal hot spots and unacceptable stresses [START_REF] Grobler | A review of aimaing strategies for central receiver[END_REF], an aiming point strategy is applied using the heliostats location optimized in SolarPILOT. This strategy is based on the TABU search [32 combined with the convolution-projection method Unizar [START_REF] Collado | An analytic function for the flux density due to sunlight reflected from a heliostat[END_REF] and is applied at the aperture of the receiver to spread the flux density on the tubes and reduce the maximum flux density.

The heliostat locations, their aiming point, and the receiver geometry are introduced as input in the ray-tracing software Solstice [START_REF] Solstice | SOLar Simulation Tool In ConcEntrating optics[END_REF] to get the flux distribution on the absorber tubes.

The flux map on each receiver surface (cavity and absorber) is introduced into a thermal model based on the net radiation method [START_REF] Grange | Thermal Performances of a High Temperature Air Solar Absorber Based on Compact Heat Exchange Technology[END_REF]. This method relies on the radiative balances in the infrared spectrum.

= . . + . ∑ . (1)
With Fij the view factor from the surface i to the surface j.

The radiative balance in the solar spectrum is achieved by the ray-tracing SOLSTICE software. The net power absorbed by the element i is given by,

= . + ∑ . - (2)
With the solar incident flux density after the radiative balance in the solar spectrum (W/m²).

The surface heat transfer from the front wall to the particles is calculated by the following equation,

= ℎ -! ,#$ (3)
With htr the heat transfer coefficient (W/m².K), Tfront the front wall temperature (K) and

Tpart,mean the mean particle temperature between the inlet and the outlet (K).

Small-fluidized particles, typically 50 μm mean diameter, are used as heat transfer fluid in the receiver tubes. The heat transfer coefficient, htr, was selected based on experimental results with a single tube achieved at the 1-MW solar furnace at the CNRS-PROMES laboratory [START_REF] Flamant | Dense suspension of solid particles as a new heat transfer fluid for concentrated solar thermal plants: On-sun proof of concept[END_REF][START_REF] Gal | Thermal analysis of fluidized particle flows in a finned tube solar receiver[END_REF].

The particle flow rate is adjusted to reach the desired particle outlet temperature. Simulations are carried out for different sun positions. Then, interpolation is used to find the appropriate particle temperature and flowrate for a given sun position.

Modeling of the FB-HEX

Typical FB-HEX is a multi-stages tube/shell heat exchanger, in which the compressed air of the GT flows into the tubes [START_REF] Gomez-Garcia | Design and performance of a multistage fluidized bed heat exchanger for particle-receiver solar power plants with storage[END_REF]. The fluidized particles in the shell-side flow from higher temperature stages to lower temperature stages. The energy balance on one stage of a FB-HEX can be expressed as,

! % $ = & ' (_ + * + , (_ (4)
With,

! % $ = -! % $ (ℎ ! % $ _ -ℎ ! % $ _ * ) (5)
& ' (_ = -& ' (_ (ℎ & ' (_ _ * -ℎ & ' (_ _ ) (6)
* + , (_ = -* + , (_ (ℎ * + , (_ _ * -ℎ * + , (_ _ )

where Q, m, and h are the energy, the mass flow rate and the enthalpy respectively, subscript "working_air" refers to the compressed air of the GT, "fluidizing_air" refers to the air used to fluidize the particles, "in" and "out" refer to the inlet and outlet of the HEX's stage respectively. The FB-HEX is supposed to be well-insulated and installed in-door so that in Equation ( 4) the heat losses to the ambient are neglected. The outlet temperature of the fluidizing air is supposed to be equal to that of the particles.

The heat gained (per unit of tube length) by the working air can be calculated by,

& ' (_ = 0 × 2 * $_ $ × 3 & ' (_ ( * $_ $ -& ' (_ _ 4$ ($ ) (8)
Dtube_inner is the inner diameter of the HEX's tubes and Uworking_air is the heat transfer coefficient of the working air. It is calculated using Gnielinski correlation [START_REF] Çengel | Fluid Mechanics Fundamentals and Applications[END_REF].

56 & ' (_ = (7 89:;<:=> /@)×( $ABCCC)×D BEB@.FG(7 89:;<:=> /@)HD

I JABK ( D LM <NOP ) C.BB (9)
where Nu is Nusselt umber, Ffriction is the friction coefficient, Re is Reynolds number, and Pr is Prandtl number.

Similarly, the heat released by the particles can be given as,

! % $ = 0 × 2 * $_ * $ × 3 ! % $ ( ! % $ _ 4$ ($ -* $_ * $ ) (10)
Where: Dtube_outer is the outer diameter of the HEX's tubes and Uparticles is the heat transfer coefficient between the particles and the tubes (a value of 800 W/m 2 .K is considered in this study), and Tparticles_average is the mean temperature of the particles.

The tube outer surface temperature (Ttube_outer) is calculated based on energy conservation on the tubes of the HEX. The conduction heat transfer, per unit of length, from the tube outer surface to the tube inner surface is,

% +*% _ * $ = 20 × R * $ (S <NOP_:>>P9 AS <NOP_=N<P9 ) TU ( W <NOP_=N<P9 W <NOP_:>>P9 ) (11)
The tube inner surface temperature is calculated by combining equations ( 6) and ( 11) while the working fluid outlet temperature is calculated by combining Equations ( 6) and ( 8). The particle outlet temperature is calculated by combining Equations ( 6) and (10).

The above model is used to predict the performance of one stage of the FB-HEX. Combining N stages in series results in a multi-stages FB-HEX.

Modeling the gas turbine cycle

The typical GT cycle consists of a compressor, a combustion chamber, and a turbine section.

A practical technique, based on the performance maps of the turbine and compressor sections, has been used to model the GT cycle. This approach enables to predict the turbine performance for part-load operations, based on similarity and dimensional analyses, and is widely used in turbomachinery [START_REF] Dixon | Fluid Mechanics and Thermodynamics of Turbomachinery[END_REF]. Indeed, the maps are graphical representations of the performance of the components (compressor and turbine sections) for a given set of operating conditions [START_REF] Semprini | Performance analysis of a micro gas turbine and solar dish integrated system under different solar-only and hybrid operating conditions[END_REF], which are themselves expressed in terms of certain corrected variables with respect to reference inlet pressure and temperature (15°C, 1 atm.). For the compressor, the maps provide the compressor ratio and isentropic efficiency as a function of the corrected mass flow and corrected shaft speed. For the turbine section, the maps provide isentropic efficiency and expansion ratio as a function of the same parameters. Figure 2 represents the components' maps of a typical gas turbine unit. Based on the inlet parameters of the compressor, the corrected shaft speed (5 % _X ) and corrected mass flow rate (-% _X ) are respectively given by,

5 % _X = Y 9=< Z _[ (12)
-% _X = - Z _[ \ _[ (13)
The index "C" refers to the compressor. θ_C and α_C are non-dimensional factors defined as,

] _X = ^S[_:> S _`W (14)
a _X = D [_:> D _`W (15)
The index "in" refers to the compressor inlet. PSTD and TSTD are respectively the pressure and temperature under standard conditions.

The compression ratio and isentropic efficiency of the compressor are functions of the corrected mass flow rate and the corrected shaft speed.

b X = c(5 % _ [ , -% _ [ ) (16
)

η e X = c(5 % _[ , -% _[ ) (17)
The following relation provides the exit air temperature of the compressor for isentropic process.

T g_hij_e = T X_ (b X ) k [ lm k [ (18)
In order to calculate the actual outlet temperature of the compressor, the following definition of isentropic efficiency is used.

η e g = n o_pqr_s An o_tu n o_pqr An o_tu (19)
The same methodology is developed to model the turbine section. To match the compressor and turbine characteristics, the actual turbine mass flow is compared to the compressor mass flow and the fuel mass flow. An iteration process on the compressor mass flow is applied to find the operation point.

It is worth noting that a closed cooling system is considered, in which a fraction of compressed air is extracted for cooling the turbine blade. This cooling air is then mixed with the exhaust gases at the outlet of the turbine section. The modeling of the cooling system in this study is based on the model developed in [START_REF] Horlock | Advanced gas turbine cycles[END_REF].

The combustion process in the gas turbine involves the fuel (the hydrocarbon) and the oxidizer (air) called the reactants, which undergo a chemical process while releasing heat to form the products of combustion such that mass is conserved. Thus, the chemical reaction can be expressed as follows,

v w x y + z % * { | }IB A| }I~x @ • + €• @ + | •I | ~I 5 @ 'ƒ → 5 X… I v• @ + 5 †@… x @ • + 5 Y I 52 + 5 … I • @ (20)
actual is the actual air coefficient, χx and Nx represent the fractions and the number of moles respectively.

The combustion chamber outlet temperature is calculated based on the following equation,

XX_ * $ = { ‡ ˆp‰Štu‹_OEt‰ A ‡ •ppŽtu‹_OEt‰ g • [[_:>•P< ' [;_:> E ‡ 'q"Ž " •• -•• E ‡ 'q"Ž g •'q"Ž ' 'q"Ž ƒ [ ‡ ˆp‰Štu‹_OEt‰ A ‡ •ppŽtu‹_OEt‰ E ‡ 'q"Ž g • [[_=N<•P< ] (21)
mcooling_air is the mass flow rate of the air that is used to cool the turbine blade, subscript CC refers to the combustion chamber, mfuel and Tfuel are the mass flow rate and the temperature of the fuel respectively, η ™™ is the combustion efficiency, and Q ™› is the fuel calorific value.

Design of the components

The design of the components has been carried out as follows. First, an off-the-shelf gas turbine that is able to work in both hybrid and solar-only modes was selected. Next, the fluidized bed heat exchanger was designed with appropriate temperature-approach. The number of stages of the FB-HEX was determined based on the temperature approach and with respect to the design cold temperature of the particles (the temperature at the outlet of the FB-HEX). The receiver-loop, comprising the receiver, the storage, and the heliostat field, was then sized to meet the heat demand of the GT cycle. Finally, the steam cycle was designed based on the exhaust gases' parameters of the GT cycle. A flowchart showing the main design steps of the complete system is given in Appendix A.

The system is supposed to be installed in Ouarzazate Province in the Drâa-Tafilalet region of south-central Morocco, at an elevation of 1100 m, in the middle of a bare plateau south of the High Atlas Mountains. The coordinates of the site are 31'N and 6.86'W. During the night, an ambient temperature of 15°C and a relative humidity of 60% are considered.

Design data and validation of the GT model

Table 1 illustrates the design data of the selected gas turbine. Siemens GT-500 17 MW-gas turbine [START_REF]SGT-500 Industrial Gas Turbine[END_REF] was chosen because it is suitable and available for the first commercial SGT with a fluidized particles-in-tube solar receiver. Low TIT and high power/weight ratio are the most important advantages of GT-500 with respect to the solar application. The former leads to optimum performance when the GT is working in solar-only mode. It is worth noting that components' performance maps are subject to confidentiality because it is the core of the GT design. Thus, the maps have been up-scaled and adapted based on those published in the literature. The practical technique used to model the GT is driven by the actual operations of the GT. The technique leads to find the operation point of the gas turbine as a function of ambient conditions and turbine load. To validate the GT model, the predicted results are compared with the rig-tests published performance [START_REF]SGT-500 Industrial Gas Turbine[END_REF].

Figure 3 illustrates a comparison between the predicted performance and Siemens' rig-tests data, for the case of the heat rate and the electricity generation, as a function of compressor inlet temperature. Good agreement has been observed thanks to components' maps, which consider the effect of ambient temperature on the compressor/expansion ratios as well as on the isentropic efficiencies of the compressor and turbine sections. As shown in Table 2, there is a minor difference between predicted performance and Siemens' rig-tests data. For the generated power, the Mean Absolute Deviation (MAD) is about 0.44 MW, which corresponds to difference of 3.4%. The deviation in the prediction of the heat rate is also negligible. The percentage is 3.3%, which corresponds to a MAD of 418 kJ/kWh. Accurate results have also been obtained for the case of the mass flow rate and the temperature of the exhaust gases. For the former, the mean deviation is 3.46% and for the latter, it is less than 1.55 %. The potential of the GT model to predict the part-load performance was examined because it is of particular interest for optimum planning of the GT operations. Figure 4 illustrates a comparison between the predicted performance and Siemens rig-tests data, for a wide range of turbine load, from 30% to 100%. For instance, at 30% load, rig-tests efficiency is 20.27% and the corresponding predicted value is 20.32%. The accurate predictions are obtained thanks to the practical technique adopted to model the gas turbine. Indeed, the model takes into account the shaft speed. For each given load and taking into account the ambient conditions, the model finds the appropriate shaft speed and matches the compressor and the turbine sections to produce the desired power. Thus, the MAD for the part-load performance is minor. It is 4.76% for the heat rate and only 3.74% for the efficiency.

These data confirm the potential of the model to predict both the full-load and the part-load performance of the gas turbine. The slight deviation of the gas turbine model is strongly related to unknown specific design details of the Siemens GT-500 that are not available. The uncertainties in the correlations used to estimate the thermo-physical proprieties of gases have a second-order impact.

Design of the FB-HEX

To design the FB-HEX, the inlet parameters of the particles and the compressed air must be defined first. The nominal inlet parameters of the compressed air correspond to the nominal outlet parameters of the compressor. For the hybrid case, particles at a nominal inlet temperature of 750°C is considered, because it is the state-of-the-art based on the past experimental results [START_REF] Benoit | On-sun demonstration of a 750°C heat transfer fluid for concentrating solar systems: dense particle suspension in tube[END_REF]. In the solar-only design, the nominal inlet temperature of the particles is supposed to be 880°C. This value is selected based on the TIT and the temperature approach of the FB-HEX.

The FB-HEX is then designed and the number of stages is determined based on a temperature approach of 30°C. The particle temperature at the outlet of the FB-HEX is fixed at 400°C to keep the particles working within the technical specifications (manufacturer technical data).

A sensitivity analysis through varying the number of stages is carried out to find the appropriate number of stages that meet the specifications cited above. Figure 5 indicates that the appropriate number of stages for the hybrid design is 15 stages while it is 19 stages for solar-only design. The effect of the number of stages on the outlet temperatures of the compressed air is presented in Figure 5 to provide a comprehensive analysis. It is apparent that the higher the number of stages, the higher the outlet temperature of the compressed air and the lower the outlet temperature of the particles because the increase in the number of stages increases the heat transfer area. Nevertheless, the higher the number of stages, the lower is the temperature increase in the high-temperature stages due to the decrease in the heat transfer coefficient (HTC). Besides, the difference in the HTC between the first stage and the last stage for the case of solar-only is 21%. Consequently, 15-stages FB-HEX provides an appropriate temperature approach (27°C) for the solar-only case. Table 3 illustrates the design data of the FB-HEX for the hybrid and solar-only GTs. The difference between the compressor mass flow rate in the hybrid case and the solar-only case is related to the pressure drop. For solar-only, the combustion chamber is by-passed (100% solar-powered), which reduces the pressure drop and consequently the mass flow rate increases. The pressure drop in the FB-HEX (pressurized-air side) is higher in the case of solar-only GT than in the case of hybrid GT because of the increase in the operating temperature. Nominal pressure drop (pressurized air) 188.4 271.0 mbar

Design of the heliostat field and the solar receiver

After the design of the FB-HEX, the solar field's components have been designed including tower, receiver, and heliostats. The tower height is set to 120 m and the heliostats have a reflective surface of 49 m² (7 m x 7 m) with a reflectance of 92% and a slope error of 0.9 mrad (high optical quality). The total reflective surface of the heliostat field is 77 518 m² and 126 812 m² for the hybrid and solar-only GT, respectively. In the latter case, the heliostat field is larger because a constant particle mass flow rate for both cases is considered. The third case (called "Solar-only GT -Low flowrate" in this paper) consists of keeping the solar field of the hybrid case and decreasing the particle mass flow rate to reach a particle outlet temperature of 880°C. The three considered cases have the same tower height and receiver geometry. Table 4 illustrates the main design data of the three considered cases. The layouts of the heliostat field for the hybrid and solar-only GTs are presented in Figure 6.

The layout of the heliostat field of the third case is similar to that of hybrid GT. The wide shape of the heliostat field is voluntary due to the geometry of the receiver shown in The solar receiver consists of 382 tubes of 7 meters height and with an outer diameter of 50 mm (black surface in Figure 7). With a nominal mass flowrate in the receiver of 90 kg/s and assuming a uniform distribution of particle in the tubes, it leads to a mass flow rate per unit surface area per tube of around 60 kg/m².s. This value lies in the range experimentally validated in [START_REF] Merchán | Modeling hybrid solar gas-turbine power plants: Thermodynamic projection of annual performance and emissions[END_REF]. The cavity is made of five refractory panels. The optical properties of the tubes and the panels are given in Table 5. The tradeoff made on the aperture between the spillage loss and the thermal efficiency leads, at solar noon, to an optical efficiency of 67.4% (12.9% of the optical losses come from the spillage) and a thermal efficiency of 83.2% for the hybrid GT. For the solar-only GT, the optical efficiency drops down to 57.6% with 20.8% of spillage loss, and the thermal efficiency is 80.2%. The optical efficiency of the third case follows the same trend as the hybrid case. The thermal efficiency is maintained around 80% during the day. At nominal conditions, the power balance is given in Figure 12. Table 6 shows the amount of hot particles collected during the vernal equinox and the winter/summer solstices and the equivalent full-load operation duration for the three cases given in Table 4. The amounts collected during summer solstice corresponds to the maximum storage capacity of the plant. Compared to the vernal equinox, there are around 15% less particles collected at the winter solstice and around 3% more particles at the summer solstice. Simulations are then carried out for different sun positions during the year, the 21 st of each month. Simulations before 8 AM and after 4 PM are performed if the DNI (average hourly DNI over the month) is higher than 150 W/m² (below this value the particle mass flow rate is too low to gain power from the wall). Between 8 AM and 4 PM, simulations are achieved every two hours with an average hourly DNI throughout the year. Then, the results are interpolated as a function of the sun azimuth and elevation. Figure 13 shows the interpolation of the particle flowrate for three cases as a function of the sun position. Similar to Figure 9, the amount of hot particles collected at any sun position is quite similar for the first two cases. For the third case, the amount of hot particles collected corresponds to approximately 70% of the amount collected in the first two cases.

Performance evaluation of the complete power plant

The previous sections described the design and performance of the system's components including GT, FB-HEX, solar receiver, and solar field. For the steam cycle, the design and performance are given in Appendix B. The present section describes the performance of the complete power plant (illustrated in Figure 1). A flowchart showing the main steps of the methodology used to calculate the performance of the complete system during typical period is given in Appendix A. The performance metrics used to evaluate the plant are the power output, the efficiency, the solar share, the specific fuel consumption, and the capacity factor.

Two operation strategies are considered: constant power production after 18h and constant power production on demand. This means that the solar plant is operated as a peaker.

Figure 14 shows the power production and the thermal efficiency of the thermodynamic cycles during 21-22 March, for the three cases (hybrid, solar-only, and solar-only lowflowrate), operating in a constant power production strategy after 18 pm, at the site located in Ouarzazate. The GT cycle produces about 14.7 MW instead of 17 MW (output under STD conditions) because of the high elevation of the site (1100 m). The operation duration for the case of solar-only low-flowrate is shorter than for the other two cases because of the low amount of the particles in the storage (see Table 6). The slight difference in the CC power production between the solar-only case and hybrid case is related to the air mass flow of the GT's compressor (see Table 3). Indeed, the compressor air mass flow of hybrid GT is slightly lower than that of solar-only GT (79.445 vs. 80.73 kg/s) because of the difference in the pressure drop. However, the decrease in the pressure drop pushes the operation point of solaronly GT far further from the nominal point, compared with that of the hybrid GT, which results in additional efficiency losses, as plotted in Figure 14b. The thermal efficiency of the GT cycle for the cases of hybrid and solar-only are 32.09 % and 32.43%, which corresponds to an output of 14.67 MW and 14.65 MW respectively. The steam turbine efficiency is the same for the three cases (34.82%) leading to a generation of 3.72 MW. Thus, the CC output reaches about 18.37 MW for the three cases with 46% efficiency. The overall efficiency (also known as solar-to-electric efficiency) is the product of the individual efficiencies of the heliostat field, the receiver, and the CC. The results illustrated, in green color, in Figure 14b corresponds to the product of the full-load efficiencies. The overall efficiency of the hybrid case is 25.80%. This value decreases to 24.70% for the case of solar-only low flowrate due to the slight decrease in the thermal efficiency of the solar receiver (from 83.2% for the hybrid case to 80%). For the solar-only case, the overall efficiency is 21.16%, which is 4.64 points lower than that of the hybrid case. This is mainly due to the low optical efficiency of the heliostat field, which drops from 67.4% (for the hybrid case) to 57.6% for the case of the solar-only. The decrease in the thermal efficiency of the receiver has a second-order impact (83.2% for the hybrid case vs. 80.2% for the solar-only case). The proposed solar power plant is modular because of its medium size. Moreover, several solar collection loops (the tower and the receiver) can be associated in parallele to increase the total power and the storage capacity of the plant. In this option, the nominal power of the combined cycle shared by the solar collection loops will increase proportionally.

Thanks to the thermal energy storage the system offers great flexibility for saving solar energy and generatind power when needed. Figure 16 shows the performance of each component for the case of solar-only operating to meet a constant power need during the periods of peak demand (18 pm-22 pm, then 6 am-8 am). The efficiency of the components (including heliostat field, receiver, and power conversion cycles) during 21-22 March is presented in Figure 16a. The power production and the state of the TES are given in Figure 16b. Flexible power production management is possible to meet a given objective. Particles can be stored during a day (8 am to 18 am) to be used when needed (18 pm-22 pm, then 6 am-8 am). Besides, the remaining energy in the storage can be saved to the next days, as shown in Figure 16b. The proposed concept is flexible enough to store energy for the next days. For instance, particles can be stored to meet the peak demand of a rainy day, which can be forecasted even one week before. Figure 17a shows the amount of electricity generated during the vernal equinox and the winter/summer solstices. The operation strategy is a constant power production after 18h. The operation duration for the case of solar-only low flowrate is short according to the amount of particles in the storage (see Table 5). Figure 17b illustrates the seasonal and annual capacity factor of the three cases. Solar-only and hybrid cases have the same storage capacity so that they have the same capacity factor. During summer, the capacity factor can reach 35.83%.

This value corresponds to the maximum capacity factor of the plant and it is 10 points higher than that of solar-only low flowrate case. Interestingly, the capacity factor during spring (autumn is similar to spring) is almost similar to that of summer for the three studied cases (the difference is less than 1%). The average capacity factor during winter is just 5 points lower than that of equinox. As a result, the annual capacity factor can reach 33.47% for hybrid and solar-only cases, and 23.75% for the case of solar-only low flowrate.

Off design performance of a solarized gas turbine

This section focuses on the part-load performance of a solarized gas turbine. Because of the similarity, the case of solar-only is considered as an example. The selected operation strategy consists of keeping the particles mass flow rate at its nominal value and reducing the power production of the GT. Consequently, the GT is subjected to the variation in the compressor mass flow rate associated with the variation of TIT. Figure 18 shows the performance of the GT as a function of the generated power (percentage of nominal load). The load of the GT is reduced from 100% to 75%. This is done by reducing the shaft speed of the GT to a value that corresponds to the desired amount of electricity produced. As Figure 18a demonstrates, a slight decrease in the shaft speed (from its nominal value) results in a significant loss in the GT performance. By reducing the GT load from 100% to 75%, the isentropic efficiency of the compressor and the turbine section is significantly reduced from its nominal value (90%, and 91.7% respectively) to 87%, which represents 5% losses. The mass flow rate of the compressor is the most affected parameter. It decreased from 80.79 kg/s, at the nominal point, to 68 kg/s, at 75% load, which represents a cut of 16%. This results in a reduction in the pressure at the compressor outlet. Therefore, the pressure drop in the FB-HEX declines (low mass flow rate results in low Reynolds number). As Figure 18b illustrates, the compressor outlet pressure decreases from 10.6 bar to 9.06 bar, which corresponds to a 14.5% loss. The diminution in the FB-HEX outlet pressure is 14.7%. As can be seen in Figure 18c, at 75% turbine load, the pressure drop is reduced from 271 mbar to 253 mbar (6.5% decrease).

At partial load, the outlet temperature of the compressor drops due to the decline in the outlet pressure. As shown in Figure 18d, the compressor outlet temperature is reduced from 341°C to 324°C. However, the TIT raises from 850°C to 876°C (but remains acceptable from the safety point of view) thanks to the selected operation strategy that kept the particles mass flow rate in the FB-HEX constant. Nevertheless, the FB-HEX is also affected by the partial load operations. When the turbine load declines from 100% to 75%, the temperature of the particles at the outlet of the FB-HEX raises from 399.38°C to 442°C, because of the low compressor air mass flow rate.

Another important point to mention is the turbine outlet temperature, which increases at partial load. This is due to the combined effect of low turbine inlet pressure (due to low compressor outlet pressure) and losses in the isentropic efficiency. The former results in a low expansion ratio while the latter induces additional entropy generation so that the turbine outlet temperature increases.

Table 7 summarizes the losses in the performance of each part of the solarized gas turbine. As can be noticed, all the parts are penalized at partial load. As a result, there are significant losses in the overall efficiency. When the turbine load reduces to 75%, the GT efficiency loses 6 points (from 32% to 26%), which represents a decrease of 18.4%. The analysis of the performance of the three studied cases (hybrid, solar-only, and solar-only low flowrate) shows that the performance of the solar part (heliostat field and receiver) has a critical influence on the overall performance of the solar power plant. In other words, the increase in the operating temperature of the receiver does not obviously improve the overall efficiency of the complete power plant. This fact can be observed when comparing the hybrid case with a solar-only low flowrate case (high operating temperature of the receiver reduces the overall efficiency by 4.26%). Besides, the high operating temperature of the receiver requires a large heliostat field (the case of solar-only), which is associated with a low optical efficiency (large heliostat field has decreased the overall efficiency from 25.80% to 21.16%, which represents a loss of 18%). Even though high operation temperature increases the efficiency of the thermodynamic cycle, it induces a decrease in the thermal efficiency of the solar receiver. On the other hand, the high operating temperature of the receiver requires a large solar field, which decreases the optical efficiency. The overall efficiency is the product of the three cited efficiencies and further studies should identify the optimum operating temperature of the receiver that maximizes the overall efficiency of the complete solar power plant.

Discussion

The proposed solar power plant offers significant improvements in the thermal efficiency with respect to the state of the art even using an off-the-shelf gas turbine. Furthermore, the designed solar part (solar field and solar receiver) can be considered as a representative module of larger plants that associate several modules sharing the same power block (heat exchanger and combined cycle). 100 MW and more solar power plants can be designed on this basis provided an additional study on particle conveying between the towers and the power block. An increase of cycle efficiency from 46% to 48.8% is expected for commercialscale solar power plants [START_REF] Valentin | Optimization of a Decoupled Combined Cycle Gas Turbine Integrated in a Particle Receiver Solar Power Plant[END_REF].

However, there is room for additional improvement using sCO2 cycles that enables 50% cycle efficiency and more with 750°C particle temperature and solar-only operation. For example, Rao et al. [START_REF] Rao | Multi-objective optimization of supercritical carbon dioxide recompression Brayton cycle considering printed circuit recuperator design[END_REF] proposed a multi-objective optimization of supercritical carbon dioxide recompression Brayton cycles that resulted in an efficiency ranging from 43 to 53.8%.

Further studies should quantify the heat losses associated with the particle conveying and the

Conclusion

The design and the performance of a medium-scale flexible solar power plant integrating a combined cycle with a fluidized particle-in-tube receiver and direct thermal storage are presented.

To provide practical technical information, a gas turbine with a low TIT has been selected to describe the performance of the solar combined cycle power plant. Besides, the components' performance maps are used to provide the actual performance of the power conversion cycles and a rigorous methodology is employed to design the solar receiver, the heliostat field, and the fluidized bed heat exchanger. Both full-load and part-load performance are evaluated and three cases are investigated: hybrid, solar-only, and solar-only low flowrate.

The evaluation of the performance under different operation strategies indicates that the solar power plant provides a great flexibility thanks to the direct energy storage. The analysis of the components and the overall plant efficiencies for typical days shows promising performance during summer, spring, and autumn leading to an annual capacity factor above 33%. The efficiency of the power block reaches 46%. The overall nominal efficiency (solar-to-electric efficiency) of the power plant showed a strong dependence on the optical efficiency of the solar field but a weak function of the receiver's efficiency. It reaches 25.80% for the hybrid case because the solar field is relatively small. For the case of the solar-only, which has a large solar field, the overall efficiency drops to 21.16%. Nevertheless, this latter value can be enhanced up to 24.7% by reducing the power of the collection loop.

This paper presents a general methodology for simulating fluidized particles-base solar thermal power plants that can be applied to the integration of different conversion cycles and be used to design large-scale commercial power plants composed of several modules sharing the same power block. Realistic overall performance calculations are provided on the basis of the use of an off-the-shelf gas turbine with known performance. It is shown that even in this case an increase of 10% relative cycle efficiency with respect to current technology is

attainable. An increase of 20 to 25% relative cycle efficiency is expected with advanced sCO2 cycles. With respect to overall plant performance, operating a solar receiver at high temperature may result in a decrease of its efficiency with respect to molten salt technology.

Nevertheless, this decrease can be limited to about 6% relative (from 90% to 85%, because an optimized design of the particle solar receiver may result in the increase of the thermal efficiency from 83% -Fig. 12-to 85%). Finally, a significant increase of overall solar power plant efficiency with respect to state-of-the-art central receiver technology is expected using the particle technology.

Figure 1 .

 1 Figure 1. Solar gas turbine combined cycle with particle-in-tube-receiver and direct thermal storage

Figure 2 .

 2 Figure 2. Performance maps of typical gas turbine unit: (a) compressor: compressor ratio and isentropic efficiency, (b) turbine: expansion ratio and isentropic efficiency [Adapted from ref. 38].

Figure 3 .

 3 Figure 3. Electrical output and heat rate as a function of compressor inlet temperature (lines are Siemens data, points are model predictions)

Figure 4 .

 4 Figure 4. Electrical output and efficiency as a function of turbine load (the line is Siemens data, points are model predictions)

Figure 5 .

 5 Figure 5. Effect of the number of stages on the performance of the FB-HEX.

Figure 6 .

 6 Figure 6. Layout of the heliostat field optimized in SolarPILOT with associated nominal optical efficiency for (a) Hybrid GT (1582 heliostats) and (b) Solar-only GT (2588 heliostats).

Figure 7 .Figure 7 .

 77 Figure 7. Indeed, Eastern and Western heliostats are required to reach the side absorber tubes.

Figure 8 Figure 8 .

 88 Figure 8 shows the results of the ray-tracing simulation with Solstice displayed in Paraview

Figure 9 .

 9 Figure 9. Particle mass flowrate during the 21 st of March for the three studied cases The particle mass flowrate follows the same evolution during the day for the hybrid GT and

Figure 10 .

 10 Figure 10. Intercepted power during the 21 st of March for the three studied cases

Figure 11 .

 11 Figure 11. Optical and thermal efficiency during the 21 st of March for both hybrid and solar GT The low thermal efficiency at 7 AM is due to a low particle mass flowrate because of a DNI

Figure 12 .

 12 Figure 12. Power balance on the solar receiver (hybrid case) the 21 st of March at noon (a) with optical loss (b) only thermal loss

Figure 13 .

 13 Figure 13. Particle mass flowrate versus sun position-(a) Hybrid GT, (b) Solar-only GT, (c) Solar-only GT -Low flowrate

Figure 14 .Figure 15 .

 1415 Figure 14. Power production and efficiency the power block during the 21 st -22 ed March operating with a constant power production after 18 pm Figure 15a shows the solar share for the three cases. The solar share of the two solar-only

Figure 16 .

 16 Figure 16. Efficiencies and solar share (a), power production and state-of-the-storage (b) of solar-only combined cycle operating on-demand

Figure 17 .

 17 Figure 17. Seasonal operation duration (left) and capacity factor (right) of the three studied cases

Figure 18 .

 18 Figure 18. Part load performance of solarized gas turbine (case of solar-only)

  parasitic power consumption due to fluidization air in both the solar receiver and the heat exchanger. Besides, a techno-economic assessment, using appropriate economic indicators, is necessary to evaluate the competitiveness of the proposed concept. For example, US DOE 2030 cost targets for CSP is 5 c$ per kilowatt-hour for a baseload plant with 12 or more hours of storage and 10 c$ per kilowatt-hour for a peaker plant with six or fewer hours of storage [45].

Table 1 .

 1 Design data of Siemens SGT-500[START_REF]SGT-500 Industrial Gas Turbine[END_REF]

	Design data	Value Unit
	Compressor
	Compressor ratio	12	-
	Isentropic efficiency (assumed)	0.91	-
	Combustion chamber
	Pressure losses (assumed)	4	%
	Turbine Inlet Temperature (TIT)	850	°C
	Combustion efficiency	0.99	-
	Turbine section
	Isentropic efficiency (assumed)	0.92	-
	Gas turbine unit
	Nominal shaft speed	3600	rpm
	Nominal output (under standard conditions) 17	MWe
	Mechanical efficiency	99	%
	Electrical efficiency	99	%

Table 2 .

 2 Statistics for the validation of the gas turbine model

	Operation mode Parameter	Mean Absolute Deviation Percentage of MAD (%)
		Power generation	0.44 [MW]	3.43
	Full-load	Heat rate	0. 42 [MJ/kW-hr]	3.30
		Exhaust gasses mass flow	2.72 [kg/s]	3.46
		Exhaust gases temperature	6.04 [°C]	1.54
	Part-load	Heat rate	0.61 [MJ/kW-hr]	4.76
		Efficiency	1.07 [%]	3.74

Table 3 .

 3 Nominal design data of the FB-HEX

	Parameter	Hybrid GT Solar-only GT Unit
	Compressor mass flow rate	79.445	80.73	kg/s
	Compressor outlet temperature	340	341.58	°C
	Compressor outlet pressure	10.07	10.06	bar
	Particles inlet temperature	750	880	°C
	Particles outlet temperature	398	399.87	°C

Table 4 .

 4 Main design data of the three considered operation modes

	Case	Hybrid GT	Solar-only GT	Solar-only GT -Low flowrate
	Tower height (m)			120
	Heliostat field (m 2 )	77 518	126 812	77 518
	Hot temperature of particles (°C)	750	880	880
	Receiver geometry	382 tubes of 7 meters height and an outer diameter of 50 mm

Table 5 .

 5 Optical properties of the tubes and refractory panels

	Parameter	Tubes	Refractory Panels
	Emissivity (-)	0.85	0.93
	Absorptance (-)	0.9	0.2

Table 6 .

 6 Amount of hot particles collected during the vernal equinox and the winter/summer solstices and equivalent full-load operation duration (see Table3)

		Winter Solstice	Vernal Equinox	Summer Solstice
	Hybrid GT (T particle at 750°C)	2027 t	2432 t	2510 t
	Solar GT (T particle at 880°C)	2060 t	2431 t	2494 t
	Equivalent full-load
	operation duration	7.1 h	8.4 h	8.6 h
	(round numbers)
	Solar GT (T particle at 880°C) -Low flowrate	1440 t	1730 t	1766 t
	Equivalent full-load operation duration	5 h	6 h	6.1 h

Table 7 .

 7 Losses in the performance of the solarized gas turbine at part load (percentage of nominal

	performance)
	Turbine	Losses in shaft	Losses in compressor	Losses in turbine	Losses in overall
	Load	speed	isentropic efficiency	isentropic efficiency	efficiency
	75%	6.76	3.46	14.48	18.37
	80%	5.58	2.81	11.39	15.48
	85%	2.89	2.45	5.33	12.57
	90%	1.73	1.95	2.95	7.86
	95%	0.92	0.94	1.56	3.37

Acknowledgments

This work has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No 727762, Next-CSP project and was supported by the French "Investments for the Future" program managed by the National Agency for Research, under contract ANR-10-LABX-22-01 (labex SOLSTICE).

Appendixes:

Appendix A:

A.1. Design methodology

The main steps in design the complete solar power plant are illustrated in Liquid water is pumped to the Heat Recovery Steam Generator (HRSG) though the Condenser pump. The HRSG consists of four main parts, i.e., an economizer, a boiler, a super-heater, and a re-heater. The superheated steam at the outlet of the HRSG is expanded through the High-Pressure Turbine (HPT). Next, this steam is preheated in the preheater. The steam is then expanded into the Low-Pressure Turbine (LPT) to the condenser working pressure. At the condenser, the steam is cold down until saturated liquid conditions. The modeling of the steam cycle is based on a practical technique similar to the one used to model the GT cycle.

B.1.2. Condenser pump

The Condenser Pump (CDP) performance map is used. Indeed, a typical pump's map provides the performance as a function of mass flow rate and rotational speed. At selected shaft speed the isentropic efficiency and head (increase in pressure) can be obtained as a function of mass flow rate.

Nrot and M are respectively the shaft speed and mass flow rate.

The outlet parameters of the CDP including pressure, temperature and enthalpy are calculated as follows. The outlet pressure is obtained by adding the increase in pressure (ΔP) to the inlet pressure (Pin).

subscripts in and out refer to the inlet, the outlet of the CDP.

The actual out enthalpy is calculated based on the concept of isentropic efficiency.

where h and ηis refer to the enthalpy and the isentropic efficiency. Subscript s refers to the isentropic enthalpy.

The water temperature at the outlet of the pump is calculated as a function of pressure and actual enthalpy.

B.1.3. Heat Recovery Steam Generator (HRSG)

The HRSG model consists of four sub-programs, i.e., preheater (economizer), evaporator, superheater, and re-heater sub-programs. The input data of the HRSG model are the exhaust gases inlet parameters (mass flow, temperature, and gases composition) as well as water inlet parameters (mass flow, pressure, and temperature).

The outlet parameters (water/steam, exhaust gases) are calculated as follows. The outlet pressure of the water/steam is less than that at the inlet due to pressure drop.

dpwater_eco is the pressure drop coefficient of cold fluid side (water) in the economizer.

To estimate the outlet temperatures of the cold and hot fluids, the NTU-Effectiveness method has been applied. The actual heat transfer (Q) in the economizer is:

where: εeco and Qmax are the effectiveness and the maximum heat transfer of the economizer respectively.

The effectiveness of the economizer is given by, A similar approach has been used to model the evaporator, the super-heater, and the re-heater.

B.1.4. Modeling the HPT and LPT sections

The steam turbine performance map is used for determining the expansion ratio and the isentropic efficiency as a function of corrected rotational speed and corrected mass flow rates.

The same methodology that has been applied for the gas turbine is used herein. The expansion ratio and isentropic efficiency of the steam turbine are a function of the corrected mass flow rate and corrected shaft speed.

The pressure at the outlet of the steam turbine section is then obtained.

In order to calculate the actual enthalpy at the outlet of the turbine section, the concept of isentropic efficiency is used.

hsteam_out_s is the enthalpy that would have obtained at the outlet of the steam turbine section if the expansion process is isentropic. This ideal enthalpy is evaluated using the outlet pressure and inlet entropy of the steam.

So that the actual enthalpy is,

The water/steam temperature at the outlet of the steam turbine section is then calculated as a function of pressure and actual enthalpy.

The steam fraction is also estimated as a function of pressure and actual enthalpy.

A similar approach has been used to model the Low-Pressure Turbine.

B.1.5. Modeling the condenser (CD)

An air-cooling condenser is considered. The model of the condenser is similar to that of the evaporator. The inlet parameters of the condenser model are the outlet parameters of the LPT.

B.2. Design of the Rankine steam cycle

The Rankine steam cycle is designed based on the temperature and the mass flow rate of the GT's exhaust gases. The difference between the parameters of the exhaust gases in the case of hybrid and solar modes are minor (solar mode: 80.73 kg/s, 386.96°C, hybrid mode: 79.45 kg/s, 887.27°C). Thus, the solar-mode is taken as a reference to size the components of the steam turbine.

The design has been carried out as follows. First, the nominal inlet pressure of the High-Pressure Turbine section (HPT) is selected based on the technical data of commercial steam turbines. A temperature approach (at the outlet of the superheated) of 10°C and a pinch temperature of 5°C is considered for the design of the HRSG. The full design input data are summarized in Table B.1. e-NTU method to used to size the heat exchangers of the HRSG as well as the re-heater. The pressure drop in the heat exchanger is neglected and a heat loss factor of 1% is considered.

The water/steam mass flow rate is then calculated based on the energy balance of the HRSG.

The re-heater is supposed to be installed at the outlet of the HRSG (after the pre-heater) so that to recover as much as possible heat from the exhaust gases. To do so, a re-heat pressure of 4 bars is chosen. An air condenser is chosen because it is more suitable for a desert climate.

The condenser working pressure is selected based on the commercial steam turbines. The performance maps of the HPT, low-pressure turbine (LPT) and the condenser pump are then generated. A modular modeling approach is used to link the models' components (HRSG, reheater, HPT, LPT, condenser pump, and condenser) together and to find the nominal operation parameters. A sliding pressure strategy is used to control the steam turbine operation. Tables B.2 illustrates the size of the components as well as the nominal operation parameters of the steam cycle.