Dr Wahiba Ramdani

Inés Rabadán González

Nora Benbakoura

Dr Mohammed Ahmar

Dr Charlie Verrier

Dr Yves Queneau

Marc Pera-Titus

Dr François Jérôme

Karine De Oliveira Vigier

] Inés

Rabadan Gonzales

Ms. I Rabadan Gonzales

Ms N Benbakoura

Prof K De Oliveira Vigier

Prof M Pera-Titus

Catalytic Aldol Condensation of 5-Hydroxymethylfurfural and its Synthesis from Concentrated Feed of Carbohydrates

Keywords: carbohydrates, aldol condensation, 5hydroxymethylfurfural, BaO, Choline chloride

come

Introduction

With the sustained increase of population on Earth and manmade activities, fossil carbon resources are expected to decline in the following decades. Thus, the synthesis of renewable and sustainable chemicals and energy sources is becoming more and more important. [1][2][3][4][5] Among renewable raw materials, lignocellulosic biomass is of interest, since a panel of valuable molecules can be obtained. For instance, 5-hydroxymethylfurfural (HMF) is a platform chemical for the synthesis of pharmaceuticals, [6] biofuels like 2,5-dimethylfuran, [7][8][9] polymers, [10] surfactants [11] Moreover, HMF can be used to produce biomass-derived jet fuel in multiple steps. [12] In this process. hydrolysis/dehydration of the a lignocellulosic carbohydrate to HMF is performed in the presence of an acid catalyst. Then, aldol condensation of HMF with a ketone (mostly acetone) leads to aldolization adducts that are hydrogenated and hydrodeoxygenated to linear alkanes. The synthesis of HMF is challenging due to its lack of stability in common solvents. Among carbohydrate-based biomass resources, fructose is the most common feedstock for preparing HMF. Over the last years, many organic and inorganic acids such as HCl, H3PO4, p-toluenesulfonic acid (PTSA) and methanesulfonic acid (MSA) have been used as homogeneous catalysts with good activity towards HMF. [13][14][15] A more available and cheaper carbohydrate that can be used for preparing HMF is glucose. Glucose can be converted into HMF via an acidcatalyzed two-step process, in which it is firstly isomerized into fructose and then dehydrated into HMF. Even better, HMF can be also obtained from cellulose by a sequence of depolymerization, isomerization and dehydration reactions. Homogeneous and heterogeneous catalysts can be used for such reactions. [16] However, in most reported examples, the feed concentration of glucose or cellulose is below 10 wt% which limits the productivity (mole of HMF produced per volume of reactor per h) of the process, and accordingly potential industrial applications. Recently, we demonstrated the association of choline chloride (ChCl) with xylose driven by hydrogen bonding through formation of a deep eutectic solvent (DES). The as-generated xyloside intermediate could afford high furfural yield from a high concentrated feed of xylose (33 to 50 wt%) in biphasic media. [17] Herein we investigated the use of ChCl to prepare HMF from more recalcitrant and challenging carbohydrates such as glucose and cellulose, starting from highly concentrated solutions, in biphasic media. Further aldol condensation with methylisobutylketone (MIBK), also used to extract HMF during its synthesis, was investigated. As a rule, aldol condensation reactions can be performed via two mechanisms (Scheme 1). Over basic catalysts, the aldol reaction progresses via an enolate mechanism, whereas over acid catalysts it occurs via an enol intermediate. The final step is the dehydration of the enol intermediate (i.e. crotonization) to the condensation product. [18] A wide range of basic catalysts have been studied for the aldol condensation of HMF with acetone, including surface-modified zeolite, [8] zirconium carbonate, [9] mixed oxides like MgZr and MgAl [10] and Cu/MgAl2O4, [11] Al0.66-DTP@ZIF-8, [12] and CO2. [13] These catalysts require long reaction times (from 6 to 24h) which decreases the productivity. Additional studies have focused on ketones, including 2-pentanone and methylisobutyl-ketone (MIBK). [6] Alkaline earth metal oxides (MgO, CaO, SrO, and BaO) have been studied as basic catalysts for isomerization reactions, [19] Claisen-Schmidt condensations, [20] Knoevenagel condensation, [21] Michael additions, [22] Tishchenko reactions, [23] aldol condensation of aromatic aldehydes, [24] and transesterification reactions. [25] One example was reported by Zhang and co-workers on CaO in the aldol condensation of HMF with promising results. [26] Here we investigated BaO as catalyst to selectively perform the aldol condensation of HMF with MIBK, and the reaction conditions, with emphasis on the solvent, were optimized. The scope of the reaction was also investigated using other ketones.

Scheme 1. Mechanisms of the aldol condensation of HMF.

Results and Discussion

Synthesis of HMF in the presence of ChCl

First, we performed the synthesis of HMF starting from 1 g of glucose and 1.2 g of choline chloride (ChCl), 2 g of an aqueous solution of HCl (pH 1.28, measured at room temperature, 250 μL of HCl 37% in 250 mL of water), 5 mol% of AlCl3 and 40 g of MIBK.

In this reaction, AlCl3 was used as Lewis acid catalyst to promote the isomerization of glucose to fructose. In our reaction system, the glucose concentration was 31 wt.% which is three times higher than common values used in the literature. The reaction kinetics was studied at 120 °C in the presence of MIBK to extract continuously the HMF produced (Figure 1). The HMF yield is 19% after 4 h at full glucose conversion, meaning that by-products are generated which are probably humins. For comparison, the reaction kinetics was also studied without ChCl in neat water. The HMF yield is lower (12%) after 4 h. A similar effect was observed in the synthesis of furfural from a high concentrated feed of xylose . [17] It can be pointed out that it takes 1h to start to observe HMF which is due to the formation of fructose before its dehydration to HMF. In order to assess the role of ChCl on the reaction mechanism, we monitored the reaction by 1 H NMR at different times (1 h, 2 h, 4 h, 7 h) (Figure 2). We compared the 1 H NMR spectra to that obtained on choline glucoside synthesized by reaction of chloroethyl glucoside with trimethylamine. [27] The 1 H NMR spectrum of the as-obtained choline glucoside exhibits a doublet centered at 4.4 ppm (β stereoisomer), while a second doublet appears at 4.9 ppm (α stereoisomer). Interestingly, both doublets appear in our reaction system after 2 h of reaction, confirming that choline glucoside is formed during the reaction. This observation points out a similar dehydration mechanism as that previously observed in the synthesis furfural from xylose. [17] One can mention that starting from fructose, choline fructoside is not observed (Figure S1). Indeed, choline fructoside could not be obtained by reaction of chloroethyl fructoside with trimethylamine, showing a radically different reactivity of fructose as compared to glucose in the presence of ChCl. We further studied the effect of the temperature. The HMF yield increases to 67% after 1 h at 180°C in the presence of ChCl, whereas the glucose conversion is complete after only 30 min. This result reinforces the benefits of the ChCl-H2O-MIBK system for preparing HMF starting from concentrated glucose. Based on these results, we investigated the conversion of cellulose to HMF. A balance is required between cellulose depolymerization and glucose degradation. To avoid glucose degradation, cellulose depolymerization was performed by ball milling, a technique that was already applied by our team. [28] Two types of acids were used, i.e. sulfuric acid and Aquivion ® PW98. Aquivion ® PW98 is a perfluorosulfonic acid ionomer with a proton exchange capacity of 1.0 mmol/g. Aquivion ® PFSA PW98 is a solid superacid with a Hammett acidity function of -12 which is similar to that of H2SO4. This solid was chosen since it is efficient in the synthesis of cellooligosaccharides from cellulose as was demonstrated previously in our group. [28] The ball milling was performed using a Retch PM 100 at 200 rpm for 24h. In the case of sulfuric acid, 10.7 wt% relative to cellulose was used. Typically, cellulose was suspended in 15 ML of diethylether with the acid solution. The solvent was removed using vacuum and the cellulose was milled. The recovered acidified low molecular weight cellooligo-saccharide (0.15 g) with H2SO4 was added to 0.2 g of ChCl, 1 g of water, 5 mol% of AlCl3 and 6 g of MIBK at 180 °C. The results were compared to the ball milled cellulose with Aquivion ® PW98 (1.5 mol% of H + ). (Figure 3). The HMF yield reaches 26% and 34%, respectively. These results demonstrate that ChCl can be used to prepare HMF starting from a high concentration feed of glucose or ball milled cellulose. One can mention that choline chloride can be recovered using the experimental procedure used for the synthesis of furfural from xylose. [17] Aldol condensation of HMF with MIBK Next, we studied the aldol condensation of a commercial HMF with MIBK over 20 wt% of BaO using different solvents, targeting the synthesis of 1-(5-(hydroxymethyl)furan-2-yl)-5-methylhex-1en-3-one (product 1, Table 1). With only MIBK, no HMF conversion is observed (Table 1, entry 1). This result shows that the aldol condensation reaction cannot be conducted just after HMF synthesis. This result matches earlier results in the solventfree condensation of cyclopentanone and furfural. [29] Indeed, weak bases cannot activate cyclopentanone to form enolate which is necessary for condensation. It is known that water preadsorption can transform Lewis basic sites into hydroxyl groups, increasing the activity for acetone self-condensation, while a very high water amount inhibits the reaction. [30] Thus, water formed during the aldol condensation reaction can condition the selectivity during the reaction by limiting the self-aldol condensation of the ketone. Different solvents were added to the HMF/MIBK solution at a HMF/MIBK/solvent weight ratio of 1:10:10 using 20 wt% of BaO (Table 1). Adding 2-propanol, full HMF conversion is achieved with 38% selectivity to product 1 (Table 1, entry 2). Using acetonitrile (MeCN), no aldol product is observed due to formation of humins (Table 1, entry 3). [31] In DMSO, full HMF conversion is obtained with 36% selectivity to 1 (Table 1, entry 4). In this case, the lack of selectivity can be attributed to the formation of byproducts such as humins. [31,32] Depending on the solvent used, the HMF can react or not with MIBK in order to produce the product 1. It can be point out that the degradation probably occur from HMF since this compound was less stable than the aldol product 1. Finally, ethanol affords 82% selectivity at full HMF conversion (Table 1, entry 5). BaO is slightly soluble in ethanol and 2-propanol, but is completely heterogeneous in the other solvents. The reaction is more selective in ethanol than in 2propanol despite the lower solubility. These results show that the BaO solubility does not directly correlate to the selectivity of product 1, but the solvent may help to control secondary reactions leading to humins. Consequently, using an appropriate solvent, the reaction can be favored over BaO and the selectivity of the reaction can be controlled.

A series of alkaline earth metal oxides were further screened at 120 °C in ethanol using 20 wt% of catalyst starting from 1 g of HMF in 20 g of MIBK/ethanol mixture at a MIBK/EtOH weight ratio of 1:1 (Table 2). Without catalyst and using CuO, MnO or ZnO, no HMF conversion is observed. Over MgO and Al2O3, the HMF conversion is 14% and 4%, respectively, with 4% and 1% yield of product 1. A much higher HMF conversion (over 97%) is achieved over CaO and BaO after 16 h and 1 h reaction, respectively. The yield of product 1 is around 80% over both catalysts. The yield of aldol product 1 evolves in the order: CuO = MnO = ZnO < Al2O3 < MgO < CaO ≈ BaO. One can mention that the catalysts were calcinated under air. However in the case of BaO and CaO the results were similar with or without treatment. That is the reason why no treatement was applied for the further studies using these two catalysts. The difference in the activity of these oxides can be ascribed to the difference in active species and basicity (see below). Given these results, the kinetic profiles of the reaction were measured over both CaO and BaO (Figure 4) (timescale in h for CaO and in min for BaO). The initial reaction rate was calculated as the number of moles of HMF converted per mole of catalyst per h at time zero. CaO shows an initial reaction rate of 0.35 h -1 , whereas that of BaO is 26.30 h -1 . Over CaO, it was necessary to extend the reaction time to 16 h to reach 97% yield of product 1.

In contrast, over BaO, the yield of 1 is above 90% after 1 h. These results confirm the higher activity of BaO compared to CaO. BaO and CaO were further characterized to rationalize their different activity. The basicity of both catalysts was measured by CO2-TPD (Figure 5). CaO exhibits a desorption band at 604 °C with a CO2 uptake of 63.4 μmolCO2/gcat ascribed to strong basic sites as reported in the literature [34]. BaO exhibits two desorption bands centered at 400 °C and 603 °C, pointing out a combination of sites with moderate and strong basicity. BaO contains a higher density of strong basic sites compared to CaO (109.3 μmolCO2/gcat at around 600 °C), with a total amount of sites about 258.4 μmolCO2/gcat, matching earlier reports. [33] A density of 149.1 μmolCO2/gcat of moderate basic sites is present for BaO whereas no moderate basic sites was observed for CaO. In this view, the higher activity of BaO can be correlated to a moderate density of strong basic sites. The specific surface area of CaO and BaO powder, as measured from N2 physisorption, is 23 m²/g and below 5 m²/g, respectively. HR-TEM shows partial agglomeration, with elementary particles in the range 20-50 nm (Figure S2). BaO shows larger elementary particles (range from 500 to 2000 nm) due to particle agglomeration (Figure S3). The BaO sample is composed of different phases (upon exposure to air), including Ba(OH)2 and BaO2 while CaO includes CaO, CaCO3 and Ca(OH)2, as inferred from XRD (Figures S4-S5). Noteworthy, under air, BaO is not carbonated, preventing deactivation of basic sites. Moreover, it is known that Ba(OH)2 is more basic than Ca(OH)2. [33] allowing better operation in the presence of water, as is the case of aldol condensation reactions since water is generated in situ during the reaction. The catalytic performance BaO was studied in a second run with or without calcination at 300°C after separation form the reaction mixture by centrifugation and filtration. Although no product 1 is observed, a total conversion of HMF was obtained (Table S1). A complex mixture of by-products was generated, which could be hardly identified. XRD analysis of the spent catalyst show that the main species present are BaO2 (Figure S5). Thus, Ba(OH)2 are no longer present on the catalyst. This can be due to the dissolution of Ba(OH)2 in the reaction media and thus these species were not recovered and the catalyst was not selective to the targeted product. We also investigated the scope of BaO for the aldol condensation of HMF with other ketones (i.e. acetone, 2-heptanone) at comparable reaction conditions. The HMF conversion is complete for both ketones with 90% yield of the aldol product after 15 min and 10 min for acetone and 2-heptanone, respectively. 3, entry 1). BaO, Cu/MgAl2O4 and Zr(CO3)x lead to full HMF conversion and a high selectivity to the aldol product, above 86% (Table 3, entries 2-4).

The remaining listed basic catalysts (MgO-ZrO2, Ni-NaY, MgZr, MgAl) exhibit lower HMF conversion and aldol product selectivity (Table 3, entries, 5-8). Among the different catalysts, BaO displays the highest activity with 159 mmol.g -1 .h -1 . Overall, these results point out that BaO is an efficient and cost-effective catalyst for the aldol condensation of HMF with ketones.

Synthesis of the aldol compound 1 from glucose

With these results in hand, we performed the synthesis of product 1 directly from glucose. To this aim, HMF was first prepared starting from 1 g of glucose, 1.2 g of ChCl, 2 g of an aqueous solution of HCl (pH =1.28), 40 g of MIBK, 180 °C and 30 min. After the reaction, HMF was recovered by extracting the aqueous phase several times with MIBK using acetone to precipitate ChCl. The recovered HMF is not pure, since humins are also formed during this reaction as shown on Figure S11. Then, MIBK was partially removed using a vacuum rotavapor pump, and 10 g of fresh MIBK and 10 g of ethanol were added together with 200 mg of BaO. Then, the aldol condensation reaction was carried out at 120 °C for 1 h. After the reaction, ethanol and MIBK were removed under vacuum rotavapor pump. The results shows that product 1 was produced with only 2% yield (GC analysis) with concomitant formation of many by-products that could be hardly identified. Moreover, slight acidity is still present after the first step that can be detrimental for the aldol condensation reaction. Traces of HCl (100 µL) was, thus, added at the beginning of the aldol condensation of commercial HMF with MIBK. 87% of 1 was obtained showing that the acidity is not the main issue of the synthesis of 1 from glucose. The addition of AlCl3 at the beginning of the aldol condensation of HMF with MIBK afforded a yield of 68% of 1. This reaction shows that AlCl3 should be fully removed in order to avoid any poisoning of BaO. An experiment was carried out using 1:2:2 synthesized HMF/MIBK/solvent weight ratio in order to increase the concentration of HMF in the solvent and 20 wt.% of BaO. The reaction of aldol condensation was carried out at 120°C for 2 h and 5% of 1 was observed. 20wt% of BaO was further added to this reaction media and we were pleased to see that 22% of aldol product 1 was obtained after 3h of reaction at 120°C which shows that BaO is poisoned by compounds that are still present in the synthesized HMF which can be some traces of AlCl3. Thus another catalyst should be used for the synthesis of HMF in order to avoid any poisoning of BaO. This will be the topic of further investigation in our group.

Conclusion

We showed along this study that ChCl can help to selectively convert highly concentration feeds of glucose and ball milled cellulose into HMF by the formation of an intermediate choline glucoside. In parallel, the aldol condensation of HMF with MIBK (and other ketones) could be performed over BaO using ethanol as a co-solvent. BaO exhibits a competitive activity and high selectivity to 1-(5-(hydroxymethyl)furan-2-yl)-5-methylhex-1-en-3-one, compared to earlier reported solid basic catalysts. The recyclability of the catalyst was not possible, and when the reaction is performed in two steps starting from a highly concentrated feed of glucose, the yield of the aldol product is about 20%. This yield was due to a poisoning of BaO by some traces of AlCl3.

Experimental Section

Chemicals Glucose (≥99%), MCC cellulose (Avicel), HCl (36.5-38.0%), Choline chloride (≥99%), HMF (99%), 4-Methyl-2-Pentanone (≥99%, FCC), AlCl3, were all purchased from Sigma-Aldrich. CaO were purchased from sigma Aldrich and BaO from Acros Organic.

General procedure for the dehydration of glucose or cellulose to HMF

A mixture of glucose or ball milled cellulose in acidified water (pH=1.28 measured at room temperature, 250 μL of an aqueous solution of HCl (37 wt.% in 250 mL of water), ChCl, AlCl3 (5 mol.%) and methylisobutylketone (MIBK) with a weight ratio of acidified water : MIBK of 1 : 20 was heated at the desired temperature in a close reactor. The reaction was performed under biphasic condition.

General procedure for aldol condensation reaction

The catalytic tests were performed in a 30-mL tubular glass reactor with a sealable arrangement on top. The reaction was performed using a mixture of 5-hydroxymethylfurfural/ketone/EtOH with a weight ratio of 1 : 10 : 10) over 20 wt.% of catalyst in a glass tube placed in a preheated oil bath (120 °C) at 600 rpm. The solvent was evaporated and a brown liquid was obtained.

Analytical methods

Glucose was quantified by external calibration at 25 °C by highperformance liquid chromatography using a Shimadzu HPLC equipped with a NH2 column, a RID detector, and using a mixture of water/acetonitrile (3:7) as mobile phase (0.8 mL.min -1 ). HMF was also analyzed by HPLC at 25 °C using an ICECoregel107H column (300 × 8 mm), a Varian Prostar UV-V detector (284 nm), and Varian Prostar pumps (model 210), and 7 mmol of sulfuric acid in water as a mobile phase (0.8 mL.min -1 ). 1-(5-(Hydroxymethyl)furan-2-yl)-5-methylhex-1-en-3-one was analyzed and quantified by gas chromatography on an Agilent GC equipped with a HP-5 capillary column with 5 wt% phenyl groups, a FID detector, and a split/splitless injector. N-dodecane was used as internal standard The calibration curve of the hydroxymethyl)furan-2-yl)-5methylhex-1-en-3-one is represented on Figure S6. NMR spectra were recorded on a Bruker Advance DPX 400 spectrometer. The samples were collected periodically for analysis. 1 H and 13 C NMR analysis were performed on all the aldol products using d-DMSO as a solvent (Figure S7-10).

Characterizations of the catalyst

Nitrogen adsorption-desorption isotherms were measured at 77 K were measured on a TriStar 3000 instrument to determine specific surface area using the multipoint Brunauer-Emmet-Teller (BET) method. Prior to the measurements, the samples were outgassed at 300 °C on a Micromeritics VacPrep061 to remove adsorbed water and vapors. CO2-TPD analysis were performed using a Micromeritrics Autochem 2910 device. Before the CO2-TPD experiment, the samples (50 mg) were treated in situ in a 50 ml/min O2 flow at 200 °C for 2 h for BaO and 700°C for 2h for CaO, cooled down to 50°C in a 50 ml/min He flow. The temperatures were chosen accordingly to the thermal analysis (Figures S5 andS6). The samples were exposed to CO2 (20 ml/min) at that temperature for 60 min, kept again in a 50 ml/min flow of He for 30 min, cooled to 50 °C, and then heated from 50 °C to 600 °C in the He gas flow at a rate of 10 °C/min -1 while monitoring CO2 desorption. HR-TEM analysis was performed on a JEOL JEM 2100 UHR equipped with a LaB6 filament with a punctual resolution of 0.19 nm. The samples were prepared in ethanol using a sonication bath, and some drops were deposited on a carbon grid (Holey Carbon grids). XRD analysis was performed using an EMPYREAN PANalytical Instrument with a CuKα as x-ray radiation within 10-80° 2Θ angle with a step of 0.02° (2Θ) in all 2s.

Figure 1 .

 1 Figure 1. Synthesis of HMF from glucose. Conditions: 1 g of glucose, 1.2 g of ChCl, 2 g of aqueous HCl (pH= 1.28), 5 mol% of AlCl3, 40 g of MIBK, 120 °C.

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra of the reaction system during the conversion of glucose to HMF. Conditions: 1 g of glucose, 1.2 g of ChCl, 2 g of aqueous HCl (pH =1.28),, 40 of g MIBK, 120 °C.

Figure 3 .

 3 Figure 3. Conversion of ball milled cellulose to HMF. Conditions: 0.15 g of ball milled cellulose, 0.2 g of ChCl, 1 g of H2O, 6 g of MIBK, 180 °C.

Figure 4 .

 4 Figure 4. Kinetic profile of the aldol condensation of HMF with MIBK. Conditions: 1 g of HMF, 1:10:10 HMF/MIBK/ethanol weight ratio, 20 wt.% of (a) CaO and (b) BaO, 120 °C.

Figure 5 .

 5 Figure 5. CO2-TPD profiles (normalized by the sample mass) of CaO and BaO.

  aldol product is 71% (Table

Table 1 .

 1 Effect of the solvent on the aldol condensation of HMF with MIBK.

	Entry	Solvent	HMF	Selectivity of	BaO solubility
			conv. (%)	1 (%)	(wt.%)
	1	-	0	0
	2	2-Propanol	100	38	8
	3	MeCN	100	0	0
	4	DMSO	100	36	0
	5	EtOH	100	82	4

[a]

Conditions: 1:10:10 HMF/MIBK/solvent weight ratio, 20 wt% of BaO, 120°C, 1 h

Table 2 .

 2 Effect of the alkaline metal oxide on the aldol condensation of HMF with MIBK.

	Entry	Catalyst	Time (h)	HMF conv. (%)	Yield of 1 (%)
	1	-	16	0	0
	2	CuO [a]	16	0	0
	3	MnO [c]	16	0	0
	4	ZnO [b]	16	0	0
	5	MgO [b]	16	14	4
	6	Al2O3 [b]	16	4	1
	7	CaO [a]	16	97	79
	8	BaO [a]	1	100	82
	Conditions: 1:10:10 HMF/MIBK/ethanol weight ratio, 20 wt% of catalyst, 120°C,
	[a] no treatment; [b] treated under air at 500 °C during 6 h; [c] treated under air
	at 600 °C during 6 h.

Table 3

 3 compares the catalytic performance of CaO and BaO with reported solid basic catalysts for the aldol condensation of HMF with acetone. CaO is slightly less active than BaO and the

Table 3 .

 3 Comparison of the catalytic properties of CaO and BaO with catalysts reported in the literature for the aldol condensation of HMF with acetone.

	Entry	Cat.	T	t (h)	HMF	Yield of	Activity
			(°C)		conv.	aldol	(mmol/
					(%)	product (%)	g/h)
	1 this study	CaO	120	5	89	63	7
	2 this study	BaO	120 0.25	100	90	159
	3 [11]	MgAl2O4	140	7	100	86	44
	4 [9]	Zr(CO3)x	54	24	100	92	0.21
	5 [8]	MgO-ZrO2 120	24	68	34	1.5
	6 [8]	Ni-NaY	120	24	51	33	1.1
	7 [10]	MgZr	50	24	68	14	0.03
	8 [10]	MgAl	50	24	32	4	0.07

Acknowledgements

The authors acknowledge the ANR for the funding of the project FurCaB (ANR-15-CE07-0016). They are grateful to the region Nouvelle Aquitaine for the CPER and EU-FEDER support, to INCREASE Federation. They also thanks the Erasmus program for the grant allocated to Inés Rabadan Gonzales.

Entry for the Table of Contents


I Zafeiratou

I Prodan

L Lefèvre

On the flat representation for a particular class of port-Hamiltonian systems

Keywords: Flatness, Bond graphs, Bicausality, Port-Hamiltonian systems, Electrical storage 1 For electrical systems, e

This paper pertains to the flat representation of a class of port-Hamiltonian (PH) systems and advocates the use of bicausality of Bond graphs for finding appropriate flat outputs. Systems which are differentially flat have several useful properties which can be exploited to generate, for example, optimal trajectories/profiles which ensure constraints satisfaction. For the special case of PH systems combining the power preserving property with the flatness properties leads to effective control strategies for multi-physical systems. Hence, the purpose of this paper is to explore the implications and features of a particular class of PH systems (which can be retrieved from a Bond Graph representation) in finding their flat output representation. We concentrate on the example of an electrical storage system of a DC microgrid to illustrate the proposed theory.

INTRODUCTION

Differential flatness is a structural property of a class of nonlinear dynamical systems, i.e., all system variables (the states and inputs) are written in terms of a set of specific variables, the flat outputs (equal in number to the number of inputs), and their derivatives [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF]).

There is a wide range of works which employ differential flatness in their control approach. Many researchers use its properties in motion planning problems [START_REF] Hervagault | Motion planning for usvs with communication guarantees: an experimental setup[END_REF] in order to validate the system's dynamics under constraints and generate optimal profiles for velocity, forces and the like. Other works employ B-Splines parametrization of the flat outputs to ensure continuoustime constraints validation at the trajectory generation level [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF]; [START_REF] Pham | Port-Hamiltonian model and load balancing for DC-microgrid lift systems[END_REF].

The idea of this paper is inspired by our recent works [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF] where we propose a flatness-based hierarchical control for power balancing in a meshed DC microgrid. Firstly, through the PH formulation, the statespace representation of the microgrid system was explicitly and structurally described. Next, the differential flatness was employed to obtain the relations of the states and control inputs in function of the flat outputs. By using flatness combined with B-splines parametrization, we were able to generate optimal profiles for the current and voltage of the energy storage (ES) system while minimizing the electricity purchase from the external grid.

However, an important issue is the calculation of the flat outputs, especially in the case of a complex dynamics with multiple states and inputs. [START_REF] Franke | On the computation of flat outputs for nonlinear control systems[END_REF] proposed an algorithm for calculating the flat outputs of nonlinear physical systems using illustrative Institute of Engineering Univ. Grenoble Alpes examples. This algorithm was implemented in [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF] to find a possible flat output set for the ES PH system. Apart from the algorithm, another method has been studied in [START_REF] Richard | Analysis of flatness using bond graphs and bicausality[END_REF]. It relies upon an important advantage of the PH systems according to which their state-space representations can be easily derived from their Bond graph representation. Therefore, through Bond graphs and a recently established notion called bicausality, the inverse dynamics of the PH systems will be studied and linked to their flat representations.

The contributions of this paper are: i) use of the bicausal Bond graphs to analyze the flat representation of the PH systems; ii) development of a method which provides directly the number of states and inputs that could be taken into account as possible flat outputs for the PH system; iii) analysis of the proposed technique over an electrical storage system.

BOND GRAPHS AND BICAUSALITY

Bond graph is a graph-oriented approach and builds upon the power-preserving interconnections amongst the elements of a physical system [START_REF] Karnopp | System dynamics: modeling, simulation, and control of mechatronic systems[END_REF]. An important characteristic of the Bond graph is the causality property which indicates the dependence among the port elements 1 characterized by pairs of efforts e and flows f as in Fig. 1. Every element has different causality structure. In here, we will focus on the storage elements (capacitors and inductors) which define the states of the electrical systems mentioned below. There are two types of causalities, the integral and the differential causality.

Integral causality:

The input is integrated to generate the output (see also Fig. 2a). For the capacitors we have Bicausality: It is an extended version of causality which allows the interpretation of the inverse dynamics [START_REF] Richard | Analysis of flatness using bond graphs and bicausality[END_REF]. In order to design a bicausal Bond graph, supplementary elements are taken into account which enable the decoupling of e -f pairs and facilitate the calculation of further quantitative variables (Fig. 4). The SS (source-sensor) elements are added, which contain the inputs and the outputs of the system (e.g., the effort sources, S e , the flow sources, S f , the effort detectors, D e , the flow detectors, D f ). The sources elements have specific assignment , which is the opposite assignment from the case of the causal Bond graph, indicating the inverse dynamics. Similarly, for the storing elements as in Fig.

2b [START_REF] Ngwompo | Bond graphbased simulation of non-linear inverse systems using physical performance specifications[END_REF].

Bicausality of a RLC circuit:

The bicausal Bond graph of the RLC circuit is introduced in Fig. 5. From the bicausal Bond graph, the following relations are obtained from the 0 and 1 junctions (keep in mind that p is the magnetic flux of the inductor and q is the charge of the capacitor)2 : e 1 = e 3 + e 4 , f 2 = f 3 = f 4 , e 4 = e 5 = e 6 , f 4 = f 5 + f 6 , e 6 = e 7 , f 6 = f 7 ,

f 5 = C de 5 dt = q, e 5 = 1 C t f 5 dt = q C , e 7 = I df 7 dt = ṗ, f 7 = 1 I t e 7 dt = p I , e R = f R R = (f 5 + f 6 )R = ( q + p I )R.
(1) Fig. 5. Bicausal Bond graph of the electrical RLC circuit.

BICAUSALITY AND DIFFERENTIAL FLATNESS

Hereinafter, the connection between bicausality and differential flatness will be analyzed.

Taking into account the definition of differential flatness (see [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF]), we show that the flatness property can be derived from the bicausal Bond graph representation of an electrical circuit detailed in Section 2. Proposition 3.1. Let us consider the state vector x(t) ∈ R 2 and inputs u(t) ∈ R derived by the bicausal Bond graph of the RLC circuit. The state and input resulted by the bicausal Bond graph is considered to be equivalent to the flat representation of the system (see [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF]).

Proof : From (1), the states and inputs are derived for the bicausal Bond graph in Fig. 5:

p = If 7 = I( q + p I -q), ( 2a
)
q = Ce 5 = C ṗ, ( 2b
)
u = v s = ( q + p I )R + ṗ, ( 2c
)
where v s is the input voltage supplied by the source.

The state vector of the system is x = [p q] ∈ R 2×1 and the input vector is u = v s ∈ R 1×1 . Regarding the relations above, (2a)-(2c), provided through the bicausal Bond graph, a physical representation is obtained similar to the flat representation of the system, considering the flat outputs in function of the states (see definition of flatness).

According to [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF], the number of flat outputs must be equal to the number of control inputs. The RLC system (2a-2c) has one control input, u = v s . Therefore, one flat output is necessary. Note that from the bicausal Bond graph (Fig. 5), we retrieve the state p and obtain (2a), which shows that p cannot be written in function of the other variables of the RLC circuit, the state q and the input v s . Consequently, a straightforward choice is the state p as a possible flat output for the system, z = p. Therefore, we obtained a flat representation of the system through its bicausal Bond graph. Proof : The proof is given over a particular example of an ES system (see the circuit in Fig. 6) which includes the KiBaM battery 3 and the Split-Pi converter.

Let us represent in Fig. 7 the causal Bond graph of the KiBaM battery, part of the ES electrical circuit presented in Fig. 6. Its corresponding state-space representation in PH form derived by its causal Bond graph has been already presented in [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF]. Developing, next, the inverse dynamics of the battery's system through its bicausal Bond graph (Fig. 8), we will derive the possible flat outputs of the battery. Therefore, the state and input of the system are give:

3 KiBaM stands for the kinetic battery model. According to Proposition 4.1, if the aforementioned system can be written in kernel representation, it composes a DS. Therefore, the kernel representation of the bicausal Bond graph in Fig. 8 is presented below:

q 1b = q2b R 2b C 1b + C 1b C 2b q 2b , (3a)
q 2b = v sc C 2b -q1b R 1b C 2b -q2b R 1b C 2b -q2b R 2b C 2b , ( 3b
)
v sc = q1b R 1b + q2b R 1b + q2b R 2b + q 2b C 2b . ( 3c
    
1 -1 0 1 0 -1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

     E      v q 1b v q 2b v R 1b v R 2b v sc      e +      0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 -1 0 1 0 1 0      F      i q 1b i q 2b i R 1b i R 2b i s      f =      0 0 0 0 0      , (4
) Hence, the bicausal Bond graph of the battery is a PH system. Indeed, by isolating the derivatives of the states on the left side in (3a)-(3c), we conclude to the PH statespace representation presented in [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF]. Afterwards, all the possible relations among the efforts and flows, derived from the 0 and 1 junctions of the bicausal Bond graph (Fig. 8), are presented below:

       q 1b q 1b q 2b q 2b v sc v sc        =           0 0 C 1b C 2b R 2b C 1b 0 0 -R 1b C 1b 0 -R 1b C 1b C 1b 0 0 1 0 0 0 0 1 0 0 0 R 1b 1 C 2b R 1b + R 2b 0 0 0 0 0 1                q 1b q1b q 2b q2b v sc      . ( 5
)
From the latter matrix representation, in the left part the states and the inputs are illustrated, which, regarding the definition of flatness, can be written in function of the flat outputs and their derivatives. Since we have one control input, v sc , we expect only one flat output [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF]. Furthermore, taking into account that the flat output can be written in function of the states, from (5) we can assume that the only possible flat output for the battery's system is z = q 2b . This conclusion comes from the fact that it is the only state not written in function of the other states or their derivatives. To look more carefully at the result, we separate the matrices, extracting the circuit parameters, C 1b , C 2b , and the dissipation, R 1b , R 2b :

           q 1b q 1b C 1b q 2b C 2b q 2b C 2b u s u s            =                  
0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1

       V +        0 0 0 R 2b 0 0 -R 1b 0 -R 1b 0 0 0 0 0 0 0 0 0 0 0 0 R 1b 0 R 1b + R 2b 0 0 0 0 0 0        W            • • q 1b C 1b q1b q 2b C 2b q2b v sc . (6a)
Therefore, from matrix V we can derive the states and control inputs which are structurally required for the flat representation. The state q 2b is included, in function of which q 1b and v sc can be derived (third column of matrix V ). We notice also that in the last row of matrix V the control input v sc appears. In W matrix, the dissipative elements are presented and are involved only with the derivatives of the states, which are not taken into consideration in the flat outputs selection. The flat representation of the battery is written below from (3a)-(3c):

q 1b = R 2b C 1b ż + C 1b C 2b z, (7a)
q 2b = z, (7b)
v sc = R 2b R 1b C 1b z + (R 1b C 1b C 2b + R 1b + R 2b ) ż + 1 C 2b z. (7c)
Following the same steps, we continue with the Split-Pi converter. Firstly, its PH state-space representation is considered (see Fig. 7) as already presented in [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF]. As aforementioned, since we have four control inputs at this case, we have to find four flat outputs.

Developing the inverse dynamics of the system through its bicausal Bond graph, we will generate the four states or inputs which can be considered as flat outputs.

                  p 1sc p 1sc p 2sc p 2sc q 1sc q 1sc q 2sc q 3sc q 3sc q 3sc v s i b                   =                            1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - I 1sc C 1sc R 1sc -I 1sc 0 0 0 I 1sc R 1sc 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 2sc 0 I 2sc 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 C 1sc 0 0 0 0 C 1sc C 2sc 0 0 0 0 -C 1sc 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -C 3sc 0 0 C 3sc C 2sc 0 0 0 0 0 -C 3sc 0 0 0 0 0 0 0 0 0 0 C 3sc R 0 0 R 1sc I 1sc 0 0 0 1 C 1sc R 1sc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C 1sc R 0 0 0 0 0                                               p 1sc ṗ1sc p 2sc ṗ2sc q 1sc q1sc q 2sc q 3sc q3sc v s i b d 1sc d 2sc                    (8)
Fig. 10 presents the bicausal Bond graph of the Split-Pi converter, composed by five storing elements, the capacitors C 1sc , C 2sc , C 3sc and the inductors I 1sc , I 2sc with their differential causalities, one effort source, S e , and the resistor R which contain the inputs (the voltage of the source, v s , and the current of the resistor R, i R ), one flow detector, D f , which is the first output (current of the source, i s ) and one effort detector, D e , which is the second output (voltage of the resistor R) of the system. Furthermore, the duty cycles, d 1sc and d 2sc , correspond to the activity of the switches, Sw 1sc , Sw 2sc , Sw 3sc , Sw 4sc . Therefore, the states are derived below:

p 1sc = I 1sc u s R 1sc - q 1sc C 1sc R 1sc -q1sc , (9a)
p 2sc = I 2sc [ q3sc + i R ] , (9b)
q 1sc = C 1sc ṗ1sc + q 2sc C 2sc (1 -d 1sc ) , (9c)
q 2sc = q 2sc , (9d)
q 3sc = C 3sc q 2sc C 2sc (1 -d 2sc ) -ṗ2sc . (9e)
Moreover, the control inputs can be deduced also in function of the states and their derivatives from the bicausal Bond graph as follows:

v s = p 1sc I 1sc + q 1sc C 1sc R 1sc + q1sc R 1sc , (10a)
i R = q 3sc C 3sc R , (10b)
d 1sc = C 2sc q 2sc ṗ1sc - q 1sc C 1sc + q 2sc C 2sc , (10c)
d 2sc = C 2sc q 2sc -ṗ2sc + q 2sc C 2sc - q 3sc C 3sc . (10d)
Next, the kernel representation of the bicausal Bond graph of the Split-Pi converter is presented, which proves that it composes a DS:

          -1 0 -1 1 -d 1sc 0 0 0 0 0 1 0 1 -d 2sc -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 1 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0           E           e p1sc e p2sc e q1sc e q2sc e q3sc e R1sc e R v s           e + (11a)
+           0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 1 1 -d 1sc -(1 -d 2sc ) 0 1 0 0 0 0 0 -1 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1           F           f p1sc f p2sc f q1sc f q2sc f q3sc f R1sc f R i s           f =           0 0 0 0 0 0 0 0           ,
Hence, the bicausal Bond graph of the Split-Pi converter is also a PH system. By placing the state derivatives on the left side in (9a)-(9e) (considering also the control inputs in (10a)-(10d)) leads to the primary PH state-space representation as in [START_REF] Zafeiratou | Flatness-based hierarchical control of a meshed dc microgrid[END_REF].

Subsequently, all the relations among the flow and the effort variables from the 0 and 1 junctions (Fig. 10) are introduced below in (8). At this point, four additional relations are obtained for the states p 1sc , p 2sc and q 2sc concerning the duty cycles, d 1sc and d 2sc , of the converter:

p 1sc I 1sc d 1sc = p 1sc I 1sc -q2sc - p 2sc I 2sc (1 -d 2sc ), (12a)
p 2sc I 2sc d 2sc = p 2sc I 2sc -q2sc - p 1sc I 1sc (1 -d 1sc ), (12b)
q 2sc C 2sc d 1sc = q 2sc C 2sc - q 1sc C 1sc + ṗ1sc , (12c)
q 2sc C 2sc d 2sc = q 2sc C 2sc - q 3sc C 3sc -ṗ2sc . ( 12d
)
These four equations will be also considered in the analysis of the flat outputs. During the operation of the converter the states p 1sc , p 2sc , q 2sc are different from 0. Contrariwise, the duty cycles, d 1sc and d 2sc , can be equal to 0. Therefore, the duty cycles cannot be placed in the denominator's position in the fractions appearing in (12a)-(12d) and cannot be included in the flat outputs selection.

The circuit's parameters and the dissipative elements are divided into two matrices as in (6a), hence (8) becomes:

p 1sc p 1sc I 1sc p 2sc p 2sc I 2sc q 1sc q 1sc C 1sc q 2sc q 3sc q 3sc C 3sc q 3sc C 3sc v s i b = (V + W ) p 1sc ṗ1sc p 2sc ṗ2sc q 1sc q1sc q 2sc q 3sc q3sc v s i b d 1sc d 2sc , (13)
where matrices V and W are developed as before, extracting the dissipative elements and placing them in W .

From (8), we can assume that a possible set of flat outputs for the Split-Pi converter system contains the states p 1sc , p 2sc , q 1sc , q 2sc , q 3sc . Concerning the state q 2sc , it must be regarded as one of the flat outputs (seventh row of matrix V ). Since it cannot be expressed in function of the other states and inputs, it is necessary for the flat representation. Additionally, because of (12a)-(12d), it is indispensable to consider the states p 1sc , p 2sc , q 2sc as part of the flat outputs since they are linked with the switching activity of the converter (12a-12d). Moreover, the d 1sc and d 2sc are given only from (12a)-(12d) and they cannot be taken into account. The control inputs, v s and i b , similarly cannot be taken into account, since they are written in function of the states and their derivatives, but not vice-versa.

To decide the proper set of flat outputs, we must consider that the number of flat outputs derivatives in the flat representation can influence the signal response and create oscillations. Thereupon, we need to select which states are appropriate from ( 13) and (8) (less derivatives create less deviations among the reference and the actual values), so that the less possible derivatives appear in the flat representation. If we look into (8), the state q 1sc is in function of more than one derivatives, while the state q 3sc can be replaced from an equation with no derivatives included. However, neither can be excluded and the decision on the appropriate flat output depends on the kind of problem to solve as referred below. Therefore, we conclude to the following possible flat outputs:

z 1 = [p 1sc p 2sc q 2sc q 1sc ] , (14)
z 2 = [p 1sc p 2sc q 2sc q 3sc ] . (15)
The flat representation of the Split-Pi converter can be written, substituting in (9a)-( 9e) and ( 10a)-(10d), each one of ( 14) and ( 15).

In Fig. 12 and Fig. 11, the control inputs from (10a)-(10d) are written in function of the two sets of flat outputs (14-15) and, after implementing the Split-Pi model in Matlab/Simulink both in PH form and in flat representation, the simulation results are obtained. The reference values of the control inputs are considered as v s = 400 V , d 1sc = 0.6, d 2sc = 0.4 and i b (the input current of the battery), which varies (blue line in the last simulation of Fig. 12 and Fig. 11). Analyzing the two figures, Fig. 12 and Fig. 11, we observe that with ( 14) we obtain more stable and less noise-affected results for v s . Contrariwise, i b signal contains more noise-affected data. In the duty cycles, slight disturbances appear for both cases. In general, both sets of flat outputs, ( 14) and ( 15 representation in our case is further combined with the Bspline parametrization, strongly dependent on the number of derivatives included. To simplify the calculations (see also (10a)), the first set of flat outputs ( 14) would be the appropriate choice, because writing v s in function of this set leads to a simpler differential equation with less derivatives.

CONCLUSION

The concept of bicausality and bicausal Bond graphs was presented. We attempted to link bicausality and Dirac structure of port-Hamiltonian systems through kernel representation. Every possible relations among the states and the control inputs were found from their bicausal Bond graphs. The idea was to find a method which provides the possible states, candidates for the flat representation of the PH systems. This method will be further extended for other port-Hamitonian physical systems.

Fig. 1 .

 1 Fig. 1. Causality in the Bond graphs indicated in red.

Fig. 2 .

 2 Fig. 2. (a) Integral causality of the storage elements. (b) Differential causality of the storage elements. e = 1 C t f dt and for the inductors f = 1 I t e dt (where C is the capacitance and I is the inductance). Concerning the dissipative elements, R, they obey to a static constitutive law which provides only algebraic relations between the effort and flow, e = f • R and f = e R . Fig. 3 depicts the causal Bond graph of the RLC circuit depicted on the left. The red dashed lines illustrate the power flow and the dependence among the port variables. Differential causality: The input's time derivative equals to the ratio of the output (see also Fig. 2b). For the capacitors wa have de dt = f C and for the inductors df dt = e I . Differential causality is used in the inverse Bond graphs, else called bicausal Bond graphs, as in Fig. 5.

Fig. 3 .

 3 Fig. 3. Electrical RLC circuit and its causal Bond graph.

Fig. 4 .

 4 Fig. 4. Relation among the e -f elements in the bicausal Bond graphs.

2 4.

 2 FROM BICAUSAL BOND GRAPHS TO DIRAC STRUCTURES In this section, we go further and analyze the connection between bicausal Bond graphs of electrical circuits and their PH formulation. Then going back, we can actually retrieve the flat representation of a PH system. In general, PH systems are determined by the interconnections among the elements included in a physical system through a Dirac Structure (DS) (see definition in (van der Schaft et al., 2014)). The DS connects all the port variables together and obeys to the power conservation among the elements (van der Schaft et al., 2014). Proposition 4.1. (Link between bicausal Bond graphs and Dirac structures): If the bicausal Bond graph admits a kernel representation (see definition in (van der Schaft et al., 2014)), then it is a Dirac structure, hence a PH system. As a consequence, in the case of the electrical circuits, the flat representation of their PH models can be analyzed through their bicausal Bond graphs.

Fig. 6 .

 6 Fig. 6. Electrical circuit of the ES system composed by the Split-Pi converter and the KiBaM battery.

Fig. 8

 8 Fig.8presents the bicausal Bond graph of the battery composed by two storing elements, the capacitors C 1b and C 2b of the battery with their differential causalities, one effort source, S e , which is the input (the voltage of the Split-Pi converter, v sc ), and one flow detector, D f , which is the output (the current of the Split-Pi converter, i sc ). Therefore, the state and input of the system are give:

Fig. 7 .

 7 Fig. 7. Causal Bond graph of the KiBaM battery.

)Fig. 8 .

 8 Fig. 8. Bicausal bond graph of the KiBaM battery.

Fig. 9 .

 9 Fig. 9. Causal bond graph of the Split-Pi converter.

Fig. 10 .

 10 Fig. 10. Bicausal bond graph of the Split-Pi converter.

Fig. 11 .

 11 Fig. 11. Actual control inputs in function of the flat outputs set of (14). v s , d 1sc and d 1sc are considered as constant in the reference PH system, while i b varies.

Fig. 12 .

 12 Fig. 12. Control inputs in function of the second set of flat outputs of (15). v s , d 1sc and d 1sc and i b have the same values as in the previous simulations (Fig. 11).

Wherever it is straightforward implied by the text, we will discard the time dependence.

ACKNOWLEDGEMENTS

This work is funded by the French National Research Agency within the framework of the project ANR-15-CE05-004-02 C 3 µ.