Marguerite Cuny

Marie-France Guerin

Barbara Petridou

Francoise Hayes

Ribosomal subunits and ribosomal proteins of Tetrahymena thermophila Effect of the presence of iodoacetamide during ribosome extraction on the properties of the subunits

 (c, FEBS 1983
Ribosomal subunits and ribosomal proteins of Tetrahymena thermophila

Effect of the presence of iodoacetamide during ribosome extraction on the properties of the subunits

Barbara PETRIDOU, Marguerite C U N Y , Maric-France G U E R I N , and FranCoise HAYES Laboratoire de Chimic Cellulairc, Institut de Biologie Physico-Chimique, Paris (Received April 11 /June 8. 1983) -EJB 830359 Proteolytic degradation of ribosomal proteins occurs during the preparation of subunits of the cytoplasmic ribosomes of the protozoa Tetrahymenu thernioplrila and the isolated subunits are inactive. Addition of 5 mM iodoacetamide to cell suspensions before extraction inhibits proteolytic activity and permits isolation of active subunits. The protein complements of these subunits have been characterized in two different two-dimensional elcctrophoretic systems, and their molecular weights have been determined.

Limited degradation of ribosomal proteins has been observed during isolation of ribosomes and ribosomal subunits from a variety of eukaryotic cells [l -61. This phenomenon was not observed in our earlier studies [7] with ribosomes of an amicronucleate Tetraliymeiza strain of unknown phenoset (Tetrahyniena 'pyr{forn~is' CGL) but has subscquently been shown to occur during preparation of ribosomes from several other amicronucleate strains [XI (and B. Petridou and F. Hayes, unpublished results). It also occurs, as shown here, during preparation of ribosomes from the micronucleate strain Tetruhymenu thermophila and leads to isolation of inactive ribosomal subunits unless protease inhibitors are present throughout the preparation process. Of a number of inhibitors tested, iodoacetamide was found to be the most effective. 40s and 60s ribosomal subunits prepared from T . thermophila in its presence are active in vitro and retain their activity during storage at low tcmperature. Alkylation of a small number of proteins in these subunits revealed by experiments with radioactive iodoacetamide appears not to impair their capacity to reassociate or to catalyse polyphenylalanine synthesis in vitro. The protein complements of active and inactive 40s and 60s ribosomal subunits of T. thernmphilu prepared in the presence and absence respectively of iodoacetamide have been characterized by polyacrylamide gcl electrophoresis in two different systems. The molecular weights of the proteins of active subunits have been estimated by SDS gcl electrophoresis.

MATERIALS AND METHODS

Strains and culture conditions

Tetrahymetza thermophila B 1868 mating types IV and VII were obtained from Dr D. Nanney (Dcpt of Zoology, University of Illinois, Urbana, Illinois 61 801, USA).

T. iherrnophilu B 1868 mating type 11 was provided by Dr P. Bruns (Section of Genetics, Physiology and Development, Cornell University, Ithaca, New York, USA) in the form of a functional heterokaryon [lo] carrying a micronucleus homozygotic for the cycloheximide-resistant mutation ChxA, and a wild type, cycloheximide-sensitive macronucleus (strain

Proteases and proteusc inhibilors

These were obtained from the following sources: papain, Sigma (USA) ; Staphylococcus uureus V 8 protease, Miles (USA); iodoacetamide, Serva (FRG); aprotinin, Sigma (USA); phenylmethylsulfonyl fluoride, Calbiochem (USA) ; Na-p-tosyl-L-lysine chloromethylketone, Serlabo (France); iod~[l-'~C]acetamide (specific activity 1047 MBq/mmol), Commissariat ti I' Energie Atomique (France).

Preparation of ribosomal subunits

All operations were carried out at 0 -4 ''C, unless otherwise indicated.

Exponentially growing cultures were harvested at cell densities of 2-3 x 105/ml by low-speed centrifugation (2000 xg, 10 min). Cells wcrc washed by centrifugation in 20 mM Tris/HCI, 100 inM KCI, 5 mM MgCI,, 6 mM 2-mercaptoelhanol, 0.25 M sucrose pH 7.6 and resuspended at a density of 10'-108/ml in the same buffer with or without addition of protease inhibitors at the following concentrations : iodoacetamide, 5 mM ; aprotinin and N-a-p-tosyl-L-lysine chloromethyl kctone, 100 pg/ml; phenylmethylsulfonyl fluoride, 3.5 mM. Nonidet P40 was then added to a final concentration of 0.24 (w/v) and lysis was allowed to proceed for 10 min in an ice bath, after which lysate volumes were doubled, their ionic composition was adjusted to 20 mM Tris/HCl, 2.5 mM MgCI,, 6 mM 2-mercaptoethanol, and protease inhibitors were added, when present, to maintain their initial concentrations. Diluted lysates were centrifuged at 10000 x g for 20 min and the supernatants, supplemented with GTP (0.2 mM) and puromycin (0.1 mM), were incubated for 45 min at 28'C and reccntrifuged at 10000 x g for 10 min (small volumes) or 6000 x g for 30 min (large volumes). Ribosomal subunits in the final supernatant were separated by centrifugation at 21 000 rev./min for I6 h (rotor SW 27) on 10-3OX sucrose gradients prepared in 20 mM Tris/HCl, 250 mM KCI, 2.5 mM MgCI,, 6 mM 2-mercaptoethanol or at 23 000 rev./min for 18 h (zonal rotor B14) on 7-30 o/, sucrose gradients prepared in the same buffer. Subunits were concentrated from pooled gradient fractions either by centrifugation at 80000 x g for 20 h after raising the Mg2+ concentration to 5 mM or by precipitation with poly(ethy1ene glycol) (final concentration 12 'x w/v) at 0 °C for 20min, after raising the Mg2+ concentration to 10 mM. Precipitated subunits were collected by centrifugation at 30000 x g for 30 min. Pellets of subunits prepared by either method were resuspended in 20 mM Tris/HCl, 250 mM KC1, 5 mM MgCI,, 6 mM 2-mercaptoethanol at a concentration of 100-175 units/ml and stored at -70 ' C.

Preparation of' ribosomal prolein5

Protein samples for one-dimensional gel electrophoresis were prepared by incubating suspensions of 40s or 6 0 s subunits (0.75 and 1A260nm unit respectively) in 0.05 ml of 0.05 M Tris/HCl pH 6.8, SDS 2 o/, (w/v), glycerol 7.5 "/, (w/v), 2-mercaptoethanol 1 "/, (v/v) at 100 -'C for 3 min. Samples were cooled and their 2-mercaptoethanol concentration was raised to 5 % (v/v) before loading onto gels. Samples for twodimensional gel electrophoresis were prepared by acetic acid extraction and acetone precipation according to Barritault et al. [12].

Assay of ribosome functions

To assay reassociation of ribosomal subunits in vitro, samples of purified 40s and 6 0 s particles (0.5 and 1.0 A260nm unit respectively) were dissolved together at 0 'C in 150 pl of 20 mM Tris/HCl, 10 mM MgCI,, 100 mM KCI, pH 7.6. The solutions were then incubated at 28 "C for 30 min, cooled to O"C, and 100-p1 samples were analysed by sedimentation at 19000 rev./min for 16 h at 4 "C on 3.6-m15 -20 "/, (w/w) linear sucrose gradients prepared in 20 mM Tris/HCl, 10 mM MgCl,, I00 mM KCI. The distribution of 260-nm-absorbing material in the centrifuged gradients was recorded.

Poly(U)-directed polyphenylalanine synthesis and peptidyl transferase activity were measured as described previously [7] with the minor modification that in the latter assay A c [ ~H ] L ~u -C-A-C-C-A was replaced by f13H]Met-tRNA (spec. act. 5600 counts min-' pmol methionine-') which was prepared by standard methods.

Polyacrylamide gel electrophoresis of ribosomal proteins

The procedure used for one-dimensional SDS/polyacrylamide gel electrophoresis was essentially that described by . Gel slabs prepared by pouring a 5 % stacking gel (1 mm x 30 mm x 370 mm) onto a 12.5-20 "/, separation gel (1 mm x 270 mm x 370 mm) were used without preelectrophoresis. Electrophoresis was carried out at 4 "C for 1 h at 100 V followed by 17 h at 200 V. Gels were stained with Coomassie brilliant blue R 250.

For two-dimensional gel electrophoresis the procedures of Kaltschmidt and Wittmann [I41 (first dimension pH 8.6, second dimension pH 4.5) and Zinker and Warner [15] (first dimension pH 8.6, second dimension SDS) were used with the following modifications. With both methods, the protein samples contained 5 % (v/v) 2-mercaptoethanol [16], the first dimension electrophoresis was carried out in cylindrical gels (4 mm x 125 mm) at 150 V for 16 h and separate first-dimension gels were uscd to fractionate acidic and basic ribosomal proteins.

In the Kaltschmidt and Wittman procedure, the second dimension electrophoresis was performed in gel slabs (2 mm x 135 mm x 230 mm) at 250 V for 16 h (first-dimension gels of acidic and basic proteins bonded to separate second-dimension gels) or in larger gel slabs (2 mm x 300 mm x 300 mm) at 300 V for 16 h (first-dimension gels of acidic and basic proteins bonded to the same second-dimension gel). Also, the firstdimension upper chamber buffer contained 0.02 "/, (w/v) cysteamine chloride.

In the Zinker and Warner procedure, the second-dimension electrophoresis was carried out in the larger gel slabs (2mm x 300 mm x 300 mm) at 250 V for 15 h (first-dimension gels of acidic and basic proteins bonded to thc same seconddimension gel). After electrophoresis, the gels were stained with Coomassie brilliant blue 250 R.

The molecular weights of ribosomal proteins were estimated by electrophoresis in polyacrylamide/SDS gel slabs according to Laemmli [17]. Samples excised from stained spots in two-dimensional Kaltschmidt and Wittmann gels were dialysed three times for 60 min against 10 ml of 62 mM Tris/HCl, 0.5 (v/v) 2-mercaptoethanol pH 6.8, placed in the sample wells of 15 "/, acrylamide/O.l "/, SDS (w/v) gel slabs and immobilized by filling the wells with 1 "/, agarose (w/v). Proteins used as molecular weight standards were placed in sample wells in solution since preliminary experiments showed that their electrophoretic mobilities were the same whether they were used in this way or in the form of 15 % polyacrylamide gels.

Samples of standard proteins contained either 2 pg of single chromatographically purified Escherichia coli 3 0 s ribosomal proteins or lysozyme or a mixture of 2 pg each of four nonribosomal proteins (excluding lysozyme) in 50 p1 of sample buffer.

Standard proteins and their molecular weights were as follows. For 30s ribosomal proteins, M , values calculated from primary sequences [18,19] were rounded off to the nearest 100 : S2, 26600; S3, 25900; S4, 23100; S5, 17500; S7K, 19700; S10, 12300; S12, 13600. For non-ribosomal proteins, M , values were : bovine serum albumin, 68000; ovalbumin, 45000; trypsin inhibitor, 21 500; egg white lysozyme, 14400; cytochrome c, 12500.

The relativc mobilities ( R F ) of Tetvahymem ribosomal proteins were calculated using trypsin inhibitor as standard. Molecular weights of proteins with mobilities higher than that of E. coli 30s protein S2 were calculated from the linear relationship observed between log RF and molecular weight for E. coli 30s proteins S2, S3, S4, S5, S7K, S10, S12, lysozyme and cytochrome c. For proteins with mobilities lower than that of E. coli 30s protein S2, thc linear relationship observed between R, and log(mo1ecular weight) for bovine serum albumin, ovalbumin, trypsin inhibitor, lysozyme and E. coli 30s ribosomal proteins S10 and S12 was used. Calculated values are given to the nearest 500.

RESULTS

Protein synthesis in Tetrahymena thermophila in vivo

Measurement of the rate of incorporation of radioactive amino acids into acid-insoluble material in growing cultures of amicronucleate strains of Tctrahyrnena has shown [8] that the rate of protein synthesis in vivo in these strains increases during early exponential growth, reaches a maximum at cell densities of 2-3 x 105/ml and then falls rapidly by a factor of about 2 Polyphen ylalanine-syn thesizing act ivit ic,s qf'ribosonzul .subun its isoluted in the presence and absence of protease inhibitors Polyphenylalanine synthesis was carried out in viiro as dcscribed previously [7] except that reaction mixtures were incubated at 28 'C. All results listed are averages of values obtained using 0.5 and 1.0 unit of equimolar mixtures of 40s and 60s subunits prepared in the presence of the indicated inhibitors. Control reaction mixtures without ribosomes incorporated a total of 95 counts/min. Incorporation by the iodoacetamide sample corresponds to polymerisation of 2-phenylalanine residues pcr ribosome although cell multiplication continues for a further one or two generations. Very similar observations have been made with T. tkrrvutophila (results not shown) and for this reason cells used for ribosome preparation were harvested from early exponential phase cultures containing 2-3 x 105 cells/mI.

Protrolytic degradation ofribosomalproteiizs during preparation o j ribosomal subunits

Attempts to prepare ribosomal subunits from T. thermophilu by the procedure previously used successfully [7] with an amicronucleate strain of Tetrahymena gave satisfactory yields of particles sedimenting at 40s and 60s. However, thesc products reassociated very poorly in vitro and possessed little or no polyphenylalanine-synthesizing activity. Electrophoretic analysis of their protein complements gave results which suggested the presence of protein degradation products and the effect of carrying out ribosome preparation in the presence of protease inhibitors was therefore tested. Table 1 lists the results of assays of the polyphenylalanine-synthesizing activity of ribosomal subunits prepared from T. thermophila B 1868 IV in the absence and presence of protease inhibitors used at the concentrations indicated in Materials and Methods. Of the four inhibitors tested iodoacetamide was the most effective and was used in all subsequent work.

Comparison qf the properties of ribosomal subunits prepared in the absence and presence of iodoacetamide

To compare the properties of ribosomal subunits isolated in the presence and absence of iodoacetamide two preparations were made in parallel from cells harvested at the same time from the same culture. Identical results were observed with all three strains of T. thermuphila used; only those obtained with strain B 1868 IV are presented.

Assays of the polyphenylalanine-synthesizing activity of subunits isolated in the presence and absence of iodoacetamide gave the results listed in Table 2. Subunits prepared in the presence of this inhibitor are highly active whereas, as expected, those isolated in its absence are almost completely inactive. Experiments with the two possible combinations of subunits Table 3. Peptidyltransferose activity of 6 0 s subunits i,rolutetl in thr presence (+) and absence (-) of iodoacetamide Peptidyltransferase assays were carried out as described previously prepared in the presence and absence of iodoacetamide shows that loss of polyphenylalanine-synthesizing activity is associated mainly, although not entirely with degradation of 40s subunit proteins (Table 2, expts 7 and 8). In agreement with this result, data in Table 3 show that 60s subunits prepared in the presence and absence of iodoacetamide possess the same peptidyltransferase activity, i. e. that this 60s subunit function is not sensitive to proteolytic degradation during subunit isolation. In contrast to these results, assays of the capacity or 40s and 60s subunits prepared in the presence and absence of iodoacetamide to reassociate in vitro (Fig. 1) shows that this function is considerably more sensitive to proteolytic degradation in the 60s than in the 40s subunit. Although the experimental conditions used to obtain the data in Table 2 and Fig. 1 differ (absence of poly(U), tRNA and cytosol in subunit reassociation mixtures) comparison of these results shows that 80s ribosomes formcd in high yield from 40s subunits prepared in the absence of iodoacetamide and 60s subunits prepared in its presence (Fig. 1C) possess very low phenylalanine-synthesizing activity (Table 2, expt 7). The small phenylalanine incorporation observed may in fact be due to-contamination of the 60s subunits prepared in the presence of iodoacetamide with traces of 40s subunits (compare expts 2 and 7 of Table 2). In contrast 80s subunits obtained in low yield by reassociation of 40s subunits prepared in the presence of iodoacetamide and 60s subunits isolated in its absence 2, expt 8). We conclude that proteolytic degradation during isolation of ribosomal subunits from T. thrrnioplzila affects mainly 60 S subunit ribosomal proteins involved in subunit interaction and 40s subunit proteins related to a process o r processes in protein synthesis later than SOS ribosome formation.

Protein conzyltvnents qf 40S and 60 S subunits isolritrd in tlzc prcwricc atid cibscncc. of ioclou (,etmiidr Analysis of the proteins of 40 S and 60 S subunits isolated from strains B1868 IV and B186X VII in the presence and absence of iodoacctamide by electrophoresis in one-dimensional SDS/polyacrylamide gradient gels [I 31 gave the results shown in Fig. 2. The migration patterns of 40s and 6 0 s subunit proteins of both strains arc identical and are clearly sensitivc to the presence or absence of iodoacctamide during subunit isolation, the differences in the two patterns being more important in the case of the small than the large subunit. (Identical results, which are not shown, were obtained with strain CU329.)

The resolving power of one-dimensional SDS gel electrophoresis is not sufficient to separate all the proteins present in either the 40s o r thc 6 0 s ribosomal subunit and so does not permit assessment of the total number of proteins which are subject to proteolytic degradation. In order to obtain this information and identify thc sensitive proteins, twodimensional gel electrophoresis according to Kaltschmidt and Wittmann [14] was used.

I n Fig. 3, which shows the results of these analyses, the numbering system presented in the next section is used to ~~ Fig. 2. One-dimensional .sepamtion patterns of proteins of' 40s and 60 S riho.sornul suhunits prepurid in ihc presence (+) und a h r i c e (-) u/' iodouwturxidi,. Ribosomal proteins of 40s and 60s subunits of T. thewnopliilu B 1 868 IV and B1868 VII isolated in the presence and abaence of iodoacetamide, were prepared and analyzed by electrophoresis on 12.5 -20 "r;, linear polyacrylamide gradient gels in the presence of SDS a s described in Materials and Methods. Samples for analysis contained 30 pg of 40s and 40 pg of 60s subunit proteins. After electrophoresis gels were stained with Coomassie brilliant bluc and photographed. The arrow shows the band corresponding to protein S1 (Fig. 3) Fig. 3. Two-dimensional electroplzorc,tic analyyis ofacidic and basic prot&s of40S and 60 S ribosomul subunits o f T . thermophila BIVprepared in h e prescwe and absence ofiodoacc,tamide. Ribosomal proteins were prepared for two-dimensional electrophoresis as described in Materials and Methods. Acidic and basic ribosomal proteins were fractionated on separate first-dimension gels each of which was then bonded to a separate second-dimension gel slab. Samples for analysis of basic proteins contained 150 pg and those for analysis of acidic proteins 300 pg of total proteins. Electrophoresis was carried out as described in Materials and Methods and gel slabs were then stained and photographed Ribosomal subunits were isolated in the presence of iodoacetamide. Electrophoresis was carried out as described in the legend to Fig. 3 using separate gels for first-dimension separation of acidic and basic proteins and a single gcl slab for the second-dimension separalion. After electrophoresis gel slabs were stained and photographed (A, C). TypIcal migration patterns of40S and 6 0 s ribosomal proteins; (B, D) schematic drawings of 40s and 60s ribosomal protein migration patterns identify proteins in migration patterns obtained with samples prepared from ribosomal subunits isolated in the presence of iodoacetamide. Spots in the migration patterns of proteins of subunits isolated in the absence of iodoacetamide are not numbered according to this system since one-to-one relationships between spots in the two types of pattern cannot be deduced from the data in Fig. 3.

The following major differences between the proteins of ribosomal subunits isolated in the presence and absence of iodoacetamide are visible in Fig. 3.

40s subunits; cicidic proteins. Of the four acidic proteins (S2,SlI,S17,S23)inFig. 3A, three(S2,S17,S23)arepresent with unaltered electrophoretic mobilities in Fig. 3 B. Spot S 11 is not seen in Fig. 3 B, showing that this protein has been lost or degraded during isolation of 40s subunits in the absence of iodoacetamide. A further difference between Fig. 3A andB is the presence of four additional spots labelled a, b, c, d in the latter. These spots presumably represent degradation products of ribosomal proteins. Although spots b, c and d could represent degradation products of the missing spot S 11 since they have higher second-dimension mobilities and thus probably contain proteins with lower molecular weights than S 11, they could also be fragments of degraded basic proteins. The additional acidic spot a must presumably be a fragment of a basic 40s protein since no acidic protein with lower seconddimension mobility than spot a is absent in Fig. 3B.

40s subunit; basic proteins. The electrophoretic properties of many or all of the upper group of proteins in Fig. 3C (S3 -S 8) are clearly altered in Fig. 3 D. Alterations are also evident in the properties of the central group of proteins in Fig. 3C Fig. 5. Ti~n-di/neff.~in~al electinplzoietit analysis of'total proteins oj 4 0 s und 60s ribosnmul subunits qf T. thermophila B I868 IV nccording 10 ZinkrJr and Warner/ IS]. Preparation of ribosomal subunits and first-dimension electrophoresis were as described in the legend to Fig. 3. Seconddimension electrophoresis was as described in Materials and Methods. After electrophoresis gel slabs were stained and photographed. (A, C) Typical migration patterns of40S and 60s ribosoinal proteins; (B, D) schematic drawings of40S and 60s ribosomal protein migration patterns (spots L9, L 15 and L24 in C and D are split due to breakage of the first-dimension gel) (S9-S26). InFig. 3DspotsS14andS16 appeartocomigrate; spot S24 has either disappeared or comigrates with S26; and spots S22 and S25 have approached each other. In addition several extra spots (e, f, g, h) are visible in Fig. 3D. 60Ssubunits; acidicproteins. Comparison of Fig. 3 E and F shows that proteins prepared from 60s subunits isolated in the absence of iodoacetamide lack acidic 60s protein L1 but contain normal amounts of acidic proteins L 10, L 16 and L20, with unchanged electrophoretic properties.

6OS .subunit ; basic. proteins. The differences between the basic proteins of 6 0 s subunits prepared in the presence and absence of iodoacetamide are relatively limited. Protein L25, visible as a strong spot in Fig. 3G, is present only in traces in Fig. 3H. However, all other stained spots in Fig. 3G appear undiminished in Fig. 3 H. Other differences between the migration patterns of proteins in Fig. 3G and H are the appearance of additional faint spots, e. g. a, b, c, and duplication of certain spots, e. g. L3. They suggest that several 60s subunit proteins in addition to L25 are partially modificd during subunit preparation in the absence of iodoacetamide.

Although alkylation of proteins by iodoacetamide is not expected to cause large alterations in their electrophoretic properties, the possibility that some of the differences in the migration patterns of proteins of ribosomal subunits prepared in the presence and absence of iodoacetamide might be due to alkylation could not be excluded a priori. Ribosomal subunit preparation was therefore carried out in the presence of radioactive acetamide, subunit proteins were fractionated by two-dimensional gel electrophoresis, and their radioactivity was measured. Significant amounts of radioactivity were found in protcin L28; minor amountsin proteinsS4, S5, S 8 , S15, L4, L12 and L33, and traces in S28. Comparison of these observations with the results in Fig. 3 shows that the only proteins labelled by iodoacetamide which behave differently in samples isolated from subunits prepared in the presence and absence of this inhibitor are S 5 and L4. We conclude that Fig. 6. Electrophoretic rinalvsis qfbasic proteins oj40 S ribosoniul subunitsprc,parrd,froin exponentially growing and starwdT. thermophila B1868 IV. Ribosomal subunits were prepared in the presence of iodoacetamide from exponentially growing and starved cells (Materials and Methods) and thcir basic proteins were analyzed by two-dimensional gel electrophoresis according to Kaltschmidt and Wittmann [ 141 and Zinker and Warner [I 51. The migration patterns of proteins S3 ~ S9 in the four gel slabs are shown in Fig. 6. The migration patterns of the remaining basic proteins of 40s subunits of growing and starved cells were identical. (A) Growing cells and (B) starved cells, electrophoresis according to Kaltschmidt and Wittmann, (C) growing cells and (D) starved cells, electrophoresis according to Zinker and Warner. Spots 7' and 7" contain modified forms of protein S7 alkylation of ribosomal proteins by iodoacetamide does not contribute significantly to the differences between the electrophoretic migration patterns of proteins prepared from ribosomal subunits isolated in the presence and absence of this in hibit or.

Migration patterns of ribosomal proteins of T. thermophila in two two-dimensional electrophoresis systems

Because the electrophoretic migration patterns of acidic and basic ribosomal proteins in Fig. 3 were obtained using independent first and second-dimension gels they do not permit correlation of the electrophoretic mobilities of these two subgroups of ribosomal proteins. This correlation can be achieved by carrying out second-dimension electrophoresis of acidic and basic proteins in the same gel slab after firstdimension migration in the same or different gels. The latter alternative permits differential loading of gels to overcome the low staining efficiency of acidic proteins without loss of resolution of basic proteins [16]. Fig. 4 contains migration patterns representative of those obtained in analyses of this type carried out using the electrophoresis conditions of Kaltschmidt and Wittmann [I41 and schematic drawings which summarize the results of a large number of analyses of independent preparations of ribosomal proteins. Twodimensional analysis were also carried out according to Zinker and Warner [I 51 (first-dimension electrophoresis as in the procedure of Kaltschmidt and Wittmann [14], second dimension electrophoresis in the presence of SDS). Migration patterns typical of those obtained in these experiments and corresponding schematic drawings are shown in Fig. 5. Nomenclature of spots in Fig. 5 The same numbering system is used in Fig. 4 and 5. Spots in gel slabs of the types shown in these figures were correlated by comparison of two-dimensional patterns obtained using first dimension gels prepared under the same conditions with the same protein preparations. Correlation was facilitated in some cases by use of the measured molecular weights of proteins separated by the Kaltschmidt and Wittmann electrophoresis procedure (see below).

40s subunit proteins in sturved cells

Kristiansen and coworkers showed [I 1,201 that phosphorylation of a 40s ribosomal protein occurs when exponentially growing Tetrahymena is transferred to non-nutrient medium and numbered it S6 on the basis of its position in twodimensional gel electrophoretic analyses carried out according to Zinker and Warner [15]. Protein S6 in Fig. 2 9 of [20] corresponds in position to protein S 7 in Fig. 5 A and B. Results summarized in Fig. 6 confirm that the electrophoretic properties of protein S7 are altered in starved cells in a manner consistent with the introduction of acidic substituents.

Molecular weights qf ribosomul proteins q f T . thermophila B1868 IV Third-dimension electrophoresis ( Materials and Methods) showed that almost all (70 of 71) samples cut out of stained spots in two-dimensional gels such as those shown in Fig. 4A and 5A contained a single protein. Exceptions were the 40s 3) and firstdimension mobility as protein L28 (Fig. 4C andD) comigrates with the latter (Fig. 5C andD). These results show that twodimensional gel electrophoresis according to Kaltschmidt and Wittmann resolves all 40s subunit proteins and, with one exception, all 60s subunit proteins of T . thermophila.

DISCUSSION

Characterisation of the cytoplasmic ribosomal subunits of Tetrahymenu thermophila and of their proteins was undertaken to obtain information required for studies, now in progress, of the structure of these particles. For such studies it is essential to use functionally and therefore structurally intact ribosomal subunits. Although active subunits can be prepared from the amicronucleate strain Tetruhymena 'pyrformis' CGL [7] without precautions against proteolytic degradation, inactive subunits containing degraded proteins are obtained under such conditions from many other amicronucleate strains [8] and, as shown here, from T. thermophila. These observations confirm and extend earlier reports of proteolytic degradation of ribosomal proteins during the isolation of ribosomes from rabbit reticulocytes [2] and rat liver [1,3,51, from an amicronucleate strain of Tetrahymena [5], and from T. therrnophila [4]. Since ribosomal protein degradation in extracts of T. thermophilu i s inhibited by iodoacetamide it seems probable that one or more of the extensively studicd thiol-requiring neutral proteases of Tetrahymma [21-271 is involved in this process. Although, as judged by thc polyphenylalanine-synthesizing activity of ribosomal subunits prepared in i t s presence (Tables 1,2,3) and the absence of degradation products in the electrophoretic migration patterns of their proteins (Fig. 2), iodoacetamide is a highly efficient inhibitor of proteolytic activity in extracts of T. thermophila, these results do not show that no degradation of ribosomal proteins takes place in its presence. It has, for example, been observed that detectable changes in the protein complements of rat liver ribosomes can be produced by limited protease digestion in vitro without effect on their proteinsynthesizing activity in v i m [l].

Proteins in both 40s and 60s subunits are alkylated by iodoacetamide when this inhibitor is present during ribosome isolation. Substitution of a small number of SH groups in ribosomal proteins should not alter their electrophoretic properties significantly but could affect their contributions to ribosomal functions. The fact that subunits prepared from T. thermophila in the presence of iodoacetamide are active in polyphenylalanine synthesis shows that any such effect i s limited in extent. We have shown elsewhere [8] that ribosomal subunits prepared in the presence or absence of iodoacetamide from two amicronucleate strains of Tetrahymena in extracts of which proteolytic degradation of ribosomal proteins is not observed, possess the same polyphenylalanine-synthesizing activity, i. e. that alkylation of the proteins of these subunits by iodoacetamide docs not affect their activity in vitro.

We believe that all the ribosomal proteins of T. thermophilu are represented in the schematic drawings in Fig. 4 and5. No acidic proteins with first-dimension electrophoretic mobilities greater than those of S l l and L6 (e.g. the highly acidic phosphorylated proteins L35 and L36 of yeast ribosomes [15]) and no basic proteins with higher mobilities than S28 and L40 were detected by reducing the duration of first-dimension electrophoresis to 4 h. In addition, no protein with an isoelectric pH of 8.6, i. e. that of the first-dimension in the Kaltschmidt and Wittmann system, was revealed by raising this pH to 9.5 or lowering it to 7.

Finally, although acidic proteins S 2 (40s) and L I (60s) possess very similar mobilities in both two-dimensional electrophoresis systems used and very similar molecular weigths (Table 4), suggesting that they are the same protein which is partitioned between the subunits upon dissociation of 80s ribosomes, their non-identity was established by analyses of their proteolytic degradation products according to Cleveland et al. [25] using papain and Stuphylococcus aureus V 8 proteases. In both cases the two proteins gave different patterns of digestion products (results not shown).

Revision of the protein catalogues in Fig. 4 and 5 and Table 4 may, however, be necessary for three reasons. a) As mentioned earlier, the assumption that active ribosomal subunits isolated in the presence of iodoacetamide contain full complements of intact ribosomal proteins, though plausible, is not entirely convincing. b) Only those spots seen reproducibly in two-dimensional electrophoretic analyses of all protein samples are shown in Fig. 4B, D and 5B, D. Additional spots observed in variable amounts in analyses of some but not all protein samples were provisionally classified as non-ribosomal contaminants and are indicated as unnumbered dotted outlines in Fig. 4B, D; 5B, D. c) Further work will be necessary t o confirm the status of the high-molecular-weight protein S 1 which is seen reproducibly but always in small amounts in two-dimensional electrophoretic analyses of 40s subunit proteins prepared in the prcsence of protease inhibitors (a sharp band corresponding to S1 is visible, indicated by a n arrow in Fig. 2, in onedimensional electrophoretic analyses of proteins of 4 0 s subunits of T. thermophilu BIV and BVIl prepared in the presence but not in the absence of iodoacetamide).

With these reservations we conclude that cytoplasmic ribosomes of T. thermophila contain 72 different proteins (Table 4). Their average molecular weight, 23 100, is close to the values obtained for higher eukaryotic organisms [2, 29-311 and significantly different from the value of 18 900 reported for the lower eukaryote Succhuromyces carlshergensis [32, 331.

  [7] using q3H]Met-tRNA in place of AC[~H]L~U-C-A-C-C

Fig. 1 ,

 1 Fig.1, Rrussociatiori q f purifkd 40 S and 60 S riho.soriiu/ .ruhuriir~s prepared in the presence (+) or ubscnce (-) of iodoaci~/urnide. 40s and 60 S subunits werc prepared in the presence (+) and absence (-) of iodoacetamide as described in Materials and Methods, from two samples of cells harvested from a single of T. tlii~rmophilu B 1868 IV. Mixtures containing thc four possible combinations of the isolated subunits were prepared in 20 m M TrisiHCI, 10 mM MgCI,, 100 mM KCI (total concentration 10 AZhOnm units/ml, AZbOnm ratio 40S:hOS = 1 :2), incubated at 28 C and analysed by scdiinentation for 16 h at 19000 rev./min on 3.6-ml 5-20 ";,linear sucrose gradients (for further dctails see Materials and Methods). The figurc shows the four sedimentation profiles obtained

Fig. 4 .

 4 Fig. 3. Two-dimensional electroplzorc,tic analyyis ofacidic and basic prot&s of40S and 60 S ribosomul subunits o f T . thermophila BIVprepared in h e prescwe and absence ofiodoacc,tamide. Ribosomal proteins were prepared for two-dimensional electrophoresis as described in Materials and Methods. Acidic and basic ribosomal proteins were fractionated on separate first-dimension gels each of which was then bonded to a separate second-dimension gel slab. Samples for analysis of basic proteins contained 150 pg and those for analysis of acidic proteins 300 pg of total proteins. Electrophoresis was carried out as described in Materials and Methods and gel slabs were then stained and photographed. The photographs were combined to produce the results shown. (A, C ) Acidic and basic proteins of 40s subunits prepared in the presence of iodoacetaniide; (B, D) acidic and basic proteins of 40s subunits prepared in the absence ofiodoacetamide; (E, G) acidic and basic proteins of 60s subunits prepared in the presence of iodoacetamide; (F, H) acidic and basic proteins of60S subunits prepared in the absence of iodoacetamide

Table

  1

Table 2 .

 2 Comparison ofthe activities of ribosomal subunits isoluted in the presence (+) and absence I-) of' iodoacetuinide Polyphenylalanine synthesis was carried out in v i m as in the legend to Table1and was measured after 45 min. Control reaction mixtures without ribosomes incorporated a total of 35 counts/min. Incorporation by 40 S (+) + 60 S (+) corresponds to polymerisation of3.8

	phenylalanine residues per ribosome
	Expt Subunits	Amount	[' 4C]Phenylalanine
				incorporation
			AZbOnm units	counts/min
	1.	40 S (+)	0.3	210
	2.	60 S (+)	0.6	780
	3.	40 S (+) + 60 S (+)	0.3 0.6 +	11 360
	4.	40 S (-)	0.3	50
	5.	60 S (-)	0.6	80
	6.	40 S (-) + 60 S (-)	0.3 0.6 +	360
	7.	40 S (-) + 60 S (+)	0.3 0.6 +	1000
	8.	40 S (+) + 60 S (-)	0.3 0.6 +	5 500

Table 4 .

 4 Molecular kceights of' ribosomal proteins qf T. thermophila Molecular weights were estimated, as described in Materials and Methods, by SDS/Polyacrylamide gcl clcctrophoresis

	40s subunit		6 0 s subunit
	Protein	Mr	Protein	M,
	s 1	71 000	L1	45 000
	s 2	48 000	L2	56000
	s 3	48 000	L 3	56000
	s 4 ss	40 000 48000	L 4 L5	45000 41 000
	S6	43 000	L6	39 000
	s 7	40 000	L7	2.5000
	S 8	21 500	L8	25000
	s 9 s 10	24000 20 000	L9 L 10	22000 1 5000
	s11	19000	L11	21 500
	s 1 2	21 SO0	L12	24 500
	S13	28000	L13	2s 000
	S14	17 SO0	L14	22 SO0
	SIS	17 000	L15	19500
	S 16	17 SO0	L16	15000
	S17	14000	L17	21 500
	518	17 500	L18	20 000
	S19 s 20	18000 18 500	L 19 L 20	17 SO0 14 500
	s21 s 22	1s 500 1 7000	L21 L 22	17000 22 500
	S 23	12000	L 23	1 8 000
	S 24	18000	L 23'	17500
	s 2 s	16 000	L 24	1s SO0
	S 26	17 SO0	L 2s	21 000
	S 27	17 SO0	L 26	17 500
	S28	17 000	L 27	1 6 000
	S 29	8 SO0	L 28	17500
	s30	16 000	L 29	1 6 000
	S31	9 SO0	L30	1 7 500
			L31	1s 500
			L 32	17000
			L33	17 000
			L 34	I S SO0
			L3S	17 SO0
			L 36	1 5 000
			L 37	1 5 000
			L38	15 SO0
			L39	I0 000
			L 40	1 I SO0
	spot S6 which contained traces of S7 and the 60s spot labellcd
	L23/L23' which was resolved in third dimension gels into two
	components L23 and L23', ncithcr of which had the third-
	dimension mobility of any of the spots neighbouring L23/L23'
	in two-dimensional gels, i.e. L18, L21, L22, L25, L26, L27.
	Resolution of spot L23/L23' into two components during
	electrophoresis according to Kaltschmidt and Wittmann can be
	achieved by increasing the duration of second-dimension
	electrophoresis. L23 and L23' are resolved during electro-
	phoresis according to Zinker and Warner, but in this case L23'
	which has the same molecular weight (Table

Table 4

 4

lists the molecular weigths of the proteins present in the 71 numbered spots in

Fig. 4 B and D.

The work described here has been supported by grants from the Centre Nutional de la Recherche Scientifique, the Fondution Nutionale pour la Recherche Midieale Fruncaise, and the North Atluntic Treaty Organization. One of us, Barbara Petridou thanks the French Goveriimcnt for a fellowship.


Yijing Chen
email: yijing.chen@lss.supelec.fr

Gilney Damm
email: gilney.damm@ibisc.fr

Abdelkrim Benchaib
email: ab-delkrim.benchaib@alstom.com

Franc ¸oise Lamnabhi-Lagarrigue
email: francoise.lamnabhi-lagarrigue@lss.supelec.fr

F Lamnabhi-Lagarrigue

Analysis of Stability and Dynamic Limitations of a VSC Terminal with DC Voltage Droop Control

This paper addresses the problem of analyzing the dynamic limitations of a VSC terminal with DC voltage droop control. Thanks to its fast response, DC voltage droop control is usually used to adjust the DC voltage in case of power imbalance. To ensure the stability of the VSC terminal, the dynamics between the dq currents (i l,dq ) and the DC voltage (uc) must be well distinguished. Therefore, the choice of the droop control gain Ku and the dq current control gains K dq is still a challenging problem. First, we assume that there exist Ku and K dq such that a slow and a fast dynamics can be imposed on uc and i l,dq , respectively. Then, we propose a novel methodology based on separation of dynamics to prove the existence of these control gains. Subsequently, a detailed stability analysis is carried out via Lyapunov theory, from which a necessary condition and a sufficient condition are derived for the control gains. To verify this methodology, numerical simulations are carried out in MATLAB/SIMULINK.

I. INTRODUCTION

Voltage-source converter high voltage direct current (VSC-HVDC) systems have been greatly promoted in today's electrical networks due to their flexibility, controllability and efficiency. They are specially suitable for long-distance power transmission, interconnection between asynchronous AC networks, underground/undersea cable transmission systems, etc. In order to enlarge power transmission capacity and realize power exchange between multi-suppliers and multi-consumers, the use of multi-terminal HVDC (MT-HVDC) systems has been increasingly widely used [START_REF] Long | Consideration for implementing multiterminal DC systems[END_REF], [START_REF] Chen | A detailed study on a DC-voltage-based control scheme using a multi-terminal HVDC system for frequency control[END_REF]. However, the control design for MT-HVDC systems is still a major problem. The control scheme should be strong enough to maintain the system in normal operation in case of small-signal perturbations [START_REF] Kalcon | Smallsignal stability analysis of multi-terminal VSC-based DC transmission systems[END_REF] or loss of some terminals [START_REF] Liang | Control of multi-terminal VSC-HVDC transmission for offshore wind power[END_REF]. In general, a master controller and local controllers compose the control system where the master one provides information such as the reference values for DC voltage, the active/reactive power to coordinate the operations of the local ones. In an MT-HVDC system, at least one terminal is in charge of regulating the DC-bus voltage level, and hence various control configurations are proposed such as masterslave control scheme, voltage margin method [START_REF] Nakajima | A control system for HVDC transmission by voltage sourced converters[END_REF].

In this paper, we consider a very popular control technique called DC voltage droop control. It is characterized by the change of power flow in proportion to the change of the DC voltage. The main advantage of the DC voltage droop control is that it involves more than one terminal in regulating the DC-bus voltage based on their predetermined DC voltage droop characteristics such that more terminals share the duty.

As a matter of fact, the use of DC voltage droop control has been widely studied in the literature [START_REF] Haileselassie | Impact of DC line voltage drops on power flow of mtdc using droop control[END_REF], [START_REF] Prieto-Araujo | Methodology for droop control dynamic analysis of multiterminal VSC-HVDC grids for offshore wind farms[END_REF]. Actually, these control structures belong to the family of vector current control since two loops are usually involved, i.e. an inner current (i) loop and an outer loop. The DC voltage droop control considered as the outer loop is used to regulate the DC voltage (u c ) and provide a current reference i * to the current loop. Then, different current loop controllers [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF], [START_REF] Chen | Nonlinear control design for a multi-terminal VSC-HVDC system[END_REF], [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF] are proposed to make i converge to i * . It is very important to notice that the vector current control is derived under the assumption that the current loop has much faster dynamics than the outer one. Thus, the dynamics of i * is neglected when designing the current control scheme. Unfortunately, there is no theoretical demonstration of this assumption. Moreover, it is still unknown which dynamic limitations should be imposed on the current and the DC voltage. This paper is devoted to the stability and dynamic limitation analysis of a VSC station equipped with DC voltage droop control. A sufficient condition for selecting the droop and the current control gains is derived based on singular perturbation theory and Lyapunov theory.

The paper is organized as follows. An averaged statespace model is introduced in Section II. The vector current control scheme with DC voltage droop control is presented in Section III. Section IV gives a detailed stability analysis from which the dynamic limitations on the current and the DC voltage are clarified. A necessary condition and a sufficient condition for choosing the control gains are derived and the steady-state analysis of the system is also presented. The theoretical analysis in this paper is verified by numerical simulations in Section V. Conclusions are drawn in Section VI.

II. MODELING

Figure 1 depicts a simplified configuration of a VSC terminal connected to AC system where R l + jL l represents the phase reactor and C is the DC capacitor used to reduce the ripple of the DC voltage.

An averaged state-space model established in a synchronous dq reference frame for the VSC terminal is given by [START_REF] Blasko | A new mathematical model and control of a three-phase AC-DC voltage source converter[END_REF]

di ld dt = - R l L l i ld + ωi lq + u c 2L l M d - v ld L l ( 1
)
di lq dt = - R l L l i lq -ωi ld + u c 2L l M q - v lq L l ( 2
)
du c dt = - 1 C [i c + 3 2
v ld i ld + v lq i lq u c ] (3)
where v l,dq , v c,dq and i l,dq are the AC system voltages, the converter voltages and the currents through the phase reactor, respectively, M dq are the modulation indices considered as the control inputs and ω is the frequency of AC system. By convention, i c is positive if the power flows from left to right as shown in Fig. 1.

Remark 1: Due to the physical limitations of the converter itself, the modulation indices should always satisfy M 2 d + M 2 q ≤ 1 and the domain of operation for u c is restricted to a narrow area, i.e. u c ∈ [u min , u max ].

Remark 2: In the present paper, we want to investigate the local behavior of the converter and hence, for convenience, we consider that the discussed terminal is connected to a controlled current source represented by i c . It can be either negative (as a supplier) or positive (as a consumer).

Remark 3: For the sake of simplicity, the dq reference frame is chosen such that v lq = 0 and v ld = V l,rms .

The instantaneous active and reactive powers on the AC side of the converter are

P l = 3 2 (v ld i ld + v lq i lq ) = 3 2 v ld i ld (4)
Q l = 3 2 (v lq i ld -v ld i lq ) = - 3 2 v ld i lq (5)

III. CONTROL STRUCTURE

An inner current loop and an outer loop compose the vector current control system. Various outer loops exist for different control objectives [START_REF] Xu | Multi-terminal DC transmission systems for connecting large offshore wind farms[END_REF], [START_REF] Haileselassie | Multi-terminal VSC-HVDC system for integration of offshore wind farms and green electrification of platforms in the North Sea[END_REF]. In this paper, the control goals are to regulate u c in its normal operation region and at the same time, to make Q l follow its reference Q * l . Thus, the considered VSC station operates in DC voltagereactive power mode. By solving (5), the q current reference is obtained as

i * lq = - 2Q * l 3v ld (6)
As to the d current reference, it is directly given by the DC voltage droop control

i * ld = K u (u c -u * c ) (7)
where K u is the droop gain and u * c is the DC voltage reference.

Remark 4: The reference values Q * l and u * c are considered constant in this paper and actually, they are provided by a higher level control called secondary control.

The objective of the current loop is to make i l,dq converge to their references i * l,dq . Different current controllers have been proposed [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF], [START_REF] Chen | Nonlinear control design for a multi-terminal VSC-HVDC system[END_REF], [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF]. In this paper, the conventional current control in [START_REF] Thomas | Analysis of a robust DCbus voltage control system for a VSC transmission scheme[END_REF] is selected. The control signals applied to the converter contain the traditional PI controllers and some compensation terms as

M d = 2L l u c (v d + R l L l i ld -ωi lq + v ld L l ) (8)
M q = 2L l u c (v q + R l L l i lq + ωi ld + v lq L l ) (9)
where v dq are

v dq = -K p,dq (i l,dq -i * l,dq ) -K i,dq (i l,dq -i * l,dq ) (10)
Finally, the vector current control scheme is built based on ( 6)- [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF].

Remark 5: Note that the dynamics of i * ld is neglected in the current control design. Substituting ( 8)-( 10) into the current subsystem leads to

di l,dq dt = v dq (11)
where v dq act as additional control inputs to regulate the behaviors of i l,dq . It is obvious that the autonomous system (11) is locally exponentially stabilized at i * l,dq if i * l,dq are constant or varying quite slowly. However, as seen in ( 7), i * ld is given by the DC voltage droop control, which means that its dynamics is regulated by K u . If K u is quite large, i * ld would have fast dynamics. In this case, the system (11) may become unstable. Therefore, it is necessary to consider the dynamic effect of i * ld on the system stability. In the following section, we will investigate what dynamics i l,d and i * ld should have so that the VSC terminal can operate properly.

IV. STABILITY ANALYSIS

As seen in ( 7)- [START_REF] Blasko | A new mathematical model and control of a three-phase AC-DC voltage source converter[END_REF], the dynamics of i * ld and i l,dq mainly depend on K u and K dq . In this section, we investigate the feasible region for these control gains. Before going further, let us study the steady-state condition first.

A. Steady-state condition

The steady-state values of the system (1)-( 3) can be obtained by solving the following algebraic equation

0 = - 1 C [i c + 3 2 v ld K u (u c -u * c ) + v lq i * lq u c ] (12)
which leads to

ūc = 3v ld K u u * c 3v ld K u + 2i c = u * c - 2u * c i c 3v ld K u + 2i c ( 13
) īld = - 2u * c i c K u 3v ld K u + 2i c (14)
The equilibrium value of i lq , denoted by īlq , is equal to i * lq . Taking the practical feasibility into account, i.e. ūc ∈ [u min , u max ], K u must satisfy the following condition Condition 1:

K u > Max(- 2i c 3v ld , 2i c u max 3v ld (u * c -u max ) , 2i c u min 3v ld (u * c -u min ) ) (15
) Remark 6: Notice that Condition 1 is a necessary condition for K u . It is obvious that the equilibrium point heavily depend on the choice of the droop gain K u and the DC voltage setpoint u * c . As mentioned in the previous section, u * c is provided by the secondary control and in the present paper, u * c is considered as a known constant parameter that is not used to regulate ūc . Therefore, if a smaller deviation between ūc and u * c is desired, a larger value for K u should be chosen.

B. Analysis of dynamic limitations and system stability

In this section, in order to state the main results, two steps are carried out. We first prove that there exist control gains such that the original system (1)-( 3) can present a multitime-scale behavior. Subsequently, a sufficient condition for choosing K u and K dq is given to guarantee that the system can be locally stabilized around the equilibrium point in Section IV-A.

Theorem 1: Consider the nonlinear system (1)-(3). For any K u satisfying Condition 1, there exists K min such that for any K dq > K min , the proposed control strategy ( 6)- [START_REF] Matas | Feedback linearization of direct-drive synchronous wind-turbines via a sliding mode approach[END_REF] enables the system to be stabilized around the equilibrium point ( īld , īlq , ūc ). Moreover, a multi-time-scale behavior can be found, i.e. a slow and a fast transients.

Proof: To demonstrate this claim, let us substitute the developed control laws ( 6)-( 10) into the actual system (1)-(3). We then get the following closed-loop system

di ld dt = -K pd [i ld -K u (u c -u * c )] + γ d (•) (16) di lq dt = -K pq (i lq -i * lq ) + γ q (•) (17)
du c dt = - 1 C [i c + 3 2
v ld i ld + v lq i lq u c ] (18)
where γ dq (•) represent the integral parts and for convenience, we set

K pd = K pq = K d . Let us introduce a new variable • K d = 1 (19)
Then, the system (16)-(18) becomes

di ld dt = -[i ld -K u (u c -u * c )] + γ d (•) (20) di lq dt = -(i lq -i * lq ) + γ q (•) (21)
du c dt = - 1 C [i c + 3 2
v ld i ld + v lq i lq u c ] (22)
When is small enough, the derivatives of i l,dq are quite large. By setting = 0 in (20) and ( 21), i l,dq quickly enter their manifolds i * l,dq and then a reduced model can be deduced from the DC-voltage subsystem as

du r c dt = - 1 C [i c + 3 2 v ld K u (u r c -u * c ) + v lq i * lq u r c ] (23)
Remark 7: The reduced model ( 23) is locally exponentially stabilized at its equilibrium point u r c = ūc . Since the reduced model is deduced when i l,dq converge to i * l,dq , it is also called a quasi-steady-state model. i * l,dq are the quasi-steady states of i l,dq . It is expected that the behavior of the original DC voltage system can be approximated by its reduced model due to its simpler form. Therefore, this requires that i l,dq converge to i * l,dq during the initial interval. In order to analyze the problem more conveniently, let us introduce some new variables:

ĩl,dq = i l,dq -i * l,dq (24) ũc = u c -u * c (25)
that shift the quasi-steady states of i l,dq to the origin. In these new variables, the full system problem becomes:

d ĩld dt = -ĩld + K u [ 3K u 2C u * c ūc v ld ūc + ũc ũc + 3v ld 2C ĩld ūc + ũc ] + γ d f d (26)
d ĩlq dt = -ĩlq + γ q f q (27) dũ c dt = - 3K u 2C u * c ūc v ld ūc + ũc ũc - 3v ld 2C ĩld ūc + ũc f u (28)
Now we set

d ĩl,dq dt = d ĩl,dq dτ (29)
where

dτ dt = 1 (30)
τ is the introduced new time variable which can be approximated by τ = t . In the τ time scale, the current subsystem is expressed as

d ĩld dτ = -ĩld + γ d + K u C [ 3v ld 2u c K u (u c -u * c ) + i c + 3v ld 2u c ĩld ] (31) d ĩlq dτ = -ĩlq + γ q (32)
The variables t and ũc are considered to be varying slowly in the τ time scale. Setting = 0 restricts t and u c to their initial values, and then the current subsystem is reduced to the following autonomous system

d ĩl,dq dτ = -ĩl,dq (33)
which is called the boundary-layer model. It is expected that the solutions of (33) can be close to the origin during the boundary-layer interval (t ∈ [0, t b ]), and that they can still remain around the origin for the future time (t > t b ) when t and u c slowly vary. This can be achieved since we have: Remark 8: The boundary-layer model ( 33) is exponentially stable at the origin, uniformly in (t, ũc ).

By combining Remarks 7-8 and referring to Theorem 11.4

in [START_REF] Khalil | Nonlinear Systems[END_REF], there exists min such that for any 1 K d = < min , the system (20)-( 22) is locally asymptotically stable around the origin. Thus, there exists K min = 1 min . This concludes the proof. Remark 9: From the above proof, i ld exhibits a two-timescale behavior. It starts with a fast transient of i ld moving from the initial value towards i * ld . Then, i ld remains close to i * ld . Finally, i ld and i * ld converge to their equilibrium īld together. In the simulation section, this two-time-scale behavior will be clearly illustrated.

In the following section, we carry out a further study on the relationship between K u and K d . Investigation of feasible regions of K u and K d is made by applying Lyapunov theory. A Lyapunov function is selected as

V = d 2 V 1 + (1 -d) 2 V 2 (34)
with d ∈ (0, 1) where V 1 and V 2 are derived from the boundary-layer model and the reduced model, respectively.

V 1 = ĩ2 ld + ĩ2 lq + K id φ 2 d + K iq φ 2 q (35) V 2 = ũ2 c (36)
where φ dq = ĩl,dq . Then, the derivative of V can be deduced from ( 26)-( 28)

V = -a 1 ĩ2 ld -d 1 ĩ2 lq -a 2 ũ2 c + a 3 ũc ĩld (37)
with

a 1 = d( 1 - 3K u 2C v ld ūc + ũc ) (38)
a 2 = (1 -d) 3K u 2C u * c ūc v ld ūc + ũc (39)
a 3 = d 3K 2 u 2C u * c ūc v ld ūc + ũc -(1 -d) 3 2C v ld ūc + ũc (40) Rewrite V as V = -ĩld ũc P ĩld ũc -d 1 ĩ2 lq (41)
where

P =   a 1 - a 3 2 - a 3 2 a 2   (42)
Lemma 1: If is selected such that the matrix P is positive definite, the system (20)-( 22) can asymptotically converge to the origin.

Remark 10: The above lemma is obvious since a positive definite matrix P would result in a negative V except at the origin. By applying Barbalat's lemma or LaSalle theorem, it can be proved that the system is asymptotically stable. In fact, the sufficient condition to be given later is mainly based on Lemma 1. It is known that a symmetric n × n real matrix P is positive definite if and only if the determinants of all leading principal minors of P are positive. Thus, in order to get a positive definite P , we need

a 1 > 0 (43) a 1 a 2 -( a 3 2 ) 2 > 0 (44)
These two inequalities indicate that if K u and K d satisfy the following condition Condition 2:

K d = 1 > 3K u 2C v ld ūc + ũc (45)
K d = 1 > a 2 3 4da 2 + 3K u 2C v ld ūc + ũc (46)
the system (20)-( 22) is asymptotically stable at the origin. Remark 11: It is clear that heavily depends on ũc . If the system is expected to be stable in a larger region, a smaller value should be assigned to , i.e. a larger value for K d . It is also found that a small (a large K d ) makes the system have fast responses. However, due to some physical consideration, we cannot improve the responses of the system unlimitedly.

Remark 12: It is worth noticing that the Lyapunov function V is deduced from the reduced model and the boundarylayer model. When K u is fixed, a large value of K d will make V quickly converge to the origin. It also implies that i l,dq enter their manifolds i * l,dq rapidly.

V. SIMULATION In this section, numerical simulation results are presented to verify the theoretical analysis and Conditions 1-2 derived from the previous section. The parameters of the VSC terminal and the base quantities used in the per-unit system are listed in Table I and Table II, respectively.

For all the simulations, the VSC terminal is supposed to be connected to a controlled current source with i c = -0.7 p.u.. The setpoints u * c and Q * l provided by the secondary control are set to 1 p.u. and 0 p.u., respectively. The converter starts from the initial point (i ld0 = 0.5 p.u., i lq0 = 0.1 p.u., u c0 = 0.95 p.u.). Taking Condition 1 into consideration, K u must be greater than 0.2075 in order to ensure that the converter acts in its domain of operation, i.e. u c ∈ [0.9, 1.1] p.u.. In the following section, K u = 0.3 is applied. As seen in the previous analysis, K d can be determined once K u is fixed. Hence, according to Condition 2, a feasible region for K d is given, i.e. K d > 985.7. In order to demonstrate the main results derived from the theoretical analysis, several simulation scenarios are carried out.

Case 1: We choose K d = 23.5 and K i = 221.6. Obviously, K d is beyond the given region. is then equal to 0.043. Figures 2(a) and 2(b) illustrate the behaviors of u c and i l,dq , respectively. We find that these state variables diverge with time. It is pointed out that in this case, the current controller cannot follow the reference trajectory given by the DC voltage droop control. As seen in ( 26)-( 27), the dynamics of i * ld can be considered as a perturbation to its boundarylayer model (33). The divergence of the system variables indicates that the current controller is not strong enough to tolerate this perturbation. It also reveals that not all current controllers can separate the system dynamics.

Case 2: The control gains are selected as K d = 36.2 and K i = 340.9. In this case, K d is sill less than 985.7, and is equal to 0.028. The trajectories of all the system variables converge to their equilibrium values. It implies that the asymptotic stability of the system can be achieved despite a K d that does not satisfy Condition 2. This is true because our condition is derived from a specific Lyapunov function and that it is not a necessary condition but a sufficient condition. Therefore, it is possible that there exists a K d not satisfying Condition 2 but still ensuring the stability of the system. However, the transient performances of the system variables are not good. Looking at the response of u c in Fig. 3(a), the VSC terminal operates beyond its region of normal operation ([0.9, 1.1] p.u.) during the interval t ∈(0, 1.5) s, which may damage electronic devices. On the other hand, the response time of the system is shown to be unsatisfactory.

Case 3: The current control gains are chosen as K d = 986 and K i = 9.28 × 10 3 satisfying Condition 2. In this case, we have = 0.001.

The responses of i l,d and u c are shown in Fig. 4. Let us first focus on the response of i ld in Fig. 4(a). The trajectory of i ld displays a two-time-scale behavior. In contrast to i ld starting from i ld0 = 0.5 p.u., its quasi-steady state i * ld does not start from the same initial value but from i * ld0 = K u (u c0 -u * c ). Hence, a large discrepancy between i ld0 and i * ld0 exists at the initial instant, as depicted in Fig. 4(a). It is found that the trajectory of i ld starts with a very fast transient from i ld0 to i * ld . After this transient, i ld remains close to i * ld . For example, the error between i ld and i * ld remains less than 0.114 p.u. after t = 0.005 s. Subsequently, i ld and i * ld converge to the equilibrium value (the red curve) together. As illustrated in Fig. 4 here) shows that i lq converges to its reference value, i.e. 0 p.u., after a short transient. Case 4: As mentioned in Remark 12, K d has a great effect on the convergence speed of the system. In order to evaluate this influence, K d = 1972 ( = 5 × 10 -4 ) and K i = 1.86 × 10 4 are selected by comparison with Case 3. By making the comparison between Case 3 and Case 4, we point out that K d ( ) can heavily influence the behaviors of the system. The simulation results show that we can improve the system transient performances by increasing K d . This confirms our theoretical analysis since the convergence speed of the system can be accelerated by a large K d .

VI. CONCLUSIONS

In this paper, we consider the problem of regulating the DC-bus voltage with DC voltage droop control. First, we present the conventional vector current control and indicate that the control strategy enables the system to be stabilized only when the dynamics between the current and the DC voltage are well distinguished. We then analyze the steadystate condition of the system and find that the current control gain K d has no effect on it. In addition, the droop gain K u and the setpoint u * c determine the steady-state values together. Furthermore, a necessary condition for K u is deduced such that the VSC terminal acts in the region of safe operation. Subsequently, singular perturbation theory is applied to analyze the dynamic limitations of the system. Then, we prove that there exist K u and K d such that the system variables exhibit a multi-time-scale behavior. Finally, we derive a sufficient condition for the control gains by means of Lyapunov theory to guarantee the stability of the system. We establish the Lyapunov function based on the boundary-layer model and the reduced model. It is pointed out that a large K d can improve the system transient performances.

The sufficient condition is verified by carrying out four sets of simulations. It is shown that the derived conditions indeed guarantee the asymptotic stability and make the behaviors of the system well regulated. In order to understand the effects of K d on the dynamics of the system, a comparison is made and the simulation results confirms the theoretical analysis.

In the present paper, the studied VSC is connected to a controlled current source. In the future, it is proposed to consider the VSC terminal in the context of an MT-HVDC system and analyze the effects of the control gains on the problem of stability.

Fig. 1 .

 1 Fig. 1. A VSC terminal connected to AC system.

2 V

 2 I b ≈ 19.67 A DC side S dc,b = 10 kVA V dc,b = 730 V I dc,b ≈ 13.70 A

  p.u.) (a) uc response.

  p.u.) i lq (p.u.) (b) i l,dq response.

Fig. 2 .

 2 Fig. 2. Simulation results with = 0.043, K d = 23.5 and K i = 221.6.

  p.u.) (a) uc response.

  p.u.) i lq (p.u.) (b) i l,dq response.

Fig. 3 .

 3 Fig. 3. Simulation results with = 0.028, K d = 36.2 and K i = 340.9. Simulation results are displayed in Figs. 3(a) and 3(b).The trajectories of all the system variables converge to their equilibrium values. It implies that the asymptotic stability of the system can be achieved despite a K d that does not satisfy Condition 2. This is true because our condition is derived from a specific Lyapunov function and that it is not a necessary condition but a sufficient condition. Therefore, it is possible that there exists a K d not satisfying Condition 2 but still ensuring the stability of the system. However, the transient performances of the system variables are not good. Looking at the response of u c in Fig.3(a), the VSC terminal operates beyond its region of normal operation ([0.9, 1.1]

Fig. 4 .

 4 Fig. 4. Simulation results with = 0.001, K i = 9.28 × 10 3 and K d = 986 corresponding to Condition 2.

  .u.) (a) i ld response.

  uc response.

Fig. 5 .

 5 Fig. 5. Simulation results with a large K d = 1972, K i = 1.86 × 10 4 and a small = 5 × 10 -4 .The simulation results are shown in Fig.5. The behavior of i ld in this case is similar to that in Case 3. The trajectory

Fig. 6 .

 6 Fig. 6. Comparison between Case 3 and Case 4.

TABLE I THE

 I VSC TERMINAL PARAMETERS.

	R l	L l	V line	C	f
	9.9 mΩ	3.2 mH 415 V 680 µF	50 Hz
			TABLE II
	BASE QUANTITIES APPLIED IN THE PER-UNIT SYSTEM.

This work is supported by WINPOWER project (ANR-10-SEGI-016).