Arnaud Fontaine

Antoine De Monte

He ´le `ne Touzet

MAGNOLIA: multiple alignment of protein-coding and structural RNA sequences

MAGNOLIA is a new software for multiple alignment of nucleic acid sequences, which are recognized to be hard to align. The idea is that the multiple alignment process should be improved by taking into account the putative function of the sequences. In this perspective, MAGNOLIA is especially designed for sequences that are intended to be either proteincoding or structural RNAs. It extracts information from the similarities and differences in the data, and searches for a specific evolutionary pattern between sequences before aligning them. The alignment step then incorporates this information to achieve higher accuracy.

INTRODUCTION

More and more newly sequenced genomes are becoming available every week. Tiling arrays are also gaining popularity for detecting novel transcripts in sequenced genomes. In this context, sequence annotation is an essential step in understanding the genome and the transcriptome of a species. Comparative genomics has proven to be a promising approach to address this problem. Large-scale comparisons of prokaryotic and eukaryotic genomes reveal thousands of conserved regions obtained by homology or synteny. These regions might be proteincoding sequences [START_REF] Zhu | Comparative genomics search for losses of longestablished genes on the human lineage[END_REF] or non-coding RNA genes [START_REF] Washietl | Mapping of conserved RNA secondary structures predicts thousands of functional noncoding RNAs in the human genome[END_REF][START_REF] Torarinsson | Comparative genomics beyond sequence-based alignments: RNA structures in the ENCODE regions[END_REF][START_REF] Washietl | Structured RNAs in the ENCODE selected regions of the human genome[END_REF]. Annotation by comparative analysis typically involves two steps: first aligning the sequences, then analysing the multiple alignment to detect an evolutionary pattern that is representative of the selection pressure. This idea is exploited in Exoniphy [START_REF] Siepel | Computational identification of evolutionarily conserved exons[END_REF] for exon detection, in RNAz [START_REF] Washietl | Fast and reliable prediction of noncoding RNAs[END_REF] or Evofold [START_REF] Pedersen | Identification and classification of conserved RNA secondary structures in the human genome[END_REF] for structural RNA prediction or in Qrna [START_REF] Rivas | Noncoding RNA gene detection using comparative sequence analysis[END_REF], that implements both a coding and an noncoding model for RNAs. In this computational protocol, high-quality sequence alignment is an essential prerequisite step. This task, however, is difficult because sequence similarity is often reduced at the nucleic level.

Regarding protein coding genes, nucleic acid sequences exhibit a much larger sequence heterogeneity compared to their encoded amino acid sequences due to the redundancy of the genetic code. It is well known that the combination of nucleic acid and amino acid sequence information leads to improved alignments [START_REF] Stocsits | Multiple sequence alignments of partially coding nucleic acid sequences[END_REF][START_REF] Morgenstern | Dialign 2: improvement of the segment-tosegment approach to multiple sequence alignment[END_REF]. The same situation holds for non-coding RNA genes. The spatial structure evolves slower than its primary structure. So puresequence-based multiple alignment tools perform poorly on low-homology datasets of structural RNAs [START_REF] Gardner | A benchmark of multiple sequence alignment programs upon structural RNAs[END_REF]. In this article, we present the MAGNOLIA website, whose objective is to provide an advanced tool for aligning nucleic acid sequences. The idea is to get rid of the dichotomy between aligning and predicting the function. If we assume that sequences are either protein-coding or structural RNAs, it is possible to incorporate some functional information into the alignment algorithm to improve the result. The multiple alignment can then be used as a starting point to refine the comparative analysis or to carry out further predictions, such as motif finding or phylogeny reconstruction.

METHODS

The method has two steps. First, it tries to predict the function of the sequences according to the substitution pattern between sequences. Second, a multiple alignment is built based on the putative function of the sequences. If the sequences are recognized as coding sequences, then the multiple alignment uses the amino acid sequences. If the sequences are recognized to contain a conserved secondary structure, then the mutiple alignment takes into consideration long-range base pair interactions.

MAGNOLIA includes three specific modules: Protea for protein coding sequences, caRNAc and gardenia for structural RNAs. The overall scenario is summed up in Figure 1.

Protea implements an evolutionary model for proteincoding sequences [START_REF] Fontaine | Computational identification of protein-coding sequences by comparative analysis[END_REF]. Here the idea is that the selection pressure tends to preserve the encoded amino acid sequence, and it is possible to identifies coding sequences by looking for a global conservation of common *To whom correspondence should be addressed. Tel: (33) 3 59 57 79 16; Fax: (33) 3 59 57 78 50; Email: Helene.Touzet@lifl.fr ß 2008 The Author(s) This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/ by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. reading frames. The method first identifies best potential reading frames from each pair of sequences, and then incorporates this information into a frame graph from which a coding significancy score is calculated. By doing so, it also predicts the associated reading frame for each sequence. If the sequences are classified as proteincoding sequences, then the multiple alignment of nucleic acid sequences is built from the hypothetical amino acid sequences using ClustalW (13), Dialign2 [START_REF] Morgenstern | Dialign 2: improvement of the segment-tosegment approach to multiple sequence alignment[END_REF] or T-coffee [START_REF] Notredame | T-Coffee: a novel method for fast and accurate multiple sequence alignment[END_REF]. caRNAc is for structural RNA genes [START_REF] Touzet | CARNAC: folding families of related RNAs[END_REF]. In this model, the selection pressure tends to preserve the secondary structure of the molecule, and mutations should retain the ability to form base pairs into energetically favorable stems.

caRNAc is able to recover a conserved secondary structure from a set of unaligned sequences. This idea is also present in refs. [START_REF] Kiryu | Murlet: a practical multiple alignment tool for structural RNA sequences[END_REF][START_REF] Will | Inferring non-coding RNA families and classes by means of genome-scale structure-based clustering[END_REF], that fold and align several sequences at the same time, for example. But these programs are still computationally demanding. We circumvent the problem by using a heuristics approach. The algorithm uses a Sankoff-based dynamic programing approach to identify conserved strutures for all pairs of sequences. Then all pairwise foldings are combined into a graph-theoretical structure called the stem graph. Only frequent common stems that correspond to highly connected subgraphs in the stem graph are retained.

Gardenia is used to build the multiple alignment for potential structural RNA sequences. The method takes into account both the nucleic sequence and the putative common secondary structure predicted by caRNAc. It relies on the dynamic programming algorithm for pairwise comparison proposed in ref. [START_REF] Blin | How to compare arc-annotated sequences: the alignment hierarchy[END_REF]. RNA sequences are encoded as arc-annotated sequences, and a multiple alignment for a set of arc-annotated sequences is a nested common supersequence. The edit scheme incorporates evolutionary operations concerning free bases (base substitution, base deletion) and base pairs (arc-mismatch, arc-removing, arc-breaking, arc-altering), originally defined in ref. [START_REF] Jiang | A general edit distance between RNA structures[END_REF]. It is easy to show that this problem is NP-hard. We take a heuristic approach and use a progressive method. The method starts with constructing all pairwise alignments to determine the degree of similarity for each pair of sequences. Then it combines sequences into a multiple alignment by an ascending hierarchical clustering. Pairwise alignment of supersequences rely on the same algorithm as pairwise alignments for arc-annotated sequences. This is made possible because supersequences can be viewed as a nested arc-annotated sequences on an extended alphabet. The score of one node is its SP (sum-of-pairs) score. Lastly, the space search of the dynamic programming alignment is pruned using constraints coming from the caRNAc output. This provides a significant speed up.

WEB SERVER

Input

MAGNOLIA requires as input data a set of RNA or DNA sequences in the standard FASTA format. This set should contain at least two distinct sequences and at most ten sequences. It can be stored in a file to be uploaded to the server, or pasted directly in the text box.

Output

A typical run of MAGNOLIA takes a few seconds. Upon completion of a job, MAGNOLIA displays the result on a new web page. The job assigned a unique identifier that can be used to retrieve results for one week. All results are available for download in Clustal and bracket-dot format.

If input sequences are annotated as coding sequences, then two multiple alignments are displayed. The first alignment is built on the putative amino acid sequences obtained by virtual translation using the predicted reading frame, and the second alignment is the corresponding alignment on nucleic acid sequences obtained by reverse translation, allowing for frameshifts. Codons in the nucleic acid sequences are put in color: two base triplets coding for the same amino acid bear the same color. The color choice is inspired from the RasMol amino acid color scheme. Figure 2 shows an example of MAGNOLIA output obtained with a family of protein-coding sequences.

If input sequences are annotated as structural RNA genes, then a multiple alignment taking into consideration the primary structure accompanied by the secondary structure is displayed. Concerning the secondary structure, base pairings are indicated in bracket-dot format: Each base-pair is represented by a pair of matching brackets and unpaired bases are represented by dots. The lack of pseudoknots in the secondary structure ensures that this notation defines a unique folding. Moreover, stems in the alignment are highlighted in colors. Figure 3 shows an example of output obtained with a family of non-coding RNAs. For each sequence, the individual putative secondary structure is also provided in five formats: CT, JPEG, PS, bracket-dot format and as a list of constrained base pairings. JPEG and PS files are automatically produced from the CT file using the NAview layout program [START_REF] Bruccoleri | An improved algorithm for nucleic acid secondary structure display[END_REF]. Some data sets are not identified as coding RNAs nor as non-coding RNAs. The first possibility is that the sequences might have an alternative function, such as untranslated regions in messenger RNAs, promoter elements, etc. The second possibility is that the sequences are highly conserved. In this context, the comparative analysis approach used by MAGNOLIA is not suitable. The evolutionary signal is too weak and the sequences do not exhibit any significant mutational bias towards any model. This is an intrinsic limitation of the method. But this limitation is harmless for practical purposes, because standard multiple sequence alignment tools usually yield good results on high-identity data sets. So when the average identity percentage is greater than 90%, the server outputs a warning message and provides a default multiple alignment constructed directly with ClustalW on the initial data set.

One final point worth mentioning is that the classification is not mutually exclusive. Some sequences might contain conserved secondary structure elements within a coding region. Two such examples are the cis-acting regulatory element from the human rhinoviruses, that is located in the open reading frame of the capsid proteins [RFAM -RF00220 ( 21)], or the Hepatitis C stem-loop VII structure found in the coding region of the RNAdependent RNA polymerase gene NS5B [RFAM -RF000468 [START_REF] Griffiths-Jones | RFAM: an RNA family database[END_REF]]. In such cases, MAGNOLIA releases two multiple alignments.

EXPERIMENTAL RESULTS

We evaluate the accuracy of the method on two large data sets: Pandit [START_REF] Whelan | PANDIT: an evolution-centric database of protein and associated nucleotide domains with inferred trees[END_REF] and BRaliBase II [START_REF] Gardner | A benchmark of multiple sequence alignment programs upon structural RNAs[END_REF]. Pandit is a registry of families of homologous protein domains, accompanied by curated RNA sequence alignments. BRaliBase II is a set of non-coding RNA families that has been used to establish a benchmark of multiple sequence alignment programs upon structural RNAs. It is composed of four families: Group II introns, 5S rRNA, tRNA and U5 spliceosomal RNA.

Results on pandit database

For each family, we selected a subset of four sequences at random. It remains 6491 families, whose average sequence length is 604 bp. 6122 (94.3%) families are correctly classified as coding sequences, among them more than 99% with the correct reading frame predicted for each sequence. Less than 3% of the families are classified as structural RNA. To estimate the quality of the alignments, we used the sum-of-pairs score (SPS) of the Baliscore software [START_REF] Thompson | BA1-iBASE: a benchmark alignment database for the evaluation of multiple alignment programs[END_REF]. The SPS is calculated such that it increases with the number of sequences correctly aligned. We compared MAGNOLIA with ClustalW, T-coffee and Dialign2 on the same nucleic acid sequences. Results are displayed in Figure 4.

Results on BRaliBase II benchmark data

This benchmark contains 388 alignments, that are classified into high, medium and low identity data sets. MAGNOLIA failed to identify a structural evolutionary pattern for 20% of them and falsely assigned a protein coding function for 7% of them. Following ref. [START_REF] Gardner | A benchmark of multiple sequence alignment programs upon structural RNAs[END_REF], we use the structure conservation index (SCI) to assess the accuracy of alignments. This score provides a measure of the conserved secondary structure information contained within the alignment. Results for MAGNOLIA are reported in Figure 5, together with results for other alignment tools used in the benchmark. We also evaluated the accuracy of the secondary structure found by MAGNOLIA and compared it to two recent structural alignment programs: Murlet [START_REF] Kiryu | Murlet: a practical multiple alignment tool for structural RNA sequences[END_REF] and Mlocarna [START_REF] Will | Inferring non-coding RNA families and classes by means of genome-scale structure-based clustering[END_REF]. For each software and for each identity class, we computed the percentage of correct base pairings amongst the set of predicted base pairings. Results are shown in Table 1. It appears that MAGNOLIA has similar performances as Murlet and outperforms Mlocarna. Furthermore, the total runtime is more than 12 times faster for MAGNOLIA than for the two other methods (< 20 min for the whole data set, compared to more than 4 hours). MAGNOLIA appears to be the closest curve to the reference curve for identity percentage ranging from 40% to 80%. For higher identity percentages, all methods show similar performances.

Figure 1 .

 1 Figure 1. MAGNOLIA scenario.

Figure 3 .

 3 Figure3. Alignment for five tRNA sequences (BRaliBase II -aln 13). The average length is 76 nt and the average identity percentage is 51.0%. MAGNOLIA finds three common stems (in blue, green and orange). All pairings are correct, and the multiple alignment is consistent with the reference alignment available in RFAM-RF00005[START_REF] Griffiths-Jones | RFAM: an RNA family database[END_REF].

Figure 2 .

 2 Figure 2. Alignment for Zn-finger in Ran binding proteins (PFAM PF00641). The average length is 92 nt and the average identity percentage is 45.1%. Triplets are colorized according to the encoded amino acid. The reference alignment provided in Pandit is almost identical (20).

Figure 4 .Figure 5 .

 45 Figure 4. MAGNOLIA alignments on Pandit. The x-axis represents the average identity percentage and the y-axis represents the SPS value. For MAGNOLIA, we tried all possible combinations concerning the alignment tool in the amino acid alignment step: ClustalW, T-coffee and Dialign2. For Dialign2, we selected the appropriate option Translation of nucleotide diagonals into peptide diagonals when comparing the nucleic acid sequences. MAGNOLIA clearly outperforms other methods.

Table 1 .

 1 Accuracy percentage for MAGNOLIA, Murlet and Mlocarna for reference secondary structures of BRaliBase II

			Identity class
		Low	Medium	High
	MAGNOLIA	72.0%	76.3%	87.0%
	Murlet	78.1%	76.2%	77.8%
	Mlocarna	68%	71.1%	78.9%

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ACKNOWLEDGEMENTS

Funding to pay the Open Access publication charges for this article was provided by CNRS (Centre National de la Recherche Scientifique).

Conflict of interest statement. None declared.


Ark Dev

Vincent Léchappé

Mrinal Kanti Sarkar

Prediction-Based Super Twisting Sliding Mode Load Frequency Control for Multi Area Interconnected Power Systems with State and Input Time Delays using Disturbance Observer

Keywords: Delay system, disturbance observer, robust control, super twisting control, load frequency control, predictive-based control

A shift in paradigm from dedicated to open communication channels in power systems have made them prone to constant and time varying delays. This paper tackles a novel load frequency control (LFC) problem in the presence of constant and time varying delays in state and control input under load disturbances. The presence of time delays can deteriorate the performance of the controller or even destabilize the system. The above problem is addressed through a prediction-based super twisting sliding mode control (ST-SMC) using a state and disturbance observer. The proposed design achieves finite time convergence of frequency and tie line power deviation. The said design is validated under ramp and random step disturbance, with nonlinearities like generation rate constraints and governor deadband, with an integration of renewable energy and also with IEEE 39 bus power system. The closed loop stability is proven thanks to candidate Lyapunov function and verified by simulations.

Introduction

Time delay is a common phenomenon in many engineering problems such as biological systems, chemical processes, communication networks, metallurgical processes, fluid transmission systems and network control systems (NCS). A system is said to be delayed in time when the rate of variation in system state depends on the past state [START_REF] Zhong | Robust control of time delay systems[END_REF][START_REF] Wu | Stability analysis and robust control of time delay systems[END_REF]. Time delay systems (TDS) are often known as systems with aftereffect or deadtime or hereditary systems. Multi area interconnected power systems are classic example of NCS.

Ever increasing demand of energy has led to the modernization of power systems. With advantages like high efficiency, reliability and sustainability of supply and distribution also lies their disadvantages. Modernization of power systems have made the control and transmission unit more computer dependent. Use of open communication channels in NCS have resulted in time delays. The presence of time delays in power systems can also be attributed to intruders, cyber-attacks, communication latencies, computational delays, delays due to sensor measurements and even sometimes naturally [START_REF] Li | Sliding mode observer-based load frequency control of multi area power systems under delayed inputs attack[END_REF][START_REF] Sargolzaei | Preventing time delay switch attacks on load frequency control in distributed power systems[END_REF]. Systems with time delays exhibit oscillations, poor performance and often are difficult to stabilize and control. Thus, the presence of time delays and their effects are inevitable and cannot be ignored during the modelling of physical systems. In power systems, time delays can be present in sensing loop or control loop or both. To the best knowledge of authors, LFC with time delays in both state and control input is still open for research. Therefore, in this article, a novel LFC problem is formulated with time delays present in both state and control input.

There exists a considerable amount of work reported on the control of time delay systems. Stability analysis of linear time varying delay system based on linear matrix inequalities (LMI) using Bessel-Legendre inequalities is reported in Seuret and Gouaisbaiut (2018). [START_REF] Fridman | Tutorial on Lyapunov based methods for time delay systems[END_REF] reported a tutorial on Lyapunov based methods for time delay systems. [START_REF] Lechappe | New predictive scheme for the control of LTI systems with input delay and unknown disturbances[END_REF] proposed a new predictive scheme for the control of linear time invariant (LTI) system with constant and known delays in the input and unknown disturbances. Predictive control with an application to a DC motor in the presence of external disturbance, unknown uncertainties and an unknown input delay is proposed in [START_REF] Lechappe | Delay estimation and predictive control of uncertain systems with input delay: application to a DC motor[END_REF]. Design of sliding mode predictor observer for state estimation and fault reconstruction for output time delay systems using a novel time shift approach is presented in Pinto et al. (2018a). [START_REF] Loukianov | Robust control for uncertain linear delay systems via sliding mode control[END_REF] reported a predictor based robust control for uncertain linear delay systems via sliding mode control (SMC) with delays in both state and control loop. A predictor-based observer using a partial differential equation backstepping method for linear systems with output time varying delay is reported in [START_REF] Pinto | Sliding mode control for disturbance rejection in systems with measurement delay using PDE backstepping predictor[END_REF]. In the field of LFC, the presence of state and input delays are very challenging and has attracted the attention of research community over the last few decades. Most of the articles reported on LFC have either ignored the presence of time delays or have considered a controller that is robust against time delays.

Ignorance of time delays in the field of LFC can have a catastrophic effect on the controller design.

Delay dependent stability for LFC with constant and time varying delays based on calculation of delay margin using Lyapunov theorem and LMI is reported in [START_REF] Jiang | Delay dependent stability for load frequency control with constant and time varying delays[END_REF]. Delay dependent robust LFC for time delay power systems using proportional integral derivative controller is reported in [START_REF] Zhang | Delay dependent robust load frequency control for time delay power systems[END_REF]. H-infinity based nonlinear SMC in interconnected power system with constant and time varying delays using delay dependent stability criteria is reported in [START_REF] Purwar | H-infinity based nonlinear sliding mode controller for frequency regulation in interconnected power systems with constant and time-varying delays[END_REF]. [START_REF] Yu | Event-triggered load frequency control for multi area power system with communication delays[END_REF] reported event triggered LFC for power systems with communication delays via an event triggered control method. LFC for two area thermal power systems based on estimated time delays and packet loss probability using the markovian approach is proposed in [START_REF] Singh | Communication time delay estimation for load frequency control in two area power system[END_REF]. A novel sliding mode estimation-based controller to predict time delays and state estimation and to reject the disturbance of estimation error is reported in [START_REF] Yan | A novel sliding mode estimation for microgrid control with communication time delay[END_REF].

Sliding mode LFC for multi area time delay power systems with wind power integration is proposed in [START_REF] Mi | Sliding mode load frequency control for multi area time delay power system with wind power integration[END_REF]. Robust H-infinity sliding mode LFC of multi area power systems with time delays taking into account stochastic disturbances induced by integration of renewable energies is proposed in [START_REF] Sun | Robust H  load frequency control of multi area power system with time delay: A sliding mode control approach[END_REF].

The main objective of the LFC scheme is to maintain the system frequency and the interchange power between control areas at the scheduled level [START_REF] Pandey | A literature survey on load-frequency control for conventional and distribution generation power systems[END_REF][START_REF] Pappachen | Critical research area on load frequency control issues in a deregulated power system: A state-of-the-art-of-review[END_REF]. In this article, load frequency problems with time delays in state and control loop under uncertain load conditions is investigated using predictionbased ST-SMC. It is also well known that the design of any state feedback controller requires information of all the system states. However, not all the system states are available or are easily measurable. In such cases, system states are estimated using a process called observation. This is done by observers which act a replacement to physical sensors in practice. It should be noted that differentiating a state to obtain other should be avoided as it leads to low signal to noise ratio. The prediction-based ST-SMC is based on estimated state and disturbance. The higher order sliding mode observer (HOSMO) with inherent advantages of robustness, finite time convergence and insensitivity to disturbances estimate system states from the output and control signal. Compared to modern state of the art, observer-based controller designs are advantageous in practice as they minimise the use of physical sensors, unwanted noise, design cost etc. Another component of the proposed design is the disturbance observer (DO). The DO based controller mitigates wide deviations in the frequency under load disturbances. This is done by estimating the unknown disturbance and designing a sliding surface dependent on the estimated disturbance [START_REF] Li | Disturbance Observer Based Control-Methods and Applications[END_REF].

The prediction-based ST-SMC uses the estimated system states and ensures finite time convergence of frequency and tie line power deviation under load disturbance with time delays. Unlike higher order SMC, the supremacy of super twisting control (STC) lies in the fact that it only acts on the first-time derivative of sliding surface to counter chattering without any loss in control accuracy. This eliminates the need of increased information demand and also memory to store the information about the higher order time derivatives of sliding surface to attenuate chattering. Furthermore, the robustness of the proposed design is validated under ramp and random step load disturbances, power system nonlinearities, integration of renewable energy resource and also with IEEE 39 bus large power system popularly known as New England 39 bus power system. To summarise, the main contributions of this work are:

• Formulation of a novel LFC problem with time delays in state and control input under uncertain load conditions.

• A prediction based chattering free controller is designed to compensate the effect of time delays and to confirm finite time convergence of frequency and tie line power deviations under load disturbance.

• To guarantee robustness of the proposed design under time varying delays, wide range of uncertainties, typical nonlinearities of power systems, with an integration of renewable energy resource and with IEEE 39 bus large power system.

The remainder of the article is outlined as follow: Section 2 deals with the dynamic model of multi area power systems with time delays in state and control loop. Section 3 investigates the design of state and disturbance observers. In section 4, prediction-based ST-SMC is designed based on estimated state and disturbance. In section 5, simulation results are discussed. Finally, the article ends with conclusion and scope for future work in section 6.

Dynamic model of multi area power systems with time delays Figure 1. Load frequency control model of power systems with time delays

The foremost step in the design and analysis of a control system is the mathematical modelling of the system. In LFC, linearized model of multi area power system is permissible as only small changes in load are expected during its normal operation (Sarkar et al., 2018;[START_REF] Prasad | Non-linear sliding mode load frequency control in multi-area power system[END_REF][START_REF] Mi | Decentralized sliding mode load frequency control for multi area power systems[END_REF]. Each control area is an assembly of its own governor, turbine, generator and load. The Load frequency model of multi area power systems with time delays in system state and control input is depicted in fig. 1.

The dynamic model for the th i area is given as,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i d i i i i i ij j i jN ji i i i x t A x t
T T i i i i G G i i E i E B sij ii jN ji K K K T T T TT A R T T K K K K          -   -         -     =   - -                 0 0 1 0 0 i i G B T     =         1 0 0 0 0 i C = (3) 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i i d G A T         - =            
; 0000 2 0 0 0 0 0 0 0 0 0 0 0000 2 0 0 0 0 0

P i sij P i ij E i sij K K T E K K             =       -       ; 0 0 0 0 P i P i i K T F  -    =       (4) sij K (p.u. MW)
is the interconnection gain between area i and j (i≠j) and if there is no exchange of power between i and j, 0. sij K = For the above defined time delay system, following assumptions are made:

Assumption 1: The incremental change of load disturbance ( )

d i Pt  in system (1) is slow
changing and bounded such that

( ) io wt
, where o  is a positive constant, and its time derivative is also bounded and satisfies ( ) [START_REF] Liao | A robust load frequency control scheme for power systems based on second order sliding mode and extended disturbance observer[END_REF].

lim 0 i t wt → = (
Assumption 2: The function ( ) i t
 is differentiable and bounded, i.e. there exist

0 i    , such that, ( ) 0 , 0 i i tt       . ( 5
max 0 ii d t d  (6) ( ) max 0 ii h t h  (7)
Remark 1: Assumption 2 also implies that ( )

i t
 is strictly increasing then the function

( ) i t
 is invertible and its inverse ( )

1 i t  -
is computable [START_REF] Barragan | Robust Control for Perturbed Linear Systems with Time-Varying Delay via Sliding Mode Control[END_REF].

The proposed control scheme for the time delay system (1) consist of dividing ( ) i ut into two components with each term having a specific purpose [START_REF] Loukianov | Robust control for uncertain linear delay systems via sliding mode control[END_REF].

( ) ( ) ( ) ii i o d u t u t u t =+ (8)
The component ( ) Substituting equation ( 8) in (1) results,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i i i d i i i o i d i ij j i jN ji x t A x t A x t B u t u t E x t w t       = + + + + + (9)
Selecting a matrix

ii i mn d A   such that ii d i d A B A =
and substituting it in equation ( 9),

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i i i i o i d i d i i ij j i jN ji x t A x t B u t u t A x t E x t w t       = + + + + + (10) Now, selecting ( ) ( ) ( ) i i d d i i u t A x t  =-
, where ( ) ( ) ( )

1 i i i tt    - =
and substituting it in equation ( 10) yields [START_REF] Barragan | Robust Control for Perturbed Linear Systems with Time-Varying Delay via Sliding Mode Control[END_REF],

( ) ( ) ( ) ( ) ( ) () i i i i i o i ij j i jN ji x t A x t B u t E x t w t     = + + + (11)

Design of state and disturbance observer

As discussed earlier, design of any multivariable state feedback control requires information of all the system states. However, not all the system states are available or are easily measurable. In such cases, estimation of unmeasurable state variables is done by observer and this process is commonly known as observation. State observers serve as a replacement for physical sensors for obtaining unavailable quantities [START_REF] Liu | Disturbance-Observer-Based Control for Air Management of PEM Fuel Cells Systems via Sliding Mode Technique[END_REF].

In practice, observer-based control design is advantageous as it minimizes the number of sensors required, unwanted noise with the sensors and also design cost. However, observation of system states in the presence of unknown input is very challenging problem in modern control theory. The design of such observers is based on a condition that only output is available without its derivatives [START_REF] Fridman | Observation of linear systems with unknown inputs via high-order sliding-modes[END_REF]. In this article, state estimation is done using HOSMO. The HOSMO provide global observation of the states and the output under sufficient and necessary condition of strong observability or strong detectability [START_REF] Fridman | Higher order sliding mode observer for state estimation and input reconstruction in nonlinear systems[END_REF][START_REF] Fridman | Higher order sliding mode observer for linear system with unknown input[END_REF]. The sliding mode observer (SMO) based on step by step state vector reconstruction require filters at each step due to imperfections of analog devices. However, the use of HOSMOs eliminates the use of filters at each step [START_REF] Fridman | Higher order sliding mode observer for state estimation and input reconstruction in nonlinear systems[END_REF].

The following assumption is necessary for the design of state observer which is discussed in the sequel.

Assumption 4: The pair (

)

,
ii CA is observable.

The estimated states are used for the design of prediction-based ST-SMC. Transforming equation ( 11) using

i i i z T x , where 1 .. T n i i i i i i T C C A C A -  =  . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i i o i ij j i jN ji i i i z t z t u t z t t y t z t      =  +  +  +  = ( 12
)
where

1 i i i i T AT - = , i i i TB = , 1 ij i ij j T E T - = , ( ) ( ) i i i t T w t = and 1 i i i CT  - = .
The HOSMO takes the form of a fundamental Luenberger observer plus higher order sliding mode differentiator [START_REF] Fridman | Observation of linear systems with unknown inputs via high-order sliding-modes[END_REF][START_REF] Yan | Robust sliding mode observer design for interconnected systems[END_REF])

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) î i i i i o i ij j i i i i i i jN ji z t z t u t z t L C e t e t w t     =  +  +  + + + (13)
where ( ) ˆi zt is the estimation of ( ) i zt, i L is the designed gain matrix, ( ) ( ) ( )

î i i e t x t x t =-
and ( ) ˆi wt is the estimated disturbance whose dynamics will be given in the sequel. Using

ˆî i i x T z - = as, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆˆˆî i i i i o i ij j i i i i i i jN ji x t A x t B u t E x t L C e t e t w t     = + + + + + (15)
Considering ( ) i wt as a state vector, the augmented model of system (11) can be rewritten in the following form (Wang et al., 2016),

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 00 ij jN j ii ii ji ii i ii i E xt x t x t A I B ut wt w t w t wt                 = + + +                         (16)
The HOSMO along with disturbance observer in the matrix form can be written as,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ˆ0 0 0 ˆ0 00 0 ij jN i j i ii ji ii i i ii ii i i E xt xt xt A I B ut wt wt LC et et                = + +                     ++      (17)
Reminding that ( ) ( ) ( )

î i i e t x t x t =-
and defining ( ) ( ) ( )

î i i w t w t w t =-, the observer error system is obtained as, ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 i i i i i i i i i ii A L C I e t e t e t wt w t w t    -       = + -       -         (18)
Defining 0

i i i i i A L C I A  -  =  - 
and based on assumption 4, the observer gain matrices i L and i  are selected such that there exist a matrix 0 0

i i i S H   =  
which is the solution of Lyapunov equation ( 19), with positive definite matrices 0 i S  and 0 i   .

T i i i i i A H H A R + = -where 0 i R  . (19)
In the following, using a candidate Lyapunov function, we ensure that the error dynamic equation ( 18) is asymptotically stable and is independent of the unknown disturbance ( ) i wt.

Theorem 1: The error dynamic equation ( 18) is asymptotically stable and is independent of the unknown disturbance ( ) i wt provided the roots of i A have negative real parts or lie at the left half of complex plane.

Proof: Combining the state and disturbance estimation error as

( ) ( ) ( ) i i i et t wt   =  
, we consider a candidate Lyapunov function as, ( ) ( ) ( )

T i i i i V t t H t  = (20)
Taking the derivative of (20) along the trajectories of ( 18),

( ) ( ) ( ) ( ) ( ) TT i i i i i i i V t t H t t H t     =+ (21)
Substituting ( )

i t  from equation (18) in (21), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) Based on assumption 1, equation ( 22) can be rewritten as follow, ( )

0 0 0 0 T T ii TT i i i i i i ii T i i i i i et V t
( )   ( ) ( ) ( ) ( ) ( ) ( ) ( ) 00
T i i i i T T T i i i i i i i i i i i e t e t V t t A H H A t H t t H          = + - -             (23)
Using ( 19), equation ( 23) can be rewritten as,

( ) ( ) ( ) ( ) ( ) ( ) 2 0 ii TT i i i i i i et V t t R t t H      = - -    (24)
Reminding ( ) ( ) ( )

i i i et t wt   =   and 0 0 i i i S H   =  
, equation ( 24) can be reduced to the following form, ( Vt . This shows that the observation error is asymptotically stable. This completes the proof.

2 T i i i i i i i i V t t R t e t S e t    = - - ) ( ) ( ) ( ) ( ) ( )

Prediction based super twisting controller using disturbance observer Figure 2. Block diagram of prediction-based ST-SMC using DO

The proposed prediction based STC using estimated states and disturbance is shown in fig. 2. The said control scheme results in equation ( 15) which is perfectly free from state time delays. In order to compensate the time delay in the control input i.e. (  ) i oi u t h t -, a delay predictor is designed of the form [START_REF] Loukianov | Robust control for uncertain linear delay systems via sliding mode control[END_REF][START_REF] Lechappe | Prediction based control for LTI systems with uncertain time varying delays and partial state knowledge[END_REF][START_REF] Lechappe | New predictive scheme for the control of LTI systems with input delay and unknown disturbances[END_REF],

( ) ( ) ( ) ( ) ( ) ( ) î i i i i t A h t A t i i i o t h t t e
x t e B u d

    - -  =+ (26)
where ( )

i n i t
  depicts the predictor state compensating input delay. The computation of ( )

i t
 is generally difficult to perform because of an integral term. The integral in ( 26) can be approximated in real time by various numerical quadrature rules like rectangular, trapezoidal, Simpson's 1/3 rd rules etc. In this case, the integral is discretised in finite number of points using backward rectangular rule which can be referred from [START_REF] Zhong | Robust control of time delay systems[END_REF].

The derivative of ( 26) is obtained as, (

ˆ1 i i i i i i ii i i i A h t A h t A h t i i i i i i o i i o i t At i i ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

o t h t t e x t A h t e x t B u t h t e B u t h t

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
ˆ1 i i i i i i i i i i i i i i i i i i i ii i ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

A h t A h t A h t A h t i i i i o i ij j i i i jN ji A h t A h t A h t i i i i i i i o t A h t A t i i o i i i o t h t t A e x t e B u t h t e E x t e L C e t e e t e w t A h t e x t B u t h t e B u t h t

A e B u d       - -   = + - + + + + + + -- - + (28) Substituting ( ) ( ) ( ) ( ) ( ) ( ) î i i i i t A h t A t i i i i i i o t h t A t A e x t A e B u d     - -  =+ in the above equation (28), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) î i i i i i i i i i i i i i A h t A h t A h t i i i i o ij j i i i i i jN ji A h t A h t A h t i i i i i i o i t A t B

u t e E x t e L C e t e e t e w t A h t e x t h t e B u t h t

      = + + + + + + + - (29)
Design of SMC involves two fundamental steps: first is the design of sliding surface and second is the design of control law that drives the system states from any initial condition to the sliding surface in finite time and forces the system states to remain on it thereafter.

In this work, a PI type sliding surface based on predictor state ( ) i t  and an estimated disturbance ( ) ˆi wt is used to improve the overall dynamic performance under mismatched uncertainties and nonlinearities [START_REF] Mi | Decentralized sliding mode load frequency control for multi area power systems[END_REF]. The following assumption is necessary for the design of controller which is discussed in the sequel.

Assumption 5: The pair ( )

,
ii AB is controllable. It implies there exists a matrix i K such that ( )

i i i A B K -
is Hurwitz and this ensures existence of a symmetric positive matrix i P , which is the solution of Lyapunov equation, ( ) ( )

T i i i i i i i i i A B K P P A B K Q - + - = -where 0 i Q  . ( 30
)
A PI type sliding surface based on predictor state and estimated disturbance is selected as follow,

( ) ( ) ( ) ( ) ( ) 0 t i i i i i i i i i t G t G A B K d w t       = - - + (31)
where ( )

i t  is the th i area sliding surface, matrix i G is selected such that ii GB is non- singular.
In this work, we select i G for the th i area as, 0 0 1 0

0 i i G G T   =   .
Based on assumption 1, differentiating equation ( 31) and solving further, ( ) ( ) ( ) ( )

i i i i i i i i t G t G A B K t    = - - (32)
Substituting equation ( 29) in ( 32), ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ˆî i i i i i i i i i i i i i A h t A h t A h t i i i o ij j i i i i jN ji ii A h t A h t A h t i i i i i i i o i i i i i i A t B
        + + + +  =   + + + -  -- (33) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) î i i i i i i i i i i ii i A h t A h t i i i i i i i o i ij j i i i i jN ji A h t A h t A h t i i i i i i i i i A h t i i i o i t G B K t G B u t G e E x t G e L C e t

G e w t G e e t G A h t e x t G h t e B u t h t

     = + + + + + + +- (34)
The equivalent control law on the sliding mode is obtained by substituting ( )

0 i t  = in the above equation (34), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 î i i i i i i i i i i i i i A h t A h t A h t i i i i i ij j i i i i i i i jN eq ji o ii A h t A h t A h t i i i i i i i i i o i G B K t G e E x

t G e L C e t G e e t ut GB G e w t G A h t e x t G h t e B u t h

t        + + +   - =     + + + -   (35)
The above formulated equivalent control compensates the time delay in the input loop and also ensures finite time convergence of the system states. The prediction based-STC

( ) i o
ut in equation ( 8) which is main focus of this article to achieve chattering free control and also to overcome the uncertainties in the system consist of three parts namely, the equivalent control law

( ) i eq o
ut designed by substituting ( )

0 i t  =
, the continuous state function and the discontinuous input with an integrator [START_REF] Chalanga | Implementation of super twisting control: Super twisting and higher order sliding mode observer-based approach[END_REF][START_REF] Heng | Design of super twisting algorithm for chattering suppression in machine tools[END_REF]. Thus,

( ) i o
ut in equation ( 8) is achieved as, ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0.5 1 2 0 1 î i i i i i i i i i i i i i A h t A h t A h t i i i i i ij j i i i i i i i jN ji o ii A h t A h t A h t i i i i i i i i i o i t i i i G B K t G e E x t
              + + +   - =     + + + -   - - (36)
where 1  and 2  are the positive constants. Reminding

0.5 1 2 0 1 ˆî i i i i i i i i i i i i i A h t A h t A h t i i i i i ij j i i i i i i i jN ji i ii A h t A h t A h t i i i i i i i i i o i t i i i d i i G B K t G e E x t G e L C e t G e e t ut GB G e w t G A h t e x t G h t e B u t h t t sign t sign t d A x t                + + +   - =     + + + -   - - - ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Rewriting the closed loop equation of system in terms of predictive states by substituting

( ) i o
ut from equation (36) in equation ( 29). Upon solving, ( )

i t  is reduced to the following form, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
0.5 12 0 t i i i i i i i i i i t A B K t B t sign t B sign t d          = - - - (38)
Rewriting the dynamics of (32) after substituting (38) we get, (

.5 12 0 t i i i i i i i i t G B t sign t G B sign t d         = - - ) ( ) ( ) ( ) ( ) ( ) 0
It should be noted that the proposed prediction-based control scheme results in a closed loop system (38) which is free from any state and input time delays. Therefore, unlike delay dependent stability conditions for closed loop systems with time delays, in this article the asymptotic stability of the closed loop system can be proved by delay independent Lyapunov functional candidate.

Theorem 2: If Assumptions 2 and 3 hold true, the trajectory ( ) i xt of the th i area tends to zero provided ii GB is non-singular and the super twisting controller gains 12 0, 0   .

Proof: Considering a candidate Lyapunov function as,

( ) ( ) 2 0.5 ii V t t  = (40)
Taking the time derivative of (

i i i V t t t  = 40), ( ) ( ) ( )
Substituting equation ( 39) in (

  0.5 12 0

t i i i i i i i i i V t t G B t sign t G B sign t d         = - - (42)
With proper selection of matrix i G as mentioned in section 4 of the article and also ensuring that ii GB is non-singular which in this article is 1

ii GB = , we can conclude that, ( )

0 i Vt . (43)
Equation ( 43) means that the sliding surface ( )

i t
 is asymptotically stable and thus tends to zero. Since the sliding surface ( )

i t  is a function of predictor state ( ) i t  , thus it can be deduced that ( ) i t
 also tends to zero. This implies that ( ) ˆi xt to zero and since the error dynamic equation is stable (from theorem 1), we can conclude that ( ) i xt tends to zero. This completes the proof.

Results and Discussion

In order to illustrate the effectiveness and feasibility of the proposed control design, three area interconnected power system model is considered with time delays in state and control loop. The basic parameters of three area interconnected power systems are given in Table 1 [START_REF] Mi | Decentralized sliding mode load frequency control for multi area power systems[END_REF]. A. Performance analysis with constant time delays In this case the proposed control scheme is validated with different constant time delays in the system state and control input. The time delays considered in system state in area 1, 2 and 3 are 0.05 second, 0.1 second and 0.15 second respectively. The input time delays considered in area 1, 2 and 3 are 0.03 second, 0.08 second and 1 second respectively. Scenario 1: A three-area interconnected power system model is considered with initial condition ( )   0 0.01 0.01 0.01 0.01 0.01

T i x = .
The frequency deviation is found to be within acceptable limit in each area with chattering free control signal (see figs. In practical applications, the assumption of constant time delays become too restrictive.

In this case, the proposed control design is validated in the presence real time operating conditions. This is achieved by incorporating time varying delays in each area of power systems. These time varying delays have a chance of varying from area to area. However, with same network protocols and measurement technique in different areas and the transmission medium together with the physical transmission distance to be almost same, this would guarantee all delays vary in the same manner in each area [START_REF] Yang | Further improvement on delay dependent load frequency control of power systems via truncated B-L inequality[END_REF].

The time varying delays in state and input considered in each area can be seen from figs. GRC is a physical constraint that limits the rate of change of generator output due to physical limitations of turbine. GDB can be defined as a band of input signal for which governor may not immediately react. In this case GRC is modelled using closed loop modelling method as shown in fig. 13(a) and can also be referred from [START_REF] Tan | Load frequency control of power systems with non-linearities[END_REF]. cheapest and eco-friendly resources of energy. However, wind power is usually dependent on the speed of the wind. Fluctuation in wind speed results in frequency deviation from its schedule value. The output of the wind turbine depends on the wind speed and is proportional to the cube of the wind speed [START_REF] Mi | Sliding mode load frequency control for multi area time delay power system with wind power integration[END_REF]. Neglecting nonlinearities in wind turbine system involving pitch control mechanism, the transfer function of wind turbine can be approximated by first-order lag function as in [START_REF] Lee | Small-Signal Stability Analysis of an Autonomous Hybrid Renewable Energy Power Generation/Energy Storage System Part I: Time Domain Simulations[END_REF], T is the time constant of wind generator. The values of the parameter can be referred from [START_REF] Lee | Small-Signal Stability Analysis of an Autonomous Hybrid Renewable Energy Power Generation/Energy Storage System Part I: Time Domain Simulations[END_REF]. The frequency deviation with the integration of renewable energy resource is found to be satisfactory and within acceptable limit (see figs. The 39-bus system can be sub grouped in 3 areas. Generators G1, G2 and G3 in area 1, Generators G8, G9 and G10 in area 2 and generators G4, G5, G6 and G7 are grouped in area 3 (Sarkar et al., 2018;[START_REF] Bevrani | Intelligent Power System Frequency Regulation Concerning the Integration of Wind Power Units[END_REF]. However, only one generator in each area is responsible for load frequency task; generator G1, G9 and G4 in area 1, 2 and 3 respectively (Sarkar et al., 2018;[START_REF] Bevrani | Intelligent Power System Frequency Regulation Concerning the Integration of Wind Power Units[END_REF]. The loads connected in area 1, area 2 and area 3 are 263 MW, 230 MW and 126 MW respectively. The other parameter values can be referred from [START_REF] Bevrani | Intelligent Power System Frequency Regulation Concerning the Integration of Wind Power Units[END_REF]. The frequency deviation is affected by oscillations; however, the response is found to be satisfactory and within acceptable limit with IEEE 39 bus large power system (see fig. 15(b)).

Conclusion

In this article, prediction-based ST-SMC with state and input time delays is reported.

The proposed prediction-based control scheme is used to compensate the effects of time delays in the state and input loop. The design of STC is based on the estimated state and load disturbance. The design is carried out on three area interconnected power systems under step load disturbance, ramp and random step load disturbances, with power system nonlinearities, with an integration of renewable energy resource and also with IEEE 39 bus large power system. The system response: frequency and tie line power deviations have been evaluated and found to be within acceptable limits. This is achieved with chattering free control signal which is important during experimental implementations.

Based on the present work, future work will focus LFC in the presence of multiple time delay. Control in the presence of multiple time delays is challenging and is still open for research.

  prediction based STC that compensates the time delay in the control loop.

  [START_REF] Chalanga | Implementation of super twisting control: Super twisting and higher order sliding mode observer-based approach[END_REF]

  (15) in equation (27) and solving further,

  . Now, using (8), the proposed prediction based chattering free control signal compensating both state and input delays and also the uncertainties can now be given as,

  figs. 4(a), (b), (c) & (d)). In practice, observer-based controller design is advantageous as system states are not always available for measurement or are difficult to measure. In that case observers act a replacement for the physical sensors.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Response using proposed design with initial condition

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Time varying state and input delays Scenario 1: The proposed design ensures finite time convergence of frequency and tie line power deviation with chattering free control signal under step load disturbance (see figs. 7(a), (b) & (c)). The disturbance observer estimates the disturbance over wide range and ensures frequency deviations under load disturbance to be within acceptable limit

Figure 9 .

 9 Figure 9. Response using proposed design under random step load disturbance

Figure 10 .

 10 Figure 10. Response using proposed design under random step load disturbance

Figure 11 .

 11 Figure 11. Response using proposed design under random step load disturbance

Figure 12 .

 12 Figure 12. Response using proposed design under random step load disturbance Scenario 3: In this case, constraints handling ability of the proposed design is validated in the presence of governor deadband (GDB) and generation rate constraints (GRC). A

  The value of GDB and GRC considered are 0.0006 per unit or 0.06% and 0.8  p.u.MW/sec respectively. From figs. 13(b), (c) & (d) it can be observed that the dynamic performance of the proposed design is affected by larger overshoot due to the presence of nonlinearities. However, the proposed design still ensures a satisfactory performance in the presence of nonlinearities.

Figure 13 .

 13 Figure 13. Response with nonlinearities under step load disturbance Scenario 4: In this case, fluctuation in frequency is observed with the integration of renewable energy resources in particular wind energy. Wind energy is one of the cleanest,

  change in output power, W K is the gain of wind generator, W

  14(a) & (b)).

Figure 14 .

 14 Figure 14. Response with an integration of wind energy under step load disturbance

Figure 15 .

 15 Figure 15. Response with IEEE 39 bus system under step load disturbance

  It should be noted that choice of constant gains 15 ,.......,

		/5 ii e t sign e t  1 i  
		3/4 ii e t sign e t  2 i  
	ii et 	2/3 ii e t sign e t  3 i  = 	(14)
		1/2 ii e t sign e t    4 5 i i i sign e t    
		 are not unique, however in
			ii
	in this work they have been selected as positive constants. System states can be recovered
	back using same transformation now in the form	1

TABLE 1

 1 POWER SYSTEM PARAMETER VALUES

	Area	i P T	K	i P	T T	i G T	i R	K	E	K	i B	K	sij
					i				i
	1	20	120	0.3	0.08	2.4	10	0.41	0.55
	2	25	112.5	0.33	0.072	2.7	9	0.37	0.65
	3	20	115	0.35	0.07	2.5	7.1	0.4	0.545


Jérémy Renard
email: jeremy.renard@ign.fr-cecile.duchene@ign.fr-guillaume.touya@ign.fr

Julien Gaffuri
email: julien.gaffuri@jrc.ec.europa.eu

Cécile Duchêne

Guillaume Touya

Automated Generalisation Results Using the Agent-Based Platform CartAGen

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

BACKGROUND AND OBJECTIVES

Cartographic generalisation aims at adapting the content of geographic data depending on their scale, to ensure the legibility of a map while preserving its main characteristics. Generalisation automation is a complex issue which has been studied by many researchers for twenty years and resulted in several different models and algorithms. A crucial problem is now to be able to gather all processes to perform global generalisation [START_REF] Edwardes | Creating a hydrographic network from its cartography representation : a case study using Ordnance Survey MasterMap data[END_REF]] and to open new perspectives of development.

CartAGen is a component-based research platform dedicated to generalisation and built on top of these processes that has been developed to face this problem of capitalisation. The main interface of the platform is on figure 1. As the core architecture has already been presented and discussed [START_REF] Gaffuri | Capitalisation problem in research -Example of a new platform for generalisation: CartAGen[END_REF], the aim of this paper is to present the first cartographic results of CartAGen.

APPROACH AND RESULTS

One of the aims of CartAGen is to gather three agent-based generalisation models developed in the past ten years : AGENT, CartACom and GAEL, each of them dedicated to different types of conflicts [START_REF] Duchêne | Combining three multi-agent based generalisation models: AGENT, CartACom and GAEL[END_REF]].

These three models imported in CartAGen already gave satisfying results, nevertheless they needed to be partly re-engineered to improve their efficiency and their quality. The results automatically computed on CartAGen are quite interesting, some examples being shown on figure 2. Then many software components have been integrated to help generalisation, for instance geometric transformation algorithms, spatial analysis treatments or new generalisation methods. A knowledge revision component is also currently added. To perform components interoperability, a generic geographic data schema dedicated to generalisation has been created, covering a wide set of spatial objects needed for generalisation, including meso level objects.

Figure 3 -Dual carriageway collapsing (left) and building alignments detection (right)

As examples of generalisation treatments that have been added to the platform, we should talk about dual carriageway collapsing, using a method adapted from hydrographical generalisation [Regnauld & Mackaness 2006] and described on figure 3. This method opens perspectives for further research on road interchanges generalisation. On another hand, generalisation of a whole city is also almost performed, including street selection in the city centre and typification through the detection and the contextual simplification of building alignments.

CONCLUSION AND FUTURE WORK

After a first step of building and completing the core of the platform architecture in order to have a strong basis, now we are able to concentrate on the development of new processes based on CartAGen, and we can see that it offers new ranges of technical possibilities.

The imported processes allow performing generalisation on many cases, but other treatments could be considered to solve new original issues. The future improvements then consists in :

 importing new models and algorithms to open perspectives, e.g. the elastic beams for network generalisation [START_REF] Bader | Cartographic displacement in generalization : Introducing elastic beams[END_REF]],  improving past results to enhance generalisation level, e.g. the achievement of a perfect urban generalisation,  integrating the concept of collaborative generalisation [START_REF] Touya | Collaborative generalisation: Formalisation of generalisation knowledge to orchestrate different cartographic generalisation processes, 6 th international conference on Geographic Information Science[END_REF]],  deploying CartAGen as generalisation web services [START_REF] Edwardes | Creating a hydrographic network from its cartography representation : a case study using Ordnance Survey MasterMap data[END_REF]].

With such components, CartAGen progressively becomes a complete platform to solve a wide set of generalisation problems with a high quality level.

Figure 1 -

 1 Figure 1 -CartAGen GUI

Figure 2 -

 2 Figure 2 -Urban generalisation at 25K using AGENT (up left), rural generalisation at 25K using CartACom (up right), road generalisation at 100K & 200K using AGENT (down)


R Émi Bignalet-Cazalet
email: bignalet@dima.unige.fr

PLANE POLAR CREMONA MAPS OF ARBITRARILY LARGE DEGREE IN POSITIVE CHARACTERISTIC

A result of I.V. Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic.

Introduction

Given a homogeneous polynomial f ∈ k[x 0 , . . . , x m ] over a field k, the polar map Φ f : P m k P m k of f is the rational map defined by the linear system ∂f ∂x0 , . . . , ∂f ∂xm . The polynomial f is called homaloidal if ∂f ∂x0 , . . . , ∂f ∂xm has no fixed component and Φ f is birational.

It was established by I.V. Dolgachev [Dol00, Theorem 4] that if k = C, the homaloidal polynomials in three variables are either of degree 2, defining a smooth conic in the projective plane, or of degree 3, defining either a union of three lines in general position or a union of a smooth conic with one of its tangents. These polynomials remain homaloidal when the base field k has characteristic greater than 2. This leads to the following question which is a generalisation of [DHS12, Question 3.7].

Problem 1. Over a field k of positive characteristic, are there other homaloidal polynomials than the ones in Dolgachev's classification?

In [START_REF] Bignalet-Cazalet | Torsion of a finite base locus[END_REF]Proposition 4.6], a first example of a homaloidal polynomial of degree 5 over a field of characteristic 3 was produced, answering negatively both Problem 1 and [DHS12, Question 3.7]. Very recently, the following example of a homaloidal curve of degree 5 in characteristic 3 was also described.

Example 2. In characteristic 3, the polynomial f = x 0 (x 2 1 + x 0 x 2 )(2x 2 1 + x 0 x 2 ), whose zero locus is the union of two conics intersecting with multiplicity two in two distinct points with the tangent at one intersection point, is homaloidal.

The negative answer to Problem 1 leads to the following question. Problem 3. Does there exists homaloidal polynomials in three variables of arbitrary large degree over fields of arbitrary large characteristic? Date: October 2, 2019. 2010 Mathematics Subject Classification. 13D02, 14E05, 14B05, Key words and phrases. rational maps, homaloidal hypersurfaces, homaloidal curves, naive graph, torsion of the symmetric algebra, Milnor number, Swan conductor.

The author was funded by the European Union's Horizon 2020 research and innovation program as a Marie Sk lodowska-Curie fellow of the Istituto Nazionale di Alta Matematica "Francesco Severi" grant No. 713485.

In this note, we answer negatively this question, see Theorem 2.4 for our expanded result.

Theorem A. Let n ≥ 2 be an integer and let k be a field of characteristic p that divides n. Then the polynomial

f n ∈ k[x 0 , x 1 , x 2 ] whose zero locus F n = V(f n ) in P 2
k is the union of n distinct lines through a fixed point z 0 and any other line not passing through z 0 is homaloidal.

This result provides an answer to Problem 3. For instance, let n ∈ N >0 be such that n ≡ 0 mod 5 and let k be a field of characteristic 5. Then the polynomial f n in Theorem A has degree n + 1 and is homaloidal. Moreover, given a prime number p, a field k of characteristic p and a positive integer m, Theorem A gives a homaloidal polynomial of degree mp + 1, so homaloidal polynomials exist in arbitrarily high degree and in any prime characteristic.

Remark that the base ideal of the polar map Φ f , i.e. the ideal generated by the partial derivatives of a polynomial f defines the singular locus of the zero locus V(f ) of f in P 2 k . In this direction, the proof given by I.V.Dolgachev about the classification of homaloidal complex polynomials relies on the Jung-Milnor's formula over C relating several invariants of singularities [Dol00, Lemma 3]. In contrast, our proof of Theorem A relies on the study of the torsion of the symmetric algebra of the base ideal of Φ f , an approach that fits in line with previous works such as [START_REF] Russo | On birational maps and jacobian matrices[END_REF], [START_REF] Dória | A characteristic free criterion of birationality[END_REF], and [START_REF] Bignalet-Cazalet | Torsion of a finite base locus[END_REF]. We emphasize that all the polynomials we consider in this note define free curves (a curve being free if, by definition, the base ideal of the polar map is determinantal [Dim17, Def 2.1]). We specially focus on this case since, when the base ideal has a linear syzygy, free curves are the curves whose singular schemes have maximal length [Dim17, Cor 1.2].

Contents of the paper. In the first section, we recall the relations between the symmetric algebra of the base ideal of a rational map Φ and the graph of Φ. Birationality of a map can be checked via its graph which explains the strategy of detecting a birational map via the symmetric algebra of its base ideal.

The second section constitutes the heart of our work. As a central idea, one can study the reduction modulo p of the presentation matrix of the base ideal in order to predict a drop of the topological degree of the polar map, see Subsection 2.1. The next step is then to evaluate this drop. We carry on this evaluation by describing the generic fibre of the naive graph and thus describing the generic fibre of the graph itself. This implies in particular that the polynomials f n are homaloidal (Lemma 2.3). We end this section by providing another example of a polynomial of degree 5 which is homaloidal in characteristic 3. Its zero locus in P 2 k is the union of the unicuspidal ramphoïd quartic and the tangent cone at its cusp (Example 2.6).

In the third section, we return to the description of the base locus of the polar map as the ideal of the singular locus of a curve. Theorem A provides in particular examples of arbitrarily big Swan conductors (Proposition 3.1).

The explicit computations given in this paper were made using basic functions of Macaulay2 and the Cremona package [START_REF] Stagliano | A Macaulay2 package for computations with rational maps[END_REF] associated. The corresponding codes are available on request.

Graph and naive graph

In this note, all the fields are assumed to be algebraically closed and denoted by the same letter k.

1.1. Multidegree of a subscheme of P 2 × P 2 , projective degrees. For this presentation, we follow [Dol11, 7.1]. Given l ∈ {0, 1, 2}, we denote by H l a general codimension l linear subspace of P 2 k . For a subscheme

X ⊂ P 2 k × P 2 k of codimension 2, the multidegree d 2 (X), d 1 (X), d 0 (X) is defined by : (1.1) d i (X) = length X ∩ p -1 1 (H i ) ∩ p -1 2 (H 2-i ) where p 1 : P 2 k × P 2 k → P 2 k and p 2 : P 2 k × P 2 k → P 2
k are the first and second projection respectively.

Consider now a rational map Φ = (φ 0 :

φ 1 : φ 2 ) : P 2 k P 2 k with base ideal I = (φ 0 , φ 1 , φ 2 ) ⊂ R = k[x 0 , x 1 , x 2 ]
where φ 0 , φ 1 , φ 2 are homogeneous polynomials of the same degree that do not share any common factor. The graph Γ of Φ is by definition the closure of {(x, Φ(x)), x ∈ P 2 k \V(I)} ⊂ P 2 k × P 2 k in the Zariski topology. It is an irreducible variety of codimension 2. We define the projective degrees

d 2 (Φ), d 1 (Φ), d 0 (Φ) of Φ as the multidegree d 2 (Γ), d 1 (Γ), d 0 (Γ) of Γ.
Since Φ is birational if and only if d 2 (Φ) = 1, this last quantity has a special importance and is called the topological degree of Φ.

Decomposition of the naive graph

. Let Φ = (φ 0 : φ 1 : φ 2 ) : P 2 k P 2 k be a rational map with base ideal I = (φ 0 , φ 1 , φ 2 ) in the coordinate ring R = k[x 0 , x 1 , x 2 ] of P 2 k and let R(I) := ⊕ i≥0 I i t i ⊂ R[t]
ideal I = (φ 0 , φ 1 , φ 2 ) ⊂ k[x 0 , x 1 , x 2 ]
of height at least 2 and where φ 0 , φ 1 , φ 2 have the same degree d. Note that, in this setting, the condition height(I) ≥ 2 is equivalent to the fact that (φ 0 , φ 1 , φ 2 ) do not share any common factor, a property which we always assume in the following. Assume moreover that Φ is determinantal, i.e. that the polynomials φ 0 , φ 1 , φ 2 are the 2-minors of a given 3 × 2-matrix such that all its entries in the first column are homogeneous of degree a and all its entries in the second column are homogeneous of degree b (hence a + b = d). The Hilbert-Burch theorem [Eis95, Theorem 20.15] implies that I has a free resolution of the form:

0 R 2 R 3 I 0. M (φ0 φ1 φ2)
where M is a 3 × 2-matrix with entries in R. Hence M is a presentation matrix of I and P(I) is the intersection of two divisors of P 2 k × P 2 k of bidegree (a, 1) and (b, 1) respectively. Considering the case where I is of linear type, since Γ has codimension 2, P(I) = Γ is a complete intersection and the projective degrees of Φ are given by Bézout's theorem:

     d 2 (Φ) = d 2 (P(I)) = ab d 1 (Φ) = d 1 (P(I)) = a + b d 0 (Φ) = d 0 (P(I)) = 1.
Assume that Φ : P 2 k P 2 k is a determinantal rational map with base ideal

I ⊂ R = k[x 0 , x 1 , x 2 ] of

Contribution of the torsion

In this section, following the situation described in Example 1.4, we illustrate first on an example how to estimate the drop of the topological degree in positive characteristic compared to characteristic 0. We analyse then in greater generality how this modification impacts the computation of the topological degree of Φ.

Reduction of the presentation matrix modulo p.

In what follows, for a homogeneous ideal I of R = k[x 0 , x 1 , x 2 ] and an integer t, we denote by I t the homogeneous piece of I of degree t. Let n ∈ N >1 and let F n be the union of n distinct lines through a point z 0 ∈ P 2 k with any other line not passing through z 0 . We can reduce to the situation where V(x 2 ) is the latter line and two lines among the n th firsts are V(x 0 ) and V(x 1 ) so that z 0 = (0 : 0 : 1). We can consequently assume without loss of generality that an equation of F n reads f n = x 0 x 1 l 2 • • • l n-1 x 2 where, for all i ∈ {2, . . . , n -1}, l i belongs to (x 0 , x 1 ) 1 (set f n = x 0 x 1 x 2 if n = 2). The ideal I of partial derivatives of f n is then equals to

I = (x 1 l 2 • • • l n-1 x 2 +x 0 x 1 ∂ ∂x 0 (l 2 • • • l n-1 )x 2 , x 0 l 2 • • • l n-1 x 2 + x 0 x 1 ∂ ∂x 1 (l 2 • • • l n-1 )x 2 , x 0 x 1 l 2 • • • l n-1 ).
Lemma 2.1. A minimal presentation matrix of I reads

M =   x 0 0 x 1 x 1 l 2 • • • l n-1 -nx 2 -l 2 • • • l n-1 x 2 -x 1 ∂ ∂x1 (l 2 • • • l n-1 )x 2   Proof.
The ideal I has depth 2 for otherwise the first two generators of I would be divisible by either x 0 , x 1 or l i for i ∈ {2, . . . , n -1} which is excluded by the assumption that all the lines in F n are distinct. A direct computation shows that

     x 1 x 2 l 2 . . . l n-1 + x 0 x 1 x 2 ∂ ∂x0 (l 2 . . . l n-1 ) = M 1 x 0 x 2 l 2 . . . l n-1 + x 0 x 1 x 2 ∂ ∂x1 (l 2 . . . l n-1 ) = M 2 x 0 x 1 l 2 . . . l n-1 = M 3
where, given j ∈ {1, 2, 3}, M j stands for (-1) j times the minor obtained from M by leaving out the j th row (in order to check these equalities, remark that, for any i ∈ {2, . . . , n -1},

l i = x 0 ∂li x0 + x 1 ∂li x1
). Hence I is a determinantal ideal given by the 2-minors of M . Since it has the expected depth, the Hilbert-Burch theorem asserts that a free resolution of I reads:

0 R 2 R 3 I 0.
M Moreover, since it does not have constant entries, M is a minimal presentation matrix of I.

Proposition 2.2. Let n ∈ N >1 and k be an algebraically closed field such that char(k) does not divide n. Then I is of linear type and the polar map Φ fn of f n has multidegree (n -1, n, 1).

Proof. Since M has one column of linear entries and one column of entries of degree n, the naive multidegree is (n -1, n, 1), see Example 1.2. Moreover Fitt 2 (I) = (x 0 , x 1 , x 2 ) is not supported on any point of P 2 k so, by (1.2), I is of linear type. Hence the graph Γ and the naive graph P(I) coincide, so the projective degrees and the naive projective degrees of Φ fn coincide.

We consider now the case where char(k) divides n in a more general situation. 2.2. Weight of the torsion. In all the section, we put n ∈ N >1 , k be any field unless otherwise specified, R = k[x 0 , x 1 , x 2 ] and we consider the ideal I generated by the 2-minors of the matrix

M =   λ 0 p 0 λ 1 p 1 λ 2 p 2   with entries in R = k[x 0 , x 1 , x 2 ]
such that for all j ∈ {0, 1, 2}, λ j belongs to (x 0 , x 1 ) 1 and p j belongs to (x 0 , x 1 ) n-2 n-1 , the homogeneous piece of degree n -1 of the ideal (x 0 , x 1 ) n-2 . We assume moreover that I has height 2 and that there exists j ∈ {0, 1, 2} such that p j ∈ (x 0 , x 1 ) n-2 n-1 \(x 0 , x 1 ) n-1 n-1 . In S = R[y 0 , y 1 , y 2 ], consider now the ideal

I P(I) = (λ 0 y 0 + λ 1 y 1 + λ 2 y 2 , y 0 p 0 + y 1 p 1 + y 2 p 2 )
generated by the entries of the matrix y 0 y 1 y 2 M . Under the conditions imposed on the entries of M , the naive graph P(I) = V(I P(I) ) ⊂ P 2

x × P 2 y has multidegree (n -1, n, 1) and is the union of a torsion part T supported on V(x 0 , x 1 ), see (1.2), and of the graph Γ = P(I)\V(x 0 , x 1 ) of the map Φ defined by the 2-minors of M . The next result is a consequence of [BCS18, Theorem 5.14] but we will give a self-contained proof.

Lemma 2.3. Under the previous conditions on M , the torsion component T of P(I) has multidegree (n -2, 0, 0) and the graph Γ of Φ has multidegree (1, n, 1).

Proof. We analyse separately each element of the multidegree d 2 (Γ), d 1 (Γ), d 0 (Γ) .

• Case i = 2. Take a general point y ∈ P 2 k as the intersection of two general lines H 1 , H 2 of P 2 k and consider the intersection P(I) y = P(I) ∩ p -1 2 (H 1 ) ∩ p -1 2 (H 2 ) ⊂ P 2 y . Under our assumptions, this intersection is a complete intersection of a line and a curve of degree n -1 in P 2 y . Moreover, since λ j ∈ (x 0 , x 1 ) 1 for all j ∈ {0, 1, 2} and since there exists j ∈ {0, 1, 2} such that p j ∈ (x 0 , x 1 ) n-2 n-1 \(x 0 , x 1 ) n-1 n-1 , this complete intersection decomposes as the union of the point V(x 0 , x 1 ) y ∈ P 2 y with multiplicity n -2 and of another point with multiplicity 1. By the generality assumption on y, we can assume that the subscheme P(I) ∩ P 2 y is defined by the ideal x 1 , x n-2 0 (x 0 + α) for some α ∈ k\{0}. Since Γ = P(I)\V(x 0 , x 1 ) is defined by the saturation [I P(I) : (x 0 : x 1 ) ∞ ] of the ideal I P(I) of P(I) by the ideal (x 0 , x 1 ) = Fitt 2 (I), see (1.2), the only points of P 2 k over which the fibre of P(I) contributes to d 2 (Γ) are those different from V(x 0 , x 1 ). Thus the other point is the only

element in Γ ∩ p -1 2 (H 1 ) ∩ p -1 2 (H 2 ). Hence d 2 (Γ) = length Γ ∩ p -1 2 (H 1 ) ∩ p -1 2 (H 2 ) = 1 and d 2 (T) = d 2 (P(I)) -d 2 (Γ) = n -2.
• Case i = 1. Since I has height 2, the linear system defined by the 2-minors of M does not have fixed components so d 1 (Γ) = d 1 P(I) = n and thus d 1 (T) = 0. • Case i = 0. The torsion component T being supported over V(x 0 , x 1 ), the intersection T ∩ p -1 1 (H 1 ) ∩ p -1 1 (H 2 ) of T with inverse images of general lines in P 2 k is empty so d 0 (T) = 0 and d 0 (Γ) = d 0 (P(I)) -d 0 (T) = 1. To sum up, T has multidegree (n -2, 0, 0) and Γ has multidegree

(n -1, n, 1) -(n -2, 0, 0) = (1, n, 1).
We have the following extension of Theorem A.

Theorem 2.4.

(1) Let n ∈ N >1 and assume that p = char k divides n, then the union F n = V(f n ) of n distinct lines in the pencil through a given point z 0 with another line not passing through z 0 is homaloidal.

(2) Let n ∈ N >1 and assume that p = char k divides n(n-1)-1, then the curve

G n = V x 0 x 1 (x n-1 1 + x n-2 0 x 2 ) is homaloidal.
Proof.

(1) By Lemma 2.1, a presentation matrix of the ideal I of partial derivatives of f n verifies the conditions of Lemma 2.3. Hence Φ fn is birational and since the associated linear system has no fixed component, the polynomial f n is homaloidal.

(2) Let n ∈ N >1 . The ideal

I = (x n 1 + (n -1)x n-2 0 x 1 x 2 , nx 0 x n-1 1 + x n-1 0 x 2 , x n-1 0 x 1 ) of partial derivatives of g n = x 0 x 1 (x n-1 1 + x n-2 0 x 2 ) has presentation matrix M =   nx 0 0 -x 1 x n-2 0 x 1 -(n(n -1) -1)x 2 -nx n-1 1 -x n-2 0 x 2   , Indeed, I has height 2 for otherwise x 0 or x 1 would divide x n 1 + (n - 1)x n-2 0 x 1 x 2 and nx 0 x n-1 1 + x n-1 0 x 2 which is not the case. Moreover      x n 1 + (n -1)x n-2 0 x 1 x 2 = M 1 nx 0 x n-1 1 + x n-1 0 x 2 = M 2 x n-1 0 x 1 = M 3 ,
where given j ∈ {1, 2, 3}, M j is equal to (-1) j times the minor obtained from M by leaving out the j th row. Hence I is a determinantal ideal and, by application of Hilbert-Burch theorem, M is a minimal presentation matrix of I. Now, if p divides n(n -1) -1, the matrix M verifies the conditions of Lemma 2.3. So, in this case, Φ gn is birational and g n is homaloidal.

Remark 2.5. The method of reduction modulo p we just described also applies to Example 2 and to the quintic Q 5 = V x 0 (x 2 1 + x 0 x 2 )(x 2 1 + x 0 x 2 + x 2 0 ) described in [START_REF] Bignalet-Cazalet | Torsion of a finite base locus[END_REF].

2.3. Limits and perspectives. The fact that the presentation matrix of the jacobian ideal reduces well modulo p does not always occur, as illustrated by the following example.

Example 2.6. Let h = x 2 (x 4 1 -2x 0 x 2 1 x 2 + x 2 0 x 2 2 -x 1 x 3 2 ) ∈ k[x 0 , x 1 , x 2 ]. Its zero locus in P 2 k is the union of the unicuspidal ramphoïd quartic with the tangent cone at its cusp, see [START_REF] Moe | Rational cuspidal curves[END_REF]. Over a field k of characteristic 0, a computation with Macaulay2 shows that a presentation matrix of the ideal of partial derivatives of h reads:

  15x 2 1 + 3x 0 x 2 72x 0 x 1 + 15x 2 2 8x 1 x 2 2x 2 1 + 30x 0 x 2 -2x 2 2 -8x 1 x 2   .
We can a priori not expect to apply Lemma 2.3 after reduction modulo p. However, after reducing modulo 3, a presentation matrix of the reduction of I modulo 3 reads

  0 x 3 1 -x 0 x 1 x 2 -x 3 2 x 1 x 0 x 2 2 -x 2 -x 1 x 2 2   .
This implies that the polar map of h is birational by Lemma 2.3 (here, remark that the torsion is supported on V(x 1 , x 2 ) and that the maximal power of x 0 is 1 is the second column). By application of Hilbert-Burch theorem, we also have that the induced linear system does not have fix components so h is actually homaloidal.

  be the Rees algebra of I. By [Dol00, 7.1.3] the blow-up Proj R(I) of P 2 k with respect to I is the graph Γ of Φ. Moreover, R(I) is the epimorphic image of the symmetric algebra S(I) of I via the epimorphisms I ⊗i ։ I i . Hence, the ideal of the graph Γ ⊂ P 2 k × P 2 k of Φ contains the ideal of P(I) = Proj S(I) ⊂ P 2 k × P 2 k generated by the entries of the matrix y 0 y 1 y 2 M S where S = R[y 0 , y 1 , y 2 ] stands for the coordinate ring of P 2 k × P 2 k and M S stands for a presentation matrix of I ⊂ R tensored with S [BC18, Subsection 1.1] (note that we use the same notation M and M S from now on). The ideal I is said to be of linear type if Γ = P(I) [Vas05, 1.1 Ideals of linear type]. Definition 1.1. The naive graph of Φ is the projectivization P(I) = Proj S(I) of the symmetric algebra of I. Example 1.2. Let Φ = (φ 0 : φ 1 : φ 2 ) : P 2 k P 2 k be a dominant rational map with base

  height 2 and denote by M the 3 × 2-presentation matrix of I. Then the complement subscheme T = P(I)\Γ of Γ in P(I) is supported on (1.2) x∈V Fitt2(I) {x} × P 2 k where Fitt 2 (I) is the second Fitting ideal of I [Eis95, Corollary-Definition 20.4], by definition generated by the entries of M , see [BC18, Corollary 1.4] for a reference. Definition 1.3. Let Φ = (φ 0 : φ 1 : φ 2 ) : P 2 k P 2 k be a determinantal rational map with base ideal I = (φ 0 , φ 1 , φ 2 ) of height 2. The naive projective degrees d 2 (Φ), d 1 (Φ), d 0 (Φ) of Φ are defined by the multidegree d 2 (P(I)), d 1 (P(I)), d 0 (P(I)) of its naive graph P(I). Example 1.4. If I is not necessarily of linear type in Example 1.2, we still have that the naive projective degree of Φ are (ab, a+b, 1) because P(I) is still a complete intersection. However, if there is an extra part T = P(I)\Γ in P(I) with support as in (1.2), we have then d 2 (P(I)), d 1 (P(I)), d 0 (P(I)) = d 2 (Γ), d 1 (Γ), d 0 (Γ) + d 2 (T), d 1 (T), d 0 (T) . So the topological degree d 2 (Φ) is strictly smaller than d 2 (Φ) because d 2 (T) ≥ length V Fitt 2 (I) is non zero. The quantity d 2 (Γ) = d 2 (P(I)) -d 2 (T) depends moreover on the scheme structure of T and is the object of Subsection 2.2.

Acknowledgements. I warmly thank Adrien Dubouloz and Daniele Faenzi for useful remarks and suggestions about early versions of this note.

Milnor numbers and Swan conductors

Let us now focus on the singularities of the curves we considered in Theorem 2.4. We follow here the presentation in [START_REF] Dolgachev | Polar Cremona transformation[END_REF] and [START_REF] Nguyen | Invariants of plane curve singularities and Plücker formulas in positive characteristic[END_REF]. Fix an algebraically closed field k.

Let f ∈ k[x 0 , x 1 , x 2 ] be homogeneous of degree d and consider the ideal I = (f 0 , f 1 , f 2 ) of the partial derivatives of f . Assume moreover that the linear system defined by the f i does not have fixed components. Given z ∈ Z = V(I) ⊂ P 2 k , define the local Milnor number µ f,z of f at z as Denoting by Φ f the polar map of f defined by f 0 , f 1 , f 2 , we have that

Define also the conductor invariant δ(f, z) as the length of the quotient module O f,z /O f,z , where O f,z is the normalization of the local ring O f,z . Over the field of complex number k = C, Jung-Milnor formula, [Mil68, Th. 10.5] establishes a relation between the local Milnor number µ f,z at z ∈ Z and δ(f, z), namely:

where r(f, z) denotes the number of local branches of F = V(f ) at z. The classification of complex homaloidal curves of I.V. Dolgachev in [START_REF] Dolgachev | Polar Cremona transformation[END_REF] relies on this formula which provides constraints on the type of singularities of a homaloidal complex curve.

In characteristic p > 0, P. Deligne showed that one has to add a corrective term depending on p for the formula to hold (cf. [START_REF] Deligne | La formule de Milnor. SGA 7 II[END_REF], [START_REF] Melle-Hernández | Pencils of curves on smooth surfaces[END_REF]), that is:

where Sw p (f, z) denotes the Swan conductor of f at z (see [Del73, 1.7, 1.8] for the definition). By [Ngu16, Corollary 3.2], Sw p (f, z) = 0 if p > d(d -1). Hence, given a polynomial f ∈ Z[x 0 , x 1 , x 2 ] of degree d, if the characteristic p is big enough with respect to d, the global Milnor number of F = V(f ) is the same as if one considers the equation of F over k or C. For big enough primes p, the topological degree of the polar map Φ f is thus invariant after reduction of f modulo p. On the other hand, if p is smaller than d(d -1), the line arrangements

Proof. Let n ∈ N >1 . As explained at the beginning of Subsection 2.1, we can reduce to the case where F n has equation x 0 x 1 l 2 • • • l n-1 x 2 and z 0 = (0 : 0 : 1). Hence, the singular locus Z of F n is supported on the union

where for i ∈ {1, . . . , n}, (α i : β i : 0) are the intersection points of V(x 2 ) with V x 0 x 1 l 2 • • • l n-1 . Because the intersection between V(x 2 ) and the n th firsts lines is transverse, each local Milnor number µ Fn,(αi:βi:0) is equal to 1.

Moreover, by (3.1), we have that:

Now, if p = 0 then d 2 (Φ fn ) = n -1 by Proposition 2.2 and since Sw(f n , z 0 ) = 0, we have 2δ(f n , z 0 ) -r(f n , z 0 ) + 1 = (n + 1) 2 . Thus, by Theorem 2.4, if p divides n, we have d 2 (Φ fn ) = 1 = n 2 -n -(n -1) 2 + Sw p (f n , z 0 ) so Sw p (f n , z 0 ) = n -2.