Rachid Tahiri

Lijun Shen

Boris E Malyugin

S Fyodorov

Cedric Schweitzer
email: cedric.schweitzer@chu-bordeaux.fr

Mateusz Kecik

Femtosecond laser-assisted cataract surgery: Update and perspectives

Keywords: cataract, cataract surgery, femtosecond laser (fs), femtosecond laser assisted cataract surgery (FLACS), phacoemulsification

Cataract surgery is among the most frequently performed surgical procedures worldwide and has a tremendous impact on patients' quality of life. Phacoemulsification (PCS) is accepted as a standard of care; its technique has continuously evolved and already achieved good anatomical, visual, and refractive outcomes. Lasers in ophthalmology are widely used in clinical practice, femtosecond lasers (FSLs) for corneal surgery in particular. It was natural to assess the usefulness of FSL in cataract surgery as this technology was within reach. Indeed, precise and reproducible cuttings provided by FSL platforms could improve standardization of care and limit the risk associated with the human element in surgery and provide a step toward robot-assisted surgery. After docking and planning the procedure, femtosecond lasers are used to perform corneal incisions, capsulorhexis, lens fragmentation, and arcuate incisions in an automated manner. A well-constructed corneal incision is primordial as it o ers safety during the procedure, self-seals afterward, and influences the refractive outcome. Capsulorhexis size, centration, and resistance to shearing influence the surgery, intraocular lens (IOL) centration and stability, and posterior capsular opacification formation. Lens fragmentation is where most of the energy is delivered into the eye, and its amount influences endothelial cell damage and potential damage to other ocular structures. The arcuate incisions o er an additional opportunity to influence postoperative astigmatism. Femtosecond laser-assisted cataract surgery (FLACS) has been a topic of research in many studies and clinical trials that attempted to assess its potential benefits and cost-e ectiveness over PCS and is the subject of this mini-review.

Introduction

Cataract remains the leading cause of blindness and the second cause of moderate and severe visual impairment in 2020, affecting 15.2 and 78.8 million people, respectively, worldwide. Cataract surgery is the most routinely performed surgical procedure, with 7 million surgeries performed per year in Europe, 3.7 million in the United States, and 20 million worldwide [START_REF] Rossi | Cataract surgery practice patterns worldwide: a survey[END_REF]. Since its first introduction in 1967 by Kelman and continuous development of machines and intraocular lenses, phacoemulsification cataract surgery (PCS) has become the standard of care [START_REF] Day | The royal college of ophthalmologists' national ophthalmology database study of cataract surgery: report 1, visual outcomes and complications[END_REF](3)[START_REF] Gale | Benchmark standards for refractive outcomes after NHS cataract surgery[END_REF][START_REF] Kugelberg | Factors related to the degree of success in achieving target refraction in cataract surgery: Swedish national cataract register study[END_REF][START_REF] Lundstrom | Risk factors for refractive error after cataract surgery: analysis of 282,811 cataract extractions reported to the European registry of quality outcomes for cataract and refractive surgery[END_REF].

The recent development of near-infrared lasers with ultrashort pulse duration of the femtosecond domain (10 -15 s) has opened new insights in ophthalmic surgery. Femtosecond lasers (FSLs) had their first clinical application in ophthalmology in corneal treatment in the late 1990s; they were then evaluated for their role in cataract surgery [START_REF] Krueger | First safety study of femtosecond laser photodisruption in animal lenses: tissue morphology and cataractogenesis[END_REF], with first treated patients in 2009 [START_REF] Nagy | Initial clinical evaluation of an intraocular femtosecond laser in cataract surgery[END_REF] and received Food and Drug Administration approval in 2010. During FLACS, after the docking procedure and owing to non-invasive and precise tissue treatment [START_REF] Chung | Surgical applications of femtosecond lasers[END_REF], FSL can be used to perform corneal incisions, capsulotomy, lens fragmentation, and arcuate incisions. Eliminating the human element from the first steps of the surgery has been suggested to improve the safety and reproducibility of surgical outcomes [START_REF] Chee | Clinical outcomes in the first 2 years of femtosecond laser-assisted cataract surgery[END_REF][START_REF] Scott | Comparison of vitreous loss rates between manual phacoemulsification and femtosecond laser-assisted cataract surgery[END_REF]. However, wellconducted and sufficiently powered randomized clinical trials comparing both techniques were needed to provide robust evidence and recommendations [START_REF] Day | Laser-assisted cataract surgery vs. standard ultrasound phacoemulsification cataract surgery[END_REF].

In this article, we report on the principles of FLACS and review current scientific evidence on PCS and its potential perspectives.

Technical performances and principles of FLACS

Femtosecond laser is a near-infrared (1,053 nm) solid-state laser achieving photodisruption in ocular tissues; it is capable of delivering ultrashort femtosecond pulses 200-800 fs (1 fs = 10 -15 s) of low overall energy and high peak power. It leads to more precise impact, less energy per pulse, and, in turn, less damage in collateral tissue than longer pulse lasers, e.g., nanosecond lasers [START_REF] Vogel | Mechanisms of intraocular photodisruption with picosecond and nanosecond laser pulses[END_REF]. To achieve photodisruption, a certain degree of energy threshold must be surpassed, depending on the nature of treated tissue and different laser features: numerical aperture (NA) and wavelength. A larger NA decreases the volume of the focal spot and, together with a shorter wavelength (e.g., in the UV range for new prototype femtosecond lasers), lowers the energy threshold. A shorter pulse laser enables a decrease in the total energy needed to achieve this threshold. Photodisruption consists of three consecutive events: plasma formation, shock wave generation, and cavitation. Rapidly expanding plasma leads to the creation of a focal shock wave, which is atraumatic for the surrounding tissue and leaves behind only a small residual gas bubble [START_REF] Heisterkamp | Nonlinear side effects of fs pulses inside corneal tissue during photodisruption[END_REF]. The mechanism of tissue separation achieved by FSL is twofold: ablation and cleaving. Gas bubbles represent directly ablated tissue by plasma formation within the laser focus, while rapid expansion and cavitation achieve further tissue separation by cleaving. Laser parameters determine which of those two mechanisms plays a chief role in tissue separation. At higher pulse energies, the radius of disrupted tissue is greater than the spot size, and tissue is separated by mechanical forces of expanding gas bubbles. Lower energies generate less cavitation and ablate a smaller volume of tissue; they allow for less traumatic tissue separation but require much tighter spot placement and call for higher pulse frequencies.

The cataract laser faces challenges different from the corneal platforms. Treatment depths in the former are usually between 100 and 130 µm, as with LASIK, while cataract surgery demands flexibility up to 7,500 µm. Such a difference generates the need for increased spot size and higher pulse energy with the cataract lasers. In addition, 5-10 times greater total energy and spot placements are required for lens surgery, in comparison to corneal surgery [START_REF] Nagy | Initial clinical evaluation of an intraocular femtosecond laser in cataract surgery[END_REF].

In FLACS, complicated and flexible laser patterns are needed, as each step of cataract surgery requires its proprietary treatment and energy profile. Laser capsulotomy requires an adjustable circular laser incision (achieved with postage stamp perforations) at a considerable depth with good accuracy. Lens fragmentation enables the surgeon to choose between different patterns of lens softening: cubes, spheres, pies, etc. So far, no ideal pattern enabling a low cumulative dissipated energy (CDE) has been described, but some lens softening patterns may be better than others in different grades or types of cataract [START_REF] Kaur | Effect of lens fragmentation patterns on phacoemulsification parameters and postoperative inflammation in femtosecond laser-assisted cataract surgery[END_REF].

The patient interface (PI) is essential to connect the laser to the eye and therefore must provide adequate stability without distortion of eye structures, be well tolerated by the patient, and not interfere with laser treatment. In FLACS, either a curved lens applanation or a fluid interface is used. Studies comparing docking with a curved vs. fluid PI showed no significant differences in capsulotomy regularity [START_REF] Harthi | Comparison of the anterior capsulotomy edge created by manual capsulorhexis and 2 femtosecond laser platforms: scanning electron microscopy study[END_REF], but corneal folds, sometimes occurring with applanating PI, may induce incomplete capsulotomies with subsequent anterior capsular tears [START_REF] Talamo | Optical patient interface in femtosecond laser-assisted cataract surgery: contact corneal applanation vs. liquid immersion[END_REF]. The non-applanating designs, using a suction ring attached to the sclera and a fluid immersion chamber, do require more corneal exposure while docking but offer a larger treatment area, better compensation for lens tilt, and a much lower induced IOP rise during the treatment [START_REF] Talamo | Optical patient interface in femtosecond laser-assisted cataract surgery: contact corneal applanation vs. liquid immersion[END_REF]. Speculation exists whether the anterior movement of the lens, anterior hyaloid, and vitreous base during suction can predispose patients to develop vitreoretinal pathology, but no evidence from corneal-only FSL exists [START_REF] Kanclerz | Does corneal refractive surgery increase the risk of retinal detachment? A literature review and statistical analysis[END_REF]. The docking procedure is paramount in achieving good laser treatment but is an extra step compared to PCS and can be time-consuming [START_REF] Feldman | Femtosecond laser will not be a standard method for cataract extraction 10 years from now[END_REF][START_REF] Bali | Early experience with the femtosecond laser for cataract surgery[END_REF]. A potential complication is the loss of suction leading to incomplete laser treatment. In an early study by Bali et al. (20), the authors experienced suction loss requiring abandoning the procedure in 2.5% of cases, but the study demonstrated a learning curve for the first 100 cases. Manning et al. [START_REF] Manning | Femtosecond laser-assisted cataract surgery vs. standard phacoemulsification cataract surgery: study from the European registry of quality outcomes for cataract and refractive surgery[END_REF] indicated that the laser procedure had to be abandoned in 0.1% (3 eyes). Zhang et al. [START_REF] Zhang | Performance of femtosecond laser-assisted cataract surgery in Chinese patients with cataract: a prospective, multicenter, registry study[END_REF] found the completion of capsulotomy, lens fragmentation, and corneal incisions to be at 98.6, 98.5, and 97.6%, respectively. In the first studies, the most frequently reported complication related to docking was transient conjunctival redness or hemorrhage in up to 34% of cases [START_REF] Nagy | Complications of femtosecond laser-assisted cataract surgery[END_REF].

Finally, unlike FSL corneal procedures, precise and reliable imaging is needed for FLACS procedures. OCT is the most frequent imaging technology, but three-dimensional confocal structured illumination imaging is also used by one platform. Indeed, by enabling a three-dimensional reconstruction of anterior eye structures, it helps define capsulotomy margins, corneal incision placement, and safety zones within the lens itself.

Clinical results

Corneal incisions

Incisions are a crucial step in completing cataract surgery with clear corneal incisions being the standard for PCS and FLACS. Laser-created incisions were found to be reproducible and stable and did not significantly change the high-order aberrations over time [START_REF] Alio | Femtosecond laser cataract incision morphology and corneal higher-order aberration analysis[END_REF][START_REF] Masket | Femtosecond laser-assisted cataract incisions: architectural stability and reproducibility[END_REF][START_REF] Chaves | Architecture evaluation of the main clear corneal incisions in femtosecond laser-assisted cataract surgery by optical coherence tomography imaging[END_REF][START_REF] Mastropasqua | Femtosecond laser vs. manual clear corneal incision in cataract surgery[END_REF]. Users can also skip this step and elect to perform them with the microkeratome; this seemed to be the common procedure among surgeons as indicated in the 2016 study from the European Registry of Quality Outcomes for Cataract and Refractive Surgery, since in only 34.7% of FLACS cases, the incisions were performed by laser [START_REF] Manning | Femtosecond laser-assisted cataract surgery vs. standard phacoemulsification cataract surgery: study from the European registry of quality outcomes for cataract and refractive surgery[END_REF]. The reason might be due to the fact that laser incisions take significantly longer to perform. The laser-created incisions show a saw-tooth pattern, increased cell apoptosis, and are less smooth than manual incisions [START_REF] Mayer | In vitro immunohistochemical and morphological observations of penetrating corneal incisions created by a femtosecond laser used for assisted intraocular lens surgery[END_REF]. They may be less stable in the early postoperative period as well [START_REF] Rodrigues | Corneal incision architecture: VICTUS femtosecond laser vs. manual keratome[END_REF].

Diakonis et al. [START_REF] Diakonis | Comparison of surgically induced astigmatism between femtosecond laser and manual clear corneal incisions for cataract surgery[END_REF] did not show any significant difference in surgically induced astigmatism between FLACS and PCS using a multivariate vectorial analysis.

Capsulotomy

The main advantage of FLACS is the ability to create a perfectly circular, centered capsulotomy with an accurate and reproducible diameter [START_REF] Mastropasqua | Scanning electron microscopy evaluation of capsulorhexis in femtosecond laser-assisted cataract surgery[END_REF][START_REF] Friedman | Femtosecond laser capsulotomy[END_REF](33). Anterior capsulotomy conditions both the remainder of cataract surgery and the refractive outcome achieved by patients. Inaccurate prediction of effective lens position (ELP), which is, in turn, affected by capsulotomy size and morphology, is one of the biggest factors leading to residual refractive error contributing to 35% of total residual error [START_REF] Norrby | Sources of error in intraocular lens power calculation[END_REF]. Studies have confirmed femtosecond laser capsulotomies to be rounder, more precise, better centered, and more stable over time as compared to manual capsulorhexis. However, this advantage has not been translated into better visual acuity or effective lens position [START_REF] Panthier | Change of capsulotomy over 1 year in femtosecond laser-assisted cataract surgery and its impact on visual quality[END_REF].

There are, however, disadvantages to a laser capsulotomy, mainly its strength. Electron microscopy images demonstrate that the laser impacts do not produce a smooth capsulotomy rim as in a manual rhexis but instead produces one with numerous notches that could be more prone to tearing [START_REF] Lua | Superior rim stability of the lens capsule following manual over femtosecond laser capsulotomy[END_REF], resulting in an overall lower tensile strength [START_REF] Daya | Comparison of anterior capsulotomy techniques: continuous curvilinear capsulorhexis, femtosecond laserassisted capsulotomy and selective laser capsulotomy[END_REF] and, in turn, a higher frequency of anterior capsular tears (up to 4% in early published studies) [START_REF] Nagy | Complications of femtosecond laser-assisted cataract surgery[END_REF][START_REF] Abell | Anterior capsulotomy integrity after femtosecond laser-assisted cataract surgery[END_REF]. Completeness of anterior capsulotomy can also be impeded by eye tremor during laser delivery, laser misfires, or corneal opacities such as scars or radial keratotomies [START_REF] Abell | Anterior capsulotomy integrity after femtosecond laser-assisted cataract surgery[END_REF][START_REF] Trinh | Outcomes of femtosecond laser-assisted cataract and refractive lens surgery in patients with prior radial keratotomy[END_REF], but the latter might be remedied by adjusting laser settings [START_REF] Trinh | Outcomes of femtosecond laser-assisted cataract and refractive lens surgery in patients with prior radial keratotomy[END_REF][START_REF] Dick | Comment on: Outcomes of femtosecond laser-assisted cataract and refractive lens surgery in patients with prior radial keratotomy[END_REF]. Laser settings seem to be key even in routine FLACS, as different lasers produce microscopically different capsulotomy edges which might explain the variability of capsulotomy completeness and anterior radial tears in literature (0.1-4%) [START_REF] Nagy | Complications of femtosecond laser-assisted cataract surgery[END_REF][START_REF] Mastropasqua | Scanning electron microscopy evaluation of capsulorhexis in femtosecond laser-assisted cataract surgery[END_REF][START_REF] Day | Efficacy of anterior capsulotomy creation in femtosecond laser-assisted cataract surgery[END_REF].

Intraoperative miosis following docking and laser procedure has also been reported in one-third of cases [START_REF] Nagy | Complications of femtosecond laser-assisted cataract surgery[END_REF]. This is mainly related to prostaglandin release during laser treatment of the anterior capsule [START_REF] Schultz | Changes in prostaglandin levels in patients undergoing femtosecond laser-assisted cataract surgery[END_REF][START_REF] Schultz | Prostaglandin release during femtosecond laser-assisted cataract surgery: main inducer[END_REF]. Factors associated with intraoperative miosis are older patient age, longer duration of laser treatment, smaller laser capsulotomy-pupil margin distance, shallower anterior chamber, and smaller preoperative pupil size [START_REF] Popiela | Factors influencing pupil behaviour during femtosecond laser assisted cataract surgery[END_REF][START_REF] Jun | Pupil-size alterations induced by photodisruption during femtosecond laser-assisted cataract surgery[END_REF]. Some preoperative treatment regimens have been proposed to reduce intraoperative miosis onset or severity: mainly topical nonsteroidal anti-inflammatory drugs 1-4 days before surgery and the day of the surgery [START_REF] Schultz | NSAID pretreatment inhibits prostaglandin release in femtosecond laser-assisted cataract surgery[END_REF][START_REF] Anisimova | Effect of NSAIDs on pupil diameter and expression of aqueous humor cytokines in FLACS vs. conventional phacoemulsification[END_REF][START_REF] Kiss | One-day use of preoperative topical nonsteroidal anti-inflammatory drug prevents intraoperative prostaglandin level elevation during femtosecond laser-assisted cataract surgery[END_REF].

Femtosecond laser-assisted arcuate keratotomy (FSAK)

Femtosecond laser-assisted cataract surgery has an advantage of an optional step of performing FSAK to correct low-to-moderate astigmatism during cataract surgery. FSAK has been found to be effective, safe, and stable over time [START_REF] Roberts | Refractive outcomes after limbal relaxing incisions or femtosecond laser arcuate keratotomy to manage corneal astigmatism at the time of cataract surgery[END_REF][START_REF] Vickers | Femtosecond laser-assisted keratotomy[END_REF][START_REF] Visco | Femtosecond laser-assisted arcuate keratotomy at the time of cataract surgery for the management of preexisting astigmatism[END_REF]. They also offer some flexibility as the surgeon can elect to open them at the slit-lamp during the postoperative period for a greater corrective effect.

Clinical studies have found FSAK to be comparable to toric IOLs in mild-to-moderate, central, and with-the-rule astigmatism correction but less predictable in moderate-to-high, against-therule, or limbus-to-limbus astigmatism [START_REF] Yoo | Femtosecond laser-assisted arcuate keratotomy vs. toric IOL implantation for correcting astigmatism[END_REF][START_REF] Noh | Comparison of penetrating femtosecond laser-assisted astigmatic keratotomy and toric intraocular lens implantation for correction of astigmatism in cataract surgery[END_REF]. A recent study by Hernandez et al. (54) found the toric IOL to be superior over FSAK in the correction of moderate (1.25-3.0 diopter) astigmatism. Nevertheless, FSAK is more predictable in comparison to manual limbal relaxing incisions, and the development of new nomograms in the future may improve their efficacy [START_REF] Roberts | Refractive outcomes after limbal relaxing incisions or femtosecond laser arcuate keratotomy to manage corneal astigmatism at the time of cataract surgery[END_REF].

Endothelial cell loss, ultrasound energy delivered in the eye, and di cult cases

A potential advantage of FLACS over PCS could be that the lens softening allows for the reduction in effective phacoemulsification time (EPT) and cumulative dissipated energy (CDE) and thus may result in less damage to ocular tissues, particularly the corneal endothelium. An early study by Abell et al. ( 55) demonstrated a 96.2% reduction in EPT in the optimized FLACS group associated with a 36.1% reduction in endothelial cell loss; the laser pretreatment enabled an ultrasound-free lens aspiration in 30% of patients.

On the contrary, the FEMCAT trial did not observe any difference in central corneal thickness or corneal endothelial cell count between FLACS and PCS [START_REF] Schweitzer | Femtosecond laser-assisted vs. phacoemulsification cataract surgery (FEMCAT): a multicentre participant-masked randomised superiority and cost-effectiveness trial[END_REF], and the FACT trial even showed more cell loss in the FLACS group [START_REF] Day | Femtosecond laser-assisted cataract surgery compared with phacoemulsification cataract surgery: randomized noninferiority trial with 1-year outcomes[END_REF]. Other clinical trials seem to agree with this observation [START_REF] Bascaran | Differences in energy and corneal endothelium between femtosecond laser-assisted and conventional cataract surgeries: prospective, intraindividual, randomized controlled trial[END_REF][START_REF] Dzhaber | Comparison of changes in corneal endothelial cell density and central corneal thickness between conventional and femtosecond laser-assisted cataract surgery: a randomised, controlled clinical trial[END_REF], and the 2022 report by the American Academy of Ophthalmology concluded that the adequately powered RTCs did not demonstrate a significant difference in endothelial cell loss between FLACS and PCS [START_REF] Lin | Femtosecond laser-assisted cataract surgery: a report by the American academy of ophthalmology[END_REF].

There might, however, exist groups of patients that could benefit from the decreased EPT more than those with a routine cataract. Patients with hard nuclear cataracts or with shallow anterior chambers were shown to achieve a faster visual recovery after FLACS [START_REF] Chen | Clinical outcomes of femtosecond laser-assisted cataract surgery vs. conventional phacoemulsification surgery for hard nuclear cataracts[END_REF][START_REF] Vasavada | Comparative evaluation of femtosecond laser-assisted cataract surgery and conventional phacoemulsification in eyes with a shallow anterior chamber[END_REF]. Yong et al. [START_REF] Yong | Comparing outcomes of phacoemulsification with femtosecond laser-assisted cataract surgery in patients with fuchs endothelial dystrophy[END_REF] found a significant difference in endothelial cell loss among Fuchs' patients (mean of 15.3 ± 17.5% for PCS and 4.4 ± 25.0% for FLACS) that was even more significant in the mild cataract subgroup.

Pediatric cataracts

Cataract surgery in infants differs from age-related phacosclerosis mainly because of the elasticity of the anterior capsule, the softness of the lens nucleus, and the necessity to perform posterior capsulotomy [START_REF] Benezra | Posterior capsulectomy in pediatric cataract surgery: the necessity of a choice[END_REF]. FLACS has already been used in congenital cataract surgery with success (65-67), including challenging cases like Peters anomaly type 2 and persistent hyperplastic primary vitreous [START_REF] Hou | Image-guided femtosecond laserassisted cataract surgery in Peters anomaly type 2[END_REF][START_REF] Tereshchenko | Femtosecond laser-assisted anterior and posterior capsulotomies in children with persistent hyperplastic primary vitreous[END_REF]. Femtosecond laser use is off-label in children and requires some adaptations in the surgical procedure. It creates capsulotomies that are larger than expected because of the tissue elasticity, and corrective factors must be used in treatment planning [START_REF] Dick | Femtosecond laser-assisted pediatric cataract surgery: Bochum formula[END_REF][START_REF] Liao | Study on the enlargement index of femtosecond laser-assisted capsulorhexis in 2-6-year-old patients with congenital cataract[END_REF]. A laser-assisted posterior capsulotomy is feasible but requires redocking after lens aspiration [START_REF] Dick | Femtosecond laser-assisted cataract surgery in infants[END_REF].

Cystoid macular edema and mediators of inflammation release

Cystoid macular edema (CME) is a relatively infrequent complication of any intraocular surgery with an incidence of 0.95% after cataract surgery [START_REF] Daien | French medical-administrative database for epidemiology and safety in ophthalmology (EPISAFE): the EPISAFE collaboration program in cataract surgery[END_REF]. FLACS was seen as a potential way of limiting this complication by decreasing EPT. However, a release of prostaglandins into the aqueous humor has been demonstrated during FLACS, especially after capsulotomy [START_REF] Schultz | Changes in prostaglandin levels in patients undergoing femtosecond laser-assisted cataract surgery[END_REF][START_REF] Schultz | Prostaglandin release during femtosecond laser-assisted cataract surgery: main inducer[END_REF]. This can result in intraoperative miosis increasing risks and surgical time during cataract removal and impacting CME formation through inflammation cascade and disruption of the blood-retina barrier [START_REF] Henderson | Clinical pseudophakic cystoid macular edema. Risk factors for development and duration after treatment[END_REF][START_REF] Lobo | Pseudophakic cystoid macular edema[END_REF]. In clinical studies, this relationship was not as straightforward, as Koo et al. [START_REF] Koo | Outcomes of femtosecond laser-assisted cataract surgery compared to conventional phacoemulsification in eyes with pseudoexfoliation syndrome[END_REF] found FLACS to be associated with a lesser likelihood of the usage of a pupil expansion device in comparison with PCS (7.0 vs. 24.2%) and a faster CME resolution in FLACS in cases of pseudoexfoliation.

Posterior capsule opacification (PCO) and phimosis

FLACS produces reliable capsulotomies contributing to bag stability and elimination of lens epithelial cells (LEC) and has been shown to induce LEC apoptosis in LEC close to the capsulotomy edge, which is linearly correlated with laser energy parameters and duration [START_REF] Toto | Induced inflammation and apoptosis in femtosecond laser-assisted capsulotomies and manual capsulorhexes: an immunohistochemical study[END_REF][START_REF] Pisciotta | Anterior capsule of the lens: comparison of morphological properties and apoptosis induction following FLACS and standard phacoemulsification surgery[END_REF][START_REF] Sun | Human lens epithelial cell apoptosis and epithelial to mesenchymal transition in femtosecond laser-assisted cataract surgery[END_REF]. This effect could theoretically decrease PCO rates in FLACS. Few results are reported, and some studies have even found higher PCO rates in FLACS compared to PCS and reported early optic axis opacities [START_REF] Manning | Femtosecond laser-assisted cataract surgery vs. standard phacoemulsification cataract surgery: study from the European registry of quality outcomes for cataract and refractive surgery[END_REF][START_REF] Lundstrom | Femtosecond laser-assisted cataract surgeries reported to the European Registry of quality outcomes for cataract and refractive surgery: baseline characteristics, surgical procedure, and outcomes[END_REF], possibly in relation to the mesenchymal transformation of LEC when exposed to adjacent laser treatment.

Results of randomized controlled trials: FLACS vs. PCS

Several well-designed and rigorously conducted randomized controlled trials (RCTs) comparing FLACS to PCS have been published in the literature (56-59, 62, 78-84); their details are shown in Table 1. FEMCAT, FACT, and St. Thomas [START_REF] Schweitzer | Femtosecond laser-assisted vs. phacoemulsification cataract surgery (FEMCAT): a multicentre participant-masked randomised superiority and cost-effectiveness trial[END_REF][START_REF] Day | Femtosecond laser-assisted cataract surgery compared with phacoemulsification cataract surgery: randomized noninferiority trial with 1-year outcomes[END_REF][START_REF] Roberts | A randomized controlled trial comparing femtosecond laser-assisted cataract surgery vs. conventional phacoemulsification surgery[END_REF][START_REF] Stanojcic | randomised controlled trial comparing femtosecond laser-assisted cataract surgery vs. conventional phacoemulsification surgery: 12-month results[END_REF][START_REF] Day | Femtosecond laser-assisted cataract surgery vs. phacoemulsification cataract surgery (FACT): a randomized noninferiority trial[END_REF] trials evaluated corrected, uncorrected distance vision acuities, and refractive outcomes and found no statistically significant differences between both groups. Since complications in cataract surgery occur rarely, no RCT has been powered sufficiently for a meaningful analysis, and the published trials have not found differences in complications of FLACS compared to PCS, with the exception of St. Thomas trial, which found a statistically higher rate of posterior capsule rupture in PCS (3 vs. 0%) [START_REF] Stanojcic | randomised controlled trial comparing femtosecond laser-assisted cataract surgery vs. conventional phacoemulsification surgery: 12-month results[END_REF]. The endothelial cell loss was similar between FLACS and PCS in published RCTs.

Cost-e ectiveness

The FLACS technology is expensive both for the hospital in relation to the initial cost of the laser, space, and personnel and for the patient due to the cost of consumables [START_REF] Bartlett | The economics of femtosecond laser-assisted cataract surgery[END_REF]. In addition, FLACS still requires the use of a phacoemulsificationaspiration machine to remove lens pieces or viscoelastic devices. Cost-effectiveness is a key aspect of integrating a surgical procedure in the medical landscape; only a few studies examined this in regard to FLACS in published literature. The FEMCAT trial [START_REF] Schweitzer | Femtosecond laser-assisted vs. phacoemulsification cataract surgery (FEMCAT): a multicentre participant-masked randomised superiority and cost-effectiveness trial[END_REF] confirmed this hypothesis and concluded that FLACS was more expensive and less effective as compared to PCS, with an incremental cost-effectiveness ratio (ICER) of e 10,703.2 saved per patient successfully treated with PCS. While the FACT trial found FLACS to be non-inferior to PCS, FLACS was also not cost-effective [START_REF] Day | Femtosecond laser-assisted cataract surgery compared with phacoemulsification: the FACT noninferiority RCT[END_REF].

Roberts et al. ( 87) designed a model that could make FLACS cost-effective only if efficiency increased by 100% or if the patient interface cost was reduced by 70%.

Learning curve for FLACS

FLACS, similar to any new surgical technique, has a learning curve even for experienced surgeons. Studies have found that, after ∼100 cases, the complication rate drops significantly [START_REF] Bali | Early experience with the femtosecond laser for cataract surgery[END_REF][START_REF] Roberts | Surgical outcomes and safety of femtosecond laser cataract surgery: a prospective study of 1,500 consecutive cases[END_REF]. Surgeons familiar with refractive femtosecond lasers may learn FLACS faster [START_REF] Roberts | Surgical outcomes and safety of femtosecond laser cataract surgery: a prospective study of 1,500 consecutive cases[END_REF]. Improvements in training techniques and laser software can also contribute to significantly flattening the learning curve [START_REF] Sutton | Femtosecond cataract surgery: transitioning to laser cataract[END_REF]. However, a survey showed that 82.4% of surgeons felt that 10 cases or less require FLACS to be performed with the same safety as PCS [START_REF] Song | Assessment of surgeon experience with femtosecond laser-assisted cataract surgery[END_REF].

Although FLACS should also be performed by surgeons who have already mastered PCS, some studies showed that senior residents had similar surgical outcomes between PCS and FLACS [START_REF] Hansen | Outcomes of resident-performed laser-assisted vs. traditional phacoemulsification[END_REF][START_REF] Pittner | Resident surgeon efficiency in femtosecond laser-assisted cataract surgery[END_REF][START_REF] Brunin | Outcomes of femtosecond laser-assisted cataract surgery performed by surgeons-in-training[END_REF].

Discussion and perspectives

Femtosecond laser use does not have to be confined to biological tissue interaction only; they can theoretically alter properties of an already implanted IOL changing its refractive index and, in turn, overall power, toricity, inducing/erasing multifocality, or generating pinhole apertures. Such concepts have only been tested in vitro so far [START_REF] Kermani | Femto-masking: laser-generated apertures to extend depth of focus and reduce optical aberrations in intraocular lenses[END_REF][START_REF] Ford | Adjustable intraocular lens power technology[END_REF][START_REF] Nguyen | Intraocular lens power adjustment by a femtosecond laser: in vitro evaluation of power change, modulation transfer function, light transmission, and light scattering in a blue lightfiltering lens[END_REF]. These techniques might decrease rates of IOL exchange, as those are related to multifocal intolerance and lens power error by 15 and 16%, respectively [START_REF] Goemaere | Fifteen years of IOL exchange: indications, outcomes, and complications[END_REF].

Capsulorhexis-fixated IOLs can theoretically benefit from the perfectly round, well-centered, and adequately sized laser capsulotomy [START_REF] Smith | Visual performance and positional stability of a capsulorhexis-fixated extended depth-of-focus intraocular lens[END_REF]. Future studies are needed to evaluate whether this will improve ELP and refractive outcomes.

Femtosecond laser technology is likely to evolve as well. Shorter pulse durations could enable higher peak power with less energy per pulse, and a multispot laser treatment could shorten the treatment time. Robotization of the steps of lens aspiration is in its infancy in cataract surgery [START_REF] Chen | Intraocular robotic interventional surgical system (IRISS): semi-automated OCT-guided cataract removal[END_REF][START_REF] Gerber | Robotic posterior capsule polishing by optical coherence tomography image guidance[END_REF], but this technology could be integrated with FLACS to create fully automated cataract extraction platforms in the future.

Conclusion

Although FLACS offers precise and reproducible technical performances, in its current state of development, these advantages do not significantly translate into clinical practice. To date, FLACS has not been able to tackle most frequent and burdensome complications of cataract surgery such as residual refractive error [24.9 vs. 22% of eyes with an absolute refractive error >0.5D in FLACS and PCS groups, respectively, in FEMCAT (56)], CME, endothelial cell loss, or vitreous loss. Furthermore, FLACS is a technological adaptation from corneal laser surgery aiming to replace steps of PCS, but the use of phacoemulsification machines is still needed to remove lens pieces and viscoelastic devices. However, although FLACS does not seem to benefit the most typical patients with cataract, it may assist in more challenging cases such as traumatic cataracts, Fuchs's dystrophy, low endothelial cell count, and some pediatric cases. In addition, despite the precise and reproducible technical performances of FSL platforms, FLACS remains in its infancy of development. Indeed, the refinement of lasers, the development of new technologies to replace phacoemulsification machines, or new IOL technologies may help improve clinical outcomes and cost-effectiveness over the conventional PCS.

  TABLE Summary of the main randomized clinical trials comparing phacoemulsification cataract surgery (PCS) and femtosecond laser-assisted cataract surgery (FLACS).

	Year	Authors	Number of eyes included	Study design	Measured	Main significant
	published		(completed follow-up)		outcomes	findings
			FLACS	PCS
	2013	Conrad-Hengerer	73	73	• 1 eye randomized to	• ECL	• Lower endothelial cell
		et al. (78)			FLACS and the other to		loss in FLACS
					PCS
					• Single surgeon	• CCT	• Reduction in corneal
							thickness with FLACS
					• 3-month follow-up		• Less EPT in FLACS
	2014	Conrad-Hengerer	101	101	• 1 eye randomized to	• CRT	• No difference in retinal
		et al. (79)			FLACS and the other to		thickness
					PCS
					• Single surgeon	• Anterior chamber	• Less AC inflammation
						flare	in FLACS immediately
							after surgery
					• 6-month follow-up
	2018	Bascaran et al. (58)	100 (92)	100 (92)	• 1 eye randomized to	• ECL	• No differences in ECL,
					FLACS and the other to		CCT, and endothelial cell
					PCS		features
					• Single surgeon	• CCT	• Less CDE and fluid
							requirements in FLACS
					• 6-month follow-up		• Eyes with
							complications excluded
							from the analysis
	2018	Roberts et al. (82)	200 (198)	200 (198)	• 1 eye included per	• Visual acuity	• No difference in visual
		St. Thomas Study			patient: randomized to	(UCDVA, BCDVA,	acuity, refractive error,
					PCS or FLACS	pinhole)	ECL, CCT, and CFT
							between the two groups
					• Single-center, 3	• Refractive error	• More posterior capsule
					surgeons		rupture in PCS (3 vs. 0%)
					• 3-4-week follow-up	• ECL, CCT, CRT	• 7 patients (3.5%)
							unable to receive FLACS
						• Complications
	2019	Krarup et al. (80)	96 (90)	96 (90)	• 1 eye randomized to	• UCDVA, CDVA	• No difference in CDVA
					FLACS and the other to		and UCDVA
					PCS
					• Single surgeon	• ECL, hexagonality	• FLACS associated with
							a 21% lower ECL; no
							difference in
							hexagonality
					• 6-month follow-up		• Some patients with
							complications (nucleus
							drop, posterior capsule
							rupture, corneal edema)
							were excluded from the
							analysis
	2019	Vasavada et al. (62)	91	91	• Eyes with shallow AC	• CCT	• Lower CCT in FLACS
					(<2.5 mm); 1 eye		up to 1 month
					randomized to FLACS
					and the other to PCS
					• Single center	• Corneal clarity	• No difference in ECD
							loss, CDVA, and UCDVA
					• 6 months follow-up	• ECL	• Less AC flare in the
							FLACS group
						• Anterior chamber	• Clearer corneas with
						flare	FLACS on day 1
						• CDVA, UCDVA
							(Continued)
	Frontiers in Medicine					frontiersin.org

TABLE (

 (

	Continued)
	Year	Authors	Number of eyes included	Study design	Measured	Main significant
	published		(completed follow-up)		outcomes	findings
			FLACS	PCS
	2020	Hansen et al. (81)	64	71	• Patients randomized to	• BCVA	• No difference in BCVA,
					PCS or FLACS		UCDVA, and ECL
					(unilateral or bilateral)
					• Single center, surgery	• ECL	• Patient satisfaction,
					performed by resident		total intraocular
					surgeons (16)		operating time, and
							complications not
							significantly different
							between two groups
					• 3-month follow-up	• Complications
						• Patient-reported
						outcomes
						• CDE, fluid
						volume
	2020	Dzhaber et al. (59)	67	67	• 1 eye randomized to	• ECL	• difference in ECL
					FLACS and the other to		and CCT
					PCS
					• Single surgeon	• CCT	• Longer operating time,
							more Balanced Salt
							Solution use, longer
							cortex removal in FLACS
					• 3-month follow-up
	2020	Stanojcic et al. (83)	116	118	• Continuation of the St.	• Visual acuity	• No difference in visual
		St. Thomas Study			Thomas Study	(UCDVA, CDVA)	acuity, refractive error,
							ECL and CCT
					• 12-month follow-up	• Refractive error	• No difference in
							posterior capsule
							opacification
						• ECL, CCT	• More effective
							astigmatic correction
							with intrastromal laser
							incisions compared to
							manual relaxing
							incisions
						• Complications
	2020	Schweitzer et al.	704	685	• Patients randomized to	• Composite	• No difference in the
		(56) FEMCAT trial			receive FLACS or PCS	outcome measure	overall success rate of the
					(unilateral or bilateral)	including:	composite outcome/ No
							difference in BCVA,
							refractive error, corneal
							astigmatism changes,
							and complications
					• Multicenter (5) study	• BCVA	• No difference in ECL
					in France; 21 surgeons		and CCT
					• 12-month follow-up	• Refractive error	• FLACS not
						(absolute refractive	cost-effective: e10,703
						error)	saved per additional
							patient treated
							successfully with PCS
						• Change to	No significant difference
						magnitude and axis	in ECL, CMT, and
						of corneal	anterior chamber flare
						astigmatism
						• Complications
						• Incremental
						cost-effectiveness
						ratio
						• ECL, CMT, and
						anterior chamber
						flare
							(Continued)
	Frontiers in Medicine					frontiersin.org

TABLE (

 ( BCVA, best-corrected distance visual acuity; CCT, central corneal thickness; CDE, cumulative dissipated energy; CDVA, corrected distance visual acuity; CRT, central retinal thickness; ECL, endothelial cell loss; EPT, effective phacoemulsification time; UCVA, uncorrected distance visual acuity.

	Continued)
	Year	Authors	Number of eyes included	Study design	Measured	Main significant
	published		(completed follow-up)		outcomes	findings
			FLACS	PCS
	2020	Day et al. (84)	392 (353)	393 (317)	• Patients randomized to	• UCDVA, CDVA	• Similar visual acuity,
		FACT trial			receive FLACS or PCS		refractive error, and
					(unilateral or bilateral)		patient satisfaction
							outcomes
					• Multicenter (3) study	• Complications	• Similar rate of
					in the UK		intraoperative
							complications
					• 3-month follow-up	• Patient-reported	• Higher ECL in the
						outcomes	FLACS group that nearly
							reached statistical
							significance (p=
	2020	Day et al. (57)	392 (311)	393 (292)	• Continuation of FACT	• UCDVA, CDVA	• Similar visual acuity
		FACT trial			trial		and patient satisfaction
							results
					• 12-month follow-up	• ECL	• No difference in
							complication rates
						• Complications	• No difference in ECL
						• Patient-reported	• FLACS was not
						outcomes	cost-effective
	AC, anterior chamber;

Frontiers in Medicine frontiersin.org

Author contributions

MK and CS have conducted literature research and have written the manuscript. Both authors contributed to the article and approved the submitted version.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

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Vivien Desveaux
email: vivien.desveaux@u-picardie.fr

Marouan Handa
email: handa@utia.cas.cz

M Handa Lamfa

A SDP relaxation of an optimal power flow problem for distribution networks

Keywords: Electric power distribution network, optimal power flow, convex relaxation, Pareto-front

In this work, we are interested in an optimal power flow problem with fixed voltage magnitudes in distribution networks. This optimization problem is known to be non-convex and thus difficult to solve. A well-known solution methodology consists in reformulating the objective function and the constraints of the original problem in terms of positive semi-definite matrix traces, to which we add a rank constraint. To convexify the problem, we remove this rank constraint. Our main focus is to provide a strong mathematical proof of the exactness of this convex relaxation technique. To this end, we explore the geometry of the feasible set of the problem via its Pareto-front. We prove that the feasible set of the original problem and the feasible set of its convexification share the same Pareto-front. From a numerical point of view, this exactness result allows to reduce the initial problem to a semi-definite program, which can be solved by more efficient algorithms.

Introduction

The optimal power flow (OPF) study in an electrical network is an essential tool in power engineering. It is defined as an optimization problem where the cost function can represent an economic objective such as power generation cost, or/and a technical objective such as the minimization of power losses in the network. The constraints often represent some physical restrictions at each bus and at each transmission line of the network.

OPF problems are widely used in the literature for different goals, see for instance [START_REF] Elattar | Modified jaya algorithm for optimal power flow incorporating renewable energy sources considering the cost, emission, power loss and voltage profile improvement[END_REF], especially:

-The minimization of power production (or importation) costs. In this case, the objective function is quadratic. -The minimization of losses of active and/or reactive power in the transmission lines. The objective function is then non-linear, or even non-convex. -Maintaining a constant voltage profile. The objective function is then in the form of a l 1 or l 2 distance between the voltage v and a target value v t , i.e. J =∥v -v t ∥.

Some papers combine several cost functions, leading to consider multi-criteria optimization problems [START_REF] Shaheen | Solving multi-objective optimal power flow problem via forced initialised differential evolution algorithm[END_REF].

To compute the fluxes through the lines of the network, we use the power flow equations (see for instance [START_REF] Momoh | Electric power system applications of optimization[END_REF]). Those equations appear in the OPF problem as equality constraints and are non-convex. Thus, it makes the optimization problem itself non-convex and therefore very difficult and costly to solve numerically. On the other hand, there is no guarantee that an approximate solution provided by an optimization method will be a global minimum. For these reasons, it is preferable to find a convex problem that is equivalent in a certain sense.

There are several methods that can be used to convexify the problem. A first approach is to linearize the non-convex constraints under certain physical assumptions, in particular the reactive power constraints must be neglected and the voltages must be fixed. It is then possible to obtain a linear or a quadratic problem (see [START_REF] Castillo | Evaluating the effects of real power losses in optimal power flow-based storage integration[END_REF]).

Another approach relies on the convexification of the original problem via a convex relaxation. This consists in replacing the non-convex set of constraints by a convex set containing it. The convex relaxation is then said to be exact if the optimal value of the original problem is equal to the optimal value of the convexified problem. A first and natural possibility is to take the convex hull of the constraints set (see for instance [START_REF] Lavaei | Geometry of power flows in tree networks[END_REF]). However, the disadvantage of this method is the difficulty to find an algebraic representation of the convex hull that can be used in practice. An alternative is to use a larger convex set than the convex hull, and which can be represented algebraically in a simple manner.

In [START_REF] Bai | Semidefinite programming for optimal power flow problems[END_REF], the authors reformulate the objective function and the constraints of the original problem in terms of traces of positive semidefinite matrices, to which is added a rank constraint. By removing this rank constraint, they obtain a convex problem. This method is called a SDP relaxation.

The authors of [START_REF] Lavaei | Geometry of power flows in tree networks[END_REF][START_REF] Lavaei | Geometry of power flows and optimization in distribution networks[END_REF] states that the SDP relaxation is exact under certain conditions of monotony of the objective function and on the voltage phase angles. In [START_REF] Lavaei | Geometry of power flows in tree networks[END_REF], the voltage magnitudes are assumed to be fixed and to prove the exactness of the SDP relaxation, they studied the geometry of the feasible set of the problem via its Paretofront. However, in their proof the authors makes a simplifying hypothesis that is not always satisfied, which makes the proof incomplete (Lemma 4, page 581). In [START_REF] Lavaei | Geometry of power flows and optimization in distribution networks[END_REF], a generalization to variable voltage magnitudes is proposed, but the proof of exactness relies on the proof of the fixed voltage magnitude case. Therefore a complete proof is still an open problem in both the fixed and variable voltage magnitude cases.

The main goal of this paper is to provide a rigorous proof of the exactness of the SDP relaxation in the case of fixed voltage magnitudes. Whereas the variable voltage magnitude case is of practical greater interest, an exactness proof appears to be much more difficult to reach in this case without additional very restrictive conditions (for instance no lower bound limitation on the injected active power). In order to do so, we will use two main ingredients: the tree structure of the distribution network and a geometrical study of the feasible sets and their Pareto-front. Furthermore, the new techniques proposed in this paper can be useful tools to help reaching a proof in the variable voltage magnitude case in a future work.

The rest of this paper is organized as follows. In Section 2, we present the model and the physical equations governing the power distribution network. We then define the optimization problem of interest and provide different equivalent formulations of this problem. Section 3 is dedicated to the notion of Pareto-front. We give some definitions and provide some elementary properties that will be frequently used to prove the theoretical results of this paper. In Section 4, we introduce and study the convexified problem in the simplified case of a network of only one line, and we show some results that will be used in the following. In Section 5, we prove the main result of this paper which states that under some conditions, the initial problem and its convexification share the same Pareto-front, i.e. the relaxation is exact. We conclude by proving a theorem that allows to use this convexification in practice. In Section 6, we present some numerical tests to highlight the proven theoretical results. Finally, in Section 7, we give some concluding remarks and perspectives.

2 Modeling of the power distribution network

Geometry of the network

We consider a power distribution network modeled by a tree Σ = (S, E), where S = {1, • • • m}, m ≥ 2, is the set of vertices that represent the nodes of the network and E ⊂ {{i, k} ∈ R 2 , i ̸ = k} is the set of non-oriented edges that represent the transmission lines. Let us notice that since Σ is a tree, the cardinal of E is necessarily m-1. Figure 1 shows an example of such a network presented in [START_REF] Kersting | Radial distribution test feeders[END_REF]. In order to enumerate the elements of E, we consider a bijection

ψ : {1, • • • , m -1} → E.
In the following, we will need to consider flows in both directions in the transmission lines. To this end, for each non-oriented edge of E, we associate two oriented edges. Therefore, we define the set

E = (i, k) ∈ S 2 | {i, k} ∈ E ,
whose cardinal is 2(m -1). In other words, for each edge {i, k} in E, the set E contains the two couples (i, k) and (k, i).

To enumerate the elements of E, we introduce the bijection

φ : {1, • • • , 2(m -1)} → E, defined such that if ψ(ℓ) = {i, k}, with i < k, then φ(2ℓ -1) = (i, k) and φ(2ℓ) = (k, i).
It means that the two orientations of the same edge are consecutive in φ, and that the order of the edges is the same in φ and in ψ.

Physical formulation

We present here the main governing equations and constraints in a AC-power distribution system. We refer to [START_REF] Momoh | Electric power system applications of optimization[END_REF] for a more detailed presentation of power engineering equations.

Let v i be the complex voltage at node i, written as

v i = V i e jθi , (1)
where V i is the modulus of v i , called the voltage magnitude and θ i is the argument of v i in ]-π, π], called the voltage angle. We assume in this work that the voltage magnitudes are fixed and known for each node i of the network.

To each non-oriented edge {i, k} ∈ E, we associate a complex admittance y ik , defined by

y ik = g ik -jb ik ,
where g ik and b ik are the conductance and the susceptance of the line {i, k}, respectively. We assume that b ik > g ik > 0. This hypothesis is classical in the literature when addressing this type of problems (see for instance Zhang et al. ( 2014)).

For each oriented edge (i, k) ∈ E, the active power flow from node i to node k is defined by

F ik (θ ik ) = V 2 i g ik + V i V k b ik sin(θ ik ) -g ik cos(θ ik ) , (2)
where

θ ik = θ k -θ i .
The total amount of power that can flow through a transmission line {i, k} = ψ(ℓ) is limited by a physical capacity. This limit is called a thermal constraint and ensures that the equipment of each transmission line does not become overloaded or overheat [START_REF] Conti | Distributed generation in lv distribution networks: voltage and thermal constraints[END_REF]; [START_REF] Liu | Optimal energy management for formulae cars with regulatory limits and thermal constraints[END_REF]. There are several ways to formulate this constraint (see [START_REF] Madani | Convex relaxation for optimal power flow problem: Mesh networks[END_REF]), for instance

F ik + F ki ≤ F ℓ , (3)
where F ℓ > 0. The advantage of this formulation is that it can easily rewrite in terms of voltage angles. Indeed, according to (2), we have

F ik + F ki = (V 2 i + V 2 k )g ik -2V i V k g ik cos(θ ik ).
We can then distinguish three cases according to the value of F ℓ :

-if F ℓ < (V i -V k ) 2 g ik , the constraint (3) is not feasible; -if (V i -V k ) 2 g ik ≤ F ℓ < (V i + V k ) 2 g ik , the constraint (3) is equivalent to -θ ik ≤ θ ik ≤ θ ik ,
where

θ ik = arccos (V 2 i + V 2 k )g ik -F ℓ 2V i V k g ik ; -if (V i + V k ) 2 g ik ≤ F ℓ , the constraint (3) is automatically satisfied.
The first case is not interesting. Therefore in the following, the quantities F ℓ will be assumed to satisfy

F ℓ ≥ (V i -V k ) 2 g ik ,
for every edge. As a consequence, the thermal constraint rewrites as follows:

-θ ik ≤ θ ik ≤ θ ik , (4)
with θ ik ∈ [0, π]. Let us notice that θ ik does not depend on the direction of the line, thus θ ki = θ ik . In the following, the thermal constraint will be expressed indifferently in the form (3) or in the form (4).

For some technical reasons, we assume that

0 < θ ik ≤ π 2 .
This condition is equivalent to

(V i -V k ) 2 g ik < F ℓ ≤ (V 2 i + V 2 k )g ik .
Actually, an even more restrictive assumption will be made later. The physical relevance of this restriction will then be discussed. At each node i ∈ S, the injected active power P i must be equal to the generated active power minus the consumed active power. These powers can also be written in terms of power flow equations as follows (see [START_REF] Monticelli | State estimation in electric power systems: a generalized approach[END_REF])

P i = k∼i F ik , (5)
which stands as an equality constraint. Furthermore, the injected active power must satisfy the following inequality constraints

P i ≤ P i ≤ P i , (6)
which limit the quantity of generated or consumed power at each node Monticelli (2012).

There are many objective functions that can be considered to define the optimal power flow problem, such as the total loss in the network or the power generation cost (see [START_REF] Elattar | Modified jaya algorithm for optimal power flow incorporating renewable energy sources considering the cost, emission, power loss and voltage profile improvement[END_REF]). Let P = (P 1 , • • • , P m ) T ∈ R m be the vector of all the power injections defined by equation (5). In this work, we consider an objective function

J : P -→ J(P ), (7)
strictly increasing with respect to P, in a sense that we will be precised in Section 3. Let θ ∈ R 2(m-1) be the vector containing all the phase angles of the network, such that θ ℓ = θ ik for φ(ℓ) = (i, k). Let us notice that according to (5) and (2), and since the voltage magnitudes V i are fixed, the active injected power P i can be seen as a function of the phase angles θ ik . In other words, P := P (θ) is a state variable which depends on the control variable θ. Therefore, we introduce the objective function

J(θ) = J(P (θ)),
where J is the function defined by ( 7).

The optimal power flow problem of interest then states as follows

min θ∈R m J(θ), s.t. -θ ik ≤ θ ik ≤ θ ik , ∀(i, k) ∈ E, P i = k∼i F ik , ∀i ∈ S, P i ≤ P i ≤ P i , ∀i ∈ S.
(OPF 1 )

Semi-definite reformulation

The power flow equations make the problem (OPF 1 ) non-convex and therefore difficult to solve. In this section, we are going to reformulate this problem in a matrix form (see [START_REF] Lam | Distributed algorithms for optimal power flow problem[END_REF]). The non-convexity will then appear more clearly. On the other hand, this reformulation will also be helpful to build a convexification of the problem.

We start from expressing all the constraints of Problem (OPF 1 ) in terms of the vector v = (v 1 , • • • , v m ) T of the complex voltages. In order to do so, we introduce the following hermitian matrices:

for 1 ≤ i ≤ m, the matrix E i such that E i i,i = 1 and all the other entries are equal to 0; the matrix Y defined by

∀i ∈ S, Y i,i = k∼i y ik , ∀(i, k) ∈ E, Y i,k = -y ik ,
and all other entries are equal to 0; for 1 ≤ i ≤ m, the matrix B i defined by

B i = 1 2 (Y * E i + E i Y ); -for (i, k) ∈ E, the matrix B ik such that      B ik i,i = g ik , B ik i,k = -y ik 2 , B ik k,i = -y ik 2 ,
and all other entries are equal to 0.

An immediate computation shows that F ik = Tr(B ik vv * ) for all (i, k) ∈ E and P i = Tr(B i vv * ), for all i ∈ S.

Now, let us consider the following optimization problem min

W ∈Hm J(W ), s.t. P i ≤ Tr(B i W ) ≤ P i , ∀i ∈ S Tr(B ik W ) + Tr(B ki W ) ≤ F ℓ , ∀{i, k} = ψ(ℓ) ∈ E, W ⪰ 0, rank(W ) = 1, (OPF 2 )
where H m is the set of hermitian matrices of order m, the notation W ⪰ 0 means that the matrix W is positive semi-definite and

J(W ) = J(Tr(B 1 W ), • • • , Tr(B m W )).
Problems (OPF 2 ) and (OPF 1 ) can be linked by the following result whose proof is immediate.

Proposition 1 Problems (OPF 1 ) and (OPF 2 ) are equivalent in the following sense:

if θ is a solution of (OPF 1 ), then noting v = V e jθ , the matrix W = vv * is a hermitian matrix of rank 1 and a solution of (OPF 2 ); if W is a solution of (OPF 2 ), then there exists a vector v ∈ C m such that W = vv * and θ = arg(v) is a solution of (OPF 1 ).

Problem (OPF 2 ) is obviousely non-convex because of the rank constraint. A natural way to make it convex is therefore to remove this rank constraint, which leads to consider the following problem:

min W ∈Hm J(W ), s.t. P i ≤ Tr(B i W ) ≤ P i , ∀i ∈ S, Tr(B ik W ) + Tr(B ki W ) ≤ F ℓ ∀{i, k} = ψ(ℓ) ∈ E, W ⪰ 0. (OPF 2 )
This is a semi-definite program (SDP) that is much easier to solve numerically than Problem (OPF 2 ) (see for instance [START_REF] Bai | Semidefinite programming for optimal power flow problems[END_REF]). However, there is no guarantee that a solution of (OPF 2 ) is of rank 1 and therefore leads to a solution of (OPF 2 ). The following of this paper is dedicated to find some conditions that ensure that this is the case and to prove it.

Set reformulation

In this section, we propose another formulation of Problem (OPF 1 ) by considering the vector of injected active power as a decision variable and by defining the constraints in terms of sets. This formulation will be more adapted to the mathematical study that will follow.

We define the following order relation: for x, y ∈ R n , we set

x ≤ y ⇔ ∀i ∈ {1, • • • , n}, x i ≤ y i .
The fluxes defined by ( 2) are gathered in one vector F (θ) ∈ R 2(m-1) where the ℓ-th component is

F ℓ (θ) = F ik (θ ik ), with φ(ℓ) = (i, k). (8)
The injected active power flow vector P (θ) ∈ R m then writes

P i (θ) = k∼i F ik (θ ik ). ( 9
)
We are now going to define a feasible set for each constraint. Concerning the injected active power constraint, we only need to introduce

P P = P ∈ R m , P ≤ P ≤ P . ( 10
)
Next we consider the thermal constraint. For each non-oriented edge ψ(ℓ) = {i, k} ∈ E, the feasible set for the active power flows is defined by

F ℓ = (F ik (θ ik ), F ki (-θ ik )) T , -θ ik ≤ θ ik ≤ θ ik . ( 11
)
Considering the whole network, the feasible set for the active power flow reads

F = F (θ), -θ ≤ θ ≤ θ . (12)
We immediately see that the sets F and F ℓ are connected as follows

F = m-1 ℓ=1 F ℓ . (13)
Finally, the feasible set of the injected active power for the thermal constraint is given by

P θ = P (θ), -θ ≤ θ ≤ θ . ( 14
)
In order to link the sets F and P θ , we introduce the connectivity matrix A ∈ M m,2(m-1) (R), defined by

A i,ℓ = 1 if φ(ℓ) = (i, •), 0 otherwise,
where the notation φ(ℓ) = (i, •) means that there exists k ∈ S such that φ(ℓ) = (i, k) (i.e. there exist k ∈ S such that i ∼ k). We then have

P (θ) = AF (θ),
and therefore

P θ = AF. ( 15
)
Such a construction already exists in the literature, see for instance [START_REF] Lavaei | Geometry of power flows in tree networks[END_REF][START_REF] Lavaei | Geometry of power flows and optimization in distribution networks[END_REF]. This leads to consider the following optimization problem min P ∈R m J(P ) s.t. P ∈ P θ ∩ P P (OPF 3 ) Proposition 2 Problems (OPF 1 ) and (OPF 3 ) are equivalent in the following sense:

-If θ is a solution of (OPF 1 ), then P (θ) defined by (8) and ( 9) is a solution of (OPF 3 ). -If P is a solution of (OPF 3 ), then there exists θ ∈ R 2(m-1) such that P = P (θ) and θ is a solution of (OPF 1 ).

Once again, we omit the proof that is immediate.

Pareto-front

In order to convexify Problem (OPF 1 ), we are going to study the geometry of the feasible set for Problem (OPF 3 ). The main tool we will use to locate potential solutions is the Pareto-front. This section is dedicated to recall this notion and its main properties. First, we introduce the following order relation

x ≺ y ⇔ ∀i ∈ {1, • • • , n}, x i ≤ y i and ∃i ∈ {1, • • • , n}, x i < y i .
We can then define the notion of Pareto-optimality.

Definition 1 Let A ⊂ R n . A vector x ∈ A is said to be Pareto-optimal in A if y ∈ A, y ≺ x = ∅.
The set of all vectors x ∈ A which are Pareto-optimal in A is called the Pareto-front of A and is denoted by O(A).

Let us notice that the Pareto-front of a set A is always a subset of the topological frontier of A.

The following definition allows to link the Pareto-front with optimization problems.

Definition 2 A function f : A ⊂ R n → R is said to be strictly increasing on A if for all x, y ∈ A such that x ≺ y, we have f (x) < f (y).
We then have the following classical result.

Proposition 3 Let f : A ⊂ R n → R be a strictly increasing function. If x ⋆ = arg min x∈A f (x), then x ⋆ ∈ O(A).
Proof By absurd, suppose that x ⋆ / ∈ O(A). Then there exists x ′ ∈ A such that x ′ ≺ x ⋆ . Since f is strictly increasing, we have f (x ′ ) < f (x ⋆ ). That contradicts the optimality of x ⋆ .

Next, we establish a result that ensures that under certain conditions, nested sets have the same Pareto-front. Up to our knowledge, this result does not exist in the literature, although it is quite elementary. If x ⋆ does not belong to O(B), then there exists x ′ ∈ B such that x ′ ≺ x ⋆ . Hence, we have x ′ ≤ x ⋆ ≤ x and consequently x ′ ∈ C. In addition, f is strictly increasing, therefore we have f (x ′ ) < f (x ⋆ ), which contradicts the optimality of x ⋆ . As a consequence, x ⋆ ∈ O(B). Since x does not belong to O(B), we cannot have x ⋆ = x. Hence we have x ⋆ ≺ x. Now, since

Proposition 4 Let A ⊂ B ⊂ R n . 1. We have A ∩ O(B) ⊂ O(A). 2. If in addition B is a compact set and O(B) ⊂ A, then O(A) = O(B). Proof 1. Let x ∈ A∩O(B). If x / ∈ O(A), then there exists x ⋆ ∈ A such that x ⋆ ≺ x. Since A ⊂ B,
x ⋆ ∈ O(B) ⊂ O(A) ⊂ A, this contradicts x ∈ O(A). Therefore x ∈ O(B) and O(A) ⊂ O(B).

Distribution network of one transmission line

The objective of the following sections is to build a relevant convexification of Problem (OPF 3 ), then to prove that this convexification is exact under some conditions. This type of proof will be based on two main elements:

the geometrical properties of the feasible sets at the local level of a transmission line; the tree structure of the network.

In this section, we are interested in a network constituted of only one transmission line. In a first instance, we don't take into account the injected power constraints. The purpose is to introduce some notations, to prove some geometric results on the feasible set and to come up with a convexification of the problem in this very simplified case. We will then extend these results to the whole network in the following sections.

We consider a graph constituted of a single edge {1, 2} (see Figure 2). For simplicity of notations, the coefficients b 12 and g 12 will simply be noted b and g. The active power flow F 12 and F 21 will respectively be noted by F 1 and F 2 . Finally, the angle θ 12 will be noted θ. T , where

F 12 (θ) = V 2 1 g + V 1 V 2 b sin(θ) -V 1 V 2 g cos(θ), F 21 (θ) = V 2 2 g -V 1 V 2 b sin(θ) -V 1 V 2 g cos(θ). We define A = F (θ) ∈ R 2 , θ ∈ [-π, π] .
Let us notice that A is an ellipse whose axis of symmetry is the line y = x, see Figure 3 (a). The feasible set of the active power flow is defined by

F = F (θ) ∈ R 2 , θ ∈ [-θ, θ] . ( 17
)
This set is an arc of the ellipse A (see the example in Figure 3 (b), with θ = π/2).

By studying the variations of functions F 1 and F 2 , we can easily see that the Pareto-front of the set F is

O(F) = F (θ) ∈ R 2 , θ ∈ -max(θ, α), min(θ, α) , (18)
see Figure 3 (c). The purpose here is to build an exact convexification of the set F which can be used in practice. A natural idea is to consider the convex hull of F. However, the convex hull does not have a simple algebraic characterization and is therefore difficult to use in practice for our optimization problem of interest.

We will build another convexification based on a positive semi-definite hermitian matrix formulation which has a very simple algebraic characterization.

We note H + m the set of m × m positive semi-definite hermitian matrices. We introduce the set

H = W ∈ H + 2 , W 1,1 = V 2 1 , W 2,2 = V 2 2 , -tan(θ) Re (W 1,2 ) ≤ Im (W 1,2 ) ≤ tan(θ) Re (W 1,2 ) .
One can easily check that a matrix W of order 2 belongs to H if and only if there exists α ∈

[0, V 1 V 2 ] and θ ∈ [-θ, θ] such that W = V 2 1 αe jθ αe -jθ V 2 2 . ( 19
)
Moreover, such a matrix is of rank 1 if and only

if α = V 1 V 2 .
The set F can then be written in terms of hermitian matrices of H as follows

F = {Re(diag(W K)), W ∈ H, rank(W ) = 1}, (20)
where the matrix K is defined by

K = g + jb -g -jb -g -jb g + jb .
By removing the rank constraint in (20), we obtain a convexification of F, i.e. a convex set containing F, which will be denoted by

conv(F) = {Re(diag(W Y )), W ∈ H}. ( 21
)
Lemma 2 We assume that θ ≤ α. Let (x ⋆ , y ⋆ ) ∈ F and (x, y) ∈ conv(F). Then, we have

1. if x ≤ x ⋆ , then y ≥ y ⋆ ; 2. if x < x ⋆ , then y > y ⋆ ; 3. if y ≤ y ⋆ , then x ≥ x ⋆ ; 4. if y < y ⋆ , then x > x ⋆ .
Proof Since θ ≤ α, thanks to (22), we have (x ⋆ , y ⋆ ) ∈ O(conv(F)). If x ≤ x ⋆ and y < y ⋆ , then we have (x, y) ≺ (x ⋆ , y ⋆ ), which contradicts (x ⋆ , y ⋆ ) ∈ O(conv(F)). The other statements can be proved in a similar way.

Consideration of injected active power constraints

Here, we consider in addition the constraints

P i ≤ F i (θ) ≤ P i , i ∈ {1, 2}.
The feasible set is then written in the form F ∩ F P where F is defined by ( 17) and

F P = F ∈ R 2 , P ≤ F ≤ P .
Note that since F P is convex, the set conv(F) ∩ F P is a simple convexification of F ∩ F P . In order to prove that this convexification is exact, we would like to have

O(F ∩ F P ) = O(conv(F) ∩ F P ). (23)
However, we can easily see that this equality is not always satisfied. To illustrate this failure, let us consider the following setting θ = π/2, b = 5, g = 2 and V 1 = V 2 = 1. Figure 5 shows different scenarios for the set F ∩ F P , depending on the values of P i and P i . In the cases (a) and (b), the sets O(conv(F) ∩ F P ) and O(F ∩ F P ) are both equal to F ∩ F P and so (23) holds. However, it is not true in case (c), as we can see more clearly in Figure 6.

We observe that those problematic cases only appear when some elements of F are not Pareto-optimal and the values of P i are large enough. To avoid this, a classical assumption is to suppose that all the points of F are Paretooptimal in F (see for instance [START_REF] Lavaei | Geometry of power flows and optimization in distribution networks[END_REF]). This assumption is equivalent to θ ≤ α. In [START_REF] Lavaei | Geometry of power flows and optimization in distribution networks[END_REF]; [START_REF] Huang | A review of the convexification methods for ac optimal power flow[END_REF], authors explain that this assumption does always hold in practice for distribution networks.

Let us notice that under this condition, equality (23) does not necessary hold. But it does not hold only when the feasible set F ∩ F P is empty, as we can see in Figure 7. This alternative is sufficient to apply in the framework of our optimization problem.

In the following section, we generalize and prove relation ( 23) in the case of a general network of m ≥ 2 nodes. We start by generalizing the definitions and notations of Section 4.2 to the case of a whole network.

For each non-oriented edge ψ(ℓ) = {i, k}, we reproduce the convexification of the power flow feasible set F ℓ , defined by (11). We thus introduce where the set H ik is defined by

conv(F ℓ ) = {Re(diag(W K ik )), W ∈ H ik }, (24)
H ik = W ∈ H + 2 , W 1,1 = V 2 i , W 2,2 = V 2 k , -tan(θ ik ) Re(W 1,2 ) ≤ Im(W 1,2 ) ≤ tan(θ ik ) Re(W 1,2 ) ,
and the matrix K ik is given by

K ik = g ik + jb ik -g ik -jb ik -g ik -jb ik g ik + jb ik .
According to relations ( 13) and ( 15), it is natural to define the convexification of the sets F and P θ by

conv(F) = m-1 ℓ=1 conv(F ℓ ) (25) and conv(P θ ) = A conv(F). ( 26
)
The sets conv(F) and conv (P θ ) are convex as a product of convex sets and the image by a linear map of a convex set.

Those definitions lead us to consider the following optimization problem min P ∈R m J(P ) s.t. P ∈ conv(P θ ) ∩ P P , (OPF 3 ) where P P is defined by (10).

As stated previously, we obtain (OPF 2 ) as a convexification of (OPF 2 ). On the other hand, Problems (OPF 2 ) and (OPF 3 ) are both equivalent to (OPF 1 ). Moreover, (OPF 2 ) and (OPF 3 ) are equivalent as stated in the following result, whose proof is straightforward and not given here.

Proposition 5 Problems (OPF 2 ) and (OPF 3 ) are equivalent in the following sense:

-If P is a solution of (OPF 3 ), then for every edge {i, k} ∈ E, there exists α ik ∈ [0, V i V k ] and θ ik ∈ [-θ ik , θ ik ] such that we have

P i = k∼i (V 2 i g ik + α ik b ik sin(θ ik ) -α ik g ik cos(θ ik )).
The matrix W defined by

     W i,i = V 2 i , if i ∈ S, W i,k = α ik e jθ ik , if (i, k) ∈ E, W i,k =0, else, (27)
is then a solution of (OPF 2 ).

In addition, if P ∈ P θ , then the matrix W is of rank 1. -If W is a solution of (OPF 2 ), then by defining

P i = Tr(B i W ), ( 28
)
where B i is defined by (2.3), the vector P is a solution of (OPF 3 ).

In addition, if the matrix W is of rank 1, then P ∈ P θ . In this case, there exists v ∈ C m such that W = vv * and we have P = P (arg(v)).

In order to prove that the convexification introduced earlier is exact, we need the following assumption.

Assumption A 1 For every edge {i, k} ∈ E, we have

θ ik ≤ arctan b ik g ik .
As introduced in the previous section, the convexification of the feasible set P is conv (P θ ) ∩ P P , where conv(P θ ) is defined by (26). In this section, in a first instance, we establish that O (conv (P θ ) ∩ P P ) = O(P), under certain conditions that will be precised. This result is difficult to prove and will require some preliminary results. We will then deduce the main theorem of this paper, which ensures that the convexification (OPF 2 ) is exact in this framework.

As a first preliminary result, we give a characterization of the vectors of O(conv(P θ ) ∩ P P ).

Lemma 3 Let us suppose that Assumption A 1 holds. Let P ∈ O(conv(P θ ) ∩ P P ) and F ∈ conv(F) be such that P = AF .

If a vertex i ∈ S satisfies P i > P i , then for every vertex k ∈ S such that k ∼ i, we have (F ik , F ki ) ∈ F ℓ , with ψ(ℓ) = {i, k} and i < k.

Proof By absurd, we suppose that P i > P i and that there exists k ∈ S with k ∼ i, such that (F ik , F ki ) / ∈ F ℓ . We will construct a vector P ⋆ ∈ conv (P θ ) ∩ P P such that P ⋆ ≺ P . This will contradict the assumption that P belongs to the Pareto-front of this set.

We have (F ik , F ki ) ∈ conv (F ℓ ) \F ℓ , then according to Lemma 1, there exists δ > 0 such that (F ik -δ, F ki ) ∈ F ℓ and for all ε ∈ [0, δ], we have (F ik -ε, F ki ) ∈ conv (F ℓ ).

We set ε = min (δ, P i -P i ) > 0. Let F ⋆ be defined by

F ⋆ qr = F qr if (q, r) ̸ = (i, k), F ik -ε if (q, r) = (i, k),
and P ⋆ = AF ⋆ . Then we have

P ⋆ i = n∼i F ⋆ in = F ik -ε + n∼i n̸ =k F in = P i -ε
and for all n ̸ = i, we have P ⋆ n = P n . We deduce that P ⋆ ≺ P . Now, let us prove that P ⋆ ∈ conv (P θ ) ∩ P P . We have F ⋆ ∈ conv(F), therefore P ⋆ ∈ conv (P θ ) . On the other hand, since P i ≤ P i -ε = P ⋆ i < P i ≤ P i , we deduce P ⋆ ∈ P P .

Hence, we have proved that P ⋆ ∈ conv (P θ ) ∩ P P .

The following Lemma allows to find a condition for a vector of conv(F) to be in F. This result relies on the tree structure of the network.

Lemma 4 Let us suppose that assumption A 1 holds. Let F ⋆ ∈ F and F ∈ conv(F) such that for every i ∈ S, we have k∼i

F ik ≤ k∼i F ⋆ ik . ( 29
)
Then, we have F ik = F ⋆ ik , for every (i, k) ∈ E.

Proof We fix arbitrarily a root r of the tree Σ. Let us notice that for a vertex i ∈ S\{r}, of parent k, relation ( 29) writes

F ik -F ⋆ ik ≤ n child of i (F ⋆ in -F in ) . (30)
First, we show by a bottom-up induction, that for every vertex i ∈ S\{r}, of parent k, we have

F ik ≤ F ⋆ ik . Let i ∈ S \ {r}, of parent k. -If i is an external vertex of Σ (i.e. i has no child), then (30) implies directly that F ik ≤ F ⋆ ik .
-If i is an internal vertex of Σ (i.e. i has at least one child), such that for every child n of i, we have

F ni ≤ F ⋆ ni , then Lemma 2 ensures that F in ≥ F ⋆ in . Relation (30) then implies that F ik ≤ F ⋆ ik .
The property is then true for every vertex i ∈ S \ {r}. Next, we show by a top-down induction that for every internal vertex i ∈ S and for every children n of i, we have

F in = F ⋆ in and F ni = F ⋆ ni . -If i is the root r of Σ, relation (30) writes n child of i (F ⋆ in -F in ) ≥ 0.
From the above, we know that for every child n of i, we have F ni ≤ F ⋆ ni . Therefore, according to Lemma 2, we have F in ≥ F ⋆ in . We deduce that

F in = F ⋆ in . Lemma 2 then implies that F ni ≥ F ⋆ ni , therefore F ni = F ⋆ ni .
-If i is not the root and its parent k satisfies F ki = F ⋆ ki , then according to Lemma 2, we have F ik ≥ F ⋆ ik . From the above, for every child n of i, we have

F ni ≤ F ⋆ ni , therefore according to Lemma 2, F in ≥ F ⋆ in . Relation (30) then writes 0 ≤ F ik -F ⋆ ik ≤ n child of i (F ⋆ in -F in ) ≤ 0 We deduce that F in = F ⋆ in . Lemma 2 ensures that F ni ≥ F ⋆ ni , hence F ni = F ⋆ ni .
We have thus proven that for every (i, k) ∈ E, we have

F ik = F ⋆ ik .
Thanks to these lemmas, we can now prove that the feasible set P θ ∩ P P and its convexification conv (P θ ) ∩ P P share the same Pareto-front, provided the set P θ ∩ P P is non-empty.

Theorem 5 We suppose that Assumption A 1 holds. If P θ ∩ P P ̸ = ∅, then we have O(P θ ∩ P P ) = O (conv (P θ ) ∩ P P ) .

Proof If O (conv (P θ ) ∩ P P ) ⊂ P θ ∩P P , then Proposition 4 ensures that O(P θ ∩ P P ) = O (conv (P θ ) ∩ P P ) . As a consequence, it is sufficient to prove the first inclusion.

Let P ∈ O (conv (P θ ) ∩ P P ) and F ∈ conv(F) such that P = AF . By absurd, we suppose that P / ∈ P θ ∩ P P . The proof will consist in constructing a vector P ′ ∈ conv (P θ ) ∩ P P such that P ′ ≺ P , which will contradict the assumption.

Since P / ∈ P θ , there exists an edge that we note {1, 2} = ψ(1) ∈ E, such that (F 12 , F 21 ) / ∈ F 1 . Let us notice that Lemma 3 ensures that P 1 = P 1 and P 2 = P 2 .

By removing the edge {1, 2} of the graph Σ, we obtain two connected components. We note S 1 and S 2 the set of vertices of the connected component containing the vertex 1 and 2, respectively.

Finally, since P θ ∩ P P ̸ = ∅, there exists P ⋆ ∈ P θ ∩ P P and F ⋆ ∈ F such that P ⋆ = AF ⋆ . Fig. 8: Example of graph Σ and the feasible paths 1st step: there exists a vertex i ∈ S such that P i > P i . Let us suppose this is not the case, in other words that every vertex i ∈ S satisfies P i = P i . Since P i ≤ P ⋆ i , we deduce that for every i ∈ S, we have k∼i

F ik ≤ k∼i F ⋆ ik .
Lemma 4 then ensures that F ik = F ⋆ ik for every oriented edge (i, k) ∈ E. Therefore, we have (F 12 , F 21 ) = (F ⋆ 12 , F ⋆ 21 ) ∈ F 1 , which is impossible. 2nd step: determination of the part of the graph where we will modify P . We call a feasible path every path of vertices q 1 ∼ • • • ∼ q r such that q 1 ∈ {1, 2}; q 2 / ∈ {1, 2}; -P qr > P qr ; for all 1 ≤ i ≤ r -1, we have P i = P i .

We consider the sub-graph Σ = ( S, E) of Σ, connected and maximal, such that every vertex i ∈ S satisfying P i > P i belongs to a feasible path in Σ. In other words, by browsing the graph from 1 or 2, as soon as we meet a vertex such that P i > P i , we remove all its descendants. The feasible paths and the construction of the graph Σ is illustrated for a simple example in Figure 8.

Let us notice that Σ admits two types of leaves:

the leaves i satisfying P i = P i , in which case, this is a leaf from the original tree Σ; the leaves i satisfying P i > P i , in which case, this is not necessarily a leaf of the original tree Σ; according to the first step and by maximality of Σ, there exists at least one leaf of this type in Σ.

The vertices i ∈ Σ which are not leaves necessarily satisfy P i = P i . We now prove that there exists a feasible path q 1 ∼ • • • ∼ q r in Σ such that for all i = 1, • • • , r -1, we have F qiqi+1 < F ⋆ qiqi+1 . By absurd, let us suppose that for all feasible path in Σ, there exists s ∈ {1, • • • , r -1} such that F qsqs+1 ≥ F ⋆ qsqs+1 . We consider then the subgraph Σ = ( S, E) of Σ in which we have removed all the vertices beyond q s+1 in the feasible paths.

Let us consider a leaf i in Σ, whose only neighbor is noted by k. We have two possibilities:

either i corresponds to a q s+1 described previously and then satisfies F ki ≥ F ⋆ ki ; or i is also a leaf of Σ satisfying P i = P i . It is therefore a leaf from the original tree Σ. As a consequence, we have

P i ≤ P ⋆ i , thus F ik ≤ F ⋆ ik . Lemma 2 then ensures that F ki ≥ F ⋆ ki .
We consider now the vertex 1 as the root of Σ. For an internal vertex i ∈ S ∩ S 2 , of parent k, we prove by a bottom-up induction that F ik ≤ F ⋆ ik .

-If all the children of i are external vertices of Σ: since i is an internal vertex of Σ different from the root, it is not a leaf of Σ. Therefore, we have P i = P i , thus

F ik + n child of i F in ≤ F ⋆ ik + n child of i F ⋆ in ,
and then

F ik -F ⋆ ik ≤ n child of i (F ⋆ in -F in ) .
A child n of i, which is an external vertex by assumption, and therefore a leaf of Σ, satisfies according to the above F in ≥ F ⋆ in , thus we have

F ik ≤ F ⋆ ik . -If all the children n of i satisfy F ni ≤ F ⋆ ni , then Lemma 2 ensures that F in ≥ F ⋆
in and we deduce in the same fashion that

F ik ≤ F ⋆ ik .
The result is then true for all the vertices of S ∩ S 2 and in particular for the vertex 2, therefore F 21 ≤ F ⋆ 21 . The same result can be obtained by considering the vertex 2 as a root and the set S ∩ S 1 . We deduce that F 12 ≤ F ⋆ 12 . Lemma 2 then ensures that

F 21 ≥ F ⋆ 21 and F 12 ≥ F ⋆ 12 , therefore (F 12 , F 21 ) = (F ⋆ 12 , F ⋆ 21 ) , which contradicts the fact that (1, 2) / ∈ F 1 . It follows that there exists a feasible path q 1 ∼ • • • ∼ q r in Σ such that for all i = 1, • • • , r -1, we have F qiqi+1 < F ⋆ qiqi+1 and thus F qi+1qi > F ⋆ qi+1qi
according to Lemma 2. We will modify P on the vertices of this feasible path. Now, we conclude this section with a result allowing to connect the solutions of the convexified problem (OPF 2 ) to the solutions of the physiscal problem (OPF 1 ). This result has direct numerical applications, since it allows in practice to solve the convex problem (OPF 2 ) and obtain either a solution of Problem (OPF 1 ), or the non-existence of solutions for this problem.

Theorem 6 We suppose that Assumption A 1 holds. Then we distinguish three cases:

1. if (OPF 2 ) admits a solution W of rank 1, then there exists v ∈ C m such that W = vv * and arg(v) is a solution of (OPF 1 ); 2. if (OPF 2 ) admits a solution W of a rank at least 2, then (OPF 1 ) is infeasible; 3. if (OPF 2 ) is infeasible, then (OPF 1 ) is infeasible. Proof 1. If (OPF 2 ) admits a solution W of rank 1, then there exists v ∈ C m such that W = vv * and Proposition 5 ensures that P = P (arg(v)) is a solution of (OPF 3 ) and that P ∈ P θ . Therefore, we have P θ ∩P P ̸ = ∅ and according to Theorem 5, we have O(conv(P θ ) ∩ P P ) = O(P θ ∩ P P ).

Since the cost function J is strictly increasing, we have P ∈ O(conv(P θ ) ∩ P P ) and therefore P ∈ O(P). We deduce that P is a solution of (OPF 3 ) and thus arg(v) is a solution of (OPF 1 ). 2. If (OPF 2 ) admits a solution W of rank greater than 2, by absurd, let us suppose that (OPF 1 ) is feasible. Then (OPF 3 ) is also feasible, therefore P θ ∩ P P ̸ = ∅. Theorem 5 ensures that O(conv(P θ ) ∩ P P ) = O(P θ ∩ P P ).

In other words, every solution of (OPF 3 ) belongs to P θ . It follows from Proposition 5 that every solution of (OPF 2 ) is of rank 1, which contradicts the assumption. 3. If (OPF 2 ) is infeasible, by absurd, let us suppose that (OPF 1 ) is feasible.

Then (OPF 3 ) is also feasible. As a consequence, we have P θ ∩ P P ̸ = ∅. Since P θ ∩ P P ⊂ conv(P θ ) ∩ P P , we have conv(P θ ) ∩ P P ̸ = ∅, thus (OPF 3 ) is also feasible. It follows from Proposition 5 that (OPF 2 ) is feasible, which is a contradiction.

Numerical tests

We present here some numerical examples to highlight Theorem 6 proved previously and to test if this theoretical result can be extended to take into account reactive powers.

IEEE 37 node test feeder

We consider the distribution network of 37 nodes whose structure is given in Figure 9. The network and the line data are provided in [START_REF] Schneider | Analytic considerations and design basis for the ieee distribution test feeders[END_REF]. The admittance matrix is computed using the code provided in [START_REF] Group | Distribution test feeders[END_REF]. In order to make our tests more relevant, we we provide data values of the lower (upper) bound of the injected active powers, thermal constraint and of the voltages, displayed in Table 1. The F ℓ are chosen so that assumption A 1 holds. For every line {i, k}, we compute the angle α ik defined by ( 16) from admittance y ik . We then set the thermal constraint as F ℓ = F ik (α ik ) + F ki (α ki ). Note that we then have θ ik = α ik .

The computations for Problem (OPF 2 ) are carried out by the primal-dual interior points method, with the function Mosek using Yalmip toolbox on Mat-lab®, see [START_REF] Löfberg | Yalmip : A toolbox for modeling and optimization in matlab[END_REF]; ApS (2019). To solve Problem (OPF 1 ), we use the function FMINCON which relies on a SQP-BFGS type algorithm (see Waltz et al. (2006)). i=1 P i , which represents the total active power losses in the network. This function is strictly increasing with respect to P , in the sense of Definition 2.

Illustration of Theorem 6

In order to highlight Theorem 6, the values of P i , i ∈ {1, • • • , 37} \ {15} are set as in Table 1 and the value of P 15 varies from 0.6 to 2.8, in an increasing manner, in order to make this constraint more and more tight. Also, we set P 15 = 3. The obtained results are displayed in Table 2. We remark that up to the value P 15 = 0.9, the matrix W solution of (OPF 2 ) is of rank 1 and both problems share the same optimal value. For the values between P 15 = 1 and P 15 = 2.6, the matrix W is of rank greater than 1 and in this case the algorithm used for solving Problem (OPF 1 ) fails to find a feasible solution. Finally, for the values P 15 = 2.7 and P 15 = 2.8, the respective algorithms of both problems do not find a feasible solution. These results are consistent with the three cases of Theorem 6.

Consideration of the reactive power constraints

In many practical OPF problems, the consideration of reactive powers constraints is essential. We want to know if Theorem 6 can handle these additional constraints or if more assumptions are required. In this paper, we did not study this case theoretically. However, we will include these additional constraints in the following numerical experience.

The reactive power flow equation is given by

G ik = V 2 i b ik -V i V k g ik sin(θ ik ) + b ik cos(θ ik ) , (31)
where θ ik is the phase angle. The injected reactive power can be defined from (31) by

Q i = k∼i G ik .
The reactive power constraints of interests write:

Q i ≤ Q i ≤ Q i , for all nodes i.
In the following, we set the values of Q i , i ∈ {1, • • • , 37} \ {2} as in Table 1 and we vary the value of Q 2 between 0.03 and 0.48 in an increasing manner. The value of Q 2 is set to 0.6. All the other parameters are set as in Table 1. The obtained results are displayed in Table 3. For the values between Q 2 = 0.03 and Q 2 = 0.06, the solution W of (OPF 2 ) is of rank 1 and both problems share the same optimal value. For the values Q 2 = 0.09 and Q 2 = 0.10, the matrix W is of rank greater than 1 and the algorithm used to solve Problem (OPF 1 ) fails to find a feasible solution. Finally, for the values Q 2 = 0.47 and Q 2 = 0.48, the algorithms of both problems do not find a feasible solution.

These results shows a similar behavior as Theorem 6. However, to prove it theoretically, some additional assumptions may be required. For instance, in Zhang et al. (2014) a study that takes into account the constraints on the reactive power was carried out, but the amplitudes of the voltages were all equal to 1 and the Q i were assumed to be small enough.

Conclusion

We have presented an AC-OPF problem with fixed voltage magnitudes in a distribution network. This problem is known to be non-convex because of the power flow equations. In order to make it convex, we applied the method used in [START_REF] Bai | Semidefinite programming for optimal power flow problems[END_REF] which consists in reformulating the cost function and the constraints in terms of traces of semi-definite matrices. This leads to a new formulation of the problem in which the non-convexity appears as a rank constraint. By removing this rank constraint, we obtain a convex relaxation of the main problem.

In order to find the conditions under which this relaxation can be exact, we used the notion of Pareto-optimality which is a very adapted tool for our study. We started by studying the problem in the case of a one transmission line network, then extended our results to the case of a whole network.

The main result we have proved is that the constraint set of the problem of interest share the same Pareto-front of its convexified relaxation. Since the cost function is strictly increasing, the relaxation is exact. We then proved a theorem which allows to exploit the convex relaxation in practice to solve the main problem on interest. We concluded by giving some numerical tests to highlight the last theorem of this work.

Some interesting extensions of this work could be:

to extend the main result in the case of variable voltages magnitudes. Some additional assumptions might be required in order to obtain a theorem similar to Theorem 6. reactive powers constrains, i.e. to add the constraints Q i > -∞ and Q i < +∞. In the case of fixed voltage magnitudes, authors of Zhang et al. (2014) shows that the feasible set of the injected reactive power constrains is a simple linear transformation of the feasible set of the injected reactive power. Therefore, by adding an assumption equivalent to Q i = -∞, they showed in this case that Theorem 6 holds.

Waltz RA, Morales JL, Nocedal J, Orban D (2006) An interior algorithm for nonlinear optimization that combines line search and trust region steps. Mathematical programming 107(3):391-408 Zhang B, Lam AY, Domínguez-García AD, Tse D (2014) An optimal and distributed method for voltage regulation in power distribution systems. IEEE Transactions on Power Systems 30(4):1714-1726

Fig. 1 :

 1 Fig. 1: Example of the graph of a distribution network of 34 nodes

  we have x ⋆ ∈ B. This contradicts x ∈ O(B) and therefore we have A ∩ O(B) ⊂ O(A). 2. Since O(B) ⊂ A, the statement 1. reformulates as O(B) ⊂ O(A). Conversely, let x ∈ O(A) and let us suppose by absurd that x / ∈ O(B). We introduce the set C = B ∩ {y ∈ R n , y ≤ x} , which is compact as an intersection of a compact and a closed set. Let f : C → R be the function defined by f (y) = n i=1 y i . The function f is continuous on C, therefore there exists x ⋆ ∈ C such that x ⋆ = arg min y∈C f (y).

Fig. 2 :

 2 Fig. 2: Network of one transmission line

Fig. 3 :

 3 Fig. 3: (a) : the set A, (b) : the set F, (c) : the set O(F)

Fig. 5 :

 5 Fig. 5: Some possible cases for the set F ∩ F P ; the orange surface corresponds to conv(F) ∩ F P ; the green curve corresponds to F ∩ F P

Fig. 7 :

 7 Fig. 7: (a) : the relation O(F ∩ F P ) = O(conv(F) ∩ F P ) holds, (b) : the set F ∩ F P is empty

Fig. 9 :

 9 Fig. 9: IEEE 37 node test feeder

Table 2 :

 2 Comparison of the optimal values of Problems (OPF 1 ) and (OPF 2 ) for different values of P 15

		(OPF 1 )	(OPF 2 )
	P 15	optimal value	optimal value	rank of W
	0.6	0.2106	0.2106	1
	0.7	0.2711	0.2711	1
	0.8	0.3376	0.3376	1
	0.9	0.4092	0.4092	1
	1.0	infeasible	0.4861	4
	1.1	infeasible	0.5861	24
	2.5	infeasible	1.9861	35
	2.6	infeasible	2.3569	37
	2.7	infeasible	infeasible	-
	2.8	infeasible	infeasible	-

Table 3 :

 3 Comparison of the optimal values of Problems (OPF 1 ) and (OPF 2 ) for different values of Q 1

		(OPF 1 )	(OPF 2 )
	Q 2	optimal value	optimal value	rank of W
	0.03	0.0176	0.0176	1
	0.04	0.0177	0.0177	1
	0.05	0.080	0.080	1
	0.06	0.086	0.086	1
	0.09	infeasible	0.0485	3
	0.10	infeasible	0.0585	3
	0.47	infeasible	infeasible	-
	0.48	infeasible	infeasible	-

Acknowledgements This work was achieved within the project VERTPOM supported by the French Environment and Energy Management Agency (ADEME), as part of the "Future Investment Program of the French government". The revision time of the second author was supported by Czech Science Foundation project No. 22-15524S

Let us notice that thanks to the hypothesis θ ≤ π 2 , the set conv(F) is actually convex. Indeed, this set is a sector of an ellipse contained in a half-ellipse, see Figure 4. Without this hypothesis, the set conv(F) would not be convex (except in the case θ = π where conv(F) is the entire ellipse). Once again, the study of variations of F 1 and F 2 shows immediately that O(conv(F)) = O(F). This can also be seen in Figure 4. This result will be later extended to the whole network.

Technical results

We present here some technical results which will be used in the following and which are obtained by making a more restrictive assumption on θ.

First, let us notice that according to 17 and 18, if θ ≤ α, we have

Now we present a characterization of the elements of conv(F)\F.

Lemma 1 Let us assume that θ ≤ α. Let (x, y) ∈ conv(F)\F. Then there exists δ > 0 such that (x -δ, y) ∈ F.

In addition, for all ε ∈ [0, δ], we have (x -ε, y) ∈ conv(F).

Proof Since θ ≤ α, we have O(conv(F)) = F. The function F 2 is strictly decreasing on [-θ, θ], therefore there exists a unique θ ∈

The second statement is immediate by convexity of conv(F).

Finally, we prove a property that allows to reverse the sense of inequalities between the two components of vectors of F and conv(F). 3rd step: Construction of P ′ . Without loss of generality, we assume that q 1 = 1. We set

, where δ > 0 and the numbers λ i ∈]0, 1[ will be fixed in the following. For every other line {i, k} of E, we set F ′ ik = F ik and F ′ ki = F ki . We then define P ′ = AF ′ and we have

and for 2 ≤ i ≤ r -1,

The objective is to have P ′ ≺ P and P ′ ∈ P P , thus P i ≤ P ′ i ≤ P i for every i ∈ S. The vertices q i , for 1 ≤ i ≤ r -1, satisfy P i = P i , therefore we must enforce P ′ i = P i . This implies that

Let us notice that for δ small enough, λ i belongs to ]0, 1[ and can be as close to 1 as we want.

Since F ⋆ qrqr-1 -F qrqr-1 < 0, we deduce that P ′ qr < P qr . By construction, all the other vertices i ∈ Σ\ {q r } satisfy P ′ i = P i , thus we have P ′ ≺ P . 4th step: Verification that P ′ ∈ conv (P θ ) ∩ P P . Let ℓ i be such that ψ(ℓ i ) = {q i , q i+1 }. We first notice that by convexity of conv (F ℓi ) , we have

On the other hand, since (F 12 , F 21 ) ∈ conv (F 1 ) \F 1 , Lemma 3 ensures that (F ′ 12 , F ′ 21 ) belongs to conv (F 1 ) at least for δ small enough. Therefore, for δ small enough, we have P ′ ∈ conv (P θ ). To prove that P ′ ∈ P P , it is sufficient to prove that P ′ qr > P qr . We have

We have seen that by choosing δ small enough, we can make λ r-1 as close to 1 as we want. Since P qr > P qr , we have P ′ qr > P qr for δ small enough, thus P ′ ∈ P P .

We have thus proven that for δ small enough, P ′ belongs to conv (P θ )∩P P . Since P ′ ≺ P , the initial assumption is contradicted.