Olivier Glassey
email: olivier.glassey@idheap.unil.ch

WRITING A NEW CONSTITUTION FOR GENEVA: AN ANALYSIS OF PARTICIPATION MECHANISMS

Keywords: participation, constitution, eDemocracy, framework, case study

the people of Geneva voted in favour of a new Constitution to replace the current one, written in 1847 and considered by many to be out of line with today's society. In the first part of this paper we set the context of our study and we define a framework to analyse participation and eParticipation in terms of institutional, mediated and informal political communication mechanisms. In the second part we apply it to the campaign for the election of a Constituent Assembly and we provide the preliminary results of this survey. The last part describes how we will use this framework to investigate these mechanisms during the process of writing a new Constitution. Geneva was a pioneer in terms of eVoting and we want to find out if this will be the case again in the domain of eParticipation, with what could potentially become the first Wiki-Constitution ever. However our first findings indicate that ICTs are rather an extension of current participation mechanisms and that they do not radically change or renew them.

Introduction

In this first section we describe the context in which this study takes place, i.e. the election of a Constituent Assembly to write a new Constitution for the Canton of Geneva. We furthermore explain the general political participation mechanisms in Switzerland and in Geneva.

Geneva's Constitution

Geneva is a republic since 1535 when the city became the capital of the Protestant Reformation. The first Constitution adopted in 1543 was largely based on the "Edits Civils" written by John Calvin. Although Geneva was a French department between 1798 (when it was invaded by Napoleon's army) and 1813, it was mostly independent until joining the Swiss Confederation in 1815.

In 1846 James Fazy led a revolution that overthrew the conservative government and subsequently wrote the 1847's Constitution that is still ruling the Canton, although it has been modified many times over that period. This text is now the oldest of the 26 Cantonal Constitutions in Switzerland and many believed that its language, structure and content are not adequate anymore [START_REF] Schmitt | Quelques réflexions comparatives à propos de l'élection d'une assemblée constituante à Genève[END_REF].

In 1999 the parliamentary group of the "Parti Radical" proposed a bill in order to completely revise Geneva's Constitution, but without success. In 2005 an association called "Une nouvelle Constitution pour Genève" (a New Constitution for Geneva) was set up. Its front man was a famous law professor, Andreas Auer, and its members came from all political parties and from the civil society. They were ready to launch a popular initiative requiring a new Constitution as the government was reluctant to do so, but after long negotiations a vote was organised. In February 2008 the people of Geneva accepted a constitutional law allowing for a new Constitution in the Canton.

Election of the Constituent Assembly

In October 2008 the people of Geneva elected 80 members of the new Constituent Assembly. This was no easy task for citizens as there were 530 candidates and 18 lists to choose from. Half of these lists were presented by traditional political parties and the other nine lists represented heterogeneous interest groups (business associations, home-owners, women, retired people, and so on). Funding for the campaign was also very heterogeneous: from 5'000 Swiss Francs (about 3'300 Euros) for the women's list [START_REF]Les p'tits nouveaux de la politique genevoise[END_REF] to 200'000 (130'000 Euros) for the business associations' list [START_REF]La dure vie des petits candidats[END_REF].

The quorum for a list to be elected was initially 7%, but the Parliament lowered it to 3% in order to have a wider participation. However one cannot say that the members of this Assembly are really representative of Geneva's people: only 14 women were elected (although there will be 16 women in the Assembly because two elected men resigned in order to leave their position to women from the same left-wing party [START_REF]Deux femmes de plus à la Constituante[END_REF]) and the average age of members is 56. Furthermore only three lists outside traditional parties made the quorum:

• The lobby of pensioned people (Avivo) got 9 seats; it must be said that Christian Grobet, the leader of this list, was a member of various legislative and executive authorities in Geneva from 1967 until 2005, thus this list is not completely "outside" political parties.

• The g[e]'avance list represented business and employers' lobbies and it was attributed 6 seats.

• The FAGE (Federation of Geneva's Associations) is the umbrella organization of 480 associations of all types (parents, culture, human rights, ecology, Attac, pacifism, consumers, social integration, gays, development, etc.); the associations' list obtained almost 4% of the votes (with a quorum at 3%) and thus obtained 3 seats.

The participation rate being of 33 % [START_REF]élection à l'Assemblée Constituante du 19 octobre[END_REF] (about 10% less than the average participation, see below), one can conclude that giving Geneva a new Constitution was not a popular issue and that only "traditional" or "politicized" voters accomplished their electoral duties. The political balance of the Constituent assembly is also similar to what can be seen at the Parliament of Geneva [START_REF]Députés au Grand Conseil[END_REF]: in the Parliament 57% of the seats are attributed to right wing parties, 33 to left wing parties and 10 are considered as independent, whereas 43 seats of the Constituent Assembly account for right wing parties (53%).

e-Voting in Geneva

Although electronic voting was not used for the Constituent Assembly, we believe it is interesting to give some details on the Geneva e-Voting project in this introduction. Swiss citizens vote 4 to 6 times a year and in some cases the participation is as low as 30%. However it must be said that in Switzerland the participation rate is calculated on the basis of all citizens over 18 and not on the basis of registered voters as this is the case in many countries. The average participation in the last 30 years is around 42%, with a record of 78% in 1992 when the citizens had to decide on whether Switzerland should join the European Economic Area. Postal vote was designed as a solution to increase these participations figures: citizens are sent the voting material at home and they have the possibility to send it back to their cantonal or communal authorities, instead of having to go to the polling station on designated week-ends. Although postal vote was already introduced in 1957 in the Canton of Vaud, voting material was sent on request only (for each vote). Most Cantons generalized this system for all citizens during the 80's and the 90's and it was first used in Geneva in 1995. Turnout increased by 20 percent and currently up to 95% of Geneva's voters use the postal vote.

In 2001, the Swiss federal government decided to test and evaluate e-voting systems. The Cantons of Geneva, Neuchâtel and Zurich were chosen to develop three separate internet voting solutions, so that they could be assessed and tailored in order to fit the 26 different legal and organizational contexts of each Swiss Cantons [START_REF] Braun | Swiss E-Voting Pilot Projects: Evaluation, Situation Analysis and How to Proceed[END_REF]. The first e-Voting test took place in January 2003 in one commune of Geneva. Currently the system is still not fully deployed: for the polls of November 30, 2008, only nine (out of 45) communes could use the system.

However nine e-Voting sessions were organized to date and there are some interesting outcomes: between 22% and 25% of all voters used the e-Voting system, amongst them 19% are regular voters and 56% are usually abstainers [START_REF] Chevallier | Internet Voting: Situation, Perspectives and Issues[END_REF].

Participation Mechanisms

This section provides definitions of participation, eParticipation and eDemocracy. We then build on them in order to define the analysis dimensions of our survey.

Participation and eParticipation

According to [START_REF] Sanford | Characterizing eParticipation[END_REF] eParticipation is an emerging research area which lacks a clear literature base or research approach. In their review of the field, they identified and analysed 99 articles that are considered to be highly relevant to eParticipation. [START_REF] Sanford | Characterizing eParticipation[END_REF] write in their introduction that governments seek to encourage participation in order to improve the efficiency, acceptance, and legitimacy of political processes. They identify the main stakeholders of participation as citizens, non-governmental organizations, lobbyists and pressure groups, who want to influence the political system, as well as the opinion forming processes. Various information and communication technologies (ICTs) are available for eParticipation: discussion forums, electronic voting systems, group decision support systems, and web logging (blogs). However traditional methods for citizen participation (charettes, citizens' juries or panels, focus groups, consensus conferences, public hearings, deliberative polls, etc.) are still very widely used and must be taken into account when studying eParticipation.

[6] defines eDemocracy as the use of information and communication technologies to engage citizens, to support the democratic decision-making processes and to strengthen representative democracy. She furthermore writes that the democratic decision making processes can be divided into two main categories: one addressing the electoral process, including e-voting, and the other addressing citizen e-participation in democratic decision-making. [START_REF] Macintosh | Evaluating how eParticipation changes local democracy[END_REF] give a working definition of eParticipation as the use of ICTs to support information provision and "top-down" engagement, i.e. government-led initiatives, or "ground-up" efforts to empower citizens, civil society organisations and other democratically constituted groups to gain the support of their elected representatives.

There are many examples of surveys on eDemocracy, such as [START_REF] Ladner | From e-voting to smart-voting -e-Tools in and for elections and direct democracy in Switzerland[END_REF] who take the case of Switzerland where citizens are often called to the polls either to vote for parties and candidates or, even more often, to decide on direct-democratic votes at the three different political levels. In their paper on "smart-voting" they analyse what they call voting assistance applications, i.e. tools where citizens can compare their positions on various political issues to those of parties or candidates. They mention the Dutch "Stemwijzer" system, first introduced in 1998 and they provide in-depth information on the Swiss smartvote website.

Even if eParticipation is a relatively new research field, projects and tools are increasing thank to governmental support [START_REF] Tambouris | A Framework for Assessing eParticipation Projects and Tools[END_REF] and a number of research projects have been funded worldwide to pave the way, such as Demo-Net.org.

Analysis Framework

The goal of this study being to investigate communication and coordination mechanisms for participation and eParticipation, we therefore had to define an analysis framework. We adapted the approach used by [START_REF] Mambrey | From Participation to e-Participation: The German Case[END_REF] for its case study on participation and eParticipation in Germany, where he used the three arenas of political communications defined by [START_REF] Habermas | The Theory of Communicative Action[END_REF]. Table 1 shows these three communications modes and the systems or actors involved in political communication, as well as the vectors used to carry this communication. We made a distinction between traditional participation vectors and ICT-enabled vectors (eParticipation). In our survey we decided not to analyse the institutional communication modes and to concentrate on the mediated and informal ones, as there is already a large amount of research that has been done on eVoting and eConsultations. We indeed believed it would more relevant to survey the mediated and informal communication arenas during the process of writing a new Constitution for Geneva. When looking at Table 1 one can see that the traditional communication modes for mediated and informal communication are relatively well differentiated, but that we could not define which eParticipation tools are used by which actors. This is precisely what we want to study: how do parties or interest unions use websites or wikis to support their communication, how do citizens or associations make their voice heard through ICTs, are these tools complementary to existing participation mechanisms, do new political communication usages appear with ICTs, and so on.

Selection of stakeholders and Methodology

In order to study participation and eParticipation and to answer the various questions listed above, we selected a sample of actors in mediated and informal communication modes:

• The "Tribune de Genève" (TDG): we chose it because it is Geneva's main printed newspaper and it furthermore provides a blog platform to its readers; most candidates' blogs were hosted at the TDG.

• The "Parti Radical Genevois" (PRD) is a progressive right-wing party that has a long history in Geneva; it was born in 1841-42 during the first revolutionary movements in the Canton and it was led by James Fazy, the author of the 1847's Constitution that is still in effect today. We selected it because PRD is rather representative of traditional parties and political structures.

• "Les Verts" are Geneva's green party. It was founded in 1983 by various members of environmental and anti-nuclear associations. The Green party is now a wellestablished party with, amongst others, two elected members of the executive government in Geneva (which comprises seven ministers). We decided to survey them because they are a newer party, created by members of the civil society and based on a more associative operational mode.

• We already introduced the Federation of Geneva's Associations (FAGE) in section 1: it is the umbrella organization of 480 associations. We integrated them in our study because they are very typical of networked communication and participation.

• Last but not least, citizens of Geneva, for quite self-evident reasons when one is writing about bottom-up participation.

This survey is mostly qualitative and based on two investigation methods:

• Periodical review of all identified websites, blogs, forums, wikis, etc. related to the subject of the Constituent Assembly.

• Interviews with representatives of the stakeholders listed above, as well as with elected members of the Constituent Assembly.

Preliminary Results

This project comprises two distinct parts: we first studied participation and eParticipation during the campaign for the election of the Constituent Assembly (until October 2008) and we will then analyse these mechanisms during the process of writing the new Constitution.

We will not go into the specifics of the campaign here, but let us already mention that it was mostly traditional participation mechanisms and communication vectors that were used, such as interview in the press, shows on local TV and radios, meetings, debates, tract distribution in the streets and on markets, etc. The only eParticipation elements came from personal (and disjoint) initiatives. We will list the most interesting of them here:

• The Tribune de Genève (TDG) followed the campaign and opened its daily editions to all candidate lists. One journalist, Jean-François Mabut, set up a blog called "Gazette de la Constituante" where he commented the campaign and pointed to interesting blog posts or websites. Moreover selected content from various political and citizens' blogs was printed in the newspaper. However this was a personal initiative from the journalist that was accepted by the direction of the newspaper but that did not get much support.

• 78 candidates had a blog on the TDG platform, but several of them had no or very limited contributions. 18 of them were elected, so if we make a quick (and not very significant) calculation, about 23% of bloggers got elected (18 of 78) against 13% of non bloggers (62 elected of 452 candidates). Beyond these figures, it is very interesting to note that this blog platform was the only way for some small lists to voice their opinion and to be visible. As written in one of the comments on the Gazette de la Constituante, "blogging was the only way of expression that the women's' list had, regarding our limited financial means. (…) We published articles, photos, letters and wishes from all our candidates and we were able to speak in one voice by putting our differences aside and finding consensus".

• Some citizens also used blogging and electronic newsletters to comment on the campaign. A good example is Pascal Holenweg that set up a blog (carmagnole.blogspot.com) and a daily newsletter where he made irreverent but relevant comments on the campaign. However it must be said that Mr Holenweg is a former member of the legislative assembly of the City of Geneva and a well-known figure of local activism.

• Some younger candidates used Facebook for their campaign. Murat Alder is one of them and he was elected on the list of the Parti Radical (PRD).

• The PRD and the FAGE set up wikis in order to gather inputs from the citizens on the new Constitution. However neither of them listed any modification after the initial online publishing, so the "wiki-participation" amounted to nil. FAGE also provided an online form to make propositions: they received exactly 16 of them, ranging from supporting sustainable development, providing education to illegal migrants, prohibiting cars, legalizing cannabis or opening more public nudist spaces.

• Many candidates and associations also used mail, postcards and email campaigning in order to make their ideas known and as an incentive for people to go and vote.

Although the author received several of them, it is impossible to have an exhaustive view on that, as it was not out in the public domain.

• Although this last element is not supported by ICTs, we mention it anyway because the concept is interesting: the FAGE organised several "Caf'Idées" (idea cafés), a mix of brainstorming sessions, musical chairs and speed-dating. Four to five debaters would indeed be seated at a same table in a café and discuss a given topic for a limited amount of time, before making a synthesis and switching tables and partners.

The first part of this survey showed that there was no real eParticipation during the campaign, apart than using the Web, blogs or social networks to provide more visibility to candidates and, to a lesser extent, to sustain opinion forming.

Next Steps

On Thursday November 20 th , 2008 the Constituent Assembly held its first session and its work will be spread over four years. The second part of our project has just started as we write these lines and it is planned that it will last during the first year of sessions. Our aim is to analyse how elected Constituents will relay the ideas of their parties or of their associations' members, and to see how citizens will be involved in the process (if at all). We will use our analysis framework to study coordination and communication mechanisms, especially in terms of co-writing or co-creating the new Constitution. Indeed we believe that, although there was not much "e" in the election process, it will surely be more developed during the redaction phase. As Yves Lador, elected member of the FAGE puts it in his own word: "I never believed that a wiki would involve the general public in participating in the Constitution's elaboration; however I think it will be a great tool for internal use and collaboration (…)"

Conclusions

On the international level it is not very often that countries completely rewrite their Constitution. Switzerland and its 26 sovereign Cantons constitute a very interesting laboratory, as 21 of them did so since 1965. Eight Cantons revised their Constitutions through a Constituent Assembly, and the other fourteen relied on dedicated commissions assigned by either the legislative or the executive authority [START_REF] Schmitt | Quelques réflexions comparatives à propos de l'élection d'une assemblée constituante à Genève[END_REF]. However the last Constitution designed by a Constituent Assembly was that of the Canton of Basel and the process lasted from 1999 until 2005, meaning that at the time tools such as blogs or wikis were not yet as well-known by the general public as now. We therefore believe that this experience in Geneva might be a first in terms of eParticipation and we are really eager to see if ICTs only provide an extension of traditional participation mechanisms (e.g. offering more visibility during a campaign for elections) or if they have the ability to modify participation and to lead to the creation of what could be the first "Wiki-Constitution" in the world.

Table 1 :

 1 Communication Modes and Vectors for Participation

	Communication /	System / Actors	Traditional	ICT-enabled
	Coordination mode		Communication Vectors	Communication Vectors
			Elections	eVoting
	Institutional	Representative Democracy	Consultation: citizen forums, public hearings or any formal consultation	eConsultation
			procedure
		Direct Democracy	Voting, referendums, initiatives	eVoting
			Articles, opinions,
		Mass Media	interviews, editorials, readers letters, polls, phone
			calls, etc.
	Mediated		Parliamentary groups Lobbies	Websites, forums, wikis,
		Parties	Strikes	emails, chats, ePolls,
		Interest groups	Meetings, campaigns, street	webcasts, social networks,
		Trade unions	or door-to-door	mobile communications,
			communication, tracts,	Web 2.0, etc.
			mailings, negotiation
	Informal	Citizens Associations Networks	Street or door-to-door communication, tracts, free radios, local TVs, cafés, clubs, etc.


Seddik Yassine Abdelouadoud
email: seddik.abdelouadoud@mines-paristech.fr

Robin Girard

François-Pascal Neirac

Thierry Guiot

Iterative linear cuts strenghtening the second-order cone relaxation of the distribution system optimal power flow problem

Keywords: Distributed power generation, power system planning, smart grids I

We present a novel iterative algorithm to solve the distribution system optimal power flow problem over a radial network. Our methodology makes use of a widely studied second order cone relaxation applied to the branch flow model of a radial network. Several types of conditions have been established under which this relaxation is exact and we focus here on the situations where this is not the case. To overcome this difficulty, we propose to add increasingly tight linear cuts to the second-order cone problem until a physically meaningful solution is obtained. We apply this technique to a sample system taken from the literature and compare the results with a traditional nonlinear solver.

INTRODUCTION

With the aim to increase the sustainability of the electric power system, the share of renewable energies in the production mix is scheduled to increase in the future. For example, the European Union has set goals for its member states in order to attain a 20% share of renewable energy in its final energy consumption by 2020, and some countries have taken even more ambitious stances. This target will be partially met by integrating significant amounts of dispersed renewable energy generators (mainly photovoltaic (PV) and wind power) to the distribution grid. These developments will have a considerable impact on the design and operation of the electric system, both at the national and local level and so new tools will be needed to assist in the planning and operation of at least the distribution network. Indeed, as the current passive distribution network turns into an Active Distribution Network (ADN) [START_REF] Pilo | Active distribution network evolution in different regulatory environments[END_REF] with the introduction of partially and totally controllable generation and storage means, planning studies based solely on power flows for extreme load conditions will not be adapted anymore. Considering the similarities between the current transmission network and the future ADN, it is a safe bet to assume that the Optimal Power Flow, a tool first introduced in 1962 by Carpentier [START_REF] Carpentier | Contribution à l'étude du dispatching économique[END_REF] and now widely used for the planning and operation of the transmission network, will prove useful for this purpose. Consequently, the adaptation of the OPF concept and resolution algorithms to the distribution network has been the subject of numerous publications in the last decade such as [START_REF] Gabash | Active-Reactive Optimal Power Flow in Distribution Networks With Embedded Generation and Battery Storage[END_REF], [START_REF] Ahmadi | Optimal power flow for autonomous regional active network management system[END_REF], [START_REF] Swarnkar | Optimal power flow of large distribution system solution for combined economic emission dispatch problem using partical swarm optimization[END_REF], [START_REF] Dolan | Using optimal power flow for management of power flows in active distribution networks within thermal constraints[END_REF]. Parallel to these efforts, a promising path has been taken by, for example, [7] and [8], to show that the OPF problem can be cast as a Quadratically Constrained Quadratic Program (QCQP). The QCQP class of problem has already been widely studied, as many instances of engineering problems, such as principal component analysis and the combinatorial max-cut problem, are included in it. General QCQP are non convex problems and thus cannot be solved in polynomial time. To deal with this, the standard practice in the OPF literature is to relax the problem to a convex conic program that can either be cast as a Second-Order Cone Program (SOCP) or Semi-Definite Program (SDP). On the application of such techniques to radial networks, we can cite the work of [9] and [10], who prove that the SDP relaxation over tree networks is exact under certain condition, or the work of [11]-[13], who took the SOCP route applied to the branch flow model and proved that the relaxation is exact if there are no upper bounds on loads or voltage magnitude and certain conditions on the objective function are respected. In these instances, it can therefore be shown that the OPF over radial network can be solved in polynomial time. However, as has been shown in [14], such approaches may fail to produce physically meaningful solutions in realistic conditions. Such conditions are the focus of our work here, as we are proposing a methodology based on adding increasingly tight linear cuts to the initial SOCP relaxation based on the branch flow model until a satisfactory solution is obtained.

The paper is organized as follows: first we introduce the problem formulation, then we present the methodology and we finish by applying it to a case study.

II. PROBLEM FORMULATION

A. Objective function

In this paper, we are particularly interested in situations described in detail in our previous work [14] and [15] where distributed generation and storage are integrated at the distribution system level, not only to alleviate network constraints through active and reactive power control, but also to provide services at the national level, such as energy and ancillary services provision or energy time-shift in the case of storage. As the time-coupling constraints can create extremely large problems, we have proposed to separate them into master problems dealing with the multistage scheduling of active powers and single-stage slave problems ensuring the respect of network constraints, while minimizing the deviation from the master problem. We focus here on the solution of the slave problem where, therefore, the objective function is of the following form:

      K k t k c t MP k c P P 2 , , , min  
, where

t MP k c P , ,
is the set of controllable active powers resulting from the master problem solution and are the corresponding control variable in the slave problem.

B. Network and apparent power constraints

As discussed earlier, we adopt here the branch flow model first introduced in [17] and [START_REF] Baran | Optimal sizing of capacitors placed on a radial distribution system[END_REF]. Henceforth, we assume that the storage devices and distributed energy units are connected to the grid through an Advanced Power Electronic Interface (APEI), as defined in [START_REF] Wächter | On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming[END_REF], and that the substation transformer is equipped with an On-Load Tap Changer (OLTC), for which we use a simplified continuous model. Consequently, the network and apparent power equations are cast in the following form for all nodes j :

P R P j j j , , 2 2          
(2)

t j NetLoad KL kl KK kk KK kk t kk t kk t kk t kk kl t j Q Q U Q P X Q j j j , , 2 2           (3) 0 2 2 2 2 ) ( 2                  t j j t j t j f j t j j t j t j S Z U U X Q R P U (4) 2 2 2 2 2 2 2 , , t t t l l l t t j j j j j Q P S X R Z V U      (5) t j c t j Load t j NetLoad P P P , , ,   (6) t j c t j Load t j NetLoad Q Q Q , , ,   (7) max min max min , OLTC t OLTC OLTC t j V V V V V V     (8) , 2 max , 2 , 2 , j c t j c t j c S Q P   (9)
, where t j P and t j Q are, respectively, the active and reactive power entering node j from the upstream node ) ( j f , t j V and t OLTC V the voltage magnitude, respectively, at node j and downstream of the substation transformer, during time step t .

C. SOCP relaxation

As explained earlier, we inspire ourselves from the work of, for example, [11] to implement the SOCP relaxation of the OPF problem. First, we define an intermediary variable t j I for all nodes j :

t j t j t j t j U Q P I 2 2  (9)
We then replace

t j t j t j U Q P 2 2  by t j
I in Eq. ( 2) and (3). In the resulting formulation, the non convexity of the problem stems only from the equality in Eq. ( 9) which we relax by keeping only the following inequality:

t j t j t j t j U Q P I 2 2   (10)
We consequently obtain a new problem consisting of Eq. ( 10) and ( 1) through (8) that we will subsequently call R-OPF.

III. METHODOLOGY

A. Guiding principle

Our methodology is based on the fact that, in order to obtain a physically meaningful solution, the equality in Eq. (9) needs to be verified. Therefore, we add linear cuts to R-OPF that take the following form for all nodes :

j t j j t j j t j j t j d U c Q b P a I        (11)
These cuts are only valid only on a certain domain, which can be apprehended in the following way :

j t j j t j j t j j t j t j t j d U c Q b P a U Q P         2 2 (12)
Dropping either the active or reactive power variable to fit in a three-dimensional representation, such cuts and their domain of validity can be graphically represented in the following way :

        t j t j t j t j t j t j U I Q P U I 2 2
(13)

B. Calculation of the linear cut defining parameters

If we consider that the domain of validity is of the form

      max min max min max min , , , , , j j t j j j t j j j t j Q Q Q P P P U U U   
we calculate the defining parameters of the linear cuts so that the barycenter of the cutting plane is the lowest possible. This is obtained by solving the following linear problem in the variables j a , j b , j c and j d :

j j j j j j j j j j d U U c Q Q b P P a                               2 2 2 min max min max min max min (14)
To which we add constraints of the same form than Eq. (12) for each extremity of the validity domain.

C. Tightening of the validity domain

To identify the solution obtain at the k-th iteration, we add a second subscript to each variable in the following manner

         t k j j t k j k j Q Q Q Q (17)             k exp (18)
Where  is a parameter that governs the speed of the tightening of the validity domain.

IV. CASE STUDY

A. Definition of the parameters

We apply this methodology to a case study consisting in the 69-bus network described in [17], with residential-type loads connected at each node and scaled so that the annual maximal load is equal to the loads used in [17]. Reactive loads are defined, so that the power factor remains constant and equal to the one used in [17]. Storage units are deployed at 13 nodes in increasing order of driving-point impedance magnitude. The maximal apparent power of the storage units is chosen to be equal to the maximal annual active load of the node where it is connected. We select the most critical time steps in the sense that we defined in [16] to ensure that our algorithm is suitable even in extreme loading or storage power injection conditions. The results are obtained through the use of the commercial solver IBM ILOG CPLEX.

B. Convergence

For this example, we set  to 10e-5 to enforce a high precision of the result and we present hereafter the rate of convergence for a  parameter varying between 6 and 12.

Lower values of this parameter led to infeasible problems in this example.

Figure 2 : Rate of convergence as a function of Beta

We then compare the final value of the objective function with the value obtained from the nonlinear solver IPOPT [START_REF] Wächter | On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming[END_REF].

We can thus observe that the higher the value of  , the better the result will be. Moreover, we show that this methodology is suitable to obtain comparable or even better results than off-the-shelf nonlinear solvers, in this particular case study.

I. CONCLUSION

We have presented a novel iterative algorithm to solve the radial distribution system optimal power flow that relies heavily on recent breakthroughs in this field, especially those concerning the use of a second-order cone algorithm. We have then detailed the methodology employed and the parameters that influence it. We finished by applying to a specific case study, which showed that this approach is promising, especially in terms of rate of convergence and optimality. Further work has to be done that will concern testing the methodology on broader sets of network, understanding in a more systematic ways the relationships between the parameters governing the rate of convergence and the optimality of the resulting solution and linking it to the previous work done the establishment of a criticality criterion by time steps.

  of the attainable voltage downstream of the substation. Eq. (1)-(9) define the OPF problem that we propose to study.

Figure 1 :

 1 Figure 1 : Graphic concepts of the linear cuts, with arbitrary units