Carlo Carraro

Gérard Mondello
email: gerard.mondello@gredeg.cnrs.fr

Civil Liability, Safety and Nuclear Parks: Is Concentrated Management Better?

Keywords: Strict Liability, Electric Energy, Nuclear Plants JEL Classification: Q5, Q58, Q53, K23, L13, L52, L94 JEL Classification : Q5, Q58, Q53, K23, L13, L52, L94 Strict liability, Electric Energy, nuclear plants, limited liability, concentration

Ultra-hazardous risky activities as nuclear industry cannot be considered as "normal industries" i.e. industries without abnormal environmental and health risks. Consequently, the industrial organization of these specific sectors is of the utmost importance. This paper aims at studying this question. We focus on the associated costs of prevention and civil liability. We analyze how civil liability rules may contribute to extend or to discourage the expansion of nuclear parks to new operators. The paper compares the consequences of extending the management of nuclear stations to several independent operators. This question can apply to the unification process of the European electricity market in which several public and private nuclear power operators are running. The paper shows that the choice between either a monopolistic scheme (one operator managing several plants) or a decentralized one (one operator by station) depends on the condition of application of the legal civil liability regime and on the strength of the safety control exerted by the Nuclear Regulatory Authorities. It is shown that when the control is high, then the safety costs generated by the monopolistic organization are less than the same costs of a decentralized one. However, conditions on the insurance policy can mitigate this result.

Introduction

From the end of the 20 th Century to the beginning of the 21 th one, nuclear power industry developed fast because of the regular price crisis that oil and gas energies suffered during this period. Nuclear energy has been seen as a powerful and efficient complement or partial substitute to them. However, this growth has been interrupted due both to the 1979 Three Miles Island 1 accident and the 1986 Tchernobyl catastrophe that froze the nuclear impetus for long. At the beginning of the 2000 years, new oil and gas crisis and the new environmental concerns about GHG emissions and their mandatory reduction 2 helped the revival of the nuclear industry 3 . This state of matter could be considered as true until 11 th March 2011 before that the 9.0-magnitude Tohoku earthquake and its following tsunami devastated Japanese northeast coast. The subsequent catastrophic consequence has been the knocking out of the reactor cooling systems of the Fukushima Daiichi and Daini electronuclear plant. This induced leakage of irradiated contaminant that threats populations located in the plant's vicinity, the supply of fresh water, the sea resources and the agriculture products on a large scale in Japan.

As for the Three Miles Island 4 and Tchernobyl accidents, by its severity, the Fukushima disaster undermines (and will undermine) the defenders' position of the nuclear option.

Already, in Europe, some countries put into question their recent nuclear power programs as Germany and Italy for instance. Anyway, the industrial and economic consequences of this catastrophic event are still unknown because at the present time (April 2011) Japanese authorities are far from getting control on the situation: the Fukushima catastrophe is still going on and developing. Consequently, these events will spark a close questioning of the viability and the safety of the whole nuclear plants in the World. Without doubt, after that, some stations will be closed or refurbished. However, from an economic viewpoint, the present situation could be quite different from what the past showed. Indeed, in the eighties, nuclear accidents entailed either the break down or the serious slow down of the electro-nuclear projects in most Countries. By now, emergent countries are deeply involved in the development of their energy plan and going back to a fossil energy alternative could be difficult for them. Hence, in spite of the Fukushima accident, nuclear program will go on spreading, even at slower rates. 1 [START_REF] Rogovin | A Model of the Optimal Use of Liability and Safety Regulation[END_REF]. 2 See e.g. the Intergovernmental Panel on Climate Change, Fourth Assessment Report, (2007). 3 So for example the [START_REF] Ampere | Rapport de la Commission pour l'Analyse des Modes de Production de l'Électricité et le Redéploiement des Énergies (AMPERE)[END_REF] report. 4 [START_REF] Rogovin | A Model of the Optimal Use of Liability and Safety Regulation[END_REF].

This structural trend and this tragic accident put spotlights on the very sensitive question of the electro-nuclear plants safety. Even if nuclear catastrophes are deemed to occur with very low probability5 , they have far-reaching consequences for health and the Environment. Furthermore, the very recent history has shown that unexpected scenarios can develop as in Fukushima where the tsunami damaged simultaneously its four reactors.

As a consequence, giving up the nuclear sector is a long run project and more probably, nuclear programs will last several more decades. Then, it is natural and legitimate to consider that, nowadays, the industrial organization of this ultra-hazardous risky industry is becoming a major stake 6 . Initially, Governments boosted the development of electro-nuclear industry in the wake of the development of military activities. However at the present, private interests are growingly interested and involved in the development of nuclear programs. This can take the shape of public-private participation. For instance, in the United States from July 1999 to September 2008, eighteen US nuclear plants changed ownership7 . The firm Tokyo Electric Power Company (TEPCO) in Japan is private and manages three nuclear plants. Private firms are all the more encouraged to invest in this sector by the international institutional improvement of the coverage conditions of third party damages. The temptations to break State monopolies in this sector are great. Furthermore, the unification of the electricity market (as it is the case in Europe) induces several nuclear operators to live together. Taking into account these factors leads to wonder whether a greater safety is guaranteed by a centralized organization of the nuclear sector, or whether the arrival of new comers can contribute to increase the level of care and protection under a decentralization process. Since the end of the nineties this point is controversial, (see for instance [START_REF] Varley | Electricity Market Competition and Nuclear Power[END_REF], or still AEN-OCDE (2000)). The recent Japanese catastrophe reopens this debate following the Italian and German decisions.

This paper assesses on a theoretical basis how civil liability rules influence the entrance of new operators on the electronuclear production. This is done by appraising the consequences on care levels of the splitting-up of an electro-nuclear park into independent entities. Performance is evaluated by the level of safety and the global cost of liability and prevention. Choosing between centralizing and decentralizing the organization of a park of nuclear plants depends on several factors as, for instance, the nature of the legal civil liability regime, the strength of the control exerted by the Nuclear Regulatory Authorities and the nature of the insurance policy. When the control is low, we show that both structures are equivalent because of the similarity of their safety costs. However, under a high control, a centralized organization generates lesser safety costs compared to a decentralized structure though this opinion should be moderated. Indeed, the present model shows that a high insurance cover may favor the decentralized organization. This may be done, for instance, by pooling the financial resources of different operators as in the US Price-Anderson Act of 1957, or the German insurance system based on a large extension of the operators' liability. This question has policy implications. For instance, it may be applied to the unification process of the European electricity market characterized by several public and private nuclear operators8 .

For this reason, compared to the wide field of the electro-nuclear economics, this paper will deal neither with the economic benefits of nuclear energy compared to other energies (MIT (2003(MIT ( ), (2009)), Bickel and Rainer (2005)), nor with the operating conditions of the nuclear deal with price uncertainty (Gollier, Proult, Thais and Walgenwitz (2004), [START_REF] Linares | The economics of new nuclear power plants in liberalized electricity markets[END_REF]), nor with the question of decommissioning plants, and, still, nor with the reprocessing of nuclear waste. Its argument borrows some features of the debate about civil liability in the electro-nuclear industry. Most of these controversial contributions [START_REF] Dubin | Subsidy to Nuclear Power Through Price-Anderson Liability Limit[END_REF], [START_REF] Heyes | Subsidy for Nuclear Power Through the Price-Anderson Liability Limit[END_REF]), (2000a), 2000b), [START_REF] Faure | Compensating Nuclear damage: A Comparative Economic Analysis of the US and International Liability Schemes[END_REF], [START_REF] Faure | An Economic Analysis Of The Nuclear Liability Subsidy School of Law[END_REF], [START_REF] Rothwell | Further Comments on Subsidy to Nuclear Power through the Price Anderson Liability Limit[END_REF]) show that putting ceiling on the level of repairs comes at subsidizing implicitly this industry9 and tends to let unpaid the externality costs.

However, this paper will not take this road. Starting from the civil liability question, it focuses rather on understanding the best economic organization scheme that insures the best safety level.

A first paragraph explains the nature of civil liability in the electro-nuclear sector. It underlines the importance of the ceiling of the level of repairs for inducing private operators to enter the market. A second paragraph describes the model based on the comparison of the expected costs of the organization of a park of two stations managed either by a unique operator or by two operators. A third one compares each kind of organization considering the level of effort dedicated to safety. A fourth one concludes and gives some policy recommendation.

Civil liability and the development of the nuclear industry

Before 1988, Soviet Union acceded neither to any nuclear conventions nor to national nuclear liability law and, after the Tchernobyl 1986 disaster, this Country escaped to the duty to compensate damages to health, crops of national and international economies. To avoid future detrimental situations, the international community amended the existing nuclear The joint protocol promotes a strict liability regime but maintains the liability limitation principle settled by the previous international conventions11 . These last one set on a global framework that national legislation can make more, but not less, stringent. Under the Joint Protocol, the operators of civil nuclear facilities are made strictly liable for damage resulting from nuclear incidents. Furthermore, there is an obligation of insurance or financial guarantee by the operator up to the fixed liability amounts in order to guarantee the availability of funds. This disposition is submitted to the approval of the Members States. What complicates things is that the States may have signed one convention but another one. This contributes to differentiate the liability ceilings of the Members States. To put in a nutshell, we can quote the 2005 report Eurotom on the harmonization of nuclear civil liability rules in Europe.

"In sum, the protection of victims of nuclear accidents, the obligations of nuclear operators, transporters, (re-)insurers and public authorities in the EU Member States are governed by a patchwork of diverse legal regimes: (i) the liability of some operators is unlimited, whereas others have a capped liability; (ii) the operators´ insurances differ both as regards their coverage and payable fees; and (iii) the obligation to compensate victims of a nuclear accident differs both as regards the damages covered and the payable amounts."

Why do States choose the ceiling of redress rather than applying some "standard" strict civil liability regime? In fact, the hugeness of repairs due to nuclear hazards coupled with a civil strict liability regime prevents private firms to exert an activity in the nuclear field. Schwartz, (2006, p.39) summarizes the point very well: "With no protection against a liability that was potentially unlimited both in time and amount, nuclear plant owners/operators, builders and suppliers were understandably hesitant to commit to the development of the industry."

Therefore, developing nuclear industry involves relieving nuclear operators of the burden of potentially ruinous liability claims12 . The governments ceiled the amount of compensation payable to victims by liable operators in the aim at avoiding judgment proofness. Under the evolution of international agreements, the caps to liability may be subject to institutional changes and, generally, the tendency is to increase the level of the ceiling. For instance, under the Brussels Convention, in France, the operator's liability is limited to €91.5 million per nuclear accident in a facility and to €22.9 million per nuclear accident during transportation.

The State in which the accident occurred will be liable for the compensation of victims up to a maximum of €228.6 million. Above this amount, the signatory's members of the Brussels Convention contribute collectively to compensation up to a ceiling of €381.1 million.

However, under the 2004 protocol, the availability of compensation amounts is much greater and covers a greater number of victims and collateral damages. Accordingly, the operator's liability is around €700 million per nuclear accident and €80 million per nuclear accident during transportation. The responsible State of a nuclear damage will be liable for amounts above the €700 million up to a maximum amount of €1,200 million. Above this amount, the States that are a party to the Brussels Convention will be contribute to a maximum amount of €1,500 million.

In the USA, the Price-Anderson Act of 1957 rules the civil liability for damages caused by a nuclear accident. Since the 1988 amendments, nuclear power plant licensees must purchase the maximum amount of commercial liability insurance available in the private market at a reasonable price. This is currently $200 million per plant. In addition, all nuclear power plant licensees must participate in a joint-insurance pool. In the case of a nuclear accident whose costs exceed the first layer of private insurance coverage, each nuclear plant is obligated to make payments of up to $88 million to cover any additional costs up to about $9.3 billion at the present time. The compensation provision of both the first and the second layers of insurance are "no fault" and are not subject to civil liability litigation. The financial cap is fixed to $9.5 billion and beyond this limit there are no further financial obligations.

The Model

We compare the cost structure dedicated to safety and civil liability of two electronuclear power structures. The first one is centralized i.e. a unique operator (public or private) manages two perfectly similar nuclear plants. The centralized structure may be split up and the two plants will be managed by two distinct operators. Each of them will own and manage a unique plant. "Similar" plant means that each one belongs to the same generation. Putting it otherwise, each one is built on the same engineering structure and has the same production capacity 13 .

Notations and assumptions

The operators

The considered economy comprises two similar electro-nuclear plants indexed by 𝑗, 𝑗 = 1,2. The index M denotes the unique operator (sometimes called also the "monopolist") and by no letter when ambiguity is impossible. When the independent operators manage them, the index will be they will be denoted by 𝑖, 𝑖 = 1,2 (operator 𝑖 manages one plant).

Assumption 1: The operators are risk neutral.

Assumption 2: Both plants are similar.

We consider now the operators' financial capacity for redress. This capacity tallies with the total available asset needed to repair the consequences of a major accident. First, let 𝑑 be the total financial capacity of the centralized structure, and, second; let 𝑑 𝑖 be the one of the operator 𝑖, 𝑖 = 1,2. 𝑑 and 𝑑 𝑖 include two parts. For the monopoly this value is 𝑑 = 𝑦 + 𝑄, where 𝑦 is the monopolist's assets dedicated to the redress and 𝑄 is the amount that insurance 13 For engineers our economic assumptions could appear quite unrealistic. For instance [START_REF] Lochbaum | Nuclear Plant Risk Studies, Failing the Grade[END_REF] notes the over-simplifications usually made to deal with this question: i) The plants are operating within technical specifications and other regulatory requirements, ii) Plant design and construction are completely adequate, iii) Plant aging does not occur; that is, equipment fails at a constant rate, iv) The reactor pressure vessels never fail, iv) Plant workers make few serious mistakes, v) Risk is limited to reactor core damage. If this set of assumptions cannot be insured for one plant, obviously it can hardly be done for a park even if plants belong to the same generation.

companies accept to cover. Symmetrically, let 𝑑 𝑖 = 𝑦 𝑖 + 𝑄 be the financial capacity of the individual operator 𝑖, 𝑖 = 1,2 , (where 𝑦 𝑖 is the wealth of 𝑖 due to repairs).

Assumption 3: The civil liability for nuclear accident is ruled by a capped strict liability regime where 𝑐 is the amount of the legal compensation (liability cap).

If 𝐷 is the total damage due to a severe harm, the institutional cap c is such that 0 ≤ c ≤ D (with the polar cases, i) c ≡ 0, which is a no-liability regime as in the Ex-USSR situation and, ii) 𝑐 ≡ 𝐷, the standard strict liability regime as in India until 2010). For most nuclear countries, 0 < 𝑐 < 𝐷. Assumption 3 means that insurance companies will not cover fully the cap and 𝑄 < 𝑐. The share 𝑄 is identical whatever the exploitation structure, this because the cap is fixed whatever the production structure. This involves that the operators' own assets must be such that 𝑦, 𝑦 𝑖 = 𝑐 -𝑄. Then, if 𝑦 < 𝑐 -𝑄 or 𝑦 𝑖 < 𝑐 -𝑄, the operator runs the risk to go judgment-proof [START_REF] Shavell | The Judgement Proof Problem[END_REF]. This situation is excluded from the field of the analysis simply because insurance companies will not insure the facility if the operators' assets are insufficient.

The states of nature

The elementary states of nature are denoted as follows. Let, the events 𝐴 and 𝐵 be respectively A:"a major accident occurs in a nuclear plant" and B: "business as usual" or still "no accident". A major accident induces the major damage 𝐷. Because the electronuclear park is constituted of two plants, the set of potential events (accident 𝐴, or no-accident 𝐵) may be described by the following sets:

-𝐻 1 = 𝐴, 𝐴 : Major accidents occurred on both plants 1 and 2, -𝐻 2 = 𝐴, 𝐵 : A major accident occurred on plant 1 but not on plant 2, -𝐻 3 = 𝐵, 𝐴 : No accident occurred on plant 1, but a major one on plant 2, -𝐻 4 = 𝐵, 𝐵 : No accident occurred on both plants 1 and 2.

Then Ω * is the set of sample:

Ω * = 𝐻 1 , 𝐻 2 , 𝐻 3 , 𝐻 4
The electro-nuclear park is controlled by a regulatory authority. This control can be strict (noted 𝑆), or sloppy ( 𝑆). A strict control is this set of necessary mitigation measures that the regulatory authority imposes to the operator to limit the occurrence of another catastrophic event after the occurrence of a major accident. The regulator has to induce the operator to take relevant safety means to avoid the duplication of an accident in the safe facility. In fact, after a major nuclear hazard, measures have to be taken to test out stations of same vintage. This matter of fact can be checked after the Fukushima accident. Indeed, everywhere in the World, national nuclear authorities are launching checking programs. Morgan Stanley (2006, p. 46) asserts that a severe accident in one reactor can "lead to the shutdown of the facility in question and, potentially, similar facilities that may be considered to present the same risks". This matter of fact can lead to slow down or interrupt the production. Consequently, the risk of other similar failure is significantly reduced to zero. This is true for both the centralized operator and the individual ones. The effective control applies for plants of the same generation in which identical defects may be suspected. Then, we can formulate the following assumption.

Assumption 4: After the occurrence of an ultra-hazardous accident, when strong control 𝑆 is exerted, the remaining safe plant is either slowed down or stopped temporarily.

We can add that the failure of a plant of a given generation means that the similar stations "do not meet the highest safety requirements and therefore pose safety risk must be identified and their safety performance must be raised to the necessary level. This task is primarily a national responsibility, but it should be facilitated through assistance measures by the international community", [START_REF] Milenin | The Chornobyl [Chernobyl] Accident and the Future of Nuclear Energy: The Path Towards Safety and Sustainability[END_REF].

The costs of controls, the slowing down or the temporarily stop down of the activity of the plant generates costs. This is expressed by the following assumption.

Assumption 5: After the occurrence of a major event, under assumption 4, the cost of stopping or slowing down the remaining plant is equal to 𝑒, (𝑐 > 𝑒 > 0).

The cost 𝑒 should not be interpreted as a cost for care but rather as an opportunity cost such as the cost of increasing the load factor in the safe plant and checking the management of safety.

The probability distribution of a major accident

Here, identical plants allow us to consider that they share the same probability distribution of major failure. Each state of nature occurs with a given probability 14 . Hence, a major accident in a plant is reputed to occur with a probability 𝑝 𝐴 ∈ (0, 1) or still 𝑝. Let the probability of the event 𝐵 ("business as usual or no accident"), be 𝑝 𝐵 = 1 -𝑝 𝐴 = 1 -𝑝. In this industry the nature of the control, weak or sloppy, is common knowledge. 14 It is difficult to assign a probability to what is considered as a rare event. As pointed by [START_REF] Schneider | Residual Risk, An Account of Events in Nuclear Power Plants Since the Chernobyl Accident in 1986[END_REF] external and internal event may influence the probability distribution (external events are external flooding (Central of Blayes, France, 1999), Tsunami (Indian Ocean, 2004), external fires (Los Alamos 1996, 2000), tornado and Hurricanes (David Besse USA, 1998, Hurricane Andrex). Considering internal events, have to be understood the losses of coolant (Kozloduy, Bulgaria 2003), numerous turbine fires, and secondary cooling circuits. These events combine with human errors and violation of procedures. Hence, as such, the probability of the melting down of the core takes sense when considering the whole set of potential failure of the security system.

Hence, there is no uncertainty from the operators' side about the type of control15 . Both situations are clearly identified.

The costs structures: infra-comparison between strong and weak control

The nature of control as defined in 2.1.2 impacts on the expected costs of the industry:

-After a major accident, the level of the cap has to be paid deducing the amount of the insurance coverage 𝑄: 𝑐 -𝑄. Let 𝜇 the insurance premium 𝜇 this one is equal to 𝑝𝑄 = 𝜇, (Shavell (1985)).

-When a strong control is exerted then, in the case of accident, the safe plant has to stop or slow down its activity for checking and this incurs some costs in compliance with assumption 5: Now, we dispose of all elements to define the cost structure of an individual operator.

To make easier the exposition we begin by describing the cost structure of the facility owned by a unique operator.

The cost structure of an individual operator

We will examine successively both cases: the weak and the strong control.

i)

Weak control

Under a weak control, the regulatory authority does not induce the operator to reduce or to stop the activity level for some deep check after an accident and assumption 5 does not apply. As a consequence, the operator 𝑖, 𝑖 = 1,2 is not affected by the accident suffered by the other plant. He has only to pay the insurance premium 𝜇 whatever the state of nature. The entire situation may be described by the following table :  Cost structure of a strictly controlled operator

States of nature Cost

Probabilities

𝐻 1 = 𝐴, 𝐴 𝑐 + 𝜇 -𝑄 𝑝 2 𝐻 2 = 𝐴, 𝐵 𝑐 + 𝜇 -𝑄 𝑝(1 -𝑝) 𝐻 3 = 𝐵, 𝐴 𝜇 𝑝(1 -𝑝) 𝐻 4 = 𝐵, 𝐵 𝜇 (1 -𝑝) 2 Sum 1 Table 1
Hence, this situation is equivalent to be faced to only two states of nature because the plants are independent one from the other one and the sample set reduces to:

Ω i = A, B , 𝑖, 𝑖 = 1,2
Thus, for the event B, the costs incurred includes only the cost of the insurance premium, 𝜇 plus the one of the disastrous event, A, (𝜇 + 𝑐 -𝑄 ). The expected costs 𝐸𝐶 𝑖 (𝑆) , 𝑖 = 1,2 , for the operator are then:

𝐸𝐶 𝑖 𝑆 = [𝑝 2 + 𝑝 1 -𝑝 ] 𝑐 + 𝜇 -𝑄 + 𝜇[( 1 -𝑝 2 + 𝑝 1 -𝑝 ] = 𝑝 𝜇 + 𝑐 -𝑄 + (1 -𝑝)𝜇, [1]
And, after the simplification (where 𝑝𝑄 = 𝜇):

𝐸 𝑖 𝐶 𝑆 = 𝑝𝑐 [2]
Consequently, under a weak control (𝑆), the expected cost of strict capped liability is equal to the expected cost of the accident proportionally to the cap 𝑐.

ii) Strong control

As mentioned above, after an accident on one plant, the remaining safe plant will be submitted to some strong costly control and/or breaks. These operations will incur a cost 𝑒 (following assumptions 4 and 5). Under a strong control, the state 𝐻 1 = (𝐴, 𝐴) becomes unrealistic and a null probability is associated to it. This explains by the fact that the probability of a major accident is very weak, then, after the checking of the other plant, the probability of a new catastrophic event on it is still weaker. The states 𝐻 2 = 𝐴, 𝐵 and 𝐻 3 = 𝐵, 𝐴 which are not symmetric at the individual level deserves some attention.

Consequently, if the operator undergoes the accident (event 𝐻 2 ), he will have to pay 𝑐 -𝑄 for repairs, while if his plant is safe, he will incur the opportunity cost 𝑒 only. The entire situation may be described by the following Sum 1 Table 2 The expected costs for the operator, 𝑖 , 𝑖 = 1,2 , 𝐸𝐶 𝑖 (𝑆) are then:

𝐸𝐶 𝑖 𝑆 = 0 𝑐 + 𝜇 + 𝑝 𝑐 + 𝜇 + 𝑝 1 -𝑝 (𝜇 + e)+𝜇(1 -𝑝) 2

[3] and, as previously by simplifying with 𝑝𝑄 = 𝜇, we get:

𝐸𝐶 𝑖 𝑆 = 𝑝 𝑐 + 𝑒 -𝑒𝑝 2 [4]
We can check that the expected costs of a strictly controlled structure are higher or equal to a weak one, indeed:

𝐸𝐶 𝑖 𝑆 ≥ 𝐸 𝑖 𝐶 𝑆 ⇒ 𝑝 𝑐 + 𝑒 -𝑒𝑝 2 ≥ 𝑝𝑐, [5]
This is always true because after developing:

𝑒 1 -𝑝 ≥ 0.

[6]

We deduce then the following proposition:

Proposition 1: Under the assumptions 1 to 5, a unique plant managed by a single operator incurs higher costs when it is strongly controlled rather than when the control is weak.

(Proof simple arithmetic and discussion above).

We can define now the cost structure for the monopolistic situation and compare then both industrial structures.

The centralized structure

The distribution of probability corresponding to each kind of control may be summarized in the following tables considering that the insurance premium is 𝜇. As before, we will examine both cases: the weak and the strong control.

i) Weak control

Under a weak control the centralized organization can go on producing energy after a major accident by using his safe plant. Hence, as previously, the plants are fully independent one relative to the other one. We can draw then the following table:

Cost structure of a weakly controlled monopolistic operator

States of nature Cost

Probabilities

𝐻 1 = 𝐴, 𝐴 2𝑐 + 𝜇 -2𝑄 𝑝 2 𝐻 2 = 𝐴, 𝐵 𝑐 + 𝜇 -𝑄 𝑝(1 -𝑝) 𝐻 3 = 𝐵, 𝐴 𝑐 + 𝜇 -𝑄 𝑝(1 -𝑝) 𝐻 4 = 𝐵, 𝐵 𝜇 (1 -𝑝) 2
Sum 1 Table 3 The expected costs for the monopolistic operator under a weak control are then:

𝐸𝐶 𝑆 = (2𝑐 + 𝜇 -2𝑄)𝑝 2 + 2𝑝(1 -𝑝)(𝑐 + 𝜇 -𝑄) + 𝜇(1 -𝑝) 2 𝐸𝐶 𝑆 = 2𝑝(𝑐 -𝑄) + 𝜇 [7]
The expected cost of liability is equal to the expected amount that the operator has to supply for repairs. This corresponds to the level of the cap minus the amount reimbursed by the insurance company plus the premium. ii)

Strong control

Under the strong control, assumption 5 applies and the monopolistic operator has to slow down or stop the production of the safe plant. He will incur the opportunity cost 𝑒 but equally the cost of due repairs. Hence, the table of cost is reproduced in the following:

Cost structure of a strongly controlled monopolistic operator

States of nature Cost Explicit probabilities

𝐻 1 = 𝐴, 𝐴 2𝑐 + 𝜇 -2𝑄 0 𝐻 2 = 𝐴, 𝐵 𝑐 + e + 𝜇 -𝑄 𝑝 𝐻 3 = 𝐵, 𝐴 𝑐 + e + 𝜇 -𝑄 𝑝(1 -𝑝) 𝐻 4 = 𝐵, 𝐵 𝜇 (1 -𝑝) 2
Sum 1 Table 4 The event 𝐻 2 means that once an accident has occurred on one plant, the operator has to stop or slow down immediately the activity of the other facility. Then, the payoff is 𝑐 + e + 𝜇 -𝑄, which occurs with a probability 𝑝. However, considering that a specific plant 𝑗, 𝑗 = 1,2 works "well" with a probability 1 -𝑝 does not prevent, that the other one can fail with a probability 𝑝. This explains the state of nature 𝐻 3 . This may be seen in the following decision tree.

[Insert Figure 1]

A priori we do not know what plant among plant 1 and 2, will fail. Hence, because they are similar, we consider that Nature (N in the decision tree) chooses with an equal probability 𝜋 = 1/2 what plant will suffer an accident. Rigorously, the expectation cost should write:

𝐸𝐶 𝑆 = 𝜋 2𝑐 + 𝜇 -2𝑄 0 + 𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝑝 1 -𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝜇 1 -𝑝 2 +(1 -𝜋) 2𝑐 + 𝜇 -2𝑄 0 + 𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝑝 1 -𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝜇 1 -𝑝 2 [8]
This writing is justified for plants presenting different risk level. This could be useful for the analysis of different generations for instance where the older generations may be considered as riskier than the younger ones. Here, this is not the case under assumption 1

(𝜋 = 1 2
. Then, the expectation costs function writes:

𝐸𝐶 𝑆 = 2𝑐 + 𝜇 -2𝑄 0 + 𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝑝 1 -𝑝 𝑐 + 𝑒 + 𝜇 -𝑄 + 𝜇 1 -𝑝 2 = 𝐸𝐶 𝑆 = 2𝑝(𝑐 + 𝑒 -𝑄) -𝑝 2 (𝑐 + 𝑒 -𝑄) + 𝜇 [9]
If we compare 𝐸𝐶 𝑆 and 𝐸𝐶 𝑆 we can see that the advantage of a "weak-control" practice under a monopolistic situation gives not as clear results as in the previous case. Thus, we study conditions for having:

𝐸𝐶 𝑆 > 𝐸𝐶 𝑆 or 𝐸𝐶 𝑆 ≤ 𝐸𝐶 𝑆 .

Considering 𝐸𝐶 𝑆 > 𝐸𝐶 𝑆 , this means that: (With 𝑐 + 𝑒 -𝑄 > 0).

Then, 𝐸𝐶 𝑆 > 𝐸𝐶 𝑆 , this is true if 𝑐 -𝑄 > 𝑒. We can establish proposition 2:

Proposition 2: Under assumption 1 to 5, the level of the opportunity costs is essential to determine whether a strict controlled centralized organization generates higher expected costs than a decentralized structure. Consequently:

i) If 𝑝 > 2𝑒 𝑐+𝑒-𝑄
, then a weak control incurs lesser costs than a centralized organization.

ii)

If, in the opposite, 𝑝 ≤ 2𝑒 𝑐+𝑒-𝑄
, the centralized organization involves smaller costs.

The proof results from the above argument.

Decentralized vs centralized organization

This section studies the efficiency of both types of organization. This consists in comparing each costs structure according the nature of the control. This analysis borrows some feature to the instruments of the contestable market theory in [START_REF] Baumol | Contestable Markets and the Theory of Industry Structure[END_REF] because we refer to the sub-additivity of costs. However, our approach is different because we refer here to expected costs and we do not examine the sustainability question, i.e. the possibility for potential entrants to compete on prices. The question we deal with is to know whether it is more costly organizing monopolistically ultra-hazardous activities, or conversely, decentralizing it with independent operators. Here, the cost structure corresponds to the cost of civil liability.

The case of weak control

It is important to note that the insurance premium plays a fundamental role. Indeed, in what follows we shall study the conditions for which the expected cost of the decentralized structure is either higher or lesser than the monopolist one, i.e:

𝐸 𝑖 𝐶 𝑆 2 𝑖=1
≥ 𝐸𝐶 𝑆 or the reverse.

𝐸 𝑖 𝐶 𝑆 2 𝑖=1
< 𝐸𝐶 𝑆 . It is obvious that the value at which the insurer fixes the level of the premium is determinant for having either the cost of monopolistic organization higher than the decentralized one or the reverse.

Let us assume that:

𝐸 𝑖 𝐶 𝑆 2 𝑖=1 < 𝐸 𝑆 .
In what follows we consider that the level of the insurance premium of the decentralized organization is given and is equal to 𝜇 = 𝑝𝑄.

From the previous definition we deduce that :

𝐸 𝑖 𝐶 𝑆 2 𝑖=1
= 2𝑝𝑐 and consequently:

𝐸 𝑖 𝐶 𝑆 2 𝑖=1 < 𝐸𝐶 𝑆 ⇒ 2𝑝𝑐 < 2𝑝(𝑐 -𝑄) + 𝜇 or, 𝜇 > 2𝑝𝑄 = 2𝜇
Then, the condition for the monopolist operator to meet a higher expected cost of accident than under a decentralized structure is that his premium insurance should be more costly than for the individual operators. Otherwise, if 𝜇 = 2𝜇, then, both organizations, monopoly or the decentralization one are equivalent.

Proposition 3: Under a weak control, under assumptions 1 to 5, if insurance premiums are defined proportionally for each plant (𝑝𝑄 = 𝜇 for each individual operator and 𝜇 = 2𝑝𝑄, for the monopolist), then the monopolistic and the decentralized organizations generates the same expected costs:

𝐸 𝑖 𝐶 𝑆 2 𝑖=1 = 𝐸𝐶 𝑆 [11]
Proof comes from the previous argument.

We have to note that this result is similar to the case in which insurance is not compulsory and may be set up at 𝜇 = 𝜇 = 0. Then, it is equivalent to exploit a ultra-hazardous risky park of plants under a monopolistic organization (unique operator) or under a decentralized organization (two independent operators).

The case of strong control

We analyze now the cost structure of the decentralized structure highly controlled. As before, we consider:

𝐸𝐶 𝑖 𝑆 = 2 𝑖=1 2(𝑝 𝑐 + 𝑒 -𝑒𝑝 2 ) [12]
And we compare it with [9] and we determine the conditions for having:

i) Either 𝐸𝐶 𝑖 𝑆 ≥ 𝐸𝐶 𝑆 2 𝑖=1 , or ii) 𝐸𝐶 𝑖 𝑆 < 2 𝑖=1
𝐸𝐶 𝑆 .

Then, let us analyze the relationship

𝐸𝐶 𝑖 𝑆 < 2 𝑖=1
𝐸𝐶 𝑆 , this is equivalent to:

2(𝑝 𝑐 + 𝑒 -𝑒𝑝 2 ) < 2𝑝(𝑐 + 𝑒 -𝑄) -𝑝 2 (𝑐 + 𝑒 -𝑄) + 𝜇
After developing, this relationship is true if there is 𝜇 > 0, such that 16 :

𝜇 > 2𝜇 + 𝑝 2 𝑐 + 𝑒 -𝑄
Consequently, if the monopoly insurance premium is higher than the sum of the premium of the individual operators' plus 𝑝 2 𝑐 + 𝑒 -𝑄 (which is positive), then, it is only if the monopoly gets adverse insurance premium that the relationship:

𝐸𝐶 𝑖 𝑆 < 2 𝑖=1
𝐸𝐶 𝑆 , is verified.

Putting it otherwise, it is only if the insurance premiums are detrimental for the monopoly compared to the decentralized facilities that the centralized organization incurs higher costs. This is sufficient to establish proposition 4:

Proposition 4: Under strong control and under assumptions 1 to 5, if insurance premiums are defined proportionally for each plant (i.e. 𝑝𝑄 = 𝜇 for each individual operator and 𝜇 = 2𝑝𝑄, for the monopolist), then the monopolistic organization incurs lesser costs than the decentralized organization:

𝐸𝐶 𝑖 𝑆 < 2 𝑖=1 𝐸𝐶 𝑆 [13]
With a monopolist insurance premium proportionally less than the one of the individual operators (i.e. 𝜇 ≤ 2𝜇), then the advantage of the centralized situation is obvious, but this is an "unnatural" position and this result is not in the spirit of this model. 16 Remember that 2𝑝𝑄 = 2𝜇

Insurance and effort

The above results are reached under the assumption that a major accident occurs under a once for all given probability. However, by increasing or decreasing the effort dedicated to improving safety, the operators may influence the level of risk and the operators may lower the accident probability by taking due care.

To deal with this point we adopt the Shavell (1986)'s presentation. Let 𝑥 be the operator's level of safety effort, 𝑥 ≥ 0. The probability of an accident modifies as 𝑥 varies, 1 > 𝑝 𝑥 ≥ 0, 𝑝′(𝑥) < 0 and 𝑝" 𝑥 > 0. In the following, the analysis is restricted to the more realistic situation of a strong control.

The counterpart to the purchase of liability insurance is that the insurance company demands the highest level of care to the operators. Because of the specificity of the electronuclear production, it is not unrealistic to assume that asymmetric information between the operator and the insurance companies are almost impossible. Indeed, at all moment the level of safety can checked by the insurance company. Sum 1

Table 6

We assume that the monopolist spends the same amount 𝑥 for care in for the whole park.

From each one we draw the expected cost of civil liability. Consequently, for the "decentralized" operator: We can check that these functions are convex. They decrease for low values of 𝑥 and increase as 𝑥 grows because 1 ≥ 𝑝 𝑥 ≥ 𝑝(𝑥) 2 ≥ 0 and for high values of 𝑥, 𝑝 𝑥 and 𝑝(𝑥) 2 tend to 0. This element is important to determine what structure will give the highest level of safety. This can be done directly by considering that the level of the premium is identical for both organizations i.e. 𝜇 = 2𝜇 . To compare fairly between these structures, we assume that, under the decentralized structure, both plants are gathered but keep their management autonomy.

𝐸𝐶
We study then two conjectures. Concerning the first one, the insurance premium is given and the operators have to adapt 𝑥 to it. Concerning the second one, while in the second one, the insurance policies is adapted to the care level supplied by operators, i.e; 𝜇(𝑥) = 𝑝(𝑥) 𝑄 𝑛, (for 𝑛 = 1,2).

The insurance premium is given

We can set up the following proposition:

Proposition 5: Under assumptions 1 to 5, when civil liability is ruled by a capped liability regime, a monopolistic organization induces a higher level of care than a decentralized one in so far that the value of the cap is such that 𝑐 -𝑒 -𝑄 > 0. If 𝑥 𝑓 * and 𝑥 𝜑 * , represent respectively the optimum effort level of the decentralized and the centralized organization, then 𝑥 𝑓 * < 𝑥 𝜑 * .

Proof:

We replace the expected costs functions by 𝑓 and 𝜑 such that:

𝑓 𝑥 = 𝐸𝐶 𝑖 𝑥 = 𝑥 + 𝜇 + 𝑝(𝑥)(𝑐 + 𝑒 -𝑄) -𝑒𝑝 𝑥 2 ), 𝑖 = 1,2.

For the decentralized structure, and:

𝜑 𝑥 = 𝐸𝐶 𝑥 = 𝑥 + 2𝜇 -(𝑐 -𝑄)(-2 + 𝑝 𝑥 )𝑝 𝑥 for the monopolist organization.

We can check that 𝜑 𝑥 and 𝑓 𝑥 are related each other:

𝜑 𝑥 = 𝑓 𝑥 + 𝜇 -𝑐 -𝑒 -𝑄 -1 + 𝑝 𝑥 𝑝(𝑥)

Then, considering that if 𝑥 𝑓 * is that level of effort such that cancels 𝑓′ 𝑥 𝑓 * ≤ 0, with equality for 𝑥 𝑓 * > 0, then:

𝜑 ′ 𝑥 𝑓 * = 0 + 𝑐 -𝑒 -𝑄 1 -2𝑝 𝑥 𝑓 Remark :
The insurance premium question: The previous results have been reached

without specific assumptions on the level of the insurance premium. More precisely, it has been assumed that the level of insurance is identical whatever the competitive structure. This subject is becoming highly sensitive between authors who think that the level of insurance premium is considerably too high, taking account of the level of effective coverage (Faure and Fiore (2008)) and authors who consider that this premium is too low because of an insufficient level of exposure from the insurance company side (Caroll, and Froggart, (2007)).

Implicitly, we have admitted that the premium is equal to 𝜇 = 𝑝 𝑄 𝑛, where 𝑛 is the number of station as in [START_REF] Faure | An Economic Analysis Of The Nuclear Liability Subsidy School of Law[END_REF]. However, we are not led to bring some judgment about the actual level of insurance premium of real companies. Indeed, our analysis considers only that under similar and proportional insurance premium the cost of safety is lesser under a centralized system than under a decentralized one. Our results are depending on the relationships between the amount of 𝑐 , 𝑒 and 𝑄. Two of these values are depending on decisions of the Authorities, the amount of the ceiling 𝑐 and on the combined choices of the insurance market and the insurance corporations i.e. the level of 𝑄. Hence, if the lag between 𝑐 and 𝑄 is sufficiently thin such that 𝑐 -𝑒 -𝑄 < 0 then 𝜑 ′ 𝑥 𝑓 * > 0 and with the same argument than above, 𝑥 𝜑 < 𝑥 𝑓 , i.e. the decentralized structure insures a better safety. This point is important. Indeed, it means either that the insurance companies accept to increase their share in compensation or that the level of the ceiling is diminishing. We recall that this level may be such that 0 ≤ 𝑐 ≤ 𝐷. Hence, the lesser the level of the cap, the larger is the road for a decentralized organization. However, as seen in the first paragraph (see also Caroll, and Froggart, (2007)), the landslide tendency is to increase the level of caps.

The question of the negative value of (𝑐 -𝑄) -𝑒, however, has to be considered seriously. Indeed, it corresponds to the increased resort to private insurance due to the pooling of insurers as Pelzer (2007, p.55) shows it by underlining the US and the German experiences.

We can quote it: "The operators' pooling systems established in Germany and in the US prove their capacity to deploy many times the amounts required under the revised nuclear liability conventions and in particular the amounts offered by the insurance industry. The money will be provided to cover the legal liability of the operator liable without excluding certain risks from coverage".

Consequently, when the lag between 𝑐 and 𝑄 diminishes (for 𝑒 given), then the decentralized structure may become more efficient than the centralized one.

Conclusion and Policy recommendations

The public authorities aim at insuring the best safety coverage of populations against the risks of the electro-nuclear industry. The present paper has been limited at assessing the cost of safety only, letting aside the working costs of this industry. We show that a natural monopoly organization generates higher costs of care level than a decentralized management.

This involves inducing dispersed operators to cluster them together. It is then the interest of the new structure to minimize its expected global safety and damage costs. This is done by increasing the level of safety expenses (proposition 6). As a corollary, the condition for increased safety is a high level of control from the regulatory authority side. Hence, the more stringent are becoming the Regulatory Authority, the more they favor scale economies (proposition 4). The above results are independent of a privatization concern because in the nuclear sector, many ways can be invoked to privatize the production park. For instance, the stations can be privatized as separate entities, which may not be a viable competitive scheme.

The safer scheme is then to consider that privatization may be achieved through the transfer of shares of a unique portfolio to different partners. Hence, this analysis has specifically focused on the question of the advantages of splitting the exploitation of a nuclear park independently of the question of privatization of assets.

If it is impossible to maintain a centralized structure, then a decentralized organization should be accompanied by deep changes in the insurance infrastructure. Indeed, this one should favor an increase of the involvement of insurances and re-insurance companies. This is shown in the final remark of section 3. This result joins the contemporaneous trend that tends to extend the cover by insurance companies in this ultra-hazardous sector.

These proposals could bring some confusion without more explanation. Indeed, the previous results do not mean that considering concrete present situations, the safety level of decentralized structures shows inefficiencies in so far that they comply perfectly with the actual norms of the regulatory authorities. Precisely, our argument means that if, for instance, safety standards have to increase, then as a consequence, they require more care effort (this lead to diminish theoretically the probability of an accident) whereas safety costs are near the equilibrium level 𝑥 𝑓 * . Then the centralized monopolistic organization generates lower equilibrium costs compared to a decentralized one.

  conventions 10 . The 1988 Joint Protocol linked together the IAEA's Vienna Convention on Civil Liability for Nuclear Damage of 1963 and the OECD's Paris Convention on Third Party Liability 1960. Protocols amending the Paris Convention and the Brussels Convention were signed on February 12, 2004.

  2𝑝 𝑐 -𝑄 + 𝜇 > 2𝑝(𝑐 + 𝑒 -𝑄) -𝑝 2 (𝑐 + 𝑒 -𝑄) + 𝜇 Or, still after simplifying:

  𝑖 𝑥 = 𝑝 𝑥 (𝑐 + 𝑥 + 𝜇 -𝑄) + 𝑝 𝑥 (1 -𝑝 𝑥 )(𝜇 + 𝑒 + 𝑥) + (𝜇 + 𝑥)(1 -𝑝 𝑥 ) 2 = 𝐸𝐶 𝑖 𝑥 = 𝑥 + 𝜇 + 𝑝(𝑥)(𝑐 + 𝑒 -𝑄) -𝑒𝑝(𝑥) 2 [14] And for the monopolist operator: 𝐸𝐶 𝑥 = 𝑝 𝑥 𝑐 + 𝑥 + 𝑒 + 𝜇 -𝑄 + 𝑝 𝑥 1 -𝑝 𝑥 𝑐 + 𝑥 + 𝑒 + 𝜇 -𝑄 + 𝜇 + 𝑥 1 -𝑝 𝑥 2 = 𝐸𝐶 𝑥 = 𝑥 + 𝜇 -(𝑐 -𝑄)(-2 + 𝑝 𝑥 )𝑝 𝑥 [15]

table 2 :

 2 Cost

structure of a strictly controlled operator

	States of nature	Cost	Probabilities
	𝐻 1 = 𝐴, 𝐴	𝑐 + 𝜇 -𝑄	0
	𝐻 2 = 𝐴, 𝐵	𝑐 + 𝜇 -𝑄	𝑝
	𝐻 3 = 𝐵, 𝐴	𝜇 + e	𝑝(1 -𝑝)
	𝐻 4 = 𝐵, 𝐵	𝜇	(1 -𝑝) 2

Cost structure, strict control, one facility and one operator

  The expected costs functions are modified, considering that the operators are risk neutral we represent the cost structure of a facility owned by a single operator by the following table:And, the second one, the monopolistic cost structure:

	States of nature	Cost	Explicit probabilities
	𝐻 1 = 𝐴, 𝐴	𝑐 + 𝜇 + 𝑥 -𝑄	0
	𝐻 2 = 𝐴, 𝐵	𝑐 + 𝜇 + 𝑥 -𝑄	𝑝(𝑥)
	𝐻 3 = 𝐵, 𝐴	𝜇 + e + x	𝑝(𝑥)(1 -𝑝(𝑥))
	𝐻 4 = 𝐵, 𝐵	𝜇 + 𝑥	(1 -𝑝(𝑥)) 2
	Sum		1
	Table 5

Cost structure of a strongly controlled monopolistic operator

	States of nature	Cost	Explicit probabilities
	𝐻 1 = 𝐴, 𝐴	2𝑐 + 𝑥 + 𝜇 -2𝑄	0
	𝐻 2 = 𝐴, 𝐵	𝑐 + 𝑥 + 𝑒 + 𝜇 -𝑄	𝑝(𝑥)
	𝐻 3 = 𝐵, 𝐴	𝑐 + 𝑒 + 𝑥 + 𝜇 -𝑄	𝑝(𝑥)(1 -𝑝(𝑥))
	𝐻 4 = 𝐵, 𝐵	𝜇 + 𝑥	(1 -𝑝(𝑥)) 2

"It is sometimes argued that for, so-called" Damocles risks", i.e. risks with a very high damage and a low probability, the risk assessment of the public is not proportional to the risk. The occurrence of a very high damage should be avoided, even if the costs for the avoidance are much higher than the expectation value of the damage. However past attempts to quantify this effect have not been successful or accepted, so there is currently no accepted method on how to include risk aversion in such an analysis. Consequently it is currently not taken into account within the ExternE methodology. Research on how to assess this, for example with participatory approaches, is clearly needed.",Bickel and Rainer (2005), p. 226

See for instance protests against the building of a new nuclear power station in Armenia at Metzamor who sits in a seismic zone that has suffered one of the worst earthquakes in modern history. http://www.nuclearpowerdaily.com/reports/Environmentalists_decry_risks_of_new_Armenia_nuclear_reactor_9 99.html.

Source Nuclear Energy Institute. http://www.nei.org/resourcesandstats/documentlibrary/reliableandaffordableenergy/graphicsandcharts/usnuclear plantsales/

In October 1, 2010, in Europe, Russia and Ukraine included, there is a total of 195 nuclear power plant units with an installed electric net capacity of 170 GWe in operation in Europe 19 units with 16,9 GWe were under construction in six countries. (source European Nuclear Society). http://www.euronuclear.org/info/encyclopedia/n/nuclear-power-plant-europe.htm

See also the synthesis achieved by[START_REF] Carroll | Nuclear Third Party Insurance The Nuclear Sector's "Silent" Subsidy State of Play and Opportunities in Europe[END_REF].

See for instance,[START_REF] Faure | An Economic Analysis Of The Nuclear Liability Subsidy School of Law[END_REF],

Under the Joint protocol of 1988, the operator's liability is absolute, i.e. the operator is held liable irrespective of fault, except for "acts of armed conflict, hostilities, civil war or insurrection", terrorism acts do not enter in the exclusion category.

More explicit still is The "Exposé des Motifs" for the 1960 Paris Convention that considers that "unlimited liability could easily lead to the ruin of the operator without affording any substantial contribution to compensation for the damage caused" (Exposé des Motifs, Motif 45).

The question of the probability of a major accident will be considered on a larger scale than the usual question of the melting of the core of a reactor. Indeed, it is the accumulated fission products in a reactor that forms the potential of radiation hazard. The melt down of the core of the reactor induces a severe accident. Safety is to prevent the release of these radioactive products and fuel isotopes. Accidents that issue on massive rejection of such material and threat populations and natural resources are particularly rare events. Indeed, in Western plants, an airtight containment reinforced concrete building (1.2 to 2.4 meters) tends to limit the effect of a melting of the nuclear core. Probabilistic methods (Probabilistic Risk Assessment) are used since the midst seventies[START_REF] Murray | Nuclear Energy[END_REF]. The object is to determine the probability of occurrence of an undesired event such "as core damage, breach of containment, or release of radioactivity, and to determine potential causes" Murray, (2000 p.277). Considering internal relationships, a catastrophic event does not occur suddenly. It supposes a succession of failure and events trees show the probabilistic path from a current incident to the disastrous event. That justifies the use of Bayesian approaches[START_REF] Chen | estimation of Initiating Event Distribution at Nuclear Power Plants by Bayesian Procedures[END_REF]). Studies in the midst of the nineties show that the probability of the melting of the core in Europe of the nuclear plant is quite variable and depends on the generation of the power plant. The ExternE study of 1995 considered a core meltdown probability of 10 -5 . However, the probability of such an event depends on the nature of reactors that have evolved throughout time following technical progress.[START_REF] Lembrechts | Technical Report on Nuclear Accidents and other Major Accidents in Europe: an integrated economic and environmental assessment[END_REF] show that the range of probability of core meltdown is comprised between 10 -3 and 10 -6 according the reactor generation. For instance, they report the studies concerning a study on two French reactors, a 900 MW Pressurized Water Reactor (PWR) and a 1300 MW PWR, indicated the following risk probability of a major core meltdown(World Bank, 1991) respectively 4.95.10 -5 and 1,05.10 -5 .


G Latu

V Grandgirard

J Abiteboul

M Bergot

N Crouseilles

X Garbet

P Ghendrih

M Mehrenberger

Y Sarazin

H Sellama

E Sonnendrücker

D Zarzoso

Accuracy of unperturbed motion of particles in a gyrokinetic semi-Lagrangian code

Keywords: gyrokinetic, conservation laws, FSL

Inaccurate description of the equilibrium can yield to spurious effects in gyrokinetic turbulence simulations. Also, the Vlasov solver and time integration schemes impact the conservation of physical quantities, especially in long-term simulations. Equilibrium and Vlasov solver have to be tuned in order to preserve constant states (equilibrium) and to provide good conservation property along time (mass to begin with). Several illustrative simple test cases are given to show typical spurious effects that one can observes for poor settings. We explain why Forward Semi-Lagrangian scheme bring us some benefits. Some toroidal and cylindrical GYSELA runs are shown that use FSL.

Précision du mouvement de particules non perturbées dans un code gyrocinétique

Résumé : Une description imparfaite de l'équilibre peut conduire à des artéfacts numériques dans des simulations turbulentes gyrocinétiques. Aussi, le solveur de Vlasov and le schéma d'intégration en temps ont un impact fort sur la conservation de quantités physiques, notamment lorsque les simulations sont en temps long. L'équilibre et le solveur Vlasov doivent être finement choisis et paramétrés pour conserver les états constants (équilibre) et pour autoriser de bonnes propriétés de conservation en temps (en commençant par la conservation de la masse). Plusieurs cas tests illustratifs sont donnés pour montrer les problèmes numériques typiques que l'on peut observer si les choix pris ne sont pas adéquats. Nous expliquons pourquoi le schéma FSL (Forward Semi-Lagrangian) apporte une réponse à certains problèmes. Des simulations en configuration cylindrique et torique de GYSELA sont présentées qui utilisent FSL.

Mots-clés : gyrocinétique, lois de conservation, FSL

1 Introduction

Inaccurate description of the gyrokinetic equilibrium can yield unphysical excitation of zonal flow oscillations[ABH + 06]. Moreover, as stated in [IIK + 08], it is important to define the initial condition using a relevant gyrokinetic equilibrium, especially in collisionless full-f simulations.

In the following, accuracy aspects are investigated for both the gyrokinetic equilibrium and the Vlasov solver used in the GYSELA code. If proper care is not taken for Vlasov solver and gyrokinetic initial equilibrium, one can observe that some conservation properties are not satisfied, for example mass conservation.

The gyrokinetic framework of the study is explained in the next section. Then, quite experimental investigations are shown to understand numerics associated with equilibrium and mass conservation. Forward Semi-Lagrangian (FSL) scheme is presented briefly. The splitting in a linear and a non-linear part of the Vlasov equation is explained. Results of numerical simulation for 4D cylindrical test cases and toroidal test cases are presented in the last section. Forward Semi-Lagrangian scheme helps a lot to preserve good accuracy on mass and total energy.

Description of the context

Gyrokinetic Vlasov equation

The time evolution of the gyro-center coordinates ( x, v , µ) of species s is given by the collisionless electrostatic gyrokinetic equations:

dx i dt = v b * s • ∇x i + v Es GC • ∇x i + v Ds • ∇x i (1)
m s dv dt = -µ b * s • ∇B -e s b * s • ∇ φ + m s v B v Es GC • ∇B (2)
where x i corresponds to the i-th covariant coordinate of x, B is the magnetic field and B * s and b * s are defined as:

B * s = B + m s v e s B µ 0 b • J (3) b * s = B B * s + m s v e s B * s µ 0 J B (4)
The advection terms are:

b * s • ∇x i = b * i s = B • ∇x i B * s + m s v e s B * s µ 0 J • ∇x i B (5) v Ds • ∇x i = v i Ds = m s v 2 + µB e s B * s B [B, x i ] (6) v Es GC • ∇x i = v i Es GC = 1 B * s [ φ, x i ] (7) v Es GC • ∇B = - 1 B * s [B, φ] (8)
the Poisson bracket is defined by

[F,G] = b•( ∇F × ∇G).
The velocity parallel to the magnetic field is v . The magnetic moment µ = m s v 2 ⊥ /(2B) is an adiabatic invariant with v ⊥ the velocity in the plane orthogonal to the magnetic field. Vacuum permittivity is denoted µ 0 . The term v E GC represents the electric E × B drift velocity of the guiding-centers and v D the curvature drift velocity. The Jacobian in phase space is J s (r, θ)B * ||s (r, θ, v ). Some references concerning the framework we use to solve this equation can be found in [GBB + 06, GSG + 08, BCG + 11, SGA + 11, AGG + 11].

Quasineutrality equation

The quasineutrality equation and parallel Ampère's law close the self-consistent gyrokinetic Vlasov-Maxwell system. However, in an electrostatic code, the field solver reduces to the numerical solving of a Poisson-like equation [START_REF] Hahm | Nonlinear gyrokinetic equations for tokamak microturbulence[END_REF]. In tokamak configurations, the plasma Latu & al quasineutrality (denoted QN) approximation is currently assumed [GBB + 06]. Electron inertia is ignored, which means that an adiabatic response of electrons is supposed. We define the operator ∇ ⊥ = (∂ r , 1 r ∂ θ ). We note n 0 the equilibrium density, B 0 the magnetic field at the magnetic axis and T e (r) the electronic temperature. We have also B(r, θ) the magnetic field, J 0 the Bessel function of first order and k ⊥ the transverse component of the wave vector. Hence, the QN equation can be written in dimensionless variables

- 1 n 0 (r) ∇ ⊥ . n 0 (r) B 0 ∇ ⊥ Φ(r, θ, ϕ) + 1 T e (r) [Φ(r, θ, ϕ) -Φ F S (r)] = ρ(r, θ, ϕ) (9)
where ρ is defined by

ρ(r, θ, ϕ) = 2π n 0 (r) B * ||s dµ dv J 0 (k ⊥ 2µ)( f -feq )(r, θ, ϕ, v , µ). ( 10
)
with feq representing local ion Maxwellian equilibrium, and . F S (r) the average on the flux surface labelled by r.

3 Unperturbed motion of particles

Simplification of the equations

Let suppose that a collisionless equilibrium distribution function f eq is taken for the initial distribution function: f t=0 (r, θ, ϕ, v ). By definition, if the Vlasov and Quasi-neutrality equations are accurately solved, the distribution function f t should remain constant over time, i.e. equal to f eq . Also, the electric potential Φ(r, θ, ϕ) should remain equal to zero because the right hand side of the Quasi-neutrality equation, the integral of f -f eq , is zero. Then, one quantitative measure of Vlasov solver accuracy is its capability to preserve an equilibrium distribution function over time, assuming Φ = 0. In order to get a simple tool to evaluate the Vlasov solver, one can also restrict to the case µ = 0. It is worthwhile rewriting gyrokinetic equations in this reduced setting:

dx i dt = v b * s • ∇x i + v Ds • ∇x i (11)
m s dv dt = 0 (12)
It gives

dx i dt = B • ∇x i B * s v + m s v 2 e s B * s µ 0 J • ∇x i B + m s v 2 e s B * s B [B, x i ] ( 13
)
dv dt = 0 (14)
and finally, assuming that the current J is perpendicular to the poloidal plane,

dr dt = B r B * s v + m s v 2 e s B * s B [B, r] (15)
dθ dt = B θ B * s v + m s v 2 e s B * s B [B, θ] (16)
dϕ dt = B ϕ B * s v + m s v 2 e s B * s J ϕ B + m s v 2 e s B * s B [B, ϕ] (17)
Particles keep a constant parallel velocity over time. Then the trajectory of one particle in phase space remains in the 3D space dimensions.

Trajectory of particles

The GYSELA code uses a Semi-Lagrangian scheme combined with a Strang splitting.

To guarantee second order accuracy in time, the following sequence [GBB + 06] is used: ( v /2, φ/2, rθ, φ/2, v /2), where factor 1/2 denotes a shift over ∆t/2. In this sequence, v and φ shifts correspond to 1D advections along v and ϕ directions, and rθ represents a 2D advection (displacement in r and θ are strongly coupled). Let us now focus on the discretization of the 2D advection which requires specific attention.

Taylor expansion For a single rθ advection in the semi-Lagrangian approach, a displacement (δr, δθ) is computed to find the foot of the characteristics that ends at a grid node (r i , θ j ). This displacement is computed thanks to a Taylor expansion of the fields acting on particles [GBB + 06]. One can derive two fields α and β from Eqs. (15, 16, 17) such as:

δr δθ (ri,θj ) = α(r i , θ j )∆t + β(r i , θ j )∆t 2 + O(∆t 3 )
The problem is that, if δr or δθ is larger than grid element sizes denoted ∆r and ∆θ, then this approximation can be really inaccurate.

Precomputed trajectories Another method is described here to find out the foot of the characteristics associated to each grid node. Since the fields acting on particles, Eqs. (11,12), do not depends on time t, one can approximate (δr, δθ) (ri,θj ) once for all. We have used Runge-Kutta time integration scheme (RK2) with a small time step to precompute these trajectories. This approach is possible only for the linear terms of Eqs. (11,12), but not for non-linear terms, present in Eqs. (1,2), that depend on E t and Φ t . Let us choose a δt, such as M δt = ∆t with M ∈ N and M large enough. One can build a series r n and θ n as follows:

r n+ 1 2 θ n+ 1 2 = r n θ n + δt 2 α(r n , θ n ) r n+1 θ n+1 = r n θ n + δt α(r n+ 1 2 , θ n+ 1 2 )
The initial condition is set to r 0 = r i , θ 0 = θ j . After M steps of Runge-Kutta iteration, it gives

δr δθ (ri,θj ) = r M θ M
Because these trajectories can be computed only once when the simulation starts, M can be taken quite large (we will assume M = 64 in the following). These precomputations will not impact significantly the global simulation time. Other time integration schemes have been tried: RK3, RK4, and also larger values of M , but these modifications does not impact significantly the precision in our test cases.

Forward Semi-Lagrangian scheme

The Forward Semi-Lagrangian scheme [START_REF] Crouseilles | A forward semilagrangian method for the numerical solution of the vlasov equation[END_REF] (the acronym is FSL) is an alternative to the classical Backward Semi-Lagrangian scheme (BSL). It is based on forward integration of the characteristics. The distribution function is updated on an eulerian grid, and the pseudoparticles located on the mesh nodes follow the characteristics of the equation forward for one time step, and are then deposited on nearest nodes. While the main cost of the BSL method comes from the interpolation step, FSL spends most of computational time in the deposition step. The FSL scheme can be set up such as mass is preserved along time.

We present the approach taken for the 2D advection in (r, θ). In our setting, the algorithm of Forward Semi-Lagrangian is the following (with S the cubic B-spline):

• Step 1: Compute the spline coefficients w n k,l (f n is known) such that (J s B * ||s f n ) i,j = k,l w n k,l S(x i -x k )S(y j -y l )
• Step 2: Integrate forward in time the characteristics from t n to t n+1 , given as initial data the grid points (x k , y l ). We obtain the new particle positions (x k , y l ) at time t n+1 by following the characteristics.

• Step 3: Deposition of particles localized at (x k , y l )

f n+1 i,j =   k,l w n k,l S(x i -x k )S(y j -y l )   /(J s B * ||s ) i,j
Latu & al

Numerical experiments

Accuracy of conservation of invariants

An equilibrium solution of the collisionless gyrokinetic equation must satisfy some conditions.

To get an equilibrium for the Vlasov equation, it suffices to take an arbitrary function of constants of motion in the unperturbed characteristics. In an axisymmetric toroidal configuration, a gyrokinetic Vlasov equilibrium is defined by three constants of motion: the magnetic moment µ, the energy E=v 2 /2+µ B(r,θ), and the canonical toroidal angular momentum [AGG + 11] Pϕ=ψ(r)+Iv /B(r,θ). The ψ(r) function is defined thanks to the safety factor q(r) by the relation:

dψ/dr=-B0 r/q(r).

To get a good accuracy in integrating the Vlasov equation in time, and then to conserve exactly an initial equilibrium distribution function, we have to take care of the correctness of:

• the initial equilibrium setting; distribution function will be a function of the constants of motion: µ, E, P ϕ ;

• computing the displacements in the Vlasov solver (finding the foot of the characteristic); a sufficiently small time step has to be chosen for the time integration scheme for example, but also the resolution in space must be sufficient to have enough accuracy to interpolate displacement fields;

• interpolate the distribution function in the Vlasov solver with enough resolution; the number of points in the computational domain has to be tuned for this issue;

We will be able to estimate quantitatively the quality of the distribution function after several time steps by:

• measuring the difference between the initial equilibrium function and the distribution function at a given time step.

First test case: equilibrium conservation

The initial distribution function is set up to

f t=0 (r, θ, ϕ, v ) = 1 [Pϕ1,Pϕ2] (P ϕ -P ϕ1 ) m (P ϕ -P ϕ2 ) m sin(γ P ϕ ) ( 18
)
where the invariant P ϕ is a function of r, θ, v and 1 is the indicator function. We will look at

u ∞ (t) = sup{|f t -f t=0 | : ∀(r, θ, ϕ, v )} u 1 (t) = |f t -f t=0 |J s B * ||s drdθdϕdv
Let us take the following parameters : γ = 4π/(P ϕ2 -P ϕ1 ), m = 1, ρ * = 0.01. The values (P ϕ1 , P ϕ2 ) are choosen adequately in order to have an interval [P ϕ1 , P ϕ2 ] that is strictly included in the computational domain r ∈ [r min , r max ], θ ∈ [0, 2π], v = -3 v th0 . A poloidal cut of this initial function is sketched for two values of parallel velocity in Figure 1. As the initial function depends only on the invariant P ϕ , it should be conserved by time integration of the Vlasov equation. The duration of the experiment is denoted t max . The timing unit is the ion cyclotronic time Ω -1 , it will also be the case in all plots of this document. The Vlasov equation is solved during all time steps with Φ forced to zero. An ideal simulation should preserve all norms ∀p, u p (t) = 0. Practically and in the worst case, the function u p (t) will move quickly away from 0 as t grows. In the experiment of Figures 2 and3, the L 1 , L ∞ norms are drawn for some simulations. A comparison is made between the two methods presented previously that are used to calculate the feet of the characteristics. The parameter ∆t is set to 2 and 10, which is near typical values used in GYSELA simulations and t max = 100. The computational domain is quite refined in (r, θ) with N r = 512, N θ = 512. One can remark that Precomputed trajectories strategy is more accurate than Taylor expansion. Nevertheless, the difference shown here is significant for ∆t = 10, but quite negligible for small ∆t = 2. On Figure 4, the differences to the initial distribution function at final state, are shown for an arbitrary value of v = -3 v th0 . One can see here where errors are located using both strategies: Precomputed trajectories at the center, and Taylor expansion at right. One can see that the amplitude of the error is larger in the case of Taylor Expansion, and also less spatially localized. On Figures 5 and6, some L-p norms produced by three simulations are shown (Precomputed trajectories method is used). Between two consecutive simulations, the sizes of the domain (N r , N θ ) have been multiplied by a factor two. While increasing (N r , N θ ), one can notice that simulations become more and more accurate, i.e. L-p norms are reduced. The increase rate of the error depends on the factor N r N θ . The domain sizes N r and N θ are increased independantly each other in the Figures 7 and8 (Precomputed trajectories method is used). It is clear that the error is only of the order of the smallest size among N r and N θ sizes. As far as Equations (11,12) are concerned, it is then noteworthy that N r and N θ should be chosen close to each other in order to improve accuracy.

Latu & al

Second test case: shear flows

Compared to the previous experiments, the initial distribution function considered in the present section is taken with an inhomogeneity in θ. Only the Precomputed trajectories method has been used for these simulations, because it is more accurate in any case. As variable θ is not an invariant, the distribution function is expected to evolve in time. For the sake of simplicity and to reduce computational costs, the simulation is setup with One can notice the shearing that appears along time. The system dynamics tends to create filamentation in configuration space. Nevertheless, the mass f 1 = f J s B drdθdφdv and the maximum value f ∞ = sup{|f | : ∀(r, θ, φ, v )} should theoretically be conserved at any time. Finer structures than mesh element size are dynamically created (see the rightmost plot of Figure 9), and they can not be correctly captured (filamentation induces that mass can be lost or gained at each time step). As it can be seen in Figures 12 and13, with the classical Backward Semi-Lagrangian scheme f ∞ quantity does not remain constant, f 1 is well conserved at (N r = 512, N θ = 512) while (N r = 128, N θ = 128) does not lead to good conservation of f 1 . It is noteworthy, that finer resolution in spacegoing from (N r = 128, N θ = 128) to (N r = 512, N θ = 512) greatly improves the conservation of mass and of the largest absolute value. If one takes the Forward Semi-Lagrangian scheme, the Figures 12 and13 illustrate that f ∞ behaves like BSL but f 1 is better preserved, as expected. Then, FSL scheme is a good solution as far as mass conservation is required. Let suppose one wants to keep the framework of the Backward Semi-Lagrangian scheme and to avoid FSL, another solution to preserve mass could be to have an aligned mesh. If one change the variables (r, θ, ϕ, v ) to (P ϕ , θ, ϕ, v ), the two initial distribution functions we have seen are more accurately discretized. Also, the conservation of mass is improved for the following reason: advection is done only in the direction θ, there is no shift along the P ϕ direction. The displacement in the poloidal plane during one time step amounts to shift only in θ. So mass has just to be conserved on each mesh line labelled by P ϕ , and it will automatically give global mass conservation. It is easy to construct an advection operator and the associated interpolation scheme that preserve mass on each mesh line P ϕ . Another benefit is that in this new set of variables, the equilibrium function is better discretized and represented: steep gradient is only in P ϕ direction in the poloidal plane. Nevertheless, building and managing a mesh that includes P ϕ variable is much more difficult to implement.

N ϕ = 1, N v = 1. The initial distribution function is now f t=0 (r, θ, ϕ, v ) = 1 [Pϕ1,Pϕ2] (P ϕ -P ϕ1 ) m (P ϕ -P ϕ2 ) m 1 [θ1,θ2] (θ -θ 1 ) m (θ -θ 2 ) m

Extrapolation to nonlinear test cases

The first test case of section 3.3.2 is linked to what is going on at the beginning of a Gysela run. The initial distribution function is a perturbed equilibrium. One wishes to remain close to this quasi-equilibrium (if the amplitude of electric potential is very small) without beeing too much biased by some large numerical artifacts of the linear operator that we have seen: artificial increase of u 1 , u ∞ .

The second test case of section 3.3.3 is near what is happening in non-linear dynamics in which many localized spatial structures develop and grow. One expect to conserve the total particle density, even if the distribution function becomes something else than an ideal function of motion invariants. The decrease of maximal value of Figure 13 is not acceptable to perform accurate simulations. Aligned coordinates along P ϕ should be able to improve conservation of f ∞ .

4 Nonlinear gyrokinetic simulations The equations (1,2) can be split into two parts (in the same spirit that [IIK + 08]). The first part includes the nonlinear terms that depends on the electric potential. The second part comprises the other terms. One can solve these two parts one after the other by splitting.

On the first hand, the linear operator is presented in Eqs (19,20), on the second hand, the nonlinear operator is described by Eqs (21,22).

Linear operator

L dx i dt = v b * s • ∇x i + v Ds • ∇x i (19) ms dv dt = -µ b * s • ∇B (20)
Nonlinear operator N

dx i dt = v Es GC • ∇x i (21) ms dv dt = -es b * s • ∇ φ + msv B v Es GC • ∇B (22)
The linear operator exhibits large displacements at large modulus of parallel velocity, and also induces shear flows. These features can interfere with the nonlinear dynamics that eventually involves small displacements that are not of the same order of magnitude. Moreover, as the dynamics generated by the two operators are different, induced accuracy problems have possibly not the same characteristics for the two operators; and the limitations (CFL-like conditions) on the time step are also not the same. Ideally, one should be able to fix the time step of linear and nonlinear operators independantly in order to enhance accuracy.

The original formulation that uses Strang splitting in phase space for the Vlasov solver, but without a linear/nonlinear splitting, can leed to serious troubles at high parallel velocities. Indeed, directional splitting suffers from some large shifts in ϕ, r and θ at each directional advection (section 3.2). Then, for not so large ∆t, the evaluation of electric fields E (.) is not done at the right spatial position at each substep of the splitting except for the first substep of the splitting. One has to take very small time step to recover small displacements in the linear operator and then a reasonable accuracy in the evaluation of E (.) . The splitting between linear and nonlinear parts eliminates this problem. The nonlinear operator is applied alone, thus the linear operator and its large shifts does not interact badly with nonlinear solver. The proposed solution is to perform at each time step: first, a directional splitting for the linear operator L with the sequence ( v /2, φ/2, rθ, φ/2, v /2), second, a directional splitting for the nonlinear operator N with the same sequence ( v /2, φ/2, rθ, φ/2, v /2).

This approach is a liitle bit expensive in term of computational cost because it nearly doubles time to solution.

Conservation of mass

Each of the operators L, N conserves the mass independently. One can track during simulation which operator degrades or not the total mass.

Experiments

In the following GYSELA runs, the BSL and FSL schemes are compared for two test cases with µ =0 (4D only). Results for both BSL scheme and FSL scheme are shown. In the FSL version that has been run, FSL scheme is present in all advections: 1D (ϕ, v ) and 2D (r, θ). The rest of the code is unchanged compared to the BSL version. In order to simplify boundary conditions in v , the distribution function is fixed at v min and v max . Therefore, we have also taken this choice for the BSL runs presented hereafter.

Toroidal case

A small toroidal test case has been set up (ρ * = 1/40). The computational domain size is N r = 256, N θ = 256, N ϕ = 64, N v = 48. The FSL version outperforms the BSL version from the point of view of mass conservation. The conservation of total energy is far more better with FSL and this is due to improved mass conservation. A precise tracking of mass losses/gains allows us to conclude that the linear operator L is mainly responsible for mass degradation using BSL approach. An explanation is that in toroidal setting, L causes large displacements, whereas in cylindrical setting (previous subsection) this was not the case.

Figure 1 :

 1 Figure 1: Initial function f t=0 at v = -3 v th0 , ϕ = 0 (left), v = 0, ϕ = 0 (right) the norms u p (t) = f t -f t=0 p with p = 1, ∞. They are defined as

Figure 2 :

 2 Figure 2: Evolution of the L 1 norm u 1 (t) with N r = 512, N θ = 512

Figure 3 :

 3 Figure 3: Evolution of the L ∞ norm u ∞ (t) with N r = 512, N θ = 512

Figure 4 :

 4 Figure 4: distance between f t and f t=0 (modulus of the difference) at t = 100 with ∆t = 10 and Precomputed trajectories (left), distance f t -f t=0 at t = 100 with ∆t = 10 and Taylor expansion (right)

Figure 5 :Figure 6 :

 56 Figure 5: Evolution of the L 1 norm u 1 (t) for different poloidal mesh sizes

Figure 7 :Figure 8 :

 78 Figure 7: Evolution of the L 1 norm u 1 (t) for different poloidal mesh sizes

Figure 9 :

 9 Figure 9: Initial function f t=0 with v = -3 v th0 , N r = 512, N θ = 512 (left), distribution function f at t = 2000 with BSL scheme (middle), distribution function f at t = 50000 with BSL scheme (right)

Figure 12 :Figure 13 :

 1213 Figure 12: Time evolution of the mass of f for different poloidal mesh sizes

4. 1

 1 Split linear and nonlinear parts 4.1.1 Global separation of linear/nonlinear terms

Figure 14 :Figure 15 :

 1415 Figure14: Evolution of the mass for a 4D, µ = 0, cylindrical test case

Figure 16 :

 16 Figure 16: Poloidal cut of electric potential at different time steps (t = 1920 left, t = 8000 right), ϕ = 0, BSL scheme

Figure 18 :Figure 19 :Figure 20 :

 181920 Figure 18: Evolution of the mass for a 4D, µ = 0, toroidal test case

RR n°8054

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Latu & al 5 Conclusion

Few solutions have been proposed to help preserving constant states, L 1 and ∞ norms in gyrokinetic semi-Lagrangian simulations. Some GYSELA runs show that we get benefits from these improvements in term of accuracy.