Stéphane Gonzalez
email: stephane.gonzalez@univ-st-etienne.fr

Fatma Zahra

Sharing the Global Outcomes of Finite Natural Resource Exploitation: A Dynamic Coalitional Stability Perspective

Keywords: Non-renewable natural resources, Cooperative games, Strong sequential core

The article explores the implications of natural resource scarcity in terms of global cooperation and trade. We investigate whether there exist stable international long-term agreements that take into account the disparities between countries in terms of geological endowments and productive capacity, while caring about future generations. For that purpose, we build an original cooperative game framework, where countries can form coalitions in order to optimize their discounted consumption stream in the long-run, within the limits of their stock of natural resources. We use the concept of the strong sequential core that satisfies both coalitional stability and time consistency. We show that this set is nonempty, stating that an international long-term agreement along the optimal path will be self-enforcing. The presented model sets out a conceptual framework for exploring the fair sharing of the fruits of global economic growth.

Introduction

The nexus between trade and mineral resource extraction is crucial to the understanding of wealth creation dynamics. Early stages of mankind are named after the materials from which tools and weapons were made of: the Bronze Age, which arose fully around 3000 BCE, was preceded by the Chalcolithic or Copper Age. The Near East was the "kernel of the Age of Metals", but was poorly endowed with these materials. Therefore, the "valley urban societies" had to exchange with the "barbarians" (in the technical archaeological sense), before trade expanded from the Near East to Europe [START_REF] Goody | Metals, culture and capitalism: an essay on the origins of the modern world[END_REF]. Modern resource extraction and trade are of much higher orders of magnitude. Industrialization is characterized by the construction of infrastructures in the sectors of heavy industry, energy, housing, transport and communication, and therefore is inevitably associated with an increase in the consumption of raw materials.

The development of trade, necessary to meet an ever-growing demand, is directly linked to the institutional implementation of free trade agreements in free trade areas.

The debate on the "trade-environment divide" [START_REF] Daniel | Bridging the trade-environment divide[END_REF] is mostly focused on pollution issues [START_REF] Brian | North-south trade and the environment[END_REF]. The interaction between trade and materially sustainable growth [START_REF] Dupuy | Sustainability and International Trade[END_REF] was studied by [START_REF] Geir | Hartwick's rule in open economies[END_REF] who looked at the effects of opening economies on the Hartwick rule. Strategic game theory approaches were also used to explore finite resources issues [START_REF] Van | Dynamic games in the economics of natural resources: a survey[END_REF], and the relationship with trade was studied with two countries [START_REF] Kagan | Battle for climate and scarcity rents: beyond the linear-quadratic case[END_REF][START_REF] Tamasiga | Differential games approach to trade with exhaustible resources[END_REF]. However, and despite the call of some game theorists to develop cooperative game theory [START_REF] Samuelson | Game theory in economics and beyond[END_REF], the literature on material sustainability neglected to treat the issue through cooperative lenses. Trade can indeed be analyzed through three concepts: "pure competition", "coalitional power" and "fair division". [START_REF] Lloyd | Pure competition, coalitional power, and fair division[END_REF] provide a static market game for the three varieties of solution and show that, under certain assumptions, the outcomes converge. In a distinct stream of literature of cooperative games, the question of renewable and common-pool resources was studied [START_REF] Funaki | The core of an economy with a common pool resource: A partition function form approach[END_REF], mostly for fishery or river sharing problems [START_REF] Ambec | Sharing a river[END_REF][START_REF] Béal | The river sharing problem: A survey[END_REF]. However, these models do not take into account the dynamic aspects of the issue.

The aim of this article is to show that there exist stable international long-term agreements that take into account the disparities between countries in terms of geological endowments and productive capacity, while caring about future generations. In this context, countries can form coalitions and look for the best agreement to optimize their allocation. Members can break an alliance according to their interest, at any point of time. Dynamic cooperative game theory is the most fitted framework to tackle this kind of issues, but the literature on the application of such games is quite limited. Some dynamic cooperative approaches were used to tackle environmental issues, especially fisheries [START_REF] Gordon R Munro | The optimal management of transboundary renewable resources[END_REF] and pollution [START_REF] Jørgensen | Time consistent side payments in a dynamic game of downstream pollution[END_REF][START_REF] Hoofe | Linear-state differential games in partition function form[END_REF]. The question of non-renewable resource exhaustibility was raised by d'Albis and [START_REF] Albis | Fair intergenerational sharing of a natural resource[END_REF], who study cooperative allocations among overlapping generations depleting a natural resource over an infinite future. However, the authors focus on intergenerational allocations and do not include any international aspect in their model.

In order to find such long-term international agreements, the twin issues of coalitional stability and time consistency have to be handled at once. First, it is a question of designing a policy that no coalition of the present generation has an incentive to refuse. Traditionally, this issue is tackled by the concept of the core, which is the set of allocations which cannot be improved upon by any coalition of agents. However, this notion does not deal with the dynamic issues relevant to the intergenerational sharing of the benefits generated from resource extraction. It must therefore also ensure that at no point in time the policy initiated then departs from the one originally planned. Recent papers tackle the issue of dynamic core concepts [START_REF] Predtetchinski | The strong sequential core in a dynamic exchange economy[END_REF][START_REF] Kranich | Core concepts for dynamic tu games[END_REF][START_REF] Lehrer | On the core of dynamic cooperative games[END_REF], but the first studies date back to the 70's with the introduction of core concepts for production economies [START_REF] Boehm | The core of an economy with production[END_REF][START_REF] Robert | The existence of core allocations in a one sector model of capital accumulation[END_REF], as well as the notion of trust in a monetary economy [START_REF] Gale | The core of a monetary economy without trust[END_REF]. [START_REF] Robert | The recursive core[END_REF] introduced the recursive core solution concept, designed for the specific case of capital accumulation models. They defined it as the set of allocations for which no coalition can improve upon its consumption stream at any time given its accumulation of assets up to that period. This concept is a particular case of the most demanding dynamic solution concept, the strong sequential core, which captures "those situations in which at each stage the grand coalition is formed, its worth is distributed among the players and no coalition has a profitable deviation" knowing that "coalitions are allowed to deviate at any stage of the game" [START_REF] Kranich | Core concepts for dynamic tu games[END_REF]. This concept has the strong property to satisfy both coalitional stability and time consistency. It is empty-valued for large classes of economies, as argued by [START_REF] Habis | Core concepts for incomplete market economies[END_REF] who compare various dynamic core concepts. In the present article, we question the emptiness of this solution concept in the case of open and resource-dependent capitalist economies.

The novelty of our approach is to model the geographical heterogeneity of natural endowments that results from geological processes, as well as the geographical heterogeneity of capital and technological endowments that results from historical processes. For that purpose, we settle a cooperative game where countries can form coalitions in order to optimize their discounted consumption stream in the long-run, within the limits of their stock of natural resources. Trade is viewed as a market cooperative TU-game à la [START_REF] Lloyd | Pure competition, coalitional power, and fair division[END_REF] where non-renewable natural resource inputs are exchanged in such a way as to maximize the total value to be shared among the coalition. This game is cast into a traditional Ramsey-type model of intergenerational equity, where the Bellman's dynamic programming framework is used. An international long-term policy is translated into:

• An action strategy taken by a coalition S. It consists in a system of quotas for extracted natural resources, a commercial quota system and an investment plan proposed to each country at each time t. An action strategy is optimal for S if the commercial quotas optimize the current collective production of S, and if the extraction quotas and the investment plan ensure the maximization of long-term consumption.

• A distribution policy of collectively produced consumption goods. Such a policy is said to be efficient and coalitionally rational if, from some date t onward, a coalition cannot block an international agreement by undertaking a policy increasing its own long-term consumption.

In other words, the coalition decides its extraction and investment plan, and deduces the amount of resources it wishes to trade. It then distributes consumption goods to its members. Note that this definition of a policy is larger than the traditional sense in the field of macroeconomics. Here, it adds to the action the notion of sharing the fruits of economic growth.

This paper addresses the underexplored topic of how to cooperate in a dynamic environment. We show that there exists a unique optimal path of extraction and investment for each country in a coalition S, which could be interpreted as quotas proposed by a social planner. As traded quantities are also unique, we conclude that there exists a unique optimal path for collective production in a coalition S. We then can build an intertemporal cooperative game where the worth of S is the discounted sum of consumption -corresponding to the remaining part of production that is not invested -along this optimal path. We demonstrate that the strong sequential core associated to this game is nonempty. This strong result states that an international long-term agreement along the optimal path will be self-enforcing. The countries, as rational agents, will stick to this agreement, as no other coalition could offer them a better outcome, at any point in time. Furthermore, we show that this strong sequential core contains an infinity of elements, opening an avenue for the research of a fair distribution from both intergenerational and international points of view.

The article is organized as follows. The following section presents our formal apparatus and details the reasoning of the model. In the third section, our main results and their interpretation are provided. The last section is a discussion on the model and the research avenue it opens. All proofs are available in the appendix.

A Formal Statement of the Problem

Game Set-Up

Let N be a fixed and finite nonempty set of countries, who are the game players. Countries have the possibility to form a coalition which can be viewed as a free-trade area. Each country i ∈ N has a natural resource1 that can be exploited by the infinitely lived country at each period t ∈ N. Growth is constrained by natural resource availability, its costs of extraction, as well as the accumulation of capital. Countries have three decisions to make at each period: resource extraction, resource trade and investment level through savings that will increase the production capacity of the next period. The remaining created flow of wealth is consumed, and generations seek to maximize their present consumption as well as the discounted consumption of their successors. The present model is therefore a cooperative games adaptation of Ramsey's benchmark model with traded mineral resources. A coalition illustrates contemporary trade agreements between countries with the free movement of natural resources, consumption goods and capital flows (in the form of foreign investments).

At period t and for a nonempty coalition S ⊆ N , consider an amount yt (S) of a single final good, which can be viewed as Ricardo's metaphorical corn in neoclassical theory. The coalition chooses the amount η i,t each country i will invest for tomorrow (the seeds to be sowed next year) to increase the country's stock of capital K i,t . This leaves yt (S) -i∈S η i,t to the present consumption of the coalition. The dynamics of i's capital stock is given by the following law of motion, τ ∈ (0, 1) being the depreciation rate:

K i,t+1 = K i,t + η i,t -τ K i,t
The coalition also chooses the quantity e i,t of resources each country will extract from its available stock Ω i,t . The dynamics of i's resource stock is given by the following law of motion:

Ω i,t+1 = Ω i,t -e i,t
Extracted resources are exchanged in a free-trade area with no transaction cost, and country i ends up with a quantity z i,t of input for its domestic production. It means that at the end of the exchange process, i∈S z i,t = i∈S e i,t holds. The country's productive sector then uses its production capacity K i,t and z i,t to produce f i (z i , K i ) through a raw country-specific production function f i . We assume that, for each i ∈ S, f i :

R + × R + → R + satisfies: (i) f i is continuous, non-decreasing, strictly concave; (ii) f i (z i , 0) = 0 for each z i ∈ R + and f i (0, K i ) = 0 for each K i ∈ R + ; (iii) f i is differentiable with respect to K i ; (iv) lim K i →0 ∂f i (z i , K i ) ∂K i > τ for each z i ∈ R + ; (v) lim K i →∞ ∂f i (z i , K i ) ∂K i < τ for each z i ∈ R + .
Note that these conditions are weaker than the Inada conditions. The cost of extraction c i (e i,t , Ω i,t ) of i ∈ N , expressed in consumption units, is represented through a function depending both on the quantity Ω i,t available to the country i and the quantity e i,t extracted at date Figure 1: Scheme of the state and action dynamics t. We assume that this function, defined on R2 + , is continuous, convex and non negative when Ω i,t is larger than e i,t . 2

We consider that each country i has an initial endowment yi,0 and that y0 (S) = i∈S yi,0 .

The production process takes one period to be achieved, since time is necessary for natural resource transportation and capital usage. The choice of the trading strategy is made such that the total net production yt+1 (S) of each coalition S is maximized through the following static optimization problem: maximize

(z i,t ) i∈S i∈S f i (z i,t , K i,t ) -c i (e i,t , Ω i,t ) subject to i∈S z i,t = i∈S e i,t .
(1)

Trade is viewed as a market cooperative TU game à la [START_REF] Lloyd | Pure competition, coalitional power, and fair division[END_REF] where extracted resource inputs are exchanged in such a way as to maximize the total value to be shared among the coalition. Figure 1 presents a scheme of the model at t and t + 1.

Significant Sets

In this subsection, a digression is made to define the sets that will be important for the rest of the article.

Ω ∈ R + is the total and finite amount of resources on Earth. Following neoclassical arguments, the assumptions on the function f i as well as the dynamics of K i ensure that there exists K ∈ R + such that for each i ∈ S and for each t ∈ N, K i,t ≤ K. As a consequence, there exists ȳ ∈ R + such that for each t ∈ N and each S ⊆ N , yt (S) ≤ ȳ.

Let Ω(S) = (Ω i ) i∈S (or simply Ω if there is no danger of confusion) be the vector of natural resource stocks and K(S) = (K i ) i∈S (or simply K if there is no danger of confusion) be the vector of capital stocks. The state space S(S) is defined by:

S(S) := {(Ω(S), K(S), y(S)) ∈ R S + × R S + × R + | y(S) ≤ ȳ and ∀i ∈ S, Ω i ≤ Ω, K i ≤ K}.
Let e(S) = (e i ) i∈S (or simply e if there is no danger of confusion) be the vector of extraction and η(S) = (η i ) i∈S (or simply η if there is no danger of confusion) be the vector of investment. The action space A(S) is defined by:

A(S) := {(e(S), η(S)) ∈ R S + × R S + | ∀i ∈ S, e i ≤ Ω, η i ≤ ȳ}.
The set of feasible vectors of extraction e(S) for a coalition S with a stock described by Ω(S) is

E Ω (S) := {e(S) ∈ (R + ) S | e i ≤ Ω i }.
We now define the set of feasible vectors of investment η(S) for a coalition S by

I y(S) (S) := {η(S) ∈ (R + ) S | i∈S η i ≤ y(S)}.
We can therefore define the set of couples (e(S), η(S)) of feasible actions by:

Φ Ω,y(S) (S) := {(e(S), η(S)) | e(S) ∈ E Ω (S), η(S) ∈ I y(S) (S)}.
Now, the set of available resource input after the exchange z(S) = (z i ) i∈S (or simply z if there is no danger of confusion) is:

Z e (S) := {z(S) ∈ (R + ) S | i∈S z i = i∈S e i }.

Coalitional and Intergenerational Worth

At each period, countries belonging to a coalition S when the production is y(S) and the investment is η ∈ I y(S) (S) will earn a worth v y(S),η (S) that equals their consumption:

∀y(S) ∈ R + , ∀η ∈ I y(S) (S) v y(S),η (S) = y(S) - i∈S η i .
Let us now zoom up in the time scale, and describe the behavior of countries from an intertemporal viewpoint. Let δ ∈]0, 1[ be the discount factor. The intertemporal worth V Ω 0 ,K 0 ,y 0 (S) (S) of the coalition S is defined as the value of the following dynamic programming problem: maximize

(et,ηt) t∈N t∈N δ t v yt(S),ηt (S)
subject to

dynamic constraints Ω t+1 = Ω t -e t K t+1 = (1 -τ )K t + η t yt+1 (S) = maximize (z i,t ) i∈S i∈S f i (z i,t , K i,t ) -c i (e i,t , Ω i,t ) subject to i∈S z i,t = i∈S e i,t
feasibility constraints

e i,t ≤ Ω i,t , i ∈ S i∈S η i,t ≤ yt (S) (2)
Observe that the intertemporal utility t∈N δ t v yt(S),ηt (S) is finite since δ ∈]0, 1[ and that the reward function (y(S), η) → v y(S),η (S) is bounded on [0, ȳ] × [0, ȳ] S . Let us now clearly describe the actions that occur on the short and long term.

(i) On the short run: Suppose that the coalition S has chosen an extraction and investment plan e t (S), η t (S) t∈N . The level of extraction and investment is therefore fixed for each one of its members i at each period t. The decision to be made during a time period t only involves the trade of natural resources. The situation is modeled as a market game à la [START_REF] Lloyd | Pure competition, coalitional power, and fair division[END_REF]. A trading process takes place inside the coalition, and the total extracted amount i∈S e i,t is exactly redistributed among the members. Each country ends up with a level z i,t to contribute to the total net production yt (S), which is the production i∈S f i (z i,t , K i,t ) net from the extraction costs i∈S c i (e i,t , Ω i,t ). The market game consists in a choice of distribution of extracted natural resources such that this total net production yt (S) is maximized through problem (1). The level of investment η i,t being also fixed, this is equivalent to maximizing the total consumption v yt(S),ηt (S) that is the total net production minus the total investments i∈S η i,t .

(ii) On the long run: the coalition S chooses through problem ( 2) its extraction and investment plan e t (S), η t (S) t∈N for each t to infinity.3 It does so by maximizing the intertemporal worth V Ω 0 ,K 0 ,y 0 (S) (S). This worth corresponds to the discounted total consumption t∈N δ t v yt(S),ηt (S) of the coalition S. This choice of the extraction and investment plan on the long term is made while taking into account the dynamics of the state variables (Ω t (S), K t (S), yt (S)) and verifying the feasibility of the chosen plan:

• the extraction flow e t (S) must be smaller than the resource stock Ω t (S);

• the total investment i∈S η i,t must be smaller than the total net production yt (S).

This last point ensures the nonnegativity of the total consumption of S at each period.

Existence of Distribution Policies Along the Optimal Path

Optimal Action Strategy

The following proposition establishes that, given a flow of resource extracted by each country, there exists a unique optimal way to trade that can be interpreted as an optimal commercial quota system.

Proposition 1. The function (Ω, e, K) → max i∈S f i (z i , K i ) -c i (e i , Ω i ) , (z i ) i∈S ∈ Z e (S) is continuous and strictly concave over [0, Ω] S × [0, Ω] S × [0, K] S . Furthermore, for each (Ω, e, K) ∈ [0, Ω] S × [0, Ω] S × [0, K] S , there exists a unique z * ∈ Z e (S) -continuously varying with (Ω, e, K) -such that max i∈S f i (z i , K i ) -c i (e i , Ω i ) , (z i ) i∈S ∈ Z e (S) = i∈S f i (z * i , K i ) -c i (e i , Ω i ) .
The next proposition states that, given the initial endowments of each country, there exists a unique optimal way to extract and to invest that can be interpreted as an optimal extraction quota system and an optimal investment plan.

Proposition 2. For each nonempty coalition S ⊆ N , and each (Ω i,0 ) i∈S ∈ [0, Ω] S , (K i,0 ) i∈S ∈ [0, K] S and y0 (S) ∈ [0, ȳ], there exists a unique optimal path of extraction and investment to the dynamic programming problem (2). This result is valid for the dynamic structure presented in the set-up description (in particular, the initial conditions and the way the capital is exchanged). The following remarks justify the choices made and show why Proposition 1 and 2 are robust to some changes in the model structure.

Remark 1. It is conventional in macroeconomics to set the stocks (here resource and capital stocks) as state variables to which a transition function is applied. The present model faces an issue of simultaneity for strategic decisions. The extraction vector e and the investment vector η for each country are decided at the same time. But choosing the investment levels requires information on the total net production y0 , which itself depends on the flow of extraction e. The option chosen here to deal with this question is to add the flow variable y0 to the state variables. Nevertheless, it is possible to find back the standard framework for which only stocks are state variables, in the tradition of optimal growth models. This requires to change the control variables of our set-up: the extraction vector e is decided at each period t. Consequently, the investment must be a function of the extraction "e → η(e)". This convention could be adopted at the cost of more complex description of the strategy and readability of the proofs. However, it does not strongly affect our results.4

Remark 2. In our set-up, capital flows are mobile internationally, which is not the case for capital stocks (we consider that they only cross borders in the form of a flow5 ). Now, let us assume that the capital stock is also mobile. In that case, it is possible to simply modify the expression of the total net production yt+1 (•) (as defined in problem (1)) by adding a capital market game besides the natural resource one. Therefore, the capital used at each period by the technology f i of a country i to produce its contribution to the final good is a capital resulting from an exchange process -which is different from the capital resulting from the accumulation of investments η i of this country. In this alternative set-up, the existence of an optimal path of extraction and investment still holds by easily adapting the proof of Proposition 1. However, the uniqueness of the investment allocation path (η i,t ) i∈N,t∈N is not guaranteed. Several investment plans can be optimal, as long as the total capital accumulated equals the total capital exchanged. But losing this uniqueness result for the investment path does not affect the uniqueness of the other stock variables. Indeed, the investment allocation to each country and its accumulation in a capital stock does not impact the total production anymore, since the capital stocks are redistributed to produce. The optimal path for the "after-exchange" capital stock used as an input by each country is therefore still unique. For example, the investment planner could decide to allocate all the investments either to one country i or to another country j. In both cases, the maximum total production would still be the same since the capital stocks would be exchanged in order to maximize this production.

Efficient and Coalitionally Rational Distribution Policy

We showed that there exists a unique optimal path of extraction and investment for each country in a coalition S, which could be interpreted as quotas imposed by a social planner. This leads to a unique path of resource and capital stocks. As traded quantities are also unique, we can conclude that there exists a unique path for collective production in a coalition S. The consumption path of each country, for its part, is not unique and depends upon the sharing rule decided by the coalition. The following result states that an international long-term agreement along this optimal path will be self-enforcing. The countries, as rational agents, will stick to this agreement, as no other coalition could offer them a better outcome, at any point in time. In other words, this agreement within N cannot be dominated by any coalition S ⊆ N . For that purpose, we adapt the concept of strong sequential core to our framework. This concept satisfies the properties of efficiency, coalitional rationality and also tackles the crucial issue of time consistency. We now formally introduce this concept.

Let IS be the set [0, Ω] N ×[0, K] N ×[0, ȳ] of initial states. For each initial state (Ω, K, y) ∈ IS of the grand coalition N , there exists a unique optimal extraction and investment solution e * t (N ), η * t (N ) t∈N to the dynamic programming problem (2) according to Proposition 2. Let Ω * t (N ), K * t (N ), y * t (N ) t∈N be the unique state evolution corresponding to the optimal path when the initial state is (Ω, K, y). We now have the material to define the notion of international distribution policy which describes the allocation of consumption goods for each country along the international optimal path. Definition 1. An international distribution policy (or more briefly a policy), π, is a mapping which associates with each initial state (Ω, K, y) ∈ IS a path of consumption vector (π i,t (Ω, K, y)) i∈N,t∈N such that ∀t ∈ N:

i∈N π i,t (Ω, K, y) = y * t (N ) - i∈N η * i,t (N ).
We denote by Π the set of international distribution policies. A policy is said to be dominated when there exists a point in time from which a coalition can achieve by itself a better payoff than the one proposed by the policy. Definition 2. A policy π is dominated by S at date T if there exists an initial state (Ω, K, y) ∈ IS such that:

+∞ t=0 δ t i∈S π i,t+T (Ω, K, y) < V Ω * S,T (N ),K * S,T (N ),y * S,T (N ) (S),
where for each nonempty coalition S of N and t ∈ N, Ω * S,t (N ), K * S,t (N ), y * S,t (N ) denotes the projection Ω * i,t (N ),

K * i,t (N ), y * i,t (N ) i∈S of Ω * t (N ), K * t (N ), y * t (N ) .
The last definition describes the set of policies for which countries stick to the global policy, as no other coalition can offer them a better outcome, at any point in time. The strong sequential core in our setting refers to the set of undominated policies.

Definition 3. The strong sequential core C is the set of policies π ∈ Π such that ∀(Ω, K, y) ∈ IS, ∀S ⊆ N , ∀T ∈ N, +∞ t=0 δ t i∈S π i,t+T (Ω, K, y) ≥ V Ω * S,T (N ),K * S,T (N ),y * S,T (N ) (S).
The following theorem states that the strong sequential core of this set-up is nonempty whatever the initial state is. The theorem further indicates that this core contains an infinity of elements, corresponding to an infinite way of distributing consumption streams among the countries of the international coalition. Such policies have the property that no coalition can form at any point in time and design a policy that is feasible for the coalition while making all coalition members better off.6 Theorem 1. The strong sequential core C contains an infinity of elements. This theorem states that, if the countries of a coalition follow a long-term cooperation strategy in terms of extraction, investment and induced trade, then the best way to optimize their own consumption while caring about future generations is to form a global coalition. The strong sequential core is constituted of the streams of consumption to be allocated to each country. Since this core is not reduced to a singleton, it is possible to compare the different distribution policies and to select a policy among these allocations. Indeed, the strong sequential core meets a coalitional stability criterion, but several sub-solutions apply to our model that combine stability with other criteria. For instance, a fairness criterion can be investigated in the tradition of the axiomatic methodology in order to explore possible ethic consumption paths. One of the most explored solutions in the literature is the nucleolus solution concept, which always belongs to the core when it is nonempty [START_REF] Schmeidler | The nucleolus of a characteristic function game[END_REF]. This solution maximizes recursively the 'welfare' of the worst treated coalitions and can be understood as an application of the Rawlsian social welfare function [START_REF] Susan S Hamlen | The use of core theory in evaluating joint cost allocation schemes[END_REF]. Other interesting solution concepts, that also belong to the core, consider an egalitarian perspective. For instance, the egalitarian core [START_REF] Arin | Egalitarian solutions in the core[END_REF] is the set of core allocations for which no transfer from a rich country to a poor country is possible without going beyond the core boundaries. The existence of such solutions shows that there are theoretical ways for international cooperation to tackle the North-South divide and related inequality issues in a world where international regulation is delicate without implementing self-enforcing agreements.

Discussion and Concluding Remarks

Dynamic core concepts for cooperative games have drawn some attention in the literature, be it from a technical perspective in general frameworks, but also in models applied to diverse economic situations. An original motivation of this article is to question whether the classical result of the core nonemptiness in a static TU market game remains true in a dynamic framework. The extension is not immediate since, for each coalition and at each period, a static optimization problem -consisting in finding efficient trading quotas -is nested in the dynamic problem which seeks to maximize the level of consumption. It follows that the exogenous basket of goods in the seminal article is endogenized in ours and constructed as part of the coalitions' long-term strategy. This strategy itself is submitted to natural resource and capital stock constraints on the long run. The present article proposes a nonempty core solution to an economic problem modeled through a dynamic cooperative game with transferable utility. [START_REF] Avrachenkov | Cooperative markov decision processes: time consistency, greedy players satisfaction, and cooperation maintenance[END_REF] designs a general framework closely related to our setting, by introducing a discrete-time Markov chain of static games, where the utility of the coalitions is transferable and whose transition probabilities depend on the players' actions in each state. They extend the core concept to this framework and find a condition for nonemptiness. Several papers do apply the strong sequential core concept but in a NTU configuration. Among them, only a few show positive results as regards to its nonemptiness. As argued by [START_REF] Habis | Core concepts for incomplete market economies[END_REF], it is empty-valued for large classes of economies. One exception comparable to the present set-up is the paper by [START_REF] Robert | The recursive core[END_REF] designed for the simple case of capital accumulation in a closed economy with no resource dependency. The originality of our methodology is the finding of a demanding result -that is a nonempty core -embedded in complex and endogenous economic dynamics of resource exhaustion, capital accumulation and input trade, by implementing standard convex optimization techniques.

This methodology is also quite neutral with respect to the interpretation of the decision process, due to the self-enforcing feature of the agreements that the core involves. We showed that an international cooperation strategy leads to a unique optimal path of collective production since all the processes involved in this production -namely the extraction of natural resources from a finite stock, their trade and the evolution of the productive capacity -should be unique to be optimal. On the demand side, a given part of this production is optimally allocated to the investment needed to build the productive capacity of each country, in a unique manner. The remaining part of production, corresponding to the consumption of goods, can be shared among countries, this time in different manners. The optimal paths could be interpreted as resulting from quotas chosen by a benevolent planner, but this interpretation that we propose could be objected. Indeed, imposing commercial quotas is not consistent with a common definition of free trade. This objection is fully anchored in the socialist calculation debate [START_REF] John | Who won the socialist calculation debate?[END_REF] and is not the subject of this study. Note however that, by definition of optimality, agents should not deviate from these quotas even in a free market interpretation. Another alternative interpretation would be the result of a negotiation procedure between countries. In all cases, no country would have the incentive to refuse a core allocation. This last interpretation is linked to the question of the implementation of cooperative game solutions, also called "Nash program". This could be the subject of a subsequent paper.

This set-up provides a basis on which to progress on questions related to the long-term material development of countries. We support the idea that cooperative game theory gives a fresh and fairly simple framework to tackle traditional issues raised in growth theory, since it better takes into account the heterogeneity of a finite number of countries. An interesting question would be to characterize technological catch-up effects between countries and how it affects the balance of power, by adding a time-dependent production function or an endogenous knowledge capital accumulation. Moreover, recycling or pollution effects could be investigated in our framework and lead to a more complete understanding of the sustainability of the "Spaceship Earth" [START_REF] Kenneth | The economics of the coming spaceship earth[END_REF].

Let (Ω α t ) t∈N be the sequence of resource stock vectors defined by Ω α 0 = Ω 0 and for each t ∈ N by Ω α t+1 = Ω α t -e α t . A simple induction leads to Ω α t = αΩ 1 t + (1 -α)Ω 0 t for each t ∈ N. Let (K α t ) t∈N be the sequence of capital stock vectors defined by K α 0 = K 0 and for each t ∈ N by K α t+1 = (1 -τ )K α t + η α t . A simple induction leads to K α t = αK 1 t + (1 -α)K 0 t for each t ∈ N. We set yα 0 (S) = y0 (S) and for each t > 0,

yα t (S) = max i∈S f i (z i,t-1 , K α i,t-1 ) -c i (e α i,t-1 , Ω α i,t-1 ) , (z i,t-1 ) i∈S ∈ Z e α t-1 (S) .
We define for each t ∈ N the following vectors8 :

(z * 0 i,t ) i∈S = argmax (z i,t ) i∈S ∈Z e 0 t (S) i∈S f i (z i,t , K 0 i,t ) -c i (e 0 i,t , Ω 0 i,t ) , and (z * 1 i,t ) i∈S = argmax (z i,t ) i∈S ∈Z e 1 t (S) i∈S f i (z i,t , K 1 i,t ) -c i (e 1 i,t , Ω 1 i,t ) .
We also define (z α t ) t∈N as follows:

∀t ∈ N, z α t = αz * 1 t + (1 -α)z * 0 t .
It is straightforward to see that

i∈S η α i,0 ≤ y0 = yα 0 , (3)
and for each t ∈ N,

i∈S η α i,t+1 = α i∈S η 1 i,t+1 + (1 -α) i∈S η 0 i,t+1 ≤ α i∈S f i (z * 1 i,t , K 1 i,t ) -c i (e 1 i,t , Ω 1 i,t ) + (1 -α) i∈S f i (z * 0 i,t , K 0 i,t ) -c i (e 0 i,t , Ω 0 i,t ) .
By concavity of f i and -c i , we therefore have

i∈S η α i,t+1 ≤ i∈S f i (z α i,t , K α i,t ) -c i (e α i,t , Ω α i,t ) . ( 4
)
Since for each t ∈ N, (z α i,t ) i∈S ∈ Z e α t (S), it follows that

i∈S f i (z α i,t , K α i,t ) -c i (e α i,t , Ω α i,t ) ≤ max i∈S f i (z i,t , K α i,t ) -c i (e α i,t , Ω α i,t ) , (z i,t ) i∈S ∈ Z e α t (S)
= yα t+1 .

(5) By combining Eqs 3, 4 and 5, we deduce that for each t ∈ N,

i∈S η α i,t ≤ yα t . ( 6
)
On the other hand, it is clear that e α 0 ≤ Ω 0 , and for each t ∈ N,

e α t+1 = αe 1 t+1 + (1 -α)e 0 t+1 ≤ α(Ω 1 t -e 1 t ) + (1 -α)(Ω 0 t -e 0 t ) = Ω α t -e α t . (7)
Therefore, from Eqs 6 and 7, we can conclude that (e α t , η α t ) t∈N is a feasible path. It follows that

V Ω 0 ,K 0 ,y 0 (S) (S) ≥ t∈N δ t v yα t (S),η α t (S) = t∈N δ t yα t (S) - i∈S η α i,t ≥ y0 (S) - i∈S η α i,0 + t>0 δ t i∈S f i (z α i,t-1 , K α i,t-1 ) -c i (e α i,t-1 , Ω α i,t-1 ) -η α i,t . (8)
By strict concavity of f i and -c i , we have for each t > 0:

i∈S f i (z α i,t-1 , K α i,t-1 ) -c i (e α i,t-1 , Ω α i,t-1 ) > α i∈S f i (z * 1 i,t-1 , K 1 i,t-1 ) -c i (e 1 i,t-1 , Ω 1 i,t-1 ) + (1 -α) i∈S f i (z * 0 i,t-1 , K 0 i,t-1 ) -c i (e 0 i,t-1 , Ω 0 i,t-1 ) . ( 9
)
Combining Eqs 8 and 9 leads to:

V Ω 0 ,K 0 , y0 (S) (S) > α y0 (S) - i∈S η 1 i,0 + t>0 δ t i∈S f i (z * 1 i,t-1 , K 1 i,t-1 ) -c i (e 1 i,t-1 , Ω 1 i,t-1 ) -η 1 i,t + (1 -α) y0 (S) - i∈S η 0 i,0 + t>0 δ t i∈S f i (z * 0 i,t-1 , K 0 i,t-1 ) -c i (e 0 i,t-1 , Ω 0 i,t-1 ) -η 0 i,t
= αV Ω 0 ,K 0 , y0 (S) (S) + (1 -α)V Ω 0 ,K 0 , y0 (S) (S) = V Ω 0 ,K 0 , y0 (S) (S).

The strict inequality leads to a contradiction.

C(N, v) = x ∈ R N | ∀S ⊆ N, i∈S x i ≥ v(S) and i∈N x i = v(N ) .
The interpretation of the core is that no group of agents has an incentive to split from the grand coalition N and form a smaller coalition S since they collectively receive at least as much as what they can obtain for themselves as a coalition. The so-called Bondareva-Shapley theorem [START_REF] Bondareva | Some applications of linear programming methods to the theory of cooperative games[END_REF][START_REF] Lloyd | On balanced sets and cores[END_REF] provides a sufficient and necessary condition under which the core of a TU-game is nonempty. First, we introduce the concept of balanced maps. A balanced map λ : 2 N -→ [0, 1] is such that: (10)

λ(∅) = 0,
The following proposition is a direct consequence of Corollary 1 in [START_REF] Gonzalez | Autonomous coalitions[END_REF].

Proof of Theorem 1

Proof. We proceed in two steps.

Step 1: For each initial state (Ω 0 (N ), K 0 (N ), y0 (N )) ∈ IS, C(N, V Ω 0 (N ),K 0 (N ),y 0 (N ) ) ̸ = ∅ and contains an infinity of elements.

By Proposition 2, for each nonempty coalition S ⊆ N , and each initial stock vector

Ω 0 (S), K 0 (S), y0 (S) ∈ [0, Ω] S × [0, K] S × [0, ȳ],
there exists a unique optimal path e * t (S), η Let λ ∈ B(N ) be a balanced system of N . If λ(N ) = 1, it is clear that

V Ω 0 (N ),K 0 (N ),y 0 (N ) (N ) = S⊆N λ(S)V Ω 0 (S),K 0 (S),y 0 (S) (S).

Assume λ(N ) ̸ = 1. We define (e λ t (N )

) t∈N ∈ (R N + ) N and (Ω λ t (N )) t∈N ∈ (R N + ) N as follows: ∀i ∈ N, ∀t ∈ N, e λ i,t (N ) = S∋i λ(S)e * i,t (S), ∀i ∈ N, ∀t ∈ N, Ω λ i,t (N ) = S∋i λ(S)Ω * i,t (S).
Observe that for each i ∈ N and each t ∈ N, we have

Ω λ i,t (N ) ∈ [0, Ω] since S∋i λ(S) = 1 and Ω λ i,t+1 (N ) = Ω λ i,t (N ) -e λ i,t (N ).
We also can assert that for each i ∈ N and each t ∈ N,

e λ i,t (N ) ≤ Ω λ i,t (N ). ( 11
)
We define as well (

η λ t (N )) t∈N ∈ (R N + ) N and (K λ t (N )) t∈N ∈ (R N + ) N as follows: ∀i ∈ N, ∀t ∈ N, η λ i,t (N ) = S∋i λ(S)η * i,t (S), ∀i ∈ N, ∀t ∈ N, K λ i,t (N ) = S∋i λ(S)K * i,t (S).
Observe that for each i ∈ N and each t ∈ N, we have K λ i,t (N ) ∈ [0, K]. We set yλ 0 (N ) = y0 (N ) and for each t > 0,

yλ t (N ) = max i∈N f i (z i,t-1 (N ), K λ i,t-1 (N ))-c i (e λ i,t-1 (N ), Ω λ i,t-1 (N )) , (z i,t-1 (N )) i∈N ∈ Z e λ t-1 (N ) (N ) .
Observe that for each i ∈ N and each t ∈ N, we have yλ

i,t (N ) ∈ [0, ȳ]. It is straightforward to see that i∈N η λ i,0 (N ) ≤ y0 (N ) = yλ 0 (N ), (12)
and for each t ∈ N,

i∈N η λ i,t+1 (N ) = i∈N S∋i λ(S)η * i,t+1 (S) = S⊆N λ(S) i∈S η * i,t+1 (S) (13) ≤ S⊆N λ(S)y * i,t+1 (S) = S⊆N λ(S) i∈S f i (z * i,t (S), K * i,t (S)) -c i (e * i,t (S), Ω * i,t (S)) = i∈N S∋i λ(S) f i (z * i,t (S), K * i,t (S)) -c i (e * i,t (S), Ω * i,t (S)) ,
where we define, for each t ∈ N,

z * t (S) = argmax (z i,t (S)) i∈S ∈Z e * t (S) i∈S f i (z i,t , K * i,t ) -c i (e * i,t , Ω * i,t ) .
By concavity of f i and -c i , we therefore have

i∈N η λ i,t+1 (N ) ≤ i∈N f i (z λ i,t (N ), K λ i,t (N )) -c i (e λ i,t (N ), Ω λ i,t (N )) (14)
where

(z λ t (N )) t∈N ∈ (R N + ) N is defined as ∀i ∈ N, ∀t ∈ N, z λ i,t (N ) = S∋i λ(S)z * i,t (S).
The following equalities hold:

i∈N e λ i,t (N ) = i∈N S∋i λ(S)e * i,t (S) = S⊆N λ(S) i∈S e * i,t (S). Since z * t (S) ∈ Z e * t (S), it follows that S⊆N λ(S) i∈S e * i,t (S) = S⊆N λ(S) i∈S z * i,t (S) = i∈N S∋i λ(S)z * i,t (S) = i∈N z λ i,t (N ),
from which we deduce that z λ t (N ) ∈ Z e λ t (N ) (N ) for each t ∈ N. It follows that

i∈N f i (z λ i,t (N ), K λ i,t (N )) -c i (e λ i,t (N ), Ω λ i,t (N )) ≤ max i∈N f i (z i,t (N ), K λ i,t (N )) -c i (e λ i,t (N ), Ω λ i,t (N )) , (z i,t (N )) i∈N ∈ Z e λ t (N ) (N ) = yλ t+1 (N ). (15)
By combining Eqs 12, 14 and 15, we deduce that for each t ∈ N,

i∈N η λ i,t (N ) ≤ yλ t (N ). ( 16
)
We can conclude from Eqs 11 and 16 that (e λ t (N ), η λ t (N )) t∈N is a feasible path. By definition of yλ t (N ) for each t ∈ N, and since Since (e λ t (N ), η λ t (N )) t∈N is a feasible path, and that Ω λ 0 (N ) = Ω 0 (N ), K λ 0 (N ) = K 0 (N ) and yλ 0 (N ) = y0 (N ), the definition of V Ω 0 (N ),K 0 (N ),y 0 (N ) (N ) ensures that

z λ t (N ) ∈ Z e λ t (N ) (N ) for each t ∈ N, we have yλ t+1 (N ) ≥ i∈N f i (z λ i,t (N ), K λ i,t (N )) -c i (e λ i,t (N ), Ω λ i,t (N )) . ( 17
V Ω 0 (N ),K 0 (N ),y 0 (N ) (N ) ≥ t∈N δ t yλ t (N ) - Using Eqs. 20, 13 and 19, we obtain

V Ω 0 (N ),K 0 (N ),y 0 (N ) (N ) = y0 (N ) - V Ω 0 (S),K 0 (S),y 0 (S) (S).

The strict inequality holds by strict concavity and since λ(N ) ̸ = 1. By Proposition 3 stating the Bondareva-Shapley theorem, C(N, V Ω 0 (N ),K 0 (N ),y 0 (N ) ) ̸ = ∅ and by Proposition 4, C(N, V Ω 0 (N ),K 0 (N ),y 0 (N ) ) contains an infinity of elements.

Step 2: Construction of the policy.

Step 1 states that for each state (Ω, K, y), the core C(N, V Ω,K,y ) is nonempty. Hence, for each initial state (Ω, K, y) we can build a sequence ((X i,t (Ω, K, y)) i∈N ) t∈N such that for each t ∈ N: (i) i∈N X i,t (Ω, K, y) = V Ω * t (N ),K * t (N ),y * t (N ) (N ), (ii) i∈S X i,t (Ω, K, y) ≥ V Ω * S,t (N ),K * S,t (N ),y * S,t (N ) (S), where (Ω * t (N ), K * t (N ), y * t (N )) t∈N is the sequence of state vectors associated with the optimal extraction and investment path of N when the initial state is (Ω, K, y).

Let π the mapping which associates to each initial state (Ω, K, y) the allocation path for each country (π i,t (Ω, K, y)) i∈N,t∈N such that for each t ∈ N, π i,t (Ω, K, y) = X i,t (Ω, K, y) -δX i,t+1 (Ω, K, y).

Let us first show that π is an international distribution policy. For each t ∈ N, Let us now prove that π is an undominated policy. For each T ∈ N, we have We conclude that π belongs to the strong sequential core C. Since π was built for each initial state (Ω, K, y) with an arbitrary element of C(N, V Ω,K,y ), we conclude from Step 1 that C also contains an infinity of elements.

  and ∀i ∈ N, S∋i λ(S) = 1. Denote by B(N ) the set of balanced maps over N . Proposition 3. (Bondareva-Shapley Theorem) For each TU-game (N, v), C(N, v) ̸ = ∅ if and only if for each balanced map λ ∈ B(N ), the following inequality holds: S⊆N λ(S)v(S) ≤ v(N ).

Proposition 4 .

 4 If, for each balanced map λ ∈ B(N ) such that λ(N ) ̸ = 1 the following strict inequality holds: S⊆N λ(S)v(S) < v(N ), then C(N, v) contains an infinity of elements.

  i∈N π i,t (Ω, K, y) = i∈N X i,t (Ω, K, y) -δ i∈N X i,t+1 (Ω, K, y) = V Ω * t (N ),K * t (N ),y * t (N ) (N ) -δV Ω * t+1 (N ),K * t+1 (N ),y * t+1 (N ) (N )and the Bellman equation ensures that:V Ω * t (N ),K * t (N ),y * t (N ) (N ) = v y * t (N ),η * t (N ) (N ) + δV Ω * t+1 (N ),K * t+1 (N ),y * t+1 (N ) (N ).Therefore, we have for each t ∈ N,i∈N π i,t (Ω, K, y) = v y * t (N ),η * t (N ) (N ).

  i,t+T (Ω, K, y) = i∈S ∞ t=0 δ t X i,t+T (Ω, K, y) -δ t+1 X i,t+T +1 (Ω, K, y) = i∈S X i,T (Ω, K, y) ≥ V Ω * S,T (N ),K * S,T (N ),y * S,T (N ) (S).

  N and t ∈ N, z λ i,t (N ) (resp. K λ i,t (N ), e λ i,t (N ), Ω λ i,t (N )) is a convex combination of (z * i,t (S)) S∋i (resp. (K * i,t (S)) S∋i , (e * i,t (S)) S∋i , (Ω * i,t (S)) S∋i ). Combining Eqs. 17 and 18, we have: yλ t+1 (N ) >

		λ(S) f i (z * i,t (S), K * i,t (S)) -c i (e * i,t (S), Ω * i,t (S))
	i∈N S∋i
	=	λ(S)	f i (z * i,t (S), K * i,t (S)) -c i (e * i,t (S), Ω * i,t (S))
	S⊆N	i∈S
	=	λ(S)y * t+1 (S).	(19)
	S⊆N
				)
	The hypothesis of strict concavity ensures that

i∈N f i (z λ i,t (N ), K λ i,t (N )) -c i (e λ i,t (N ), Ω λ i,t (N )) > i∈N S∋i λ(S) f i (z * i,t (S), K * i,t (S)) -c i (e * i,t (S), Ω * i,t (S))

(18)

because for each i ∈

Note that this model could be extended to a finite number of resources.

Of course, this function is economically irrelevant if the stock of resources is smaller than the extracted flow: this feasibility constraint is integrated in the definition of the feasible sets.

By convention, the strategy starts at t = 0, meaning that the first pair to be chosen is (e0(S), η0).

The optimal way to trade remains unique (Proposition 1), as well as the optimal path of extraction (first part of Proposition 2). An optimal path of investment is, with this convention, a time sequence of functions that is not unique anymore. However, when applied to the optimal path of extraction, all the optimal paths of investment functions would have the same image. Therefore, the uniqueness result can be found again (using the same kind of arguments of compactness, continuity and concavity as used in our proofs) but in a much more complex form. This technical point is left to the reader.

This can be justified for both interpretations of the definition of capital as raised by the Cambridge controversy. If a physical capital is at stake, we can take the example of a factory that cannot be moved, but can only be bought or sold. If we are talking about monetary capital, accumulating capital in another country means buying or selling shares of a company, which is the exact definition of a foreign direct investment.

The feasibility constraint on investments in problem (2) ensures that v yt (S),η t (S) is always nonnegative, as well as V Ω 0 ,K 0 ,y 0 (S) (S). For example, this is true if S = {i}. So, each country can ensure a nonnegative consumption in autarchy. The nonnegativity of each country's consumption stream follows from the definition of the strong sequential core. Indeed, the level of consumption of each country is given by a policy belonging to this core which guarantees to each country a better consumption than the one secured by autarchy (or any other subcoalition).

Note that the domain of the reward function corresponds to the action space[START_REF] Rangarajan | A first course in optimization theory[END_REF], not to be confused with the feasible set for which investments cannot exceed production.

It follows from Proposition 1 that both vectors are uniquely defined.

Acknowledgments

This work was first developed as a chapter of Fatma Zahra Rostom's PhD dissertation. We are indebted to Raimund Bleischwitz, Lise Clain-Chamosset-Yvrard, Mouez Fodha, Gaël Giraud, Céline Guivarch, Francesco Ricci, Philippe Solal, Xavier Venel, Olivier Vidal and Stéphane Zuber for their fruitful comments and suggestions. We also thank the anonymous referees for their stimulating comments. All remaining errors are our own. This paper was presented at the 19th Annual SAET Conference and the International Conference of Public Economic Theory 2019. Financial support of IDEXLYON from Université de Lyon (project INDEPTH) within the Programme Investissements d'Avenir (ANR-16-IDEX-0005), "Mathématiques de la décision pour l'ingénierie physique et sociale" (MODMAD) and the Chaire " Énergie et Prospérité" are gratefully acknowledged.

Appendix A Proof of Proposition 1

Proof. Let g : [0, Ω] S × [0, Ω] S × [0, K] S × R S + → R + be the function defined as g((Ω, e, K), z)

and C : [0, Ω] S × [0, Ω] S × [0, K] S ⇒ R S + be the correspondence defined as C(Ω, e, K) = Z e (S).

By continuity and strict concavity of

and has a convex graph. We have the needed conditions to apply Berge's Maximum Theorem under Convexity (Sundaram, 1996, p.237). We conclude that max{g (Ω, e, K), z , z ∈ C(Ω, e, K)} is a continuous and strictly concave function on [0, Ω] S × [0, Ω] S × [0, K] S and C * (Ω, e, K) = argmax { g (Ω, e, K), z , z ∈ C(Ω, e, K)} is a continuous single-valued function.

Proof of Proposition 2

Proof. Existence. The existence of an optimal path of extraction and investment is a direct application of a well-established theorem of existence in dynamic optimization (see for instance Theorem 12.19 in Sundaram (1996, p.298)). Indeed,

Hence, the transition function is continuous.

(iii) The feasible action correspondence (Ω, y(S)) → Φ Ω,y(S) (S) is compact-valued and continuous.

Uniqueness. Assume by way of contradiction that there exist two different optimal paths (e 0 t , η 0 t ) t∈N and (e 1 t , η 1 t ) t∈N when the initial vector is (Ω 0 , K 0 , y0 (S)) for a coalition S. We denote by (Ω 0 t ) t∈N and (Ω 1 t ) t∈N the respective sequence of resource stock vectors defined by Ω 0 0 = Ω 0 , Ω 1 0 = Ω 0 and for each t > 0, Ω 0 t+1 = Ω 0 t -e 0 t and Ω 1 t+1 = Ω 1 t -e 1 t . We denote by (K 0 t ) t∈N and (K 1 t ) t∈N the respective sequence of capital stock vectors defined by

For each α ∈ [0, 1], we define the path of extraction (e α t ) t∈N as follows:

the path of investment (η α t ) t∈N is defined as:

Appendix B

First, let us recall some definitions, notations, and useful results of the cooperative game theory in order to demonstrate Theorem 1. Let N denote a fixed finite nonempty set with n members, who will be called agents or players. Coalitions of players are nonempty subsets of N .

A transferable utility (TU) game on N is a pair (N, v) where v is a mapping v : 2 N → R satisfying v(∅) = 0. We denote by G(N ) the set of all games over N . For any coalition S, v(S) represents the worth of S, i.e., what coalition S could earn regardless of other players.

A payoff vector is a vector x ∈ R N that assigns to agent i the payoff x i . A payoff vector is efficient with respect to (N, v) if i∈N x i = v(N ); it is coalitionally rational if i∈S x i ≥ v(S) for every possible coalition S.

The core of (N, v), denoted by C(N, v), is the set, possibly empty, of efficient and coalitionally rational payoff vectors:


Leonid Prigozhin
email: leonid@bgu.ac.il

Vladimir Sokolovsky
email: sokolovv@bgu.ac.il

Spectral method for inhomogeneous superconducting strip problems and magnetic flux pump modeling

Keywords: spectral method, high-Tc superconducting dynamo, inhomogeneous superconducting strip

We present a new spectral method for solving 2D magnetization and transport current superconducting strip problems with an arbitrary current-voltage relation, spatially inhomogeneous strips, and strips in a nonuniform applied field. The method is based on bivariate expansions in Chebyshev polynomials and Hermite functions and can be employed for numerical modeling magnetic flux pumps of different types and investigating AC losses in coated conductors with local defects. Using a realistic 2D version of the superconducting dynamo benchmark problem as an example, we show that this spectral method is a competitive alternative to finite element methods.

I. INTRODUCTION

Although finite element methods remain the most popular and versatile tool for solution of applied superconductivity problems, alternative approaches have also been proposed recently. One of them is the FFT-based methods for the 2D thin film and 3D bulk problems [START_REF] Vestgården | Modeling non-local electrodynamics in superconducting films: the case of a right angle corner[END_REF][START_REF] Prigozhin | Fast Fourier transform-based solution of 2D and 3D magnetization problems in type-II superconductivity[END_REF], another is based on the Chebyshev spectral methods derived for the problems formulated as a 1D integrodifferential equation with a Cauchy-type singular kernel or as a system of such equations (superconducting strips with or without transport current in a uniform field, stacks of strips, pancake coils) [START_REF] Sokolovsky | Chebyshev spectral method for superconductivity problems[END_REF]. Ideally, the spectral methods demonstrate an extremely fast exponential convergence. Even if this is not achieved, e.g., if the solution is not smooth enough, etc., they can still be at least as efficient as the finite element methods. Here we extended the spectral method to the 2D thin strip problems.

While Chebyshev polynomial expansions are convenient for solving 1D integral equations with the Cauchy and logarithmic type singular kernels, no similar approach has been developed for the Green-function-based 2D integral formulation. To circumvent this difficulty, we applied the Fourier transform in the along-strip direction. For each wave number, the 2D integral equation is represented in the Fourier space by a 1D singular integral equation written for the transverse coordinate; expansions in Chebyshev polynomials can be used to efficiently solve these equations. To realize this approach, one needs an efficient numerical implementation of the direct and inverse Fourier transforms. Typically, this is realized by replacing the infinite domain by a sufficiently large finite one and using the FFT algorithm on a uniform mesh. A different method we explored in our work is based on expansions in the Hermite functions which diagonalize the Fourier transform. An approximate solution to the 2D integrodifferential problem is, therefore, sought at each moment in time using bivariate Hermite-Chebyshev expansions. The method of lines is employed for integration in time.

II. PROBLEM FORMULATION

Let the superconducting strip of the width 2a be represented by a 2D domain in the plane 0:

z =   | | , | | . x y a  =   
Introducing the stream (magnetization) function ( , , )

g x y t satisfying ( , ) yx g g g =   =  - j
, where j is the sheet current density, we present first the problem formulation for the case of a nonuniform external field vanishing at infinity and zero transport current:

( ) ( ) e1 0 , d , z G g t h  -      - -  = +   r r r r e 1 0 00 (| | / ) / , | 0, | ( , ), n cc y a t e g j g j g g g x y - = = =     == e
where e ( , , ) z h x y t is the z-component of the applied magnetic field in the strip plane and g means t g  . Here we assumed the power current-voltage relation 1 0

( ) ( / ) /

n cc e j j j -= e j j with a field-dependent and, possibly, spatially inhomogeneous sheet critical current density c j which, however, becomes spatially homogeneous as . x →  Applying the Fourier transform with respect to x to the integral equation above, using the convolution theorem, and singling out the singular parts of the integral kernels, we obtain a 1D integrodifferential equation for g in the Fourier space for each wave number , k

j jj F x L L kL    =-
are used for the approximate Fourier transform of functions and its inverse; here L is the scaling factor. A standard solver is used for integrating in time the system of ordinary differential equations for the mesh node values of g . Details of our numerical scheme, including the algorithms for linear operations on the bivariate Hermite-Chebyshev expansions (interpolation, differentiation, integration, Fourier transform), can be found in [START_REF] Sokolovsky | Hermite-Chebyshev pseudospectral method for inhomogeneous superconducting strip problems and magnetic flux pump modeling[END_REF]. Here we present two numerical examples.

IV. EXAMPLES A. Inhomogeneous Strip with Transport Current

Let the transport current be harmonic, 200sin

B. Strip in a Nonuniform Field: the Dynamo Flux Pump

A realistic 2D version of the simplified 1D superconducting dynamo flux pump "benchmark" problem [START_REF] Ainslie | A new benchmark problem for electromagnetic modelling of superconductors: the high-Tc superconducting dynamo[END_REF] was solved by the finite element methods in [START_REF] Prigozhin | Two-dimensional model of a high-Tc superconducting dynamo[END_REF][START_REF] Ghabeli | 3D modeling of a superconducting dynamo-type flux pump[END_REF] and also by the FFT-based method in [START_REF] Prigozhin | Two-dimensional model of a high-Tc superconducting dynamo[END_REF]. Now we use the spectral method and compare (Fig. 2) the calculated voltages, V(t), induced in the superconducting strip (dynamo stator) by the rotating permanent magnet attached to the dynamo rotor; the open contour conditions are assumed. The rate of convergence was estimated using the relative voltage deviation (in the 1 L norm), , V  from the most accurate solution for each method (Table I); these results confirm the efficiency of the new spectral method. 2 The voltage induced during the second rotor rotation computed by the mixed finite element method [START_REF] Prigozhin | Two-dimensional model of a high-Tc superconducting dynamo[END_REF] and using the spectral Hermite-Chebyshev method.

  the dependence on the magnetic field. Numerical simulation results are shown in Fig. 1.

Fig. 1

 1 Fig. 1 Top: Sheet current density at the peak of the transport current (spatial inhomogeneity of the strip is indicated by the background colour). Bottom: Time-averaged loss power density.

Fig.

  Fig.2The voltage induced during the second rotor rotation computed by the mixed finite element method[START_REF] Prigozhin | Two-dimensional model of a high-Tc superconducting dynamo[END_REF] and using the spectral Hermite-Chebyshev method.

  1D strip problem with the same transport current, the uniform external field e () t

	g	( , , ) g x y t g y t ( , ),  =-	where	( , ) g y t
			 h	and spatially homogeneous c j .
				III. NUMERICAL METHOD
		To solve the singular integral equation (1), for each wave
	number k	we seek ( g	k	,	y	,	t	)	as an expansion in Chebyshev
	polynomials depending on		/ ya and this enables efficient
	treatment of both kernel singularities. The interpolating
	expansions in scaled Hermite functions		( j xL / )	, vanishing
	at infinity and satisfying			[	(	/	)]	2	(	i)	(	),
											a a k -	|   ln | | 22 11 , , d 2 y k y y g k y t y , , d , g k y t y Z yy     --     +  = - a a -  - 		+	(1)
											where	[] f F f =	denotes the Fourier transform,
											e1 0 z Z h  -= +	(	x y e  -	y x e	)	,	and the functions (	, k y y -	)
											and (	, k y y -	)	are regular.
											For problems with a given transport current () It and a
											nonvanishing but uniform at infinity external field
											ee  → hh (	( ) t	as	) x →  , the formulation (1) in the Fourier
											space has to be modified and written for the difference,



is the solution of a

TABLE I . Convergence of Two Numerical Methods.

 I

	Mixed finite elements [6]	Spectral method
	N of	Time/cycle
	elements	(min)


Ghazi Hajiri

Kévin Berger

Jean Lévêque

Modelling of the pulsed field magnetization of HTS bulks in solid nitrogen at 40 K

Keywords: HTS bulks, A-H formulation, phase change modelling, cryogenics

To date, the pulsed field magnetization (PFM) method is considered one of the most promising magnetization methods due to its low cost and flexibility to magnetize samples within the final system. In addition, REBCO bulks with an operating temperature below 77 K are increasingly used in electrical machine applications, where cooling is primarily provided by a closed helium cold loop. In the same context, solid nitrogen (SN2) can provide a stable cryogenic environment for magnetizing several bulks simultaneously. This paper discusses the magnetothermal study and modelling of the magnetization of several REBCO bulk samples during their magnetization. For the electromagnetic calculation, the A-H formulation is used, coupled with a thermal model where the solid-liquid phase change of nitrogen has been considered. 2D and 3D numerical results will be compared in the final paper. Furthermore, the results will be compared and validated with experimental measurements.

I. INTRODUCTION

Since the manufacturing and commercialization of superconducting bulks, several methods have been used for their magnetization, among which the so-called field cooling and zero field cooling methods. Alternatively, the pulsed field magnetization (PFM) method is considered the most common method to magnetize bulks inside a superconducting machine, due to its low cost and flexibility [START_REF] Huang | Trial Test of a Bulk-Type Fully HTS Synchronous Motor[END_REF]- [START_REF] Deng | Pulsed Field Magnetization Properties of Bulk RE-Ba-Cu-O as Pole-Field Magnets for HTS Rotating Machines[END_REF].

Within this context, several investigations have been conducted at liquid nitrogen temperature and atmospheric pressure, where the magnitude of the magnetic field trapped in the bulk was limited by the critical current density of the sample at 77 K [START_REF] Ida | Waveform Control Pulse Magnetization for HTS Bulk With Flux Jump[END_REF]. At very low temperatures, other research employs direct conduction cooling where the cooling of the system relies on the use of one or more cryocoolers for the high temperature superconducting (HTS) bulks and the magnetizing coil [START_REF] Yokoyama | Influence of the Shape of Soft-Iron Yoke on Trapped Field Performance of HTS Bulk[END_REF]. In this regard, solid nitrogen (SN2) can create both a homogeneous thermal environment for cooling and great flexibility for testing multiple and various samples. In addition, solid nitrogen can be used as a thermal buffer to improve the thermal stability of superconducting systems under transient heating circumstances [START_REF] Kim | The design and testing of a cooling system using mixed solid cryogen for a portable superconducting magnetic energy storage system[END_REF].

Recently, the stability of the 3D A-H formulation has been validated and discussed several times [START_REF] Kapek | 2D and 3D validation of a hybrid method based on A and H formulations for Pulsed Field Magnetization[END_REF]- [START_REF] Dular | What Formulation Should One Choose for Modeling a 3D HTS Magnet Motor Pole with a Ferromagnetic Material?[END_REF]. So far, the coupling of the A-H formulation with thermal equations is typically based on the use of a convection coefficient h in case the superconducting material is immersed in a liquid, or by means of a thermal resistance if a cryocooler is used. In this work where the superconducting samples are embedded in solid nitrogen, the thermal equations are based on a solid-toliquid phase change formulation and coupled with the A-H formulation. Moreover, the results of the final 3D coupled A-H formulation will be compared with experimental measurements. The cooling system and process are thoroughly described in [START_REF] Hajiri | Design and Testing of a New Cooling System using Solid Nitrogen for Pulsed Field Magnetization and Characterization of HTS Bulks[END_REF].

II. MODELLING

The set of equations was implemented and solved using COMSOL Multiphysics 5.6.

A. A-H formulation

The coupled A-H formulation is based on the resolution of a magnetic vector potential A equation in domains such as air and conductors and a magnetic field H equation in superconducting parts only. As shown in Fig. 1, the Hformulation (Hf) is used in the ΩHTS superconducting domain, where the equation to be solved can be written as follows:

𝜇 0 𝑑 𝑡 𝐇 𝐇f + ∇ × (𝜌∇ × 𝐇 𝐇f ) = 0 𝐇 𝐇f ∈ ΩHTS (1)
The A-formulation (Af) is applied in the coil ΩCoil and air ΩA domains, where the vector potential A is defined by 𝐁 𝐀f = ∇ × 𝐀 𝐀f . The equations to be solved can be written as:

∇ × (𝜇 0 -1 ∇ × 𝐀 𝐀f ) = 𝐽 Coil 𝐀 𝐀f ∈ ΩCoil (2) ∇ × (𝜇 0 -1 ∇ × 𝐀 𝐀f ) = 0 𝐀 𝐀f ∈ ΩA (3)
The coupling between the two formulations Af and Hf is provided by a vector equality of the magnetic field at the boundary between the two domains, i.e., 𝛤 HTS in the case studied. In order to ensure the continuity of the physical quantities, the Neumann conditions are used, and the coupling equations can be expressed in the form:

𝐧 × 𝐇 𝐇f = 𝐧 × (𝜇 0 -1 ∇ × 𝐀 𝐀f ) on 𝛤 HTS (4) 𝐧 × (-𝑑 𝑡 𝐀 𝐀𝐟 ) = 𝐧 × (𝜌∇ × 𝐇 𝐇f ) on 𝛤 HTS (5)

B. Thermal equation

To account for thermal effects during magnetization, the heat diffusion equation is implemented in each domain as follows:

𝛾𝐶 p 𝑑𝑇 𝑑𝑡 + ∇ • (-𝑘∇𝑇) = { 𝑄 HTS 𝑄 Coil 0 𝑇 ∈ Ω HTS 𝑇 ∈ Ω Coil 𝑇 ∈ Ω N2 (6)
with k the thermal conductivity, Cp the specific heat and 𝛾 the material density. Furthermore, in each domain ΩHTS and ΩCoil, we define the losses per unit of volume QHTS and QCoil respectively.

C. Solid-liquid phase change

The heat capacity formulation is implemented to consider the solid-liquid phase change. The principle consists of adding the latent heat L to the heat capacity when the material reaches its phase change temperature 𝑇 SN2→LN2 . Moreover, the transformation from the solid to the liquid state takes place in a temperature interval ΔT around 𝑇 SN2→LN2 . In order to account for the material property in its proper state, two smooth functions θSN2 and θLN2 are defined. As an example, θSN2 is equal to 1 before 𝑇 SN2→LN2 ΔT ⁄ 2 and 0 after 𝑇 SN2→LN2 + ΔT ⁄ 2, and inversely true for θLN2. In other words, the sum θSN2 + θLN2 is always equal to 1.

𝑘 = 𝜃 SN2 𝑘 SN2 + 𝜃 LN2 𝑘 LN2 (7)
𝐶 p = ( 1 𝛾 (𝜃 SN2 𝛾 SN2 𝐶 pSN2 + 𝜃 LN2 𝛾 LN2 𝐶 pLN2 )) + 𝐿 SN2→LN2 𝑑𝛽 N2 𝑑𝑇 (8) 𝛾 = 𝜃 SN2 𝛾 SN2 + 𝜃 LN2 𝛾 LN2 (9)
In a constant pressure phase change, 𝛽 N2 represents the mass ratios between the liquid and the solid states, it can be represented as a function of the different density and smooth function θSN2 and θLN2.

𝛽 N2 = 1 2 𝜃 LN2 𝛾 LN2 -𝜃 SN2 𝛾 SN2 𝜃 LN2 𝛾 LN2 + 𝜃 SN2 𝛾 SN2 (10)
III. RESULTS AND CONCLUSION In this section, the magnetization of a REBCO bulk is modelled within an axisymmetric 2D problem. The bulk has a radius of 15.2 mm and a height of 17 mm. The power law E(J) as well as the dependence of the critical current on the magnetic field and the temperature Jc(B, T) have been considered. The current pulse is provided by the discharge of a 5 mF capacitor and an initial charge voltage U0 that can reach up to 2000 V. To consider the electrical components of the magnetization system [START_REF] Kapek | 2-D Numerical Modeling of a Bulk HTS Magnetization Based on H Formulation Coupled With Electrical Circuit[END_REF], the electrical equations of the magnetizer were integrated within the A-H formulation. Fig. 2 shows the influence of the discharge voltage on the distribution of the trapped magnetic flux density at the top surface of the REBCO bulk placed in solid nitrogen at 40 K at the end of the PFM process, i.e. t = 1000 s. Fig. 3 shows the current pulse and the average temperature of the bulk during a PFM for different values of the capacitors charging voltage U0 of 1000 V, 1500 V and 2000 V. In addition, the average temperature of the bulk calculated by using the solid-liquid phase change formulation and under adiabatic conditions are shown. One can note that during the current rise before t = 2 ms, the temperatures of both models are similar, this is related to the fact that the thermal time constants are considerably higher than the electrical ones. This behavior can offer advantages for 3D simulations. Indeed, to decrease the computation time, during the current rise, the solving of the thermal equations can be considered only in ΩCoil and ΩHTS under adiabatic conditions. The phase change formulation in ΩN2 can then be applied after 2 ms with the appropriate initial conditions observed at the end of the adiabatic regime.

In the final version of the paper, the behavior of solid nitrogen during the application of multiple pulses will be examined. In addition, a coupled 3D model concerning the magnetisation of a set of several bulks will be detailed and compared to the experiment.

Fig 1 .

 1 Fig 1. Breakdown of the studied domains and thermal coupling: the bulk ΩHTS, the coil ΩCoil, the nitrogen ΩN2 and the air ΩA. The boundaries between the superconductor and the air are denoted ΓHTS and the boundary of the air formulation is ΓA.

Fig. 2 .

 2 Fig. 2. Distribution of the magnetic flux density norm at the top surface of a REBCO bulk placed in SN2 at 40 K at the end of the PFM process and for different values of the capacitors charging voltage U0. Fig. 3. Current pulse and average temperature of the bulk versus time for different values of U0 equal to 1000 V, 1500 V and 2000 V. Two average temperatures of the bulk are shown: T * is given according to the phase change formulation and T + is under adiabatic conditions.


Ghazi Hajiri

Anjela Koblischka-Veneva

Kévin Berger

Quentin Nouailhetas

Michael R Koblischka

Jean Lévêque

Numerical modelling of a compact trapped field magnet using an HTSc tape as energizing coil during PFM

Keywords: Trapped field magnet, pulsed magnetization, modelling, coupled T-A-H formulations

In this contribution, we present a design of a compact trapped field (TF) magnet unit. The unit consists of a (RE)BCO bulk (diameter d, height h), enclosed in a stainlesssteel ring (thickness s 0 ) to minimize cracking effects of the bulk sample like in several other TF-systems already designed in the literature [START_REF] Oka | Processing and applications of bulk HTSC[END_REF]- [START_REF] Vakaliuk | Trapped field potential of commercial Y-Ba-Cu-O bulk superconductors designed for applications[END_REF]. A short coil is wound around the bulk sample with n layers, which is energized by pulsed currents, and will be cooled together with the bulk sample to the operating temperature. Such a compact TF unit, which we may call PTF magnet hereafter, may be valuable for a variety of applications. The original idea of such a system was described by Koblischka et al. for application in a trapped field flux pinning docking interface (TF-FPDI) for satellites in space [START_REF] Koblischka | Flux Pinning Docking Interfaces in Satellites Using Superconducting Foams as Trapped Field Magnets[END_REF]. However, there are many more applications like within robot arms or crane systems to contactlessly attach and release metallic/magnetic parts or also for medical drugdelivery applications [START_REF] Price | Magnetic Drug Delivery: Where the Field Is Going[END_REF], [START_REF] Liu | A review of magnet systems for targeted drug delivery[END_REF]. All these applications have in common that the unit will be cooled by cryo-cooling systems to a temperature of 77 K, or lower in case of a space use, in order to be cost-effective and easy to handle. Furthermore, for use in satellites, the unit weight is a crucial point, and for the targeted medical applications, a hand-held system is desirable. In this point of view, it is essential to optimize all system parameters d, h, n, s 0 and the parameters of the required energizing unit and the cryo-cooler to obtain a maximum TF value with minimum material cost, dimensions and weight.

II. DESCRIPTION AND MAIN PARAMETERS OF THE SYSTEM

As shown in Fig. 1, the diameter of the bulk cannot be too small as the max. TF field depends crucially on this size, and also the bending radius of the tape is not allowed to be too small. Thus, it is clear that a lower limit for d exists. On the other hand, an upper limit is given by the magnetic field produced by the HTSc coil which must be reasonably high to allow for proper field trapping. As d + s 0 corresponds to the inner diameter of the HTSc coil, the upper limit (= max. TF) is a function of (d + s 0 ). The stainless-steel ring is required for avoiding cracking of the bulk during energizing, so we must select a proper s 0 and pulse strength and duration for our compact PTF unit. An additional constraint are the required insulation and impregnation layers, shielding the flowing currents as well as protecting the superconducting elements. We make use here of the recent developments of Refs. [START_REF] Takahashi | Numerical Simulation of Electromagnetic and Thermal Stress in REBaCuO Superconducting Ring and Disk Bulks Reinforced by Stainless Steel Ring With Various Widths During Field-Cooled Magnetization[END_REF]- [START_REF] Ainslie | Numerical modelling of mechanical stresses in bulk superconductor magnets with and without mechanical reinforcement[END_REF].

III. COUPLED 2D NUMERICAL MODELLING

In this section, we present the axisymmetric model of the complete system implemented on COMSOL Multiphysics. In order to reduce the computation time of the simulation, the solution of the electromagnetic equations is based on the coupling of three formulations, i.e., T, A and H, which is for the first time observed here to the best of the authors' knowledge. As shown in Fig. 2, the three formulations have been implemented in the different domains as follows: • The A formulation is solved in air (Ω A ), and the magnetic flux density B is obtained by B = ∇ × A.

• The H formulation is solved in the YBCO bulk (Ω Bulk ), and the current density J is obtained by the relation

J = ∇ × H.
• The T formulation is solved in the HTSc coil (Ω HTSc ), where the determination of the current density J is obtained by J = ∇ × T.

As shown in Fig. 3, the HTSc tape is described as a surface current density with I = (T 1 -T 2 ) / δ, where δ is the thickness of the superconducting layer and T 1 and T 2 are the current vector potentials at the tape edges.

The equation to be solved in the different domains are:

∇ × (𝜇 0 -1 ∇ × 𝐀) = 0 Ω A ( 1
)
𝜇 0 𝑑 𝑡 𝐇 + ∇ × (𝜌 Bulk ∇ × 𝐇) = 0 Ω Bulk ( 2
)
𝜇 0 𝑑 𝑡 𝐇 + ∇ × (𝜌 HTSc ∇ × 𝐓) = 0 Ω HTSc ( 3
)
with 𝜇 0 the permeability of the air and 𝜌 the resistivity of the considered material. For superconducting materials, the power law E(J) = E c (J/J c ) n has been considered. The critical electric field E c is equal to 1 µV/cm with n = 25 and 8 for Ω Coil and Ω Bulk respectively. The critical current density of the bulk depends on the local magnetic field at the operating temperature of the bulk while the critical current of the HTSc tape is equal to 725.6 A/cm with a tape width of 12 mm, which corresponds to SuperOx tape data at 77 K.

The coupling between the T and A formulation is achieved through the exchange of the current density J and the magnetic flux density B in the HTSc tape [START_REF] Zhang | An efficient 3D finite element method model based on the T-A formulation for superconducting coated conductors[END_REF]. On the other hand, the coupling between the A and H formulations is achieved by a vector equality of the magnetic field at the boundary between the two domains Γ Bulk [START_REF] Dular | On the Stability of Mixed Finite-Element Formulations for High-Temperature Superconductors[END_REF].

IV. RESULTS AND DISCUSSION

Fig. 4 shows a 3D view of the magnetic field trapped in the (RE)BCO bulk and in the different layers of the HTSc tape. In this example, the HTSc coil contains 80 turns with a peak current value equal to 0.9 I c , i.e., 783.65 A. In Fig. 5, the evolution of the trapped magnetic field at 77 K is plotted as a function of the number of layers of the superconducting tape for I max /I c = 0.7, 0.8 and 0.9. The increase in the number of turns has a significant impact on the value of the trapped field at the center of the top surface, and a compromise must be found between the size of the system and the desired performances.

In the final version of the paper, an optimization of the complete system will be performed. Indeed, the study will aim at maximizing the field trapped in the bulk while minimizing the losses of the magnetization system and its total weight. In our view, the coupling of the T and A-H formulations may have a strong potential in the electromagnetic simulation of electrical systems with the presence of HTSc bulk and coils such as motors. To that end, we plan to share this COMSOL model as soon as it is accepted on the HTS Modelling Workgroup website.

  Numerical modelling of a compact trapped field magnet using an HTSc tape as energizing coil during PFM Ghazi Hajiri, Kévin Berger, Michael R. Koblischka, Anjela Koblischka-Veneva, Quentin Nouailhetas, Jean Lévêque To cite this version: Ghazi Hajiri, Kévin Berger, Michael R. Koblischka, Anjela Koblischka-Veneva, Quentin Nouailhetas, et al.. Numerical modelling of a compact trapped field magnet using an HTSc tape as energizing coil during PFM. 8th International Workshop on Numerical Modelling of High Temperature Superconductors (HTS 2022), Kévin Berger (Université de Lorraine -GREEN), Jun 2022, Nancy, France. hal-03791411 I. INTRODUCTION

Fig. 1 .

 1 Fig. 1. Illustration of the compact PTF unit, consisting of a (RE)BCO bulk and several layers nt of superconducting tape wound around it. st is the thickness of the superconducting tape.

Fig. 2 .

 2 Fig. 2. (a) Representation of the different fields of study: H is calculated only in the bulk (gray), T is calculated only in the superconducting domain (blue), while A is calculated in air (light red). (b) Coupling between T and A-H formulations in the simulation. J and B are the two coupling variables.

Fig. 3 .

 3 Fig. 3. The thickness of the HTSc tape is reduced to a surface current density (1D element). The current is imposed by boundary conditions at the T1 and T2 edges of the tape. 𝛤Bulk is the boundary between the two regions (Bulk and Air).

Fig. 4 .

 4 Fig. 4. 3D view of the PTF at the end of the energization process at 77 K, where the norm of the magnetic flux density is presented both in the (RE)BCO bulk and the HTSc tape for Imax/Ic = 0.9 and 80 turns.

Fig. 5 .

 5 Fig. 5. Trapped field at the center of the top surface of the (RE)BCO bulk at 77 K as a function of the number of layers of the HTSc tape and for different values of Imax.

ACKNOWLEDGMENT

This work is part of the SUPERFOAM international project funded by ANR and DFG under the references ANR-17-CE05-0030 and DFG-ANR Ko2323-10, respectively.


Ghazi Hajiri

Kévin Berger

Rémi Dorget

Yanis Laïb
email: yanis.laib@univ-lorraine.fr

Jean Lévêque

Study of the current limiting capacity of 2G HTS tapes

Keywords: current limiting, 2G HTS tape, superconducting cable, coupled electro-thermal model

Recent years have seen an increasing interest in high temperature superconducting (HTS) cables as they provide a large current carrying capability as well as a potential current limiting feature that can increase the reliability of urban power networks. Within this context, this article investigates current limiting in HTS tapes for new current limiting DC cables by studying both the theoretical and experimental behavior of HTS tapes behavior when subjected to a sudden rise in current. The experiment consists of a sudden discharge of a capacitor bank in a coil connected through an HTS tape. This will allow the dynamic regime of the strip to be studied well above its critical current. A shunt placed in parallel with the HTS tape will allow us to vary the current in the different layers of the superconducting tape in order to strengthen both our different theoretical models and to give us a better practical knowledge of the tape which will then be used in the desired HTS cable.

I. INTRODUCTION

In recent years, the interest for high temperature superconducting (HTS) cables has grown as several cable projects are being deployed [START_REF] Stemmle | AmpaCity Project ? World's First Superconducting Cable and Fault Current Limiter Installation in a German City Center[END_REF], [START_REF] Tomita | Energy-saving railway systems based on superconducting power transmission[END_REF]. Indeed, the large current carrying capacity of HTS cables allows to increase network capacity in urban environment there is limited room. An additional interesting feature of HTS cables is the current limitation in case of fault [START_REF] Llambes | Performance of 2G HTS Tapes in Sub-Cooled LN2 for Superconducting Fault Current Limiting Applications[END_REF], [START_REF] Wojtasiewicz | Tests and Performance Analysis of 2G HTS Transformer[END_REF]. In order to properly design a current limiting DC HTS cable, it is important to properly model its behavior in limitation mode, i.e. when the current temporarily reach a current higher than the critical current Ic, and especially the current and temperature distribution between the different layers composing a tape [START_REF] Hajiri | Thermal and Electromagnetic Design of DC HTS Cables for the Future French Railway Network[END_REF], [START_REF] De Sousa | An opensource 2D finite difference based transient electro-thermal simulation model for three-phase concentric superconducting power cables[END_REF]. To this end, this article aims to compare the current limiting behavior of several commercial HTS tapes with different layer thicknesses and topologies. The present study establishes a coupled dynamic electro-thermal model of a single HTS tape considering the main layers. Once the model of the 2G HTS tape is well understood, a second step will be to model a whole cable by assessing its current limiting capabilities in its working environment.

II. STUDY OF THE CURRENT LIMITING CAPACITY OF 2G HTS TAPES

A. Experimental setup

Firstly, we aim to model and analyze the dynamic behavior of a single HTS tape. In order to create a large current rise in the tape, a discharge system is set up. As shown in Figure 1, the experimental system consists of a capacitor with a capacity of 5 mF and a supply voltage of up to 2 kV, resulting in peak current of up to 30 kA. The superconducting tape is placed in series with a coil, where the maximum value and rise time of the current depend mainly on its inductance, the value of the capacitor being fixed. Furthermore, two scenarios will be studied, the first one consists in placing only a superconducting tape in series with the coil, where the coil current Itotal and the tape current Itape are equal. In the second configuration, a conventional copper cable is placed in parallel with the tape, which we will later call "shunt". All accessible currents, i.e. those flowing through the coil and the shunt for the second scenario, are measured and analyzed to better understand the current distribution in the different layers of the tape, where Itotal = Itape + Ishunt.

A sample holder is designed to ensure electrical connection with the superconducting tape and the flow of high current. Each connection terminal consists of two solid copper plates to ensure good contact between the samples and the power supply. The contact area on both sides is equal to 50 mm × 4 mm. The whole system is immersed in a liquid nitrogen container at a temperature of 77 K and at atmospheric pressure.

B. Electro-thermal modelling of 2G HTS tapes

2G HTS tapes used in superconducting power cable applications are typically 4 mm wide. As shown in Figure 2, the first layer of the tape is the substrate, which typically contains the nonmagnetic Hastelloy C-276. Depending on the manufacturer, the thickness of this layer can vary from 50 to 150 μm to provide mechanical resistance. The second is a stack of buffer layers of about 0.5 μm, on which the superconducting layer is deposited. The thickness of the REBCO layer can also vary from 1 to 3 μm depending on the number of deposition steps. Finally, there is a silver layer of a few micrometers thick and a copper layer whose thickness depends on the tape application case.

In this part of the work, several tapes from different manufacturers will be evaluated, each sample having its own electro-thermal model and parameters considering its different layers and its geometry. As shown in Figure 3, each layer of the tape is described by electrical and thermal resistances. Both are composed of two pairs of resistors, i.e., in the x and y directions. By connecting the different elements forming the tape, the electric current and the heat flow can circulate in a 2D (xOy) plan. In addition, at each time step t+∆t, the electrical model exchanges the losses value with the thermal model, and in return, the thermal model sends the value of the temperature to the electrical model in order to consider the temperature dependence of the electrical parameters.

III. RESULTS AND DISCUSSION

Figure 4 shows the current and temperature of the HTS tape during the capacitor discharge for the two scenarios, i.e., with and without shunt and for a 10 cm in length tape immersed in liquid nitrogen with a power law index equal to 25, a critical current of 200 A and a copper layer thickness thCu equal to 25 μm. For the shunted tape scenario, one can observe that the tape temperature stays below 79 K and 82 K throughout the pulse for the two shunt resistors of 0.4 and 0.8 mΩ respectively. In contrast, without shunt, the current beyond the tape critical current must flow through the other layers of the tapes, resulting in higher losses during the pulse. As a result, the tape temperature reaches 295 K. Thus, the proposed electro-thermal model will be validated using both scenarios. The first scenario, where the shunt is present, will allow to evaluate the accuracy of the electrical model. Indeed, as shown in Figure 5, the temperature rise being limited (ΔT < 2 K), the thermal model has only a limited impact on the results. However, in the second scenario without shunt, the thermal model has a significant influence on the results. Thus, the study of the two proposed scenarios makes it possible to discriminate separately the errors or uncertainties caused by the thermal and electrical models.

In the final version of the paper, the above study and the corresponding experiment will be performed on several commercial HTS tapes with different copper thicknesses. In addition, the proposed electro-thermal model will be compared with a more detailed 2D quench model based on the T-A formulation. Besides, after the validation of the single tape model, a whole cable model will be considered, and the different elements and the behavior of the cable will be discussed.

Figure 1 .

 1 Figure 1. Circuit diagram of the pulse current supply for a HTS tape.

Figure 2 .

 2 Figure 2. View of the different layers of a typical 2G HTS tape.

Figure 3 .

 3 Figure 3. Diagram of the electro-thermal model of a 2G HTS tape.

Figure 4 .

 4 Figure 4. Tape current and temperature as a function of time, with a copper layer thickness of 25 μm, without shunt in parallel to the HTS tape and for two different values of Rshunt.

Figure 5 .

 5 Figure 5. Tape, shunt and total current and temperature of the tape versus time, for two different copper thicknesses and Rshunt = 0.4 μm.


Ghazi Hajiri

Bruno Douine

Kévin Berger

Frederic Trillaud

Study of AC losses in HTS tapes coupling FEM and electrical circuits with harmonics

Keywords: AC losses, coupled T-A formulation, cosimulation, harmonics, HTS tapes

To advance the understanding of the behavior of High Temperature Superconductors (HTS) in electrical systems involving several components, it becomes relevant to couple high-end Finite Element Models (FEM) and electrical circuits made of lumped parameters. However, such an approach is often limited by the computation cost associated with computing discrete variables especially when relatively large time scales are involved such as those encountered in power systems. Nevertheless, it is a powerful tool if the system as much as the component should be detailed to understand the impact of the material on the overall response of the system. Here, a step-bystep approach is taken to validate a coupling between FEM and electrical circuits involving HTS and its response under harmonics. First, a single tape is characterized in terms of its losses, and measurements are compared to simulation results with and without circuit coupling. The same protocol is used to cross-check the FEM-Circuit coupling on a coil that targets a more complex co-simulated system.

I. INTRODUCTION

The next step in the development of HTS models, apart from more efficient 3D models [START_REF] Zermeño | 3D modeling and simulation of 2G HTS stacks and coils[END_REF]- [START_REF] Hajiri | Design and modelling tools for DC HTS cables for the future railway network in France[END_REF], is the capability to connect HTS components to external complex electrical systems. Systems such as those found in power grids involve large scales in terms of dimensions and time constants [START_REF] Zhu | Design, dynamic simulation and construction of a hybrid HTS SMES (hightemperature superconducting magnetic energy storage systems) for Chinese power grid[END_REF]. As a first step, the inclusion of HTS technologies in the distribution grid is appealing as often they are overloaded and congested [START_REF] Maguire | Development and Demonstration of a Fault Current Limiting HTS Cable to be Installed in the Con Edison Grid[END_REF]. For such networks for which the system remains relatively small, the coupling FEM-Circuit may be a plausible modeling option [START_REF] Wang | A Coupling Simulation and Modeling Method for High Temperature Superconducting Magnets[END_REF]. It would allow studying the performance of HTS technology interacting with complex surroundings. In the following work, initial steps in that direction are taken with a clear focus on validating step by step the coupling FEM-Circuit to lay out the basis for the cosimulation of REBCO devices for stringent conditions with current signals that may include harmonics. Hence, the results of simulations with and without coupling are compared to experimental data obtained for a single tape and a coil. First, the measurements are carried out in a pure AC regime [START_REF] Statra | Contribution to the Experimental Characterization of the Electromagnetic Properties of HTS[END_REF], then in AC conditions with harmonic contents by adding a diode in series with the sample. This approach provides a thorough analysis of the impact of the choice of co-simulation parameters on the accuracy of the simulation.

II. EXPERIMENTAL SETUP AND CIRCUIT MODEL

The layout of the experimental setup is shown in Fig. 1. Two experiments are considered. The first one is without a diode to generate a pure AC signal as a reference on the AC losses. The second uses a diode in series with the sample to produce harmonics as the AC current signal is clipped out over half a power cycle. Indeed, harmonics are found in power systems and are expected to affect the HTS.

As shown in Fig. 1 (a) and (b), the experimental setup consists of a home-made step-down transformer (TF) with a transformation ratio a = 0.13 (secondary "s" over primary "p") feeding current to the sample (S). The sample is a single tape or a coil. The transformer, bathed in liquid nitrogen, operates at a constant temperature of 77 K to ensure consistency in the settings of the experimental setup over the time of data recording. Fig. 1 (c) shows the different parameters of the setup. It is the exact circuitry modeled in Simulink with the sample simulated by the FEM. The energy is extracted from an AC power supply (P) with controllable frequency and current amplitudes. The TF parameters modeled in Simulink were estimated via short circuit and open circuit tests.

III. T-A FORMULATION IN FEM

The latest T-A formulation of the Maxwell equations is implemented in COMSOL Multiphysics version 5.6 for its fast computation time with a good accuracy to model REBCO technology [START_REF] Huber | The T-A formulation: an efficient approach to model the macroscopic electromagnetic behaviour of HTS coated conductor applications[END_REF]. The current vector potential T is given by:

𝜕 𝜕𝑥 ( 𝜕𝑇 𝑦 𝜕𝑦 ) = - 𝜕𝐵 𝑦 𝜕𝑡 (1)
and the magnetic vector potential A is obtained by solving: Δ 𝐀 = 0, hence providing the magnetic flux density as 𝐁 = ∇ × 𝐀 to solve T. The current is imposed as a boundary Dirichlet condition, 𝑇 1 -𝑇 2 = 𝐼 / 𝛿, with δ the thickness of the HTS layer. The indexes 1 and 2 refer to the edges of the 1D layer. For the magnetic vector potential, the following expression is used to impose the current as a surface current, 𝐧  (∇ × 𝐇 1 + ∇ × 𝐇 2 ) = 𝜇 0 𝐊, with H1 and H2 are the magnetic field on both side of the HTS layer and 𝐧 a normalized normal vector to the flat surface of the HTS layer, and 𝛿𝐊 = ∇ × 𝐓.

IV. SIMULINK AND COMSOL COUPLING

The coupling between COMSOL and Simulink is provided by the LiveLink TM Module. The Matlab/Simulink model manages the global variables and decides the communication time steps from t to t + Δt between COMSOL and Simulink. In our study, a constant communication time step Δt was set in order to study the coupling even though it can be automatically decided by Simulink. For each time step, Simulink sends the instantaneous value of current to COMSOL, where the later determines the initial conditions of FEM model at t = t0. At the end of the communication step at t = t0 + Δt, COMSOL provides the resistance of the superconducting sample to be used in the Simulink model. Finally, the losses Q are computed in COMSOL and postprocessed in Matlab.

V. PRELIMINARY RESULTS

The AC frequencies f0 are 20, 40, 60, and 80 Hz at a fraction of the critical current Imax/Ic between 60% and 90%. The bare REBCO tape, 4 mm x 0.1 mm thick, was supplied by Shanghai. The tape has a critical current of 138.5 A and a power law index n of 22. At this stage, only the tape (20 cm long) was measured in pure AC conditions at 80 Hz. Fig. 2 shows a comparison between the experimental data, the Norris' analytical formula for strips, and the FEM results without co-simulation for the initial experimental setup without the diode, see Fig. 1(c). The Norris' formula overestimates the AC losses with an average relative error of about 75%. For the FEM, tow T-A models are considered, T-A + and T-A * , with the same mesh size (dmesh = 40 μm). The T-A + is calculated with a constant Jc + = Ic / Asc with Asc the cross-section of the HTS layer. For the T-A * , a new Jc * is computed for each experimental loss to get the best agreement with the measurements within a 1% tolerance. The T-A * model is then used to study the impact of the cosimulation parameters on the simulation results. Fig. 3 shows the calculation of the instantaneous AC losses Q computed for Imax= 122.2 A at 80 Hz as a function of the communication time step Δt. The solver ODE 15s of Simulink is used with a relative error of 10 -6 and a maximum time step equal to 10 -3 / f0 = 12.5 µs. The error on the calculation of Q increases with the increase of Δt. A compromise between calculation time and accuracy is found as shown in Fig. 4. For an error limit of 10% (5% measurement error + 5% model accuracy) the optimum time step is about 16.66 μs in the present case.

VI. CONCLUSION

The first part of the work has been carried out on a single tape. Its AC losses have been compared to simulations. The next step is the study including the diode. Once the study on the single tape has been completed, a similar work will be carried out on a coil wound with insulated REBCO tape from the same batch as the single tape. The first results show the importance of the communication time step between the two softwares. More parameters should be still studied to complete the proposed objective. This work will provide a sound basis to build more complex co-simulation on more complex systems.

Fig. 1 .

 1 Fig. 1. (a) pictures of the transformer, (b) bath of liquid nitrogen at 77 K and single tape, (c) Schematic drawing of the experimental setup and cosimulation layout. The diode is present or absent according to the experimental protocol.

Fig. 3 .

 3 Fig. 3. Instantaneous AC losses Q calculated for different communication time steps Δt between COMSOL and Simulink (Imax = 122.2 A).

Fig. 4 .

 4 Fig. 4. Relative error e (left axis, red) and computation time (right axis, black) for Imax = 122.2 A and a frequency of 80 Hz as a function of the communication time step Δt between softwares.

Fig. 2 .

 2 Fig. 2. Current density Jc * (left axis, red) and AC losses per cycle W (right axis, black). Wn corresponds to the Norris' analytical model, WT-A + corresponds to the FEM results with Jc + constant, WT-A * corresponds to a nonconstant Jc * , and Wm are the experimental data.