Student Member, IEEE Manuel Marin

David Defour

Fellow, IEEE Federico Milano

Asynchronous Power Flow on Graphic Processing Units

Keywords: Asynchronous iterations, fixed-point method, power flow analysis, graphic processing units

Asynchronous iterations can be used to implement fixed-point methods such as Jacobi and Gauss-Seidel on parallel computers with high synchronization costs. However, they are rarely considered in practice due to the low convergence rate. This paper describes an implementation on GPUs of a novel Power Flow analysis model using asynchronous iterations. We present our model for the solution of the Power Flow analysis problem, prove its convergence and evaluate its performance for a GPU execution.

I. INTRODUCTION

Power Flow (PF) analysis refers to the steady-state analysis of AC power networks, widely used in several applications for power system operation and planning [START_REF] Milano | Power system modelling and scripting[END_REF]. Traditional PF analysis tools rely on matrix factorization using highly optimized serial direct solvers [START_REF] Simulator | Version 10.0 scopf[END_REF], [START_REF] Milano | A python-based software tool for power system analysis[END_REF], [START_REF] Simulink | The mathworks[END_REF]. More recently, a parallel-friendly PF analysis tool has also been introduced, based on an analogical representation of the system [START_REF] Nagel | Analog microelectronic emulation for dynamic power system computation[END_REF]. The real power system is mapped onto a CMOS integrated circuit, which allows one to physically simulate the system at a much higher frequency. The efficiency of this approach is linked with the intrinsic parallelism of the physical system, where every element of the network is acting simultaneously. The main drawback is the lack of flexibility since for every system that one wants to simulate, a new analogue circuit has to be built.

In this paper we present a software implementation of the same idea presented in [START_REF] Nagel | Analog microelectronic emulation for dynamic power system computation[END_REF], so that any arbitrary system can be analyzed without extra effort. This allows us to propose an intrinsically parallel PF analysis model. Our approach is based on fixed-point iterations performed individually for every element in the system. Results of succesive updates are exchanged until an equilibrium point is reached, as in the concept of team algorithms [START_REF] Talukdar | Designing algorithms and assignments for distributed processing[END_REF], [START_REF] Tsitsiklis | Problems in decentralized decision making and computation[END_REF]. The approach is well-suited for parallel implementation using asynchronous iterations, which have also been applied to the network flow problem [START_REF] Bertsekas | Distributed asynchronous relaxation methods for convex network flow problems[END_REF], [START_REF] Baz | Asynchronous iterative algorithms with flexible communication for nonlinear network flow problems[END_REF].

The main reason why asynchronous iterations are not usually preferred over synchronous ones is that (i) they require stronger assumptions in order to converge, and (ii) convergence rate is smaller. However, massively parallel computing environments can benefit from the asynchronous approach M. Marin is with Université de Liège, Institut Montefiore, Liège, Belgium (e-mail: mmarin@ulg.ac.be). D. Defour is with Université de Perpignan Via Domitia, DALI and Université Montpellier 2, LIRMM, France (e-mail: david.defour@univ-perp.fr).

F. Milano is with the School of Electrical, Electronic and Communications Engineering of the University College Dublin, Dublin, Ireland (e-mail: federico.milano@ucd.ie).

whenever the cost of synchronization becomes too high. For instance, in the Graphic Processing Unit (GPU) synchronization between threads from different blocks can consume up to 50% of the total execution time [START_REF] Xiao | Inter-block gpu communication via fast barrier synchronization[END_REF]. In order to target GPU architectures, we study convergence properties of our model and provide a sufficient condition for convergence.

The remainder of the paper is organized as follows. Section II introduces fixed-point iterations and presents several alternatives for parallel implementation, including synchronous, asynchronous and partially asynchronous frameworks. In Section III, the proposed intrinsically parallel method for PF analysis is presented and its convergence properties are studied. Section IV describes the implementation and evaluation of the proposed method on a set of benchmarks as well as comparisons with existing approaches. Finally, Section V draws conclusions and outlines future work.

II. FIXED-POINT ITERATIONS ON A PARALLEL COMPUTER

Let us first recall the concept of fixed-point iterations defined as follows.

Definition 1 (Fixed-point iteration). Let f : R n → R n and x ∈ R n . Then, x is a fixed-point of f (•) if and only if x = f (x). Furthermore, let x (0) ∈ R n . Then, the iteration given by:

x (k+1) = f (x (k) ), k = 0, 1, . . . , (1)
is called a fixed-point iteration on f (•).

Examples of fixed-point iterations are the Jacobi and Gauss-Seidel methods for solving systems of linear equations. Note that these methods can be succesfully applied to PF analysis by re-writing the (non-linear) PF equations in a suitable way [START_REF] Milano | Power system modelling and scripting[END_REF]. In this article, we consider the PF problem in its original nonlinear form.

Fixed-point iterations can be implemented in parallel by splitting the set of input element among threads. Each thread updates its respective elements and communicates its results to others with or without synchronization.

A. Synchronous iterations

With synchronous iterations, all the elements are updated by block before the process moves as a whole into the next step. The above requires to place a synchronization point at the end of each iteration. As a consequence, threads that finish their updates earlier are forced to wait for the others to catch up. The reason for threads having different processing times are various. Some of them may be intrinsic to the algorithm, e.g., irregular memory access patterns, load imbalance. Some others may be related to the architecture, e.g., differences in processor frequency, communication network constraints [START_REF] Frommer | On asynchronous iterations[END_REF].

B. Asynchronous iterations

An alternative to the synchronous approach is the so-called asynchronous iteration, where individual threads do not wait for others and simply go ahead with their work. This means that at every asynchronous iteration only a subset of the elements are updated. No synchronization point is explicitly added to the program. As a consequence, some of the updates may be computed with data from several iterations back.

An asynchronous iteration is defined in terms of an update function and a delay function. The update function, noted U(•), receives the iteration counter k ∈ N, and returns a subset of {1, . . . , n} indicating the list of threads that update their elements at iteration k. The delay function, noted d(•), receives the indices i, j ∈ {1, . . . , n} and the iteration counter k ∈ N, and returns the delay of thread j with respect to thread i at iteration k. This delay represents the number of iterations performed since the value of x j was last made visible to i.

Definition 2 (Asynchronous iteration). Let x ∈ R n and f : R n → R n . For k ∈ N, i, j ∈ {1, . . . , n}, let U(k) ⊆ {1, . . . , n} and d(i, j, k) ∈ N 0 , such that d(i, j, k) ≥ 0, ∀i, j, k, (2a)
lim k→∞ d(i, j, k) < ∞, ∀i, j, (2b)
|{k : i ∈ U(k)}| = ∞, ∀i. (2c)
In addition, let x (0) ∈ R n . Then, the iteration defined by: Assumptions in (2a), (2b) and (2c) ensure that the process is well defined. Specifically, the assumption in (2a) states that only values computed in previous iterations are used in any update. The one in (2b) states that newer values of the elements are eventually used. Finally, the assumption in (2c) states that no element ceases to be updated during the course of the iteration. In practical terms, if the updates are made visible as soon as they are computed, then the third assumption implies the second. This is the case, for example, of an application that uses GPU global memory to store the data. Note that a synchronous iteration is a special case of asynchronous iteration with U(k) = {1, . . . , n} and d(i, j, k) = 0, for all i, j, k. Due to the peculiarities introduced above, asynchronous iterations converge differently from synchronous ones. The following result considers the case of a linear application.

x (k+1) i = f i (x (k-d(i,1,k)) 1 , . . . , x (k-d(i,n,k)) n ) if i ∈ U(k), x (k) i if i ∈ U(k), (3)
Theorem 1 (Necessary and sufficient condition for convergence of linear asynchronous iterations [START_REF] Chazan | Chaotic relaxation[END_REF]). Let f : R n → R n , given by,

f (x) = Lx + b, ( 4
)
where L ∈ R n×n , and b ∈ R n . Let |L| denote the matrix of absolute values of the entries of L, and ρ(|L|) its spectral radius. If

ρ(|L|) < 1, (5)
then any asynchronous iteration on f (•) converges to x * , the unique fixed-point of f (•), regardless of the selection of U(•), d(•) and x (0) . Furthermore, if ρ(|L|) ≥ 1, then there exists U(•), d(•) and x (0) such that the associated asynchronous iteration does not converge to x * .

Proof. See [START_REF] Chazan | Chaotic relaxation[END_REF].

C. Partially asynchronous iterations

In some circumstances, convergence of asynchronous iterations can be facilitated by making threads synchronize every once in a while. Definition 3 (Partially asynchronous iteration). Consider Definition 2 of an asynchronous iteration. Replace assumptions in (2b) and (2c) by the following:

∃ d ∈ N : d(i, j, k) ≤ d, ∀i, j, k, (6a)
∃ s ∈ N : i ∈ s s=1 U(k + s), ∀i, k. (6b)
d(i, i, k) = 0, ∀i, k, (6c)
Then, the iteration given by equation ( 3) is now termed a partially asynchronous iteration.

The assumption in (6a) establishes that not only newer values of the vector elements are eventually used, but each of these values is used before d iterations have passed from their calculation. The assumption in (6b), in turn, states that each element is updated at least once in every s consecutive iterations. Finally, the assumption in (6c) establishes that every element is updated using its own last calculated value. In practical terms, the first two assumptions can be met by performing a synchronization step every l iterations, where l is the minimum of d and s. The third assumption can be met by having each element assigned to only one thread.

As mentioned above, partially asynchronous iterations converge 'more easily' than totally asynchronous ones. The following result establishes a convergence criteria in the linear case, with softer assumptions than the ones in Theorem 1.

Theorem 2 (Sufficient condition for convergence of linear partially asynchronous iterations [START_REF] Tseng | Partially asynchronous, parallel algorithms for network flow and other problems[END_REF]). Let f : R n → R n , given by,

f (x) = Lx + b, (7)
where L ∈ R n×n , and b ∈ R n . Let g : R n → R n , given by:

g(x) = (1 -α)x + αf (x), (8)
where α ∈ (0, 1).

If L = (l ij ) is irreducible and n j=1 |l ij | ≤ 1, ∀i, (9)
then any partially asynchronous iteration on g(•) converges to x * , fixed-point of f (•).

Proof. See [START_REF] Tseng | Partially asynchronous, parallel algorithms for network flow and other problems[END_REF].

In [START_REF] Lubachevsky | A chaotic asynchronous algorithm for computing the fixed-point of a nonnegative matrix of unit spectral radius[END_REF], the authors presented a special case in which convergence occurs for α = 1, at a linear (geometric) rate. They also determined a lower bound for the average convergence rate (per iteration) in the long-term, τ , as follows:

τ ≥ (1 -σ r ) 1/r , (10)
with

r = 1 + d + (n -1)(s + d), (11a)
σ = min i,j + x * i l ij x * j , (11b)
where min + (•) refers to the minimum of the positive elements. This means that the error is scaled at most by a factor of (1σ r ) 1/r in any average iteration. As σ is always lower than 1, the convergence rate approaches 1 for increasing values of n, d and s. That is to say, the convergence becomes sublinear.

For illustration purposes, consider that d = s = 0 (synchronous iteration). In this case, r = 1 and τ = 1σ. In other words, the convergence rate is unaffected by the problem size n and only depends on σ. Now consider that d = s = 1. This corresponds to a partially asynchronous iteration with a synchronization step every other iteration. In this case, r = 2n, which means that convergence slows down as the problem grows. Figure 1 shows the lower bound on the convergence rate as a function of the problem size n. The different curves represent different values of σ. Observe that, unless σ is very high, convergence becomes sub-linear very quickly.

Rate of convergence, τ

Problem size, n σ = 0.1 σ = 0.5 σ = 0.9 This might be a strong reason to prefer the synchronous approach over the partially asynchronous. However, the partially asynchronous approach can be a valid alternative in some cases, e.g., if the costs of synchronization are relatively high, or when the number of iterations required to converge is relatively low.

III. PROPOSED METHOD FOR AC CIRCUIT ANALYSIS

This section presents how to perform PF analysis based on partially asynchronous iterations. The method is designed to be implemented on Single Instruction Multiple Thread (SIMT) architectures such as GPUs, by exploiting regularity. The method is presented through a top-down approach. The network model is described first, specifying how the different parts communicate and interact. Next, the models of the different elements that compose the power system are specified in their atomic behaviour. Finally, the convergence properties of the proposed method are studied.

A. Network model

The proposed methodology distinguishes two fundamental units in the power system: components and nodes. The components are the electrical devices, and the nodes are the points of interconnection. For sake of simplicity and also to facilitate illustration of the method, the following three assumptions are made:

1) Only dipole components, i.e., components connected to two nodes, are present. 2) There is no more than one component connected between any two nodes. 3) Every node is connected to at least two components. The circuit is well defined if there are components connecting all the nodes in a closed path. Node h in the circuit is noted n h . The component connected between n h and n m is noted c hm . The set of all nodes in the circuit is noted N , and the set of components, C.

In addition, a set of influencers, I h ⊂ N , is defined for each non-ground node n h ∈ N . This set contains all the nodes separated from n h by exactly one component. These are also known as the fanin nodes [START_REF] Newton | Relaxation-based electrical simulation[END_REF].

For illustration purposes, consider the circuit in Fig. 2. In this case, N = {n 0 , n 1 , n 2 , n 3 } and C = {c 01 , c 12 , c 02 , c 23 , c 03 }. The circuit is well defined, since c 01 , c 12 , c 23 and c 03 connect all the nodes in a closed path. In addition, I 1 = {n 0 , n 2 }, I 2 = {n 0 , n 1 , n 3 }, and In the proposed methodology, Ú h denotes the voltage at node n h , and Ù hm denotes the voltage at node n h 'according' to component c hm . The two are different during the course of the iteration, and become equal at convergence. In addition, to all of them. Then, the process moves into a next iteration. Figure 3 illustrates the above for component c 12 and node n 2 in the example 4-node circuit.

I 3 = {n 0 , n 2 }.
replacements c12 Ú1, ∆×1 Ú2, ∆×2 Ù12 ×12 Ù21 ×21 n1 n2 Ú × Ù × Ù × (a) c12 Ú × Ú × Ù × Ù21 ×21 n0 n1 n3 c02 c23 n2 Ú2 ∆×2 Ù20 ×20 Ù23 ×23 (

B. Component and node models

1) Component model: For the purposes of the proposed method, only three types of component are required: the generator, the branch and the shunt. The use of a universal model improves instruction regularity of the GPU implementation, as discussed in Section IV. The impedance of component c hm is noted z hm . a) Generator: It is connected between the ground and one node, say n h . The generator is there to ensure that the reactive power mismatch at the node is equal to zero. In this sense, the model assumes that generators are able to provide as much reactive power as needed in order to maintain the voltage. The slack generator also does the same for real power. Accordingly, the iteration is the following: for nonslack generators,

Im(× (k) h ) = 0. ( 12
)
And for the slack generator,

× (k) h = 0. ( 13
)
b) Branch: It is connected between two non-ground nodes, say n h and n m . It consists of a constant impedance, z hm . The branch iteration is defined as follows:

Ù (k) hm = Ú (k) h + z hm ∆× (k) h Ú (k) h * , (14a)
× (k) hm = Ú (k) h Ú (k) m -Ú (k) h z hm * . ( 14b
)
c) Shunt: It is connected between the ground and one node, say n h . It represents an impedance to ground, noted z h0 (the subscript '0' indicates the ground). The shunt iteration is similar to the branch one. The main difference is that Ú m in equation ( 14b) is replaced by zero (the ground voltage). The iteration is the following:

Ù (k) h0 = Ú (k) h + z h0 ∆× (k) h Ú (k) h * , (15a)
× (k) h0 = - |Ú (k) h | 2 (z hm ) * , (15b)
where |•| indicates the absolute value. These equations are obtained by applying the Ohm Law on component c hm , and the Kirchhoff Current Law on node n h . Note that if the power mismatch at n h is equal to zero (i.e., ∆×

(k) h = 0), then Ù (k) hm = Ú (k)
h . That is to say, the voltage according to component c hm is the same as the one according to node n h itself. This corresponds to a condition of convergence at component level.

2) Node model: The node is simply modelled as a point where different components exchange data and consolidate their results. The node iteration is defined as follows:

Ú (k+1) h = m∈I h w hm Ù (k) hm , (16a)
∆× (k+1) h = m∈I h × (k) hm , (16b)
where w hm is defined as the weight of component c hm within node n h . These weights satisfy

m∈I h w hm = 1. (17)
In other words, the voltage at each node is updated with the weighted sum of the voltages computed by each component connected to it. Note that if convergence is attained at component level (i.e., Ù

(k) hm = Ú (k) h , ∀m ∈ I h ), then Ú (k+1) h = Ú (k) h .
This happens when all the components agree on the value of the node voltage, and corresponds to a condition of convergence at node level.

A second node iteration is added to allow PQ loads to become constant impedances, if the voltage at the node is outside specific limits. This is a technique used in traditional PF analysis [START_REF] Milano | Power system modelling and scripting[END_REF]. In the proposed method, the power injected to n h by loads is noted × L,h . If the voltage is outside the limits, this variable is scaled by a factor depending on the current voltage and the violated limit. Then, the power mismatch at the node is adjusted to reflect the change in the load. In conclusion, the iteration is composed of two parts. The first updates the power injected by loads, as follows:

× (k+1) L,h =              v (k) h v min h 2 × (k) L,h if v (k) h < v min h , v (k) h v max h 2 × (k) L,h if v (k) h > v max h , × (k) L,h
otherwise, [START_REF] Collange | Design challenges of GPGPU architectures: specialized arithmetic units and exploitation of regularity[END_REF] where v min h and v max h are the voltage limits at n h . Then, the second part updates the power mismatch, as follows:

∆× (k+1) h = ∆× (k) h -× (k) L,h + × (k+1) L,h . ( 19
)
The use of the above iteration reduces instruction regularity, as it needs to be performed only by nodes. However, it may be required for convergence of the method and involves a limited number of operations which can be acceptable in a SIMT execution context.

C. Convergence study

This section investigates the convergence properties of the proposed iterative method for two different synchronization schemes.

Let n 0 be the ground-node, and N ′ the set of all nodes in the circuit minus the ground. That is to say, N ′ = N \ {n 0 }. Also, let Ú ∈ C n-1 be the vector of voltages at all non-ground nodes. The iteration function, : C n-1 → C n-1 , is given by:

h (Ú) = Ú h + m∈I h j∈I h w hm z hm Ú j -Ú h z hj , h ∈ N ′ . ( 20
)
This expression is obtained by combining equations (14a), (14b), (16a), (16b) and [START_REF]Nvidia's next generation CUDA compute architecture: Kepler GK110[END_REF]. Equation ( 20) is linear, thus, it can be written as follows:

(Ú) = LÚ, ( 21
)
where L is given by:

l hh = 1 - m∈I h j∈I h w hm z hm z hj , l hj =    m∈I h w hm z hm z hj if j ∈ I h , 0 otherwise. ( 22
)
However, one can notice that a totally asynchronous iteration on f (•), as given by [START_REF] Davis | Algorithm 907: KLU, a direct sparse solver for circuit simulation problems[END_REF], is not guaranteed to converge. This is linked with the fact that the sum of the elements in any row of L, where L is given by [START_REF] He | GPU-accelerated parallel sparse lu factorization method for fast circuit analysis[END_REF], is equal to 1, then 1 is an eigenvalue of L and ρ(L) ≥ 1. Since for every matrix

A, ρ(|A|) ≥ ρ(A), then ρ(|L|) ≥ 1.
This implies that to ensure convergence of the proposed iteration, further assumptions are needed. The following theorem establishes a sufficient condition for convergence based on partially asynchronous iterations.

Theorem 3. Let (•) defined by (Ú) = (1 -α)Ú + α (Ú),
where 0 < α < 1 and (•) given by [START_REF] Miellou | A new class of asynchronous iterative algorithms with order intervals[END_REF]. Let the weights, w hm , m ∈ I h , satisfy the following:

1 - m∈I h j∈I h w hm z hm z hj ≥ 0, ∀h. (23)
Then, any partially asynchronous iteration over (•) converges to Ú * , fixed-point of (•).

Proof. See [START_REF] Marin | Gpu-enhanced power flow analysis[END_REF].

IV. CASE STUDY

The proposed method for PF analysis has been tested on the Nvidia Kepler GPU architecture [START_REF]Nvidia's next generation CUDA compute architecture: Kepler GK110[END_REF]. It has been evaluated on a series of benchmarks, with the aim of exploring convergence, performance and scalability issues.

A. Implementation

Regularity is one of the most powerful leverages for GPU performance [START_REF] Collange | Design challenges of GPGPU architectures: specialized arithmetic units and exploitation of regularity[END_REF]. Algorithm 1 presents a regular implementation of the proposed method. In order to ensure asynchronous evaluation of each component, we set the number of blocks to be smaller that the number of blocks that can be handled in parallel by the hardware. Therefore all threads are continuously running until convergence of the system. Threads are handling the set of components in a round-robin manner, which leads to a similar amount of work per thread as components are always connected to either one or two nodes. If threads were assigned to nodes instead, then some threads would perform much more work than others, as nodes can be connected to any number of components.

Instructions 5 and 6 calculate the voltage at its component's terminals and updates the appropriate coordinates on vector Ú. Instructions 7 and 8 calculate the power injections at each terminal and updates the appropriate coordinates on vector ∆×. The process is repeated after synchronizing all threads, until convergence is reached.

Algorithm 1 Intrinsically parallel algorithm for PF analysis. Input: Vector of initial power mismatches, ∆×; component model parameters; tolerance for the error, ǫ, maximum number of asynchronous iterations, K. Output: Vector of node voltages, Ú.

1: Ú = 1∠0 # Initial guess 2: for all h, m : c hm ∈ C in parallel do

until |∆× h |, |∆× m | < ǫ
As discussed in [START_REF] Cuda | Programming guide[END_REF], coalesced global memory accesses can drastically improve GPU performance by reducing the number of memory transactions. In the proposed implementation, the above is achieved by using a 'structure of arrays' approach. In addition, performance can benefit of shared memory for recurrent memory operations. However, shared memory is only accessible to threads within one thread-block, whereas global memory is accessible to all threads. Accordingly, using shared memory in the proposed method increases the delay between threads, which affects convergence as discussed in Section III-C. The point where shared memory usage brings performance improvement is highly dependent of the problem parameters and setting this point could benefit from previous research conducted on flexible communication [START_REF] Miellou | A new class of asynchronous iterative algorithms with order intervals[END_REF].

B. Evaluation

The implementation is evaluated using randomly generated benchmarks from 1,024 to 8,192 nodes, which correspond to real size networks. The benchmarks are obtained using the random radial network generator provided by Dome [START_REF] Milano | A python-based software tool for power system analysis[END_REF]. We used a Xeon E5645 CPU, with 12 cores at 2.4Ghz (1 used), a Tesla K40c, with 2 880 cores at 0.7 Ghz and a GeForce GTX 680 with 1 536 cores at 1.06Ghz. All source code is compiled using G++ 4.8.2 and CUDA 6.5.

1) Convergence: As discussed in Section II-C, partially asynchronous fixed-point iterations typically exhibit a sublinear rate of convergence. This means that reducing the convergence tolerance below a small threshold may require a large number of iterations. Figure 4 illustrates the relation between the convergence error and the number of iterations for the proposed method using the K40c GPU. It shows the mismatch (on a logarithmic scale) as a function of the maximum number of iterations performed by any component. The different curves represent different problem sizes. Note that the behaviour is very similar on all 6 benchmarks, suggesting that convergence is independent from the problem size. However, the convergence rate for this experiment depends on the targeted accuracy. For example, reducing the error from 10 -3 to 10 -4 requires about 555 iterations in average. This corresponds to a convergence rate of 0.995. The convergence rate has an asymptotic limit of 1 when the targeted accuracy is increasing which is coherent with the theoretical study of Section II-C,

The farther the method is from the exact solution, the faster it approaches that solution. This is the opposite of the Newton-Raphson (NR) method where the convergence rate is improving at each iteration. From this point of view, it may be interesting to combine both approaches into a hybrid solution that switches methods depending on the error. For example, the proposed method can be used to obtain a first solution fast, and the NR method can be used to improve that solution afterwards.

2) Performance: We compared the performance of the proposed approach with serial NR provided by Dome [START_REF] Milano | A python-based software tool for power system analysis[END_REF]. The Dome's implementation uses KLU, a highly optimized library for sparse matrix factorization, particularly suited to deal with matrices from circuit analysis [START_REF] Davis | Algorithm 907: KLU, a direct sparse solver for circuit simulation problems[END_REF]. The proposed method is tested on the Tesla K40c GPU, and the NR on the Xeon E5645 CPU.

Figure 5 shows the execution time of both methods (on a logarithmic scale) as a function of the problem size, n, for a targeted error of 10 -4 . The NR execution time grows exponentially with the problem size, which is consistent with precedent results [START_REF] Milano | Power system modelling and scripting[END_REF]. In the proposed method, in turn, the execution time seems less affected by the size, and exhibits a more flat profile. At about 3,000 nodes, there is a turning point in which the proposed method becomes faster than NR. This turning point is dependent of many parameters (architecture, implementation, topology, ...) and it is expected to evolve favorably for the proposed method as the number of embeded cores per chip is increasing. Note that the parallelization of the NR method on GPU is out of the scope of this paper. Indeed, parallelizing the LU matrix factorization on GPU is a very complicated task due to irregular memory access patterns and strong data dependencies [START_REF] He | GPU-accelerated parallel sparse lu factorization method for fast circuit analysis[END_REF]. For this reason, comparisons with traditional PF methods implemented on GPU are left for future works.

3) Scalability: As mentioned in Section I, performance of asynchronous methods should increase with the number of computing cores. The above is confirmed by Fig. 6, which shows the execution time of the proposed asynchronous method on two Nvidia Kepler GPU architectures as a function of the problem size. The Tesla K40c GPU, embedding 2,880 computing cores, allows for a performance gain of about 1.3 over the GTX 680, embedding 1,536 cores. This means that the proposed method almost achieve strong scalability. Fig. 7 represents the number of iterations needed to converge as a function of the number of blocks launched. We compared the synchronous version (that synchronizes threads after each iteration), with the asynchronous one (that does not perform any synchronization at all). We observe that only the latter benefits from increasing number of thread-blocks. Moreover, we see a turning point in which performance does not improve and rather decays as the number of blocks keeps growing. V. CONCLUSIONS This paper presents a partially asynchronous fixed-point method for PF analysis. The method is designed to be implemented on SIMT architectures such as the GPU. A theoretical result is provided which shows that the method is guaranted to converge, although convergence is generally sub-linear. Simulation results show that: (i) the convergence rate of the proposed method tends to be better at an early stage, which makes it a good complement of the traditional NR method; (ii) the proposed method becomes faster than the traditional NR approach for problems of considerable size; (iii) the proposed method scales well with the number of computing cores in the GPU architecture. Future works will consider the improvement of the proposed CUDA implementation. In particular, the use of shared memory will be investigated. This addressing convergence of the method in the case where an increasing number of iterations are performed 'asynchronously', i.e., without communicating the results to all threads.

  is called an asynchronous iteration on f (•), with update function U(•) and delay function d(•).

Fig. 1 .

 1 Fig. 1. Bound on the convergence rate as a function of the problem size, for different values of the parameter σ.

Fig. 2 .

 2 Fig. 2. Example 4-node circuit.

b)Fig. 3 .

 3 Fig. 3. Network model: (a) Component; and (b) Node.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Numerical error as a function of the number of iterations on the K40c GPU.

Fig. 6 .Fig. 7 .

 67 Fig.[START_REF] Talukdar | Designing algorithms and assignments for distributed processing[END_REF]. Related performance of the proposed method on two GPU architectures.

  × hm denotes the power injected to n h by component c hm , and ∆× h denotes the power mismatch at n h .At every iteration k, a component c hm obtains Ú hm to it. A similar exchange occurs with node n m at the other end of the component. Next, at the same iteration k, node n h receives

	∆×	(k) h	from node n h , and returns Ù	(k) hm and × (k)	(k) h	and
	Ù (k) hj and × returns Ú (k+1) (k) hj from each component c hj connected to it, and h (k+1) and ∆× h


Claire Clivaz

Matej Durco

Marianne Huang

Executive Summary

This WP6 deliverable presents a dissemination strategy for the DARIAH-RC project, which will be updated during the project.

The mission of the DARIAH-RC project is to develop an open-source, high quality, multilingual reference curriculum for the Digital Arts and Humanities. The DARIAH Reference Curriculum (DARIAH-RC) will strengthen alliances and foster innovative teaching and learning practices among members of the ESFRI-roadmap Digital Research Infrastructure for the Arts and Humanities (DARIAH).

The aims of the dissemination plan are:

• to identify the best channels and means for communicating with each one

• to co-ordinate the dissemination activities of the partners involved to ensure that DARIAH-RC receives national, European and international coverage

• to co-ordinate and plan dissemination materials for use by the project partners

• to prepare the evaluation of dissemination activities across the project

The objectives of the dissemination plan are to ensure:

• a program of dissemination activities at national and international levels to maintain DARIAH-RC's profile

• dissemination activities are effective and timely

• the project's resources are used effectively to maximize the representation of the project at a broad range of events and in the media

• content providers within the sector are made aware of DARIAH-RC aggregation services

• a wide audience within and outside the sector is made aware of DARIAH-RC

The plan is a live document and it will be updated every six months to include reports on dissemination activities and with contributions from all partners.

Project Overview

This ERASMUS+ Strategic Partnership grant started on the 1 st of January 2015 and it will run for a duration of two-and-a-half years. DARIAH-RC will feature flexible, localized curriculum materials as well as a central, modular portal that will host, deliver and maintain those materials. The portal will also support the development of new curricula materials and the sustainability of the project after the grant period has ended, in close collaboration with DARIAH. DARIAH-RC represents the first openly accessible Digital Humanities curriculum of its kind and will serve as a model for creating and delivering open-source asynchronous online educational materials, from which other communities can benefit.

Purpose

The purpose of dissemination activities in the DARIAH-RC project are:

• to raise awareness of the DARIAH-RC project and inform international, national and regional organizations, researchers and practitioners in Europe about its activities

• to inform the DH networks about the project and its developments

• to engage with DH communities and gain feedback from them on the activities of the project

• to promote the results of the DARIAH-RC project and encourage use of its web-portal and materials in DH communities

• to promote the DARIAH-RC project in ERIC DARIAH in order to secure its long-term sustainability

The aims of the dissemination plan are:

• to identify each stakeholder group and the best channels and means for communicating with them

• to co-ordinate the dissemination activities of the project partners to ensure that DARIAH-RC receives national, European and possibly international coverage

• to co-ordinate and plan dissemination materials for use by the project partners

• to prepare the evaluation of dissemination activities across the project

The objectives of the dissemination plan are to ensure that:

• there is a regular program of dissemination activities to maintain DARIAH-RC's profile

• the dissemination activities are effective and timely

• the project's resources are used effectively to maximize the representation of the project at research events and in the media

• to organize a multiplier event in March 2017, to reach a wide audience regarding the results of the project

The dissemination strategy for DARIAH-RC will be regularly discussed by the team.

Message and Visual identity Key message:

The mission of the DARIAH-RC project is to develop an open-source, high quality, multilingual reference curriculum for the Digital Arts and Humanities. The DARIAH Reference Curriculum (DARIAH-RC) will strengthen alliances and foster innovative teaching and learning practices among members of the ESFRI-roadmap Digital Research Infrastructure for the Arts and Humanities (DARIAH).

Defining a visual identity:

The Erasmus+ logo has to be present on all the official, project communications and in press releases in particular. A visual identity and logo has been developed for the project by UNIL's Unicom service. It is reminiscent of the DARIAH flower logo but expresses the project's own identity by the use of a pixelated flower (see the top of this report). The two logos are on the project website, as well as on our Twitter account.

Communication Plan Channels

Our own communication/dissemination channels

The project's Twitter account has been active since the 22 nd of April 2015, and the project's welcoming website pages have been online since the 8 th of May 2015. The WP6 team encourages all partners to use the Twitter account to keep it updated in real time so that current news and information about project events are regularly uploaded. More formal announcements or project improvements will be reported on our website with the possibility of a RSS feed. The Twitter account is: @dariahTeach Currently the project website address is http://acdh.ac.at/wp/ but it will be replaced soon by a DARIAH web domain (www.dariah.eu/teach) with a specific email address (teach@dariah.eu). Our website will be the main communication channel used to convey information regarding project activities including events held and materials relevant to the project topics and areas of expertise. This will be a bespoke communication and dissemination platform: the Erasmus+ DARIAH-RC web-portal.

The maintenance of the web-portal is the responsibility of WP 4; WP 6 together with WP 1 are responsible for the content. A content management system will be utilized for easy administrative and editorial rights for the project team.

News will be released on a regular basis, ideally monthly depending our production output: it is expected that every member will contribute to this effort by gathering news and materials relevant to the main goals of the project.

Academic oral presentations or publications will be announced regularly. To ensure that due credit is given for academic work, the following principles will be adhered to: firstly, for more generic work on the project, all our names will be used; secondly, for more specific work, the names of the contributors will be used with the inclusion of "on the behalf of the ERASMUS+ DARIAH-RC team".

Through subscribing to the project mailing list and RSS feed, those interested in the project will be kept updated regarding the project activities. Throughout the various communication channels, the logo will be used extensively. The content submitting process will allow flexible and efficient treatment of the news: title, 2-3 lines of abstract, who, where, to whom, links to any Work packages, external and/or internal hypertext link, etc.

Communication/dissemination via other channels

Other DH related communication channels will also be utilized. These include: Isidore (via HAL-SHS), DARIAH-EU, DARIAH-VCC2, Humanistica, the Humanist Discussion Group, the French-speaking diffusion list and other national diffusion lists (such as DhD, AICUD, etc).

For official announcements, we will use the institutional communication offices of our partners, as well as public media when appropriate. See Annex 2 for media publications and press notices.

Content and Phases a) The first step: getting to know each other better and getting others to know us

Achieving visibility for the project on the web: during the first phase of the project, while the web-portal is not fully functional, communication will rely mainly on social media, notably Twitter and Facebook. The part of the web-portal that will be accessible will mainly display information regarding the project structure, goals, partners, members and future timetable.

b) The second step: communication on ERASMUS+ DARIAH-RC dissemination

This task will begin as soon as the web-portal is fully realized (after month 12) and the project's first productions are released: training material, scientific publications, tutorials, etc. Users of the website will be kept regularly updated with the ongoing progress of the project.

c) ERASMUS+ DARIAH-RC final dissemination -Web-portal (D 25, December 2015)

After the project grant comes to an end, the portal and learning materials therein will become a Work Group of DARIAH's VCC2 (Education and Training) thus sustaining its outputs.

-Lausanne multiplier event (D 32, March 2017) The final multiplier event scheduled in Lausanne in spring 2017 will be a major opportunity to disseminate the content of the project. All seven partners will of course be present and we will work together on the most interactive ways to present to a wide academic public the results of our ERASMUS+ DARIAH-RC grant.

-International meetings (dissemination activities) Participation in international meetings with workshops to present the project and train others to use the modules, notably at the DH 2016 in Krakow (application before the 1 st of November 2015). WP6 will regularly encourage project partners to apply for common workshops or papers at local and international meetings. The following is a list of potential international meetings that will be strongly considered (we will seek to apply for 2016 and 2017 editions, when the project will be more advanced): Based within the European/UK traditions of distance learning, this may be the oldest conference in the field.

o

Final report on Dissemination (D 24, May 2017)

In coordination with all the partners, WP6 is in charge of establishing a final report on dissemination. It will describe everything that has been accomplished by including the usual steps in EU dissemination reports: aims of the project, toolbox, dissemination, exploitation, impact and testimonials. The final report on dissemination will be the result of the ongoing updating of the preliminary report.

The ERAMUS+ DARIAH-RC web-portal: structure and content

General overview

The ERASMUS+ DARIAH-RC web-portal will be the main communication channel for conveying information regarding project activities as well as main events and materials relevant to the project topics and areas of expertise.

By publishing extended descriptions of the project goals, achievements and the team, we intend to make advanced students, teachers and researchers aware of the project and the communities of members behind it. This awareness of the project must include the training material produced or highlighted by the project team and devoted to the community of scholars.

The web-portal will gather two main sets of information: 1. Information pages (project structure, partners of the project, etc.), as well as communication and dissemination materials as part of the activities of the project. Other news, events and materials relevant to the project's topics must be included. 2. The project's training portal itself, storing the educational material produced within the project framework as well as other productions of interest.

Information pages

Home page

This page welcomes visitors and simply introduces them to the main goals of the project, the main institutional affiliations, the project logo, and of the ERASMUS+ program. A Twitter widget and a RSS button allow for a continuous information update. A news page informs visitors about academic communications and publications, as well as project events.

The Partners page

For each partner, an institutional description is provided with a hypertext link to the official web site of the institution, arranged in the alphabetic order after the leading institution. The main area of expertise of each partner is listed if relevant. The tasks of the project involving each partner are mentioned, as well as the team leaders and the collaborator(s) if relevant.

List of partners:

o National University of Ireland Maynooth; team leader: S.

•

  Kick-off meeting and report -Athens 22-23 March 2015 • Launch of the project twitting -April 2015 • First draft of the dissemination plan -April 2015 • Presentation of the project by partners to their organizations and signing of contracts • Announcing the start of the project in national level press releases • Presentation of the project by partners at the DARIAH-IE launch and the first academic communications • Preparation of the web-portal 1 st June 2015 -1 st November 2015 -Startup phase • Presentation of the project by partners at International conferences and events • Preparation of the modules • Technical Report on Project Development (month 9) • Report on User Requirements: Reference Curricula to be Developed and Benchmarking Key Criteria (month 10) • Release of the Project Portal for Communication Purposes (month 12) • Report on Quality Assurance and Evaluation Methodology (month 12) • Web-portal released (month 12) • Financial and project report (month 13) • Visualization of Benchmarking Key Criteria (month 15) • Second project meeting and report (month 17) • Prototype release of Project Portal for delivery of first reference curricula (month 18) • Preliminary report on exploitation & sustainability planning (month 18) • First reference curricula ready for testing in real-world situations by project partners (month 19) 1 st November 2015 -1 st January 2017 -Core phase • Presentation of the project by partners at International conferences and events • Preparation and release of the modules • Continuing development of the DARIAH-RC website • First dissemination event (month?? -deliverable 31) • First reference curricula made publicly available (month 24) 1 st January 2017 -1 st July 2017 -Final steps • Multiplier event in Lausanne (month 27) • Final report on evaluation of reference curricula and delivery portal (month 27)

  This Strategic Partnership is under the lead of Maynooth University (Ireland, NUIM) and is comprised of Aarhus University's DIGHUMLAB (Denmark, AU), the Athena Research Center (Greece, ARC), the Austrian Academy of Science (Austria, AAS), the Belgrade Centre for Digital Humanities (Serbia, BCDH), the Erasmus University Rotterdam (The Netherlands, EUR) and the University of Lausanne (Switzerland, UNIL). These partners represent countries that are either founding DARIAH countries or working towards Cooperative Partner status; they represent a pool of experience within the Digital Arts and Humanities fields.

	DARIAH-RC is the result of a working team from the VCC2 of DARIAH-RC: Research
	and Education (see http://dariahre.hypotheses.org/). VCC2 Research and
	Education promotes and supports the use of research data and ICT methods and
	technologies, including the DARIAH infrastructure. It acts as the primary contact
	between the Arts and Humanities research and teaching communities, providing the
	interface between DARIAH and researchers undertaking basic, applied and practice-
	led research. It seeks to understand Arts and Humanities research practices and
	processes, and to understand and promote the use and application of ICT-enabled
	methods and tools, with a particular emphasis on interdisciplinary understanding and
	exchange. VCC2 Research and Education is coordinated by Susan Schreibman
	(Maynooth	University, Ireland)	and	Marianne	Ping	Huang	(Aarhus
	University, Denmark).
	DARIAH-RC values: Sharing, Knowledge, Creativity
	DARIAH-RC Actions: DH Teaching, DH Learning, Digital Resources, European
	Networking

  Common guidelines will be used, in the case of the project, Europeana guidelines have been suggested:

http://group.europeana.eu/web/guest/communication-tools;jsessionid=A05219C14C4769D6BBEABBB598B825C0

•

  Third project meeting and report (month 29) • Report on impact of project work (month 29) • Final Dissemination Plan Updated and Completed (month 29) • Final report on exploitation & sustainability planning (month 29) • Financial and project report (month 30) • Creation of a post-grant editorial board (month 30) • All reference curricula made publicly available (month 30) • Deliverable of final version of Project Portal with all learning objects publicly available (month 30)

Annex 2: Detailed breakdown of Events (in development)

	Conferences and workshops
	Date	Partner Event	Audience
	…..	…..	……	……
	15/05/15	UNIL	DH conference in Geneva, poster related to the project Researchers:
				Historians
	18/05/15	BCDH	Slam presentation at the DARIAH-IE launch day	Researchers, DH community
	29/06/15	UNIL	Poster at the DH 2015 about the project	DH community
		and all
	…..	…..	…..	……
	…..	….
	….	….	…	….
	Media publications and Press notices
	Date	Partner Publication	Audience

DARIAH-RC

Grant Agreement number:

2014-1-IE02-KA203-000125

This section will be the main part of the website presenting the content and the results of the project.

We will offer at least five modules, furthermore we look forward to collaborations on supplementary modules. The five modules will be presented according to a template prepared by WP 3 (available in Annex 1). The template distinguishes between the presentation of the content (abstract, description of the module, explicit learning outcomes, prerequisites, acquired Competencies, learning object metadata), and the content itself (Units and formats).

These five modules will be presented:


Katsuhiko Naoi
email: k-naoi@cc.tuat.ac.jp

Kazuaki Kisu

Etsuro Iwama

Shota Nakashima

Yuki Sakai

Yuki Orikasa

Philippe Leone

Nicolas Dupré

Thierry Brousse

Patrick Rozier

Kisu Kazuaki

Nicolas Dupre

Wako Naoi

Patrice Simon

Ultrafast charge-discharge characteristics of a nanosized core-shell structured LiFePO4 material for hybrid supercapacitor applications

Introduction

The development of power electronics as well as the growing need for portable and mobile electronic devices highlight the need for high performance electrochemical energy storage sources. Intensive research efforts are currently made to improve battery performance by designing advanced materials based on the Li-ion 1 or new chemistries. 2 Batteries can provide high energy density, thus ensuring autonomy, but fail to deliver high power peaks needed for many applications. 3 As a result, electrochemical capacitors (ECs), also called supercapacitors, have been attracting attention due to their much greater power density and cyclability as compared to batteries. 4 Conventional ECs store the charge by ion adsorption at high surface-area porous carbon electrodes.

Such an electrostatic charge storage mechanism limits the capacitance of the carbonaceous materials, thus limiting the energy density of the devices. One way to increase the energy density of ECs is to move from an electrostatic to a pseudocapacitive charge storage mechanism.

Pseudo-capacitance 5 arises from a fast redox reaction at the surface of materials. RuO 2 , 6 MnO 2 , 7 oxides with spinel structures, 8 or some metal nitrides 9 are well-known examples of pseudocapacitive materials, which have been described in the literature. Various strategies have been proposed for improving their performance, including nanostructuration, 10 deposition onto high-surface area materials like graphene, carbon nanotubes or porous carbons 11 and the control of crystallographic structure. 12 However, most of the pseudocapacitive materials operate in aqueous electrolytes, thus limiting their practical interest for high-energy supercapacitor applications. Recently, in a non-aqueous electrolyte, a specifically designed orthorhombic Nb 2 O 5 material used as a negative electrode has shown intrinsic capacitive features with extremely high power performance. 13 Therefore Li + intercalation was achieved by the pseudocapacitive mechanism. Nano-sizing of battery materials is also known to enhance their power performance. 14 However, despite nano-sized Li 4 Ti 5 O 12 , 15 TiO 2 (B), 16 LiCoO 2 , 17 and V 2 O 5 18 demonstrating higher power and capacity than the bulk materials, a major concern is that nanostructured forms exhibit irreversible capacity loss during cycling. In addition, since the exposure of the surface to the electrolyte is a critical factor, preparation of conventional composite electrode architectures with binders is difficult to achieve because of a decrease in the surface area and a poor electrical percolation network. 10 LiFePO 4 (lithium iron phosphate, LFP) has long been investigated as a cathode material in Li-ion batteries because of its high theoretical capacity of 170 mA h g À1 , low cost and high electrochemical and thermal stabilities. 19 An electrochemical reaction of LFP proceeds through a two-phase reaction between Li-rich Li 1Àa FePO 4 (LFP) and Li-poor Li b FePO 4 (FP) 20 with Li insertion/deinsertion occurring along the b axis. 21 However, the limited diffusion kinetics of Li ions at the LFP/FP interface together with the poor electronic conductivity of the pristine olivine-LFP (10 À10 to 10 À7 O À1 cm À1 ) 22 limit the power capability of the material. Downsizing the particle size (5-100 nm) failed to enhance the power performance due to the re-aggregation of particles and the difficulty in creating efficient electron pathways. 23 The synthesis of carbon coatings onto LFP particles has also been proposed, by adding carbon precursors during the synthesis of LFP. [24][25][26] However, non-conformal amorphous carbon coatings did not show enough improvement in electrical conductivity when nano-sized LFP particles were prepared. 27 In this paper, we report on the synthesis of single nanosized LFP crystals encapsulated within hollow-structured graphitic carbons by an in situ ultracentrifugation (UC) 28,29 process for hybrid supercapacitor applications. The composite material has a core LFP (crystalline (core 1)/amorphous (core 2))/graphitic carbon shell structure that offers both high reversibility and high rate capability. The electrochemical properties and performance of the LFP/graphitic carbon material showed outstanding high rate and capacity retention with a capacity of more than 24 mA h g À1 at a 480C discharge or charge rate (7.5 seconds). Such results pave the way for designing high energy and high power materials to be used in hybrid supercapacitors. 30

Results and discussion

Structural characterization

The Scanning Electron Microscope (SEM) image of the LFP/ graphic carbon composite shows uniformly dispersed 30 nmdiameter spheres (Fig. 1a and Fig. S1a, ESI †). Transmission Electron Microscope (TEM) experiments show that they are made of a core of LFP particles with the size between 10 and 20 nm as evidenced using dark field images (white spots in Fig. 1c). These particles are encapsulated in a 5 nm thick carbon shell visible as a grey region in both bright (Fig. 1b and Fig. S1b, ESI †) and dark (Fig. 1c and Fig. S1c, ESI †) field images.

X-ray photoelectron spectroscopy (XPS) analysis of the Fe 2p 3/2 energy level was performed on samples obtained. The XPS data obtained after increasing the duration of Ar-ion etching show that above 20 s a slight intensity at 708 eV appears (Fig. S2,ESI †). This is the evidence of the presence of Fe 0 indicating that Ar-ion etching has brought about an unwanted reduction of Fe species to the metal state thus destroying the LFP compound. Therefore, we used the data within 14 s that would be reasonable to judge the valence of Fe as a function of depth of the composite (Fig. 1d). The XPS spectrum of the surface of the composite (0 s) shows a peak with a maximum at 711.2 eV, that contains signals of Fe 2+ (LFP; 710 eV) and Fe 3+ (FP; 712 eV) 31 in almost equivalent amount. The spectra obtained on samples with increasing etching duration (increasing depth in the composite) show an increase of the peak intensity and a shift of the peak position up to 710.1 eV. The increase of the intensity well agrees with the previous SEM and HRTEM observations of the encapsulation of the Fe-containing particle by a graphitic carbon hollow structure. The gradual shift of the peak evidences that the LFP core contains a Fe 2+ and Fe 3+ mixture with a Fe 3+ /Fe 2+ ratio changing from the surface to the core of the particle, the latter corresponding to pure Fe 2+ . The analysis of the valence states by Fe Mo ¨ssbauer spectroscopy (Fig. 2a and Table S1, ESI †) confirms the presence of both Fe 2+ and Fe 3+ valence states in an atomic ratio of 53/47. The deconvolution of the spectrum shows that only one contribution for Fe 3+ is observed (blue line) while two contributions have to be taken into account for the Fe 2+ one (red line) corresponding to undistorted M2 sites (14%) and the second (green line) to distorted M2 sites (40%). 32,33 An X-ray diffraction study was conducted to crosscheck the crystallographic information on the LFP/graphitic carbon composite (Fig. 2b). All diffraction peaks were indexed in the orthorhombic lattice of LFP (JCPDS card 72-7845) showing that no crystalline impurity is detected. The cell parameters, refined using the profile matching method, are a = 10.325(1) Å, b = 6.004(5) Å, c = 4.692(2) Å, and V = 290.9(1) Å 3 in accordance with the reported values for conventional LFP materials (V = 291.2 Å 3 ) 34,35 and clearly higher than the reported ones for highly defective LFP 33 (V = 286.8 Å 3 for 44 at% of Fe 3+ ). The lowering of the unit cell volume being related to the shortening of Fe-O bonds with the increase of the Fe valence state, the slight (0.1%) decrease observed compared to the high amount of Fe 3+ (47%) suggests that most of the Fe 3+ are included in an amorphous phase.

TEM experiments are in accordance with the existence of such an amorphous phase. The TEM images (see Fig. 3 and Fig. S3, ESI †) show that the sample is constituted of three parts. A crystalline core (ca. 12-15 nm diameter), an intermediate amorphous layer (ca. 2-5 nm thick) (core 2 in light purple) and an outermost shell layer (ca. 5 nm thick) composed of more or less randomly distributed units. The analysis of the crystalline core (core 1) shows an interplanar lattice distance of d = 0.42 nm corresponding to the (101) plane of the olivine LFP (Fig. 3a) and the outermost shell layer shows an inter unit distance of d = 0.35 nm a little larger than that of graphene. This confirms that the sample contains a crystalline LFP core surrounded by an amorphous phase, both being encapsulated in randomly organized (KB derived) carbon graphitic units stacked onto each other. Based on the combination of the different experiments it can be assumed that the amorphous layer contains most of the Fe 3+ ions; it will be named as ''amorphous defective LFP'' in the following. The composite contains 48% of LFP and 52% of graphitic carbon (Fig. S4,ESI †). Elemental mapping obtained from electron energy loss spectroscopy (Fig. S5,ESI †) shows that iron, phosphorus, and oxygen are localized in the whole core-shell particles, while Li is distributed over a larger area (Fig. S6, ESI †) including the carbon part. While the average composition is in accordance with the expected one for LFP/C composites, the specific distribution of Li in the whole sample accounts for the partial oxidation of Fe 2+ up to Fe 3+ . In addition, thermal gravimetric analysis (TGA) of the LFP/graphitic carbon composite shows a decrease of the decomposition temperature of the carbon (Fig. S4, ESI †), confirming a close interaction between the LFP and the graphitic carbon shell. Thus it can be expected that the combination of KB and UC treatment allows a partial oxidation of pristine LFP which would explain that Fe 3+ concentration increases from the core to the surface of the particles leading, for highly defective LFP, to an amorphization of the sample. A sample containing only LFP (without KB) was prepared under the same UC conditions. The uc-LFP material resulted in a large or agglomerated crystal as shown in Fig. S7b (ESI †) that does not exhibit such a peculiar core-shell structure (Fig. S7a, ESI †). The composite was also thermally treated to remove the graphitic carbon shell under an air atmosphere at 700 1C. This experiment resulted in substantial agglomeration and increased particle size (Fig. S7c, ESI †). Therefore, the role of the carbon shell is clearly very important to maintain the peculiar nano-sized ''core-shell'' structure to sustain ultrafast electrochemical performance (see later Section 2.3).

Combined electrochemical and in situ XAFS and XRD characterization of LFP/graphitic carbon composites

The electrochemical behavior of the LFP/graphitic carbon composite was investigated in half-cells versus Li anode at 1C rate (1C = 170 mA g À1 ) (Fig. 4). At the 1st charge, a low capacity of 54 mA h g À1 of composite is obtained, that is 87 mA h g À1 of LFP. This corresponds to 0.51 Li extracted per LFP after subtracting the double layer capacitance contribution of the UC treated KB (uc-KB) (24 mA h g À1 , Fig. S8,ESI †). After the 1st discharge process, a reversible capacity of 96 mA h g À1 per composite is obtained, corresponding to 134 mA h g À1 of LFP and 0.78 Li-ion intercalated per LFP, after removing the reversible capacity of uc-KB at the 1st discharge (63 mA h g À1 ). The presence of Fe 3+ in the as-prepared composite explains the difference between charge and discharge capacity at the 1st cycle. This confirms the previous results from Mo ¨ssbauer analysis, showing 53% of Fe 2+ and 47% of Fe 3+ in the pristine LFP/graphitic carbon composite. After the 1st charge/discharge cycle, the reversible capacity is increased to 140 mA h g À1 of LFP, corresponding to 0.82 Li exchanged per LFP. Long-life cycling performance of the composite was investigated. We have cycled more than 2000 times at 10C (Fig. 4 (inset)), 60C and 240C (Fig. S9,ESI †). The retention of the discharge capacity was 85, 92 and 96% for 10, 60 and 240C, respectively. Such cycling performance positively compares with other reports on LFP nanoparticles. 25,36,37 The electrochemical signature of the composite shows three various regions: a plateau at 3.4 V and sloping profiles above and below B3.4 V. The plateau corresponds to the conventional two-phase reaction observed for crystalline-LFP. A small polarization of only 100 mV at 1C is measured, suggesting that the presence of the amorphous defective LFP phase does not hinder Li diffusion along the [100] channels. Below 3.4 V, as mentioned previously, the potential changes with the electrode capacity. Amisse et al. have recently reported such a sloping profile with highly defective crystalline LFP phase containing 44% of Fe 3+ . 33 They attributed this signature to the redox activity of Fe 2+ and Fe 3+ in distorted M2 sites. Considering, in the present paper, that the reversible capacity of the uc-KB is only 30 mA h g À1 below 3.4 V (Fig. S8, ESI †), 75% of the capacity below 3.4 V can be assigned to the redox activity of the Fe 3+ defects in the pristine composite.

To confirm this, the redox states of Fe were investigated by in situ X-ray absorption measurements during cycling. The X-ray absorption near-edge structure (XANES) spectra of the Fe K-edge for the LFP/graphitic carbon composites are shown in Fig. 5a (insets) during the 1st cycle. The threshold energy position of the Fe K-edge absorption gives information about the valence state of the probing atom. The average valence state of Fe in the composite is determined by the deconvolution of the spectra using the contributions of Fe 2+ and Fe 3+ (see the ESI †). Fig. 5a shows the changes of the valence numbers of Fe during charge-discharge at 0.1C rate. At 3.1 V (OCV), the initial valence state of Fe is +2.5, in good agreement with the previous Mo ¨ssbauer spectroscopy results (Fig. 2a). During the 1st delithiation process, from runs 1 to 15, the valence state linearly increases from +2.50 to +2.89 in the plateau domain (up to run 10) while in the sloping region from 3.4 V to 4.2 V, the redox state of Fe is kept almost constant (runs 10 to 15). Then, during the 1st lithiation, the Fe valence state is similarly almost constant above 3.4 V (runs 15 to 18). This indicates that the charge storage mechanism beyond 3.4 V does not involve redox reaction, but double layer capacitance from the graphitic carbon shell. Further, the valence state linearly decreases from 3.4 V (run ''18'') down to 2.5 V (run ''33'') and the capacity is associated with the reduction of Fe from (+2.89) down to (+2.14). Below 2.5 V, the Fe valence is almost unchanged (runs 33 to 36), showing a major contribution of the double layer capacitance from the graphitic carbon shell. During the 2nd charge, the Fe valence state increases with increasing potential. The valence number of Fe varies symmetrically during the 1st discharge (runs 15 to 36) and 2nd charge (runs 36 to 58) cycles, from +2.89 to +2.14. The change in the valence state over one charge-discharge cycle is AE0.75 corresponding to 128 mA h g À1 for LFP, that is 75% of its theoretical capacity.

Based on the results of the XANES study, a reaction mechanism can be proposed: 1st charge: Li 0.50 FePO 4 -Li 0.11 FePO 4 + 0.39Li + + 0.39e À Subsequent cycle: Li 0.11 FePO 4 + 0.75Li + + 0.75e À 2 Li 0.86 FePO 4

The obtained number of exchanged Li from the chargedischarge test (0.82 Li per LFP) and in situ XANES measurements (0.75 Li per LFP) is slightly different. This slight difference of o10% could be attributed to the double layer capacitance associated with the surface contribution of the LFP nanoparticle.

Li intercalation in the crystalline phase was further assessed by in situ XRD measurements. Fig. 5b (left panel) shows the potential versus composition signature of LFP/graphitic carbon during the 1st discharge at 0.1C. Three selected diffraction peaks are presented in Fig. 5b (right panel). The cell was fully charged up to 4.2 V at a slow rate (0.1C) before measurement. At 4.2 V, two peaks are observed at 30.41 and 30.91 that correspond to FePO 4 (FP)( 211) and (020), respectively. During the discharge reaction, new peaks characteristic of LFP( 211) and (020) appear at 29.81 indicating the lithiation reaction in the potential plateau region. Then the peak position and intensity do not change below 3.3 V, while a decrease in the valence state of Fe is observed in this potential region (Fig. 5a) showing that a redox process involving Fe ions occurs in a non-crystalline phase. This confirms the presence of an amorphous phase in the pristine composite that contains most of the Fe 3+ ions. Accordingly, the capacity in the plateau region can be attributed to the crystalline (c-LFP) phase while the capacity below 3.3 V (sloping region) originates from the amorphous defective LFP phase. The comparison of the capacities calculated in the plateau region (crystalline LFP phase) and in the sloping domain (amorphous defective LFP phase) gives a crystalline/amorphous defective LFP phase of 31/69 by weight (Fig. S10,ESI †). In addition, the peak positions of LFP and FP do not shift during discharge. This indicates that in the crystalline LFP core, Fe 2+ are located only in undistorted M2 sites and that partial Fe 2+ lying in distorted M2 sites are located in the amorphous defective LFP phase. This is in fair agreement with the studies showing that crystalline defective LFP is reduced via a solid solution mechanism as exhibited by progressive Bragg's peak shifts reported by Amisse et al. 33 In summary, three contributions to the reversible capacity of the LFP/graphitic carbon composite have been identified as (i) c-LFP which shows a constant charge-discharge potential at about 3.4 V; (ii) amorphous defective LFP undergoing a onephase reaction with a sloping potential profile below 3.4 V; and (iii) double layer capacitance of the graphitic carbon in the entire 2.0-4.2 V potential range.

Ultrafast charge-discharge behavior

Charge-discharge tests at various current densities (1-480C) were performed (Fig. 6). The LFP/graphitic carbon composite exhibits excellent rate capability, delivering discharge capacities of 89, 60, 36 and 24 mA h g À1 at 1, 100, 300, and 480C rates, respectively, outperforming the reported data in the literature. 25,26,28,[35][36][37][38][39][40] Even more importantly, the LFP/graphitic carbon composite has an extremely high-rate capability in charge as well; it stores 60, 36 and 24 mA h g À1 at 100C, 300C, and 480C rate, respectively. Considering that the capacity for the activated carbon (AC) normally used in EDLC devices is typically 40 mA h g À1 , the obtained capacity of 60 mA h g À1 is still very interesting in the field of supercapacitors. It should be noted, however, that the discharge and charge capacities at each rate show a linear relationship, meaning that the composite can offer a high power density, regardless of the way it is charged or discharged. Such a linear relationship between the charge and discharge capacities highlights the high-power capability of the material in discharge as well as in charge, such as expected for the practical use of hybrid supercapacitors. Such high power performance for both charge and discharge processes is due to the optimized core-shell nanostructure of the LFP/graphitic carbon composites allowing fast Li + transport. This is confirmed by the calculation of Li + diffusivity in the materials. We found that the diffusion coefficient of Li + (D Li +) in the sloping potential region corresponding to the amorphous LFP phase (Fig. S11,ESI †) is 10 À11 -10 À12 cm 2 s À1 . This is up to 2 orders of magnitude higher than that of the crystalline LFP phase (D Li + = 10 À13 cm 2 s À1 ) in the plateau region. 41,42 It confirms that the creation of defects or an amorphous phase increases the diffusion coefficient of Li-ions, explaining the high-rate performances. 33 In addition, extra electrochemical characterization carried out using the uc-LFP (without KB) sample, which did not exhibit the peculiar core-shell structure (Fig. S7a, ESI †), and the composite thermally treated under an air atmosphere to remove the carbon component (Fig. S7c, ESI †) led to degraded electrochemical performances especially without high charge and discharge capacity (@100C) compared with the core-shell composite (Fig. S7d, ESI †). Therefore, the carbon shell is clearly very important to maintain the peculiar nano-sized ''core-shell'' structure to sustain ultrafast electrochemical performance. In fact, there are various synthesis parameters that tune the crystalline/amorphous ratio of the LFP phases in the composite. The encapsulated LFP cores did not change essentially by varying annealing time (5 min-2 h) or annealing temperature (700-800 1C) at a fixed ratio of carbon content (50 wt%) (Fig. S12(a), ESI †). However, decreasing the carbon content below 30 wt% (Fig. S12(b),ESI †) drastically increased the LFP particle size, which is not anymore encapsulated. It seems that the carbon content is the factor that affects the size/morphology of the entire composite. Fine-tuning of these parameters would permit designing composite structures with an optimized crystalline/amorphous ratio to get the best electrochemical performance. Systematic studies in this regard are currently under investigation for a follow-up paper.

These peculiar core-shell nanostructured materials offer new opportunities for designing high-rate positive electrodes that can be charged and discharged in a few tens of seconds, for hybrid supercapacitor applications. We have conducted an exploratory test of a full cell configured as AC (20 mm)/1 M LiPF 6 (EC + DEC)/ uc-LFP (20 mm) as shown in Fig. S13 (ESI †). The capacity ratio of the anode/cathode was set as 1/1, so the weight ratio of the AC/uc-LFP is 2/1. The voltage range tested was 0-2.7 V at a current density of 34 mA g À1 (per LFP). The hybrid full cell delivers 41 F g À1 , which is 1.3 times higher than a reference cell (AC/AC in the same electrolyte). This is a small demonstration of a hybrid cell concept and it is feasible for further testing. Note as well that the electrochemical response of the crystalline LFP phase (plateau) is observed, contributing to the improvement of cell capacitance and energy. This validates the proposed concept.

Conclusion

Highly dispersed crystal/amorphous LFP nanoparticles encapsulated within hollow-structured graphitic carbon were synthesized using an in situ ultracentrifugation process. Ultracentrifugation triggered an in situ sol-gel reaction that led to the formation of core-shell LFP/graphitic carbon composite particles. The coreshell structure contains a crystalline LFP (core 1) covered by an amorphous LFP containing Fe 3+ defects (core 2) that are encapsulated by a graphitic carbon shell. These core-shell LFP nanoparticles show improved Li + diffusivity thanks to the presence of the amorphous LFP phase. This material enables ultrafast discharge rates (60 mA h g À1 at 100C and 36 mA h g À1 at 300C) as well as ultrafast charge rates (60 mA h g À1 at 100C and 36 mA h g À1 at 300C), which offer new opportunities for designing high-energy, ultrafast hybrid supercapacitors.

Materials and methods

Materials

Fe(CH 3 COO) 2 (Aldrich), LiCH 3 COO (Wako Pure Chemicals) and H 3 PO 4 (Wako Pure Chemicals) were used as the Fe, Li, and PO 4 sources, respectively, while citric acid (C 6 H 8 O 7 ) (Aldrich) was used as the chelating agent. Hollow-structured Ketjen Black (KB; EC600JD, Ketjen Black International Company [START_REF]KETJENBLACK Highly Electro-Conductive Carbon Black[END_REF] ) with 50 nm diameter primary particles with a specific surface area (SSA) of 1270 m 2 g À1 was selected as the precursor carbon matrix for encapsulating the nanoscale LFP because of its high electronic conductivity and specific surface area. Ultra-pure water (17 O cm) was used as the medium for the entire preparation scheme.

Preparation of the LFP/graphitic carbon composite by UC treatment

Two different solutions, A and B, were prepared. Solution A contains 0.0837 g (1.00 eq.) of LiCH 3 COO, 0.2205 g (1.00 eq.) of Fe(CH 3 COO) 2 and 0.2448 g (1.00 eq.) of C 6 H 8 O 7 dissolved in 7.00 g of H 2 O. Solution B contains 0.1462 g (1.00 eq.) of H 3 PO 4 dissolved in 14.00 g of H 2 O. Solution B and 0.200 g of KB were then mixed for 30 min using ultrasonication to obtain a uniform premixture. This premixture was then treated by the first UC for 5 min to form a blackish gel. In this process, PO 4 3À uniformly covered the surfaces of the dispersed primary KB carbon particles. After the addition of solution A, a second UC treatment was performed for 5 min. After drying the product at 80 1C for 12 h in a vacuum (ultimate vacuum = 0.67 Pa), the precursor of the LFP/graphitic carbon composite was obtained. A further annealing treatment [START_REF] Kisu | [END_REF] for 8 min (heating: 3 min, holding 5 min) of 700 1C under a nitrogen flow of the precursor powder led to the obtention of the nano-crystalline LFP/graphitic carbon composite. During these drying and annealing processes, we assume that the water media containing Fe, Li, PO 4 sources, and chelating agent (C 6 H 8 O 7 ) get concentrated within the hollows of the graphitic carbon shell due to capillary effects leading to the formation of LFP particles in the inner KB instead of outside (Fig. S14, ESI †). Then during annealing at 700 1C, the media evaporates faster in outer shells than inner composite shells. Thereby, the media as well as the LFP-precursor may become more and more concentrated and confined in inner shells. During annealing process, the confined LFP-precursor gets crystallized to become the LFP core part. We also speculate that in the vicinity of outer shells the LFP-precursor has more interference with the graphitic carbons preventing full crystallization resulting in partial crystallized or amorphous phases. Therefore the resulting LFP nanoparticles are configured generally as core crystalline and amorphous outer sphere.

Physicochemical characterization of the LFP/graphitic carbon composite

Particle size distribution of LFP, the KB-derived graphitic carbon layer, and the nano-structure was carried out by high-resolution transmission electron microscopy (HRTEM, Hitachi H9500 model). X-ray diffraction (XRD, Rigaku SmartLab) was used to characterize the crystalline structure of the LFP/graphitic carbon. The stoichiometry of the composites was determined using thermogravimetric differential thermal analysis (TG/DTA, Seiko Instruments TG/ DTA6300) under a synthetic air atmosphere (O 2 : 20%, N 2 : 80%). X-ray photoelectron spectroscopy (XPS JEOL Ltd. JPS-9200) was carried out with a pass energy of 40 eV for high-resolution scans using a monochromated Mg X-ray source. 57 Fe Mo ¨ssbauer spectra were collected in transmission geometry on a constant acceleration spectrometer using a 57 Co g-ray source in a Rh matrix equipped with a cryostat. Velocity and isomer shift (IS) calibrations were performed using a-Fe as a standard at room temperature. Quadrupole splitting (QS) data for each type of Fe ion were analyzed as discrete 0.1 mm s À1 step distributions in the 0-6 mm s À1 range and then fitted with a Gaussian distribution curve.

Electrochemical characterization of the LFP/graphic carbon composites

Half-cells were assembled using a negative Li metal electrode and a positive LFP/graphic carbon electrode in 2032 coin-type cells. The electrolyte was a mixture of ethylene carbonate (EC) and diethyl carbonate (DEC) containing 1.0 M lithium hexafluorophosphate (LiPF 6 ). LFP/graphic carbon electrodes were prepared by mixing 90 wt% of the composite and 10 wt% of weight of polyvinylidene difluoride (PVdF) in N-methyl pyrrolidone (NMP).

The mixture was coated on an etched-Al foil (current collector) and dried at 80 1C in a vacuum for 12 h. The electrode density, calculated from the loading mass of the composite (2.00 mg) on 1.54 cm 2 of etched Al current collector and the thickness of the LFP/graphic carbon electrode (10 AE 1 mm), was ca. 1.30 g cm À3 .

Charge-discharge tests were performed under constant current mode between 2.0 and 4.2 V vs. Li/Li + at various current densities ranging from 0.1 to 480 C-rate, assuming that 1C-rate equals 170 mA g À1 .

In situ XAFS measurements

In situ X-ray adsorption fine structure (XAFS) measurements at the Fe K-edges for the composite samples were performed in transmission mode at the beam line BL14B2 of the synchrotron radiation facility SPring-8 (Hyogo, JAPAN). 45 Laminate-type twoelectrode cells (pouch cells) were assembled using lithium metal foil as a negative electrode and the LFP/graphitic carbon composite as a positive electrode. Charge-discharge tests were performed within a 2.0-4.2 V voltage range during the 1st cycle at a rate of 0.1C. XAFS spectra were recorded at equilibrium following a rest period of 20 min at each voltage. The obtained XAFS spectra were analyzed using the spectral fitting software REX2000 (Rigaku Corp.) to evaluate the ratio of Fe species with different oxidation states, such as Fe 2+ (LFP bulk sample) and Fe 3+ (FP bulk sample oxidized by the chemical method using NO 2 BF 4 ).

Fig. 1

 1 Fig. 1 (a) Scanning electron microscope (SEM), (b) bright-field, and (c) darkfield images of the UC-derived LFP/graphitic carbon composites represent a unique nanostructure of two phases, whereby each spherical LFP core is accommodated/encapsulated within the hollow structured graphitic carbon shells. (d) Consecutive XPS (Fe 2p core peaks) measurements with data acquisition at 1 s intervals during 0-14 s Ar ion etching, representing a depth profile of the same sample to ensure that the peculiar core-shell structure of our LFP/graphitic carbon composites was derived from ultracentrifugation.

Fig. 2

 2 Fig. 2 Structural characterization of the same LFP/graphitic carbon composite as in Fig. 1. (a) Mo ¨ssbauer spectrum fitted with three Fe environments (Fe 2+ , disordered Fe 2+ , and disordered Fe 3+ ), suggesting that the LFP/graphitic carbon composite contains amorphous LFP with Fe 3+ defects. (b) XRD patterns of the same LFP/graphitic carbon composite. The major diffraction peaks ((101), (111), (211), and (301)) of the XRD patterns are well indexed to the orthorhombic structure of LiFe(II)PO 4 with the Pnma space group (JCPDS card No. 72-7843). No peaks due to Fe(III)PO 4 appear in the XRD pattern, suggestive of the existence of an amorphous phase Fe(III)PO 4 . A broad peak observed at around 24.51 corresponds to the (002) plane of KB. This result indicates that the obtained LFP/graphitic carbon composite did not have any impurities such as FeP and Li 3 Fe 2 (PO 4 ) 3 . The Scherrer equation was applied to the strongest three peaks corresponding to (104), (110), and (116) planes for LFP. The average crystal size was determined to be 39 nm for LFP, suggesting that nano-LFP particles were synthesized by UC treatment.

Fig. 3

 3 Fig. 3 (a) Higher-magnification HRTEM image of the LFP/graphitic carbon composite featuring crystalline LFP nanoparticles (ca. 15 nm) with clear lattice fringes (d = 0.42 nm; LFP(101)) encapsulated within the random graphene fragments derived from high-surface-area KB graphitic carbons. (b) Magnified HRTEM image of the LFP/graphitic carbon composite with a marker for each component. (c) Schematic illustration of the core-shell nanostructure of the LFP/graphitic carbon composite, representing a minute structure consisting of an amorphous outer sphere of a LFP containing Fe 3+ defects and an inner sphere of crystalline LFP.

Fig. 4

 4 Fig. 4 Charge-discharge curves (1st-3rd cycle) of a half-cell consisting of Li/1 M LiPF 6 EC + DEC (LFP/graphitic carbon) composite at a 1C rate. Inset: Retention of discharge capacity over 1-2000 cycles at 10C rate.

Fig. 5

 5 Fig. 5 In situ XAFS and XRD measurements on a half-cell consisting of Li/1 M LiPF 6 EC + DEC (LFP/graphitic carbon) composites. (a) Solid bold line shows the galvanostatic charge profile of LFP/graphitic carbon during the in situ XANES measurements at 0.1C. Dashed lines correspond to the changes in the evaluated ''formal valence number'' of Fe during the lithiation-delithiation process. The formal valence number of Fe for the composites was calculated by fitting using commercial LiFePO 4 powder and FePO 4 powder oxidized using the chemical method. The XANES spectra at the iron K-edge of Li x FePO 4 rate are shown in three insets. (b) Detailed XRD patterns during the second discharge reactions for the 0.1C rate. The voltage profiles during the measurements are shown on the left.

Fig. 6

 6 Fig. 6 Plots of discharge capacity vs. charge capacity of a half-cell consisting of Li/1 M LiPF 6 EC + DEC (LFP/graphitic carbon) composite as a function of C-rate. Inset: Charge-discharge profiles at different charge C-rates from 1 to 480C.

Acknowledgements

This study was supported in part by the Global Innovation Research Organization in TUAT. The synchrotron radiation experiments were performed at the BL14B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2015A1962). Electron microscopy and EELS observations were performed by Hitachi High-Technologies Corporation. This work was supported by JSPS KAKENHI Grant Numbers JP25249140, JP15H06193. The authors would like to dedicate this paper to the memory of Dr Philippe Leone who performed the Mo ¨ssbauer measurements at IMN, Nantes.