David Bremner
email: bremner@unb.ca

Olivier Devillers

Marc Glisse
email: marc.glisse@inria.fr

Sylvain Lazard

Giuseppe Liotta
email: giuseppe.liotta@unipg.it

Tamara Mchedlidze

Sue Whitesides

Stephen Wismath
email: wismath@uleth.ca

Monotone Simultaneous Paths Embeddings in R d

We study the following problem: Given k paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension d 2, there is a set of d + 1 paths that does not admit a monotone simultaneous geometric embedding.

Introduction

Monotone drawings and simultaneous embeddings are well studied topics in graph drawing. Monotone drawings, introduced by Angelini et al. [START_REF] Angelini | Monotone drawings of graphs[END_REF], are planar drawings of connected graphs such that, for every pair of vertices, there is a path between them that monotonically increases with respect to some direction. Monotone drawings of planar graphs have been studied both in the fixed and in the variable embedding settings and both with straight-line edges and with bends allowed along edges; recent papers on these topics include [START_REF] Angelini | Monotone drawings of graphs with fixed embedding[END_REF][START_REF] Felsner | Strongly Monotone Drawings of Planar Graphs[END_REF][START_REF] Hossain | Straight-line monotone grid drawings of seriesparallel graphs[END_REF][START_REF] Kindermann | On monotone drawings of trees[END_REF].

The simultaneous (geometric) embedding problem was first described in a paper by Braß et al. [START_REF] Braß | On simultaneous planar graph embeddings[END_REF]. The input is a set of planar graphs that share the same labeled vertex set (but the set of edges differs from one graph to another); the output is a mapping of the vertex set to a point set such that each graph admits a crossing-free drawing with the given mapping. The simultaneous embedding problem has also been studied by restricting/relaxing some geometric requirements; for example, while every pair of planar graphs sharing the same labeled vertex set admits a simultaneous embedding where each edge has at most two bends (see, e.g., [START_REF] Erten | Simultaneous embedding of planar graphs with few bends[END_REF][START_REF] Giacomo | Planar and quasi-planar simultaneous geometric embedding[END_REF]), not even a tree and a path always admit a geometric simultaneous embedding (such that the edges are straight-line segments) [START_REF] Angelini | On a tree and a path with no geometric simultaneous embedding[END_REF]). See the book chapter on simultaneous embeddings by T. Bläsius et al. [START_REF] Blaäsius | Simultaneous embedding of planar graphs[END_REF] for an extensive list of references on the problem and its variants.

In this paper, we combine the two topics of simultaneous embeddings and monotone drawings. Namely, we are interested in computing geometric simultaneous embeddings of paths such that each path is monotone in some direction. Let V = 1, 2, . . . , n be a labeled set of vertices and let Π = {π 1 , π 2 , . . . , π k } be a set of k distinct paths each having the same set V of vertices. We want to compute a labeled set of points P = {p 1 , p 2 , . . . , p n } such that point p i represents vertex i and for each path π i ∈ Π (1 i k) there exists some direction for which the drawing of π i is monotone.

It is already known that any two paths on the same vertex set admit a monotone simultaneous geometric embedding in 2D, while there exist three paths on the same vertex set for which a simultaneous geometric embedding does not exist even if we drop the monotonicity requirement [START_REF] Braß | On simultaneous planar graph embeddings[END_REF]. An example of three paths that do not have a monotone simultaneous geometric embedding in 2D can also be derived from a paper of Asinowski on suballowable sequences [START_REF] Asinowski | Suballowable sequences and geometric permutations[END_REF]. On the other hand, it is immediate to see that in 3D any number of paths sharing the same vertex set admits a simultaneous geometric embedding: Namely, by suitably placing the points in generic position (no 4 coplanar), the complete graph has a straight-line crossing-free drawing; however, the drawing of each path may not be monotone. This motivates the following question: Given a set of paths sharing the same vertex set, does the set admit a monotone simultaneous geometric embedding in d-dimensional space for d 3?

Our main result is that for any dimension d 2, there exists a set of d + 1 paths that does not admit a monotone simultaneous geometric embedding in d-dimensional space. Our proof exploits the relationship between monotone simultaneous geometric embeddings in d-dimensional space and their corresponding representation in the dual space. Our approach extends to d dimensions the primal-dual technique described in a recent paper by Aichholzer et al. [START_REF] Aichholzer | Monotone simultaneous embeddings of upward planar digraphs[END_REF] on simultaneous embeddings of upward planar digraphs in 2D.

Definitions

Let v be a vector in R d and let G be a directed acyclic graph with vertex set

V . An embedding Γ of the vertex set V in R d is called v-monotone for G if the vectors in R d corresponding to oriented edges of G have a positive scalar product with v. Let V = { v 1 , . . . , v k } be a set of k > 1 vectors in R d and let G = {G 1 , G 2 , . . . , G k } be a set of k distinct acyclic digraphs on the same vertex set V . A V-monotone simultaneous embedding of G in R d is an embedding Γ of V that is v i -monotone for G i for any i. A monotone simultaneous embedding of G is a V-monotone simultaneous embedding for some set V of vectors.
If a graph is a path on n (labeled) vertices, it can be trivially identified with a permutation of [1, n]. We look at the monotone simultaneous embedding problem in the dual space, by mapping points representing vertices to hyperplanes in R d . The dual formulation of monotone simultaneous embeddings is as follows (the equivalence of these formulations is shown in the next section).

Let Π = {π 1 , π 2 , . . . , π k } be a set of k permutations of [1, n]. A parallel simultaneous v 2 H 2 : y = 2 H 4 : y = x H 3 : y = 1 2 x + 2 π 1 = (4, 3, 2, 1)
π 4 = (1, 2, 3, 4) π 3 = (1, 2, 4, 3) π 2 = (4, 2, 3, 1) H 1 H 4 H 2 H 3 L 1 L 2 L 3 L 4 v 1 1 1 v 3 v 4 2 2 H 1 : y = -x + 3
embedding of Π in R d is a set of n hyperplanes H 1 , H 2 , . . . , H n and k vertical lines L 1 , L 2 , . . . , L k such that the set of n points L j ∩ H πj (1) , . . . , L j ∩ H πj (n) is
linearly ordered from bottom to top along L j , for all j.

The Dual Problem and Non-Existence Results

The first two lemmas give duality results between monotone simultaneous embeddings and parallel simultaneous embeddings.

Lemma 1. If a set of k permutations of [1, n] admits a parallel simultaneous embedding in d dimensions, it also admits a monotone simultaneous embedding in d dimensions.

Proof. Consider the following duality between points and hyperplanes, where we denote by H the dual of a non-vertical hyperplane H:

H : x d = d-1 i=1 α i x i -α 0 , H = (α 1 , . . . , α d-1 , α 0 ).
This duality maps parallel hyperplanes to points that are vertically aligned (and vice-versa). Let (H i ) 1 i n , (L j ) 1 j k be a parallel simultaneous embedding and refer to Fig. 1By . Therefore (H i ) 1 i n is an embedding such that path π j is v j -monotone, for all j.

Lemma 2. If a set (π j ) 1 j k of k permutations of [1, n] admits a monotone simultaneous embedding in d dimensions, there is a set (π j ) 1 j k that admits a parallel simultaneous embedding in R d where, for every j, π j is either equal to π j or to its reverse.

Proof. As in the proof of Lemma 1, we consider point-hyperplane duality. Let (p i ) 1 i n be an embedding v j -monotone for π j , and (p i ) 1 i n the corresponding set of dual hyperplanes. Let H j be a hyperplane with normal vector v j , 1 j n. Define L j to be the vertical line through point H j . By construction, the points

L j ∩ p πj (i) i
appear in order on L j for one of the two possible orientations of L j . In particular, when v j points downward, L j lists the points L j ∩ p πj (i) from bottom to top and vice versa.

We now prove results of existence and non-existence of parallel simultaneous embeddings, starting with a very simple result of existence. Proposition 1. Any set of d permutations on n vertices admits a monotone simultaneous embedding and a parallel simultaneous embedding in d dimensions.

Proof. Choose d points in general position in the hyperplane x d = 0 and draw a vertical line through each of these points. For each vertical line, choose a permutation and place on the line n points numbered according to the permutation. Fit a hyperplane through all the points with the same number. By construction, this set of hyperplanes is a parallel simultaneous embedding. Going to the dual, by Lemma 1, gives a monotone simultaneous embedding. Alternatively, the monotone embedding can be seen directly by considering the rank in the i-th permutation as the i-th coordinate.

We now turn our attention to non-existence. For proving that there exists k = d + 1 permutations that do not admit a parallel simultaneous embedding in d dimensions, observe that we can consider any generic placement of the d first lines L j since all such placements are equivalent through affine transformations. We then construct permutations for n big enough that cannot be realized with any placement of L d+1 . Similarly, constructing k = d + 1 permutations that cannot be realized even up to inversion, yields the non-existence of a monotone simultaneous embedding in d dimensions by Lemma 2. We start with dimension 2, then move to dimension 3 and only then, generalize our results to arbitrary dimension. Observe that 2D results also follow from [5, Lemma 1 & Prop. 8], but we still present our proofs as a warm up for higher dimensions. Lemma 3. There exists a set of 3 permutations on {0, 1, 2} that does not admit a parallel simultaneous embedding in 2D.

Proof. Let L 1 and L 2 be two vertical lines, H 1 and H 2 two other lines, and let τ 1 = (1, 2) and τ 2 = (2, 1) be two permutations of {1, 2}. As in Fig. 2-left, if L 1 is left of L 2 and the intersections of H 1 and H 2 with L j are ordered according to τ i , we can deduce that H 1 ∩ H 2 is between L 1 and L 2 . It follows that a vertical line crossing H 1 below H 2 is to the left of that intersection point and thus to the left of L 2 . Similarly, a vertical line crossing H 1 above H 2 is to the right of L 1 . If we now consider τ 1 = τ 2 = (1, 2) we have that a vertical line crossing H 1 above H 2 is not between L 1 and L 2 (Fig. 2-center). Consider now π 1 = (1, 0, 2), π 2 = (2, 1, 0) and π 3 = (0, 2, 1). Restricting the permutations to {1, 2} gives that L 3 must be right of L 1 , restricting to {0, 2} gives that L 3 must be left of L 2 , and restricting to {0, 1} gives that L 3 cannot be between L 1 and L 2 (Fig. 2-right). We deduce that no placement for L 3 can realize π 3 . Notice that the reverse order (1, 2, 0) can be realized and thus the dual of this construction is not a counterexample to simultaneous monotone embeddings.

H 1 H 2 τ 1 = (1, 2) L 1 L 2 τ 2 = (2, 1) H 1 H 2 τ 1 = (1, 2) L 1 L 2 τ 2 = (1, 2) H 1 H 2 L 1 L 2 H 0 π 1 = (1, 0, 2) π 2 = (2, 1, 0) π 3 = (0, 2 , 1) unfeasible (1, 2, 0) feasible
Lemma 4. There exists a set of 3 permutations on 6 vertices that does not admit a monotone simultaneous embedding in 2D.

Proof. Let π 1 = (

The sub-permutations of π 1 , π 2 and π 3 on {a, b, c} are (by matching (a, b, c) to (0, 1, 2)) the 3 permutations that do not admit a parallel simultaneous embedding in the proof of Lemma 3. The same is obtained by reversing only π 1 (resp. π 2 , π 3 ) and considering sub-permutations on {a, c, d} (resp. {d, b, e}, {b, f, d}). Other possibilities are symmetric and Lemma 2 yields the result.

Lemma 5. There exists a set of 4 permutations on 5 vertices that does not admit a parallel simultaneous embedding in 3D.

Proof. As in the proof of Lemma 1 we consider 3 points 1 , 2 , 3 in general position in the hyperplane x 3 = 0 and the 3 vertical lines L 1 , L 2 , L 3 going through these points. Let L be a vertical line (candidate position for L 4 ) and its intersection with x 3 = 0. We consider the 3 permutations τ 1 = (1, 2, 3), τ 2 = (2, 3, 1), τ 3 = (3, 1, 2) defining the vertical order of the intersections of L 1 , L 2 , L 3 with hyperplanes (H i ) 1 i 3 . We denote by h i,j the projection of the line H i ∩ H j , 1 i = j 3, onto the plane x 3 = 0. Since the three planes H i , 1 i 3 meet in one point, the lines h 1,2 , h 2,3 and h 1,3 meet at the projection of that point onto the plane x 3 = 0. Refer to Fig. 3. For L to cut H 2 below H 1 , must be in the half-plane limited by h 1,2 and containing 2 , and, similarly, for L to cut H 3 below H 2 , must be in the half-plane limited by h 2,3 and containing 3 . Thus, must be in a wedge with apex h 1,2 ∩h 2,3 (Fig. 3-left). Since h 1,2 separates 2 from 1 and 3 , and h 2,3 separates 3 from 1 and 2 , the union of all wedges, for all possible positions of h 1,2 and h 2,3 , is the union, R, of triangle 1 2 3 and the half-plane limited by 2 3 and not containing 1 (Fig. 3-center). To summarize, if τ 1 = (1, 2, 3), τ 2 = (2, 3, 1), τ 3 = (3, 1, 2), and τ 4 = (3, 2, 1) then 4 (the intersection point of L 4 with the hyperplane x 3 = 0) must lie in this region R. Next, we build the permutations π 1 , π 2 , π 3 and π 4 by repeating this example as follows: π 1 = (0, 1, 2, 3, 4), π 2 = (2, 3, 4, 0, 1), π 3 = (3, 4, 0, 1, 2), and π 4 = (1, 3, 2, 0, 4). The restriction of these permutations to {0, 2, 3} yields that 4 must be in the triangle or in the half-plane limited by 2 3 and not containing 1 . The restriction to {1, 2, 3} yields that 4 must be in the triangle or in the half-plane limited by 1 3 and not containing 2 . The restriction to {0, 2, 4} yields that 4 must be in the triangle or in the half-plane limited by 1 2 and not containing Lemma 6. There exists a set of 4 permutations on 40 vertices that does not admit a monotone simultaneous embedding in 3D. Sketch of proof. The idea is to concatenate several versions of the counterexample of the previous lemma to cover all possibilities of reversing permutations. Note that the number of 40 vertices is not tight. Lemma 7. There exists a set of d + 1 permutations on 3 • 2 d vertices that does not admit a parallel simultaneous embedding in d dimensions. Sketch of proof. As in Lemma 5, the idea is to consider the simplex ( j ) 1 j d and to construct the permutations for the L i in order to prevent all possibilities for placing d+1 .

To get a result in the dual, the difficulty is that we have to prevent not only some permutations but also their reverse versions. Theorem 1. There exists a set of d + 1 permutations on 3 • 2 2d vertices that does not admit a monotone simultaneous embedding in d dimensions. Sketch of proof. As for Lemma 6 we concatenate several versions of previous counter-example to cover all possibilities of reversing permutations.

Fig. 1 .

 1 Fig. 1. Duality between monotone simultaneous embeddings and parallel simultaneous embeddings for k = n = 4 and d = 2.

Fig. 2 .

 2 Fig. 2. Non-existence of two-dimensional parallel simultaneous embeddings.

1 RFig. 3 .

 13 Fig. 3. Non-existence of 3D parallel simultaneous embeddings for 5 vertices.

  H πj (n) are collinear and therefore represent parallel hyperplanes in the dual plane. Consider the vector line v j perpendicular to these hyperplanes and pointing downward. This line crosses them in the order (L j ∩ H πj (1) ) , (L j ∩ H πj (2) ) , . . . , (L j ∩ H πj (n) ) . Since point H i lies in hyperplane (L j ∩ H i ) , points H i , 1 i n, project on v j in the order H πj (1) , H πj (2) , . . . , H πj (n)

definition, line L j crosses hyperplanes H 1 , . . . , H n in the order H πj (1) , H πj (2) , . . . , H πj (n) . The intersection points L j ∩ H πj (1) , L j ∩ H πj (2) , . . . , L j ∩

. Finally, considering {0, 1} yields that

must be outside the triangle (Fig.3right). Thus there is no possibility for placing L 4 .

Acknowledgements. This work was initiated during the 15 th INRIA-McGill-Victoria Workshop on Computational Geometry at the Bellairs Research Institute. The authors wish to thank all the participants for creating a pleasant and stimulating atmosphere.

Research supported in part by the MIUR project AMANDA "Algorithmics for MAssive and Networked DAta", prot. 2012C4E3KT 001, and NSERC.

A Proofs

Proof of Lemma 6. We consider π 1 =(0, [START_REF] Aichholzer | Monotone simultaneous embeddings of upward planar digraphs[END_REF][START_REF] Angelini | Monotone drawings of graphs[END_REF][START_REF] Angelini | Monotone drawings of graphs with fixed embedding[END_REF][START_REF] Angelini | On a tree and a path with no geometric simultaneous embedding[END_REF][START_REF] Felsner | Strongly Monotone Drawings of Planar Graphs[END_REF][START_REF] Giacomo | Planar and quasi-planar simultaneous geometric embedding[END_REF][START_REF] Hossain | Straight-line monotone grid drawings of seriesparallel graphs[END_REF][START_REF] Kindermann | On monotone drawings of trees[END_REF][START_REF] Radon | Mengen konvexer körper, die einen gemeinsamen punkt enthalten[END_REF]20,21,22,23,24,30,31,32,33,34,40,41,42,43,44,50,51,52,53,54,60,61,62,63,64,70,71,72,73,74), π 2 =(2, 3, 4, 0, 1, [START_REF] Hossain | Straight-line monotone grid drawings of seriesparallel graphs[END_REF][START_REF] Kindermann | On monotone drawings of trees[END_REF][START_REF] Radon | Mengen konvexer körper, die einen gemeinsamen punkt enthalten[END_REF][START_REF] Felsner | Strongly Monotone Drawings of Planar Graphs[END_REF][START_REF] Giacomo | Planar and quasi-planar simultaneous geometric embedding[END_REF]22,23,24,20,21,32,33,34,30,31,41,40,44,43,42,51,50,54,53,52,61,60,64,63,62,71,70,74,73,72), [START_REF] Kindermann | On monotone drawings of trees[END_REF][START_REF] Radon | Mengen konvexer körper, die einen gemeinsamen punkt enthalten[END_REF][START_REF] Felsner | Strongly Monotone Drawings of Planar Graphs[END_REF][START_REF] Giacomo | Planar and quasi-planar simultaneous geometric embedding[END_REF][START_REF] Hossain | Straight-line monotone grid drawings of seriesparallel graphs[END_REF]22,21,20,24,23,32,31,30,34,33,43,44,40,41 The idea is that we have eight groups of vertices. Group {0, 1, 2, 3, 4} restricts exactly to the example of Lemma 5 and prevents going from primal to dual without reversing any permutations. Group {10, 11, 12, 13, 14} prevents going from primal to dual reversing exactly π 4 . The other groups prevent all combinations of reversals that leave the first permutation fixed. In this example we prefer the simplicity of proof to optimizing the number of vertices. Counterexamples with less vertices can be easily obtained by sharing vertices between the different groups.

Proof of Lemma 7. As in previous lemma, we generalize Lemma 3 without trying to optimize the number of vertices in the permutations. We consider d points ( j ) 1 j d in general position in the hyperplane x d = 0 and the d vertical lines (L j ) 1 j d going through these points. Let L d+1 be a (variable) vertical line and d+1 its intersection with x d = 0. In a similar manner as in two dimensions consider τ 1 = (1, 0, 2), τ 2 = (2, 1, 0), and τ 3 = (0, 2, 1) and Π 1 ⊂ {i | 1 i d}, Π 2 = {i | 1 i d} \ Π 1 , and Π 3 = {d + 1}; then assume that τ i is the order of hyperplanes H 0 , H 1 , H 2 along L k for any k ∈ Π i . In other words, above k , we have for instance H 2 above H 1 for k ∈ Π 1 and the converse for k ∈ Π 2 ∪ Π 3 .

In projection, this means that h 1,2 = H 1 ∩H 2 separates ( i ) i∈Π1 from ( i ) i∈Π2 and that d+1 is on the side of ( i ) i∈Π2 . Thus, d+1 must be in the pink hatched part in Fig. 4. Considering h 0,2 yields similarly that d+1 must be in the blue hatched part, and consequently, there is a hyperplane through d+1 that separates ( i ) i∈Π1 from ( i ) i∈Π2 . Now we construct π 1 , . . . , π d+1 by concatenating one copy of τ 1 , τ 2 , and τ 3 with three new vertices for each possible partition of {i | 1 i d} in Π 1 and Π 2 . For any such partition, there is a hyperplane through d+1 that separates ( i ) i∈Π1 from ( i ) i∈Π2 . Points ( j ) 1 j d+1 can be seen in R d-1 (since x d = 0) and considering the partition with Π 1 = ∅ yields that there is a hyperplane (in R d-1 ) through d+1 with all ( j ) 1 j d on one side. In other words, there is a hyperplane (in R d-1 ) separating d+1 from ( j ) 1 j d . Projecting ( j ) 1 j d onto that plane (with a central projection with center d+1 ) yields d points in R d-2 , which can be partitioned in two sets, whose convex hulls intersect by Radon's theorem [START_REF] Radon | Mengen konvexer körper, die einen gemeinsamen punkt enthalten[END_REF]. For this partition, there is no hyperplane through d+1 that separates ( i ) i∈Π1 from ( i ) i∈Π2 , which is a contradiction. Hence, these d+1 permutations on 3 • 2 d vertices prevent all placements for d+1 , which concludes the proof. (Note however that this number of vertices is clearly non-optimal.) Proof of Theorem 1. A counterexample of d + 1 permutations (π j ) 1 j d with no monotone simultaneous embedding must be a counterexample of d + 1 permutations with no parallel simultaneous embedding for any set of permutations obtained from (π j ) 1 j d by reversing some of these permutations. Since there are 2 d ways of choosing which permutations are reversed, we can concatenate 2 d images of counterexamples from Lemma 7 by reversing some permutations so that the situation of Lemma 7 appears whatever choice of reversing is done.


Marie-Agathe Charpagne

Suzanne Jacomet

Julien Thebault

Nathalie Bozzolo

Critical recrystallization during sub-solvus annealing in a polycrystalline Nickelbased superalloy

Selective and exaggerated growth of some grains is a phenomenon which is likely to occur during annealing following forging operations, in Nickel-based superalloys. It has been investigated in the Rene 65 alloy in terms of metallurgical characteristics of such grains, mechanisms and kinetics. The nature of this phenomenon, which is commonly reported as abnormal grain growth, is discussed based on the obtained results and appears to be a case of critical recrystallization.

Introduction

Polycristalline Nickel-based superalloys are widely used as structural materials in the aircraft and energy industries. Their mechanical integrity, thanks to their microstructure, has to be guaranteed. Most applications require a fine and homogeneous microstructure consisting of equiaxed grains of 10-30 μm in diameter. Conventional cast and wrought superalloys are forged at high temperature, at which grain boundary migration and recrystallization are triggered. Most processes include a final annealing treatment, which aims at lowering the overall dislocation density and erasing heterogeneities in its distribution among the different grains. This treatment is performed in the subsolvus domain, so second-phase precipitates should pin the grain boundaries. However, observations have shown that some grains can grow in an exaggerated manner during that treatment, despite the presence of pinning particles. The appearance of such grains leads to a bimodal grain size distribution and the resulting heterogeneous microstructures are very similar to those having undergone abnormal grain growth. However, the exact mechanism and origin of appearance of such grains is still not well understood. Recent publications have highlighted the role of the level and distribution of stored energy on the development of such grains [1,2]. Abnormal grain growth occurs at low strains, as a limited number of recrystallization nuclei are likely to be present in the microstructure. The microstructural features of those grains (low dislocation density, front velocity and grain size profile as a function of strain) have lead some authors to propose a scenario of critical recrystallization for their formation. Those grains are thus either already present in the microstructure, as recrystallization nuclei, or either both nucleate and grow during the treatment. Agnoli et al. have mentioned the existence of an incubation time before the appearance of such grains [3]. The distribution of stored energy in the sensitive microstructures has also been shown to be favourable for rare nucleation events by grain boundary bulging [2,4]. Both observations are in favour of the second hypothesis. Under the first hypothesis, the rarity of recrystallization nuclei would make them difficult to observe. However, Charpagne et al. have recently reported the existence of a new recrystallization mechanism, occurring at low strain levels in polycrystalline Nickel-based superalloys [5][START_REF] Charpagne | Superalloys 2016[END_REF][START_REF] Charpagne | Recryst. Grain Growth VI proceedings[END_REF]. This new mechanism consists in the formation of recrystallized γ grains around primary γ' precipitates. Couples of γ' precipitateγ grains have the same crystallographic orientation [START_REF] Charpagne | [END_REF]. This recrystallization mechanism is dominant in the microstructures that are sensitive to abnormal grain growth. The aim of the present work is to study the influence of such grains on the occurrence of abnormal grain growth phenomena.

Material and procedures

The material selected for this study is Rene 65, a polycristalline γ-γ' Nickel-based superalloy designed by General Electric and ATI Allvac [START_REF] Bond | 8th Int. Symp. Superalloys[END_REF]. Its chemical composition is inspired by those of Rene 88DT, which has been adapted for cast and wrought processing : 13 Cr -16 Co -3.7 Ti -2.1 Al -4 Mo -4 W -1 Fe -0.7 Nb -0.05 Zr -0.016 B -bal Ni [START_REF] Heaney | 8th Int. Symp. Superalloy[END_REF]. The samples have undergone an annealing treatment and forging operations triggering hetero-epitaxial recrystallization, described in [START_REF] Charpagne | Recryst. Grain Growth VI proceedings[END_REF]. The deformed microstructures have been characterized by means of a coupled EDS-EBSD technique [START_REF] Charpagne | [END_REF], using the Bruker Quantax Crystalign package (XFlash 5030 EDS detector and e-Flash HR EBSD camera), mounted in a Zeiss Supra 40 FEG-SEM. The forged samples have then been submitted to the standard solution treatment, consisting in an hour exposure at T sol . The solution treated microstructures have been analyzed by means of optical and electron microscopy in order to determine the critical deformation conditions for abnormal grain growth. Based on those results, one sample has been selected for an insitu annealing experiment using the stage described in [START_REF] Bozzolo | [END_REF] mounted in a FEI XL30 FESEM, enabling to follow the microstructure evolutions step by step. EBSD maps have been post-processed using the TSL OIM 5.3 software.

Metallurgical features of the abnormally grown grains

Figure 1 shows the typical microstructure configuration after exaggerate grain growth. Fig. 1-a. shows that those grains have a high density of twin boundaries. Fig. 1-b. is arbitrarily coloured (one colour per grain), ignoring twin boundaries. This reveals the large grain size and the large amount of primary γ' precipitates located inside those grains. Those features are common with those reported in previous studies [2,4,[START_REF] Watson | Eurosuperalloys 2014 proceedings[END_REF][START_REF] Whitis | [END_REF]. The average GOS parameter of those grains is lower than 0.5°. Their stored energy is thus below the noise associated with the EBSD data collection.

Microstructure development along a strain gradient

Figure 2 reports the typical profile of evolution of the maximum and average grain size along the strain gradient of one of the samples. The maximum grain size reaches a peak (point A), corresponding to the appearance of the first overgrown grain. The microstructure is then heterogeneous and more and more large grains invade the microstructure while their size decreases (point B). As the strain increases, the grain size distribution becomes fine and homogeneous again (point C). This profile is also typical from static recrystallization processes.

Incubation time prior abnormal grain growth

Based on the profile presented on Figure 2, the strain ε crit at which the first large grain appears has been measured, for various time exposures at 1065°C. (ε crit corresponds to point A on Figure 2). This critical strain decreases as the annealing time increases and the diameter of the largest grain increases. The large grains thus appear in the most deformed areas of the microstructure and the overgrown grain front moves towards the lower strain levels as the time exposure at T so l increases. The existence of an incubation time for the phenomenon is another characteristic in favour of the first scenario.

In-situ evolution of the hetero-epitaxial grains

A sample forged at 1040°C and 0.01s -1 has been solution treated in-situ in a SEM, by steps of 2 to 5 seconds. All hetero-epitaxial grains have been identified in the initial microstructure and their evolution has been followed step by step in all EBSD maps. Several large grains have developed on the surface of the sample. None of them results from the growth of any of the hetero-epitaxial grains. Instead, the precipitates that were initially surrounded by hetero-epitaxial grains are rather located at the boundaries of the large grains: over 75% of them are located at the grain boundaries of the overgrown grains.

Conclusion

Those results suggest that the phenomenon of abnormal grain growth observed in Rene 65 during solution treatment is more likely to be a case of critical recrystallization, in which a limited number of nuclei form and invade the microstructure. The newly discovered hetero-epitaxial recrystallization mechanism does not seem to promote the formation of those large grains but seem however to limit their growth by pinning their boundaries more efficiently. Their role in this phenomenon remains to be further investigated.

Figure 1 .

 1 Figure 1. EBSD maps of ovegrown grains in a sample forged and annealed at T sol for one hour.

Figure 2 .

 2 Figure 2. Maximum and average grain sizes in a sample forged at 1040°C, 0.001s -1 , water quenched and then annealed at T sol for one hour.

Table 1 .

 1 Critical strain for the appearance of the first abnormal grain and corresponding maximum grain size after forging at 1040°C and 0.001 s -1 .

	t (min)	0 10	20	30	40	60
	ε crit	-0.19 0.17 0.15 0.14 0.05
	D MAX (μm) -	40	55	70	75	170


Adeel Khan
email: adeel_khan18a@yahoo.com

Muhammad Iftikhar Khan

Mubasher Hussain

Engr Muhammad

Reliability Comparison of Five 220 KV Substations in Pakistan Power System

Keywords: National Transmission and Dispatch Company, Event tree analysis, Fault tree analysis

In modern electrical power system, the reliability is considered to be an important factor in power system operation. The term reliability defines as the ability of power system to deliver an adequate supply of electrical energy to the consumers. Fault tree analysis and event tree analysis are the two methods used here for reliability analysis. To find the failure probability fault tree is used. To classify whether the system is safe or not event tree is used. The reliability of five 220kV National Transmission and Dispatch Company (NTDC) substations of Pakistan with respect to protection failures evaluates in this paper. The main method used in this study is Event Tree Analysis. Fault tree analysis helps to find the input for event tree.

INTRODUCTION

In electrical power system, the grid substation is the strongest point, it still contains a chance of loss of load due to weak point in the system. Therefore, it is important to determine the substation reliability as it helps to find the point of failure that may be causing unreliability to whole system.

The three factors on which the substation reliability mainly depends that leads to protection failure are: incorrect system design, incorrect component installation and deterioration of service [START_REF] Roy Billinton | Reliability Evaluation of Power Systems[END_REF]. To evaluate the reliability of substation different techniques and computational simulations are used. Substation connectivity is mainly focused in most of the studies. Protection system reaction is fully neglected. In substation arrangements to select the failure modes and in outage categories, minimum cut set techniques are used.

Event tree analysis (ETA) [START_REF] Adebayo | A Reliability Assessment of Power Transmission And Distribution Systems; Ibadan South-West Nigeria As a Case Study[END_REF][START_REF] Hong | Reliability Assessment of Protection System for Switchyard Using Fault-Tree Analysis[END_REF], fault tree analysis (FTA) [START_REF] Adebayo | A Reliability Assessment of Power Transmission And Distribution Systems; Ibadan South-West Nigeria As a Case Study[END_REF][START_REF] Hong | Reliability Assessment of Protection System for Switchyard Using Fault-Tree Analysis[END_REF], state enumeration and Monte Carlo simulation [START_REF] Dong | Reliability of various Industrial substations[END_REF][START_REF] Rausand | System Reliability Theory: Models, Statistical Methods, and Applications[END_REF] are the methods that are mostly used. Which technique is to be used varies on the method of evaluation. FTA, state enumeration and Monte Carlo simulation used to study the static reliability. ETA and sequential Monte Carlo methods are used for dynamic reliability.

Design and connection faults are neglected in this paper and only considered criteria is the service deterioration. Protection system failures and failures of associated circuit breaker (CB) are the two types of failures of protection that are considered in this study. FTA is combined with ETA in this study. The method used in this study is ETA and to get the event tree input FTA output is used [START_REF] Haarla | A method for analyzing the reliability of a transmission grid[END_REF][START_REF] Megbowon | A Reliability Assessment of the South-Western Transmission Lines of the Nigeria national Grid[END_REF].

METHODOLOGY

To minimize the quantity of elements in event trees, components are combined into one unit that are in the similar protection zone. The protection scheme will respond if any element in a zone fails. Consequently, to evaluate this "unit" unavailability FTA is required. The result for the ETA will be the probability of initiating event.

Based on the back up protection, primary protection, and CBs functionality ETA has four states [START_REF] Wang | Reliability Evaluation of Substations Subject to Protection Failures[END_REF].

State 1: There will be no failure of protection if the circuit breakers and primary protection both function well and successfully it will clear the fault.

State 2: If the fault sense by the primary protection but associated CBs fails to respond, it will have energized the failure function of the CB. The circuit breaker failure function will trip the associated CBs.

State 3: If primary protection fails to detect the fault, after some time delay the fault will be cleared by the back-up protection.

State 4: The chance of failure of both the back-up protection and primary protection are not considered. The back-up CBs in this paper are supposed to be perfect.

The probability can be simply calculated of event trees with probabilities products along the tree branch, if the events are independent of each other. Thus, by multiplying the tree branch"s probability of each event, the event tree of four states probability can be evaluated,

( ) ( ) ( ) ( ) ( )
where Uav(ini) represents the initial event unavailability, Ava(pri) represents the primary protection availability and Ava(CB) represents the circuit breaker availability.

If the initiating event input data changed to failure frequency from its unavailability while keeping the protection failure and CB failure conditional probabilities, it will also change the tree output to a failure frequency,

( ) ( ) ( ) ( ) ( )
where fre(ini) represents the initial event failure frequency. The probability of failure and unavailability concept pairs are comparatively the similar. Availability/unavailability are the used parameters in this paper.

Before analyzing substation reliability, first made several assumptions.

• Components that are connected to the substations such as isolators, circuit breakers, bus bar, step-up transformers, stepdown transformers, potential transformers, current transformers, cables and overhead lines are having a chance to failure.

• Substation reliability not effect by surge arrestors and earthing switches within substation so they are ignored.

• Assumed that back-up circuit breakers system and the systems back-up protection are to be ideally perfect.

• Assumed that neighboring substations are working properly.

A. Reliability Effect on Substation Lines

To understand the reliability effect on substation lines, consider a "S/S 1" substation as shown in Fig. 1. "S/S 2"is the neighboring substation that comprise of circuit breakers that will be perfect, i.e. S/S 2 circuit breaker will never fail. 3) is calculated in hours, but MTTF and MTTR both in equation ( 4) are measured in years.

B. Reliability Data

The different components of substation i.e. voltage transformer, current transformer, CBs, and other components of switchgear, failure statistics are unavailable for power system of Pakistan. Thus, various sources are used to get the statistics from [START_REF] Wang | Reliability Evaluation of Substations Subject to Protection Failures[END_REF][START_REF] Sadegh | Fault Tree Analysis Approach in Reliability Assessment of Power System[END_REF]. The substation configuration data and transmission line lengths are carried from the Pakistan"s national grid operator, NTDC.

C. Reliability Evaluation of NTDC Substations

Pakistan"s national grid operator i.e. NTDC operates and maintains twenty-nine 220 KV and twelve 500 KV grid stations, [10]. Reliability assessment of five 220 kV substations that are Tarbela, Burhan, Islamabad, Peshawar and Daudkhel are compared in this paper. Detail analysis for 220 kV Tarbela substation is presented.

Distance protection, differential protection, and busbar protection in substation coordinates with each other. The fault location mainly determines which of these protection schemes respond to fault. For the components, the effect on the protection will be the similar that are within one protection zone. Hence, the 220 kV Tarbela grid substation can be divided into three zone types.

a. Zone-1: Bus Bar

As in Fig. 2, the initiating fault occurred in busbar protection zone, it should react and isolates the D3Q3, D2Q2 and D1Q2 circuit breakers. The failure function of CB will be activated and trip Burhan-1, D1Q1, Burhan-3, D2Q1, Mardan-1and D3Q1 if any of these CBs does not respond. In substation, there is no back-up protection. If the protection of busbar fails to respond, then fault in this substation will not be sensed by the distance protection. It will be seen by the nearby substation"s distance protection that will clears the fault. Therefore, it is assumed that it will shut down the whole substation.

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In fig. 3 the event tree for busbar (BB2) fault zone is shown. There will be no lines isolated if both the associated CBs and busbar protection (primary protection) works effectively. The CB failure function will isolate the next stage of CBs if busbar protection successfully works, but fails to trip one of the associated CBs. In this case, the corresponding lines Burhan-1/2 and Burhan-3, Mardan-1/2 and Islamabad will be isolated. If the busbar fault fails to detect by the busbar protection it will shut down the entire substation. The condition is not considered where back-up protection and primary protection both fails, since in this study the back-up protection is supposed to be ideally perfect. In fig. 6 the zone for step-down transformer is presented. In this case if fault occurs on line, then field differential of line will response first. If it is line fault, the differential protection of line will act as primary protection. If the fault is in the zone of transformer, it is cleared by the field differential protection of transformer. The similar CBs will trip by these various modes of primary protection systems, therefore, combined these three various zones into one.

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In fig. 7 the event tree for Burhan-2 fault zone is shown. If this zone detects an initiating fault, the CBs: D1Q2, D1Q20, D1Q3 will be trip by the primary differential protection and isolates only Burhan-2. If one of these CBs: D2Q2, D3Q2 and D1Q1 fails to response, the circuit breaker failure function will isolate Burhan-1. It will also isolate Burhan-2. The distance protection should response due to the failure of primary protection and it will isolate Burhan-1.

RESULTS & ANALYSIS

The frequency failures and resulting probabilities of lines can be calculated from event tree. By using a small fault tree, the initiating events unavailability and failure frequency is computed first. The zone is considered as failed if any of the components inside a zone fails. As such, each zone reliability can be evaluated as it is considered as a small series associated system.

TABLE I: ZONE RELIABILITY OF TARBELA S/S

In Table I the Tarbela substation zone reliability is presented.

Above calculations are the outcomes of Fault Tree Analysis and used for Event Tree Analysis input. By applying the technique discussed in section III, the losing lines frequency and probability can be evaluated by adding up the same end states failure statistics. In Table II Due to any fault in the components the line"s frequency failure and unavailability shows the probability that this substation line is isolated. Using event trees these are calculated. In reliability assessment, it is observed that transmission lines are the dominant element. The longer the line, greater the possibility that the fault occurs in line.

The longest transmission lines are Mardan-1 and Mardan-2: 67 km. So, these are the highest failure frequency lines. The MTTF shows that due to a fault in the component it takes average time before the specific line isolated. The MTTR shows that the line takes average time to detect and clear the fault, and then placed that isolated line in return to function.

In terms of highly reliable zone, a data of the five NTDC 220 kV substations and the highest probability zone to fail on behalf of all five substations are showed in Table III By adding up the different zone"s failure frequency and unavailability, the overall failure frequency and unavailability is computed. Then by using Eqs. ( 3) & (4) MTTF is computed. From figure, the highest unavailability substation is Daudkhel which has of slightly over 0.0050, followed by an unavailability of Tarbela which is over 0.0030. Burhan and Islamabad substations have same level of unavailability as they are near the Islamabad, the capital city. Tarbela grid substation has the highest MTTF.

With lowest MTTF and highest frequency failure the Daudkhel substation is most vulnerable. Failure frequency and MTTF of Islamabad substation will seriously affect operational safety and design of the proposed new substation of 220 kV at Mansahra.

CONCLUSION

Reliability of five NTDC 220kV substations are examined and their frequency failure, unavailability and MTTF is compared in this paper. This study will help the NTDC to understand the area of grid station where most faults occur. By proper maintenance of components in substation reliability can be enhanced. During the study, it has been observed that the transformers and line/cables are the major part in the calculation of the system"s reliability, due to their large failure frequency the lines with similar location and parameters will have similar statistics of failure. All the substation components are supposed to be working properly in this paper, which shows that maintenance is not included. In future studies maintenance might be included.

Figure 1 .

 1 Figure 1. Reliability effect on substation lines

Figure 2 .Figure 3 .

 23 Figure 2. 220 kV Tarbela substation Busbar Zone

Figure 4 .

 4 Figure 4. 220 kV Tarbela substation Line zone

Figure 6 .

 6 Figure 6. 220 kV Tarbela grid station Step-down transformer zone

Figure 8 .

 8 Figure 8. Reliability Tables of five 220kV substations: (a) failure frequency, (b) MTTF (c) unavailability

  results are presented.

TABLE II :

 II FAILURE STATISTICS OF SPECIFIC LINES

  . The highest reliable zone is the one which has least failure frequency or highest MTTF and the zone which have highest frequency failure or lowest MTTF has the highest possibility of failure.

		Failure	Unavailability of
		Frequency of	Zone
	Zone	Zone
	Burhan1	0.08701	0.00007936
	Burhan2	0.08701	0.00007936
	Burhan3	0.08766	0.00008006
	Islamabad	0.1472	0.00013446
	Mardan1	0.1573	0.00015705
	Mardan2	0.1573	0.00015705
	Substation	Zone:	Zone:
		Most Reliable	Most
			Unreliable
	TARBELA	Burhan1 and 2	Mardan1 and 2
	ISLAMABAD Burhan	Mansahra
	PESHAWAR	Sheikh	Ghazi Barotha
		Muhammadi1 and 2
	BURHAN	Sangjany	Tarbela3
	DAUDKHEL	CHASNUPP1 and 2 Sheikh
			Muhammadi

Vol. 4, Issue 10, PP. 173-183, October 2017 ISSN: 2409-2770

TABLE III :

 III ZONE RELIABILITY OF GRID SUBSTATION

	Overall frequency failure, MTTF and unavailability of the
	five NTDC 220 kV substations are shown in figures 8(a) to
	8(c).

ACKNOWLEDGMENTS

We would to acknowledge all the faculties and students of Electrical Engineering Department, UET Peshawar for their assistance, anticipation and resources extended to us.


Maryam Shahjehan

Waleed Shahjehan

Muhammad Naeem Arbab

Grid-Tied PV System with Energy Optimization

Keywords: Smart grid, Solar system, Grid connected system, Renewable energy, Continuous supply of energy. I

Electricity that is generated from coal, natural gas and fossil fuel has an impact on human health and also causes global warming. The integration of renewable energy sources with the grid is a good solution to these problems. This approach is known as smart grid. Sources of renewable energy such as wind or PV are not able to provide a continuous supply of energy to the load due to periodic or seasonal variations. Connecting these renewable sources to the grid can help in overcoming these problems as the grid will ensure continuous supply of power to the load. The smart grid can help in reducing the impacts on human health and effects on environment and can also help in increasing efficiency, reliability and security of the grid. Simulations were carried out on Matlab software and their results shows that the use of solar system with the grid helps in reduction of the electricity charges.

INTRODUCTION

For attaining some goals such as balance between customer load demand and generation, reducing the cost of electricity, reducing the burden on the grid and increasing the amount of power from renewable sources, smart grids have played an important role [START_REF] Sun | Adoption of Solar Grid-Tied PV-System Adopted in a Residential Building[END_REF]. Intelligent devices are installed and used with components at different distribution and transmission levels in order to enhance the reliability and security of the main grid. Smart grids are a part of efficient, safe and environment-friendly energy system. The accuracy of nonlinear calculation of consumption of power has been improved through system of smart metering. The purpose of recent initiatives of smart grid is to enhance the efficiency of the grid by decreasing its environmental cost [START_REF] Lee | Renewable integration algorithm to compensate PV power using battery energy storage system[END_REF]. Renewable sources of energy can be used in islanded mode or can be integrated to the grid. Grid-tied renewable sources of energy can be used for which the grid behave as a dispatchable source and supply of power to the load is made continuous.

II. SOLAR PHOTOVOLTAIC SYSTEM

Efficiency of solar cells is a very important factor which is currently 18.3% in commercial market but the efficiency of module is a little bit lower than the cell efficiency due to spaces between cells in a module. The efficiency of the system takes into consideration all the system component's performance.

Several factors like electrical performance of solar cell, factors of degradation, the capability of producing power of an array, environmental factors and operating temperature of a cell determine the performance of an array.

A. Ouput Power of PV Array

It is the amount of power that a PV array will produce and is expressed in watts which is either specified as peak watts or average power for a day.

B. Output Energy of PV Array

It is the amount of energy that a PV array will produce and is expressed in Wh.

C. Characteristics of PV array a. Effect of Irradiance

The incident irradiance affects the output power of PV. Fig. 1 shows the effect of irradiance on current and voltage of the PV module. The current in case of short circuit of the PV increases linearly with increase in irradiation, whereas there is an exponential increase in open circuit voltage with the increasing irradiation [START_REF] Sa'd Basha | Analysis and evaluation tools development of photovoltaic modules and system performance under Jordanian and German climatic conditions[END_REF].

b. Effect of Teperature

The ambient temperature highly affects the temperature of the module. Fig. 2 shows the effect of ambient temperature on current and voltage of the PV module. When the temperature of the PV module rises above 25ºC, the current for short circuit increases. However, the effect on open circuit voltage is enormous. It means that the increase in current is relatively lower than the decrease in voltage. So the PV output power is reduced [START_REF] Sa'd Basha | Analysis and evaluation tools development of photovoltaic modules and system performance under Jordanian and German climatic conditions[END_REF].

ISSN: 2409-2770

c. Maximum Power Point

It is the point where the electrical power obtained from the PV is maximum. In Fig. 3, the maximum power point (MPP) is represented by the I-V curve's knee. It is obtained by multiplying the current at the MPP and voltage at the MPP. It is the maximum power possible at standard conditions [START_REF]Guide To Interpreting I-V Curve Measurement of PV Array[END_REF]. The usable output power depends on efficiency of the module.

d. Fill Factor (FF)

It is an essential parameter for PV module which represents the largest rectangle area that fits in I-V (current-voltage) curve. It is shown in Fig. 4. The output power magnitude depend on fill factor (FF). If the value of fill factor is high, the output power will also be higher. It is actually the ratio of two rectangular areas. Equation (1) shows the formula for fill factor. A unity fill factor (ideal case) shows that both the rectangles are similar [START_REF]Guide To Interpreting I-V Curve Measurement of PV Array[END_REF].

(1)

Where

is the current at the maximum power point and is the voltage at the maximum power point. is the open circuit voltage and is the short circuit current. e. Efficiency of Module The efficiency of the PV module or cell is its ability of converting sunlight into electricity. Mathematically, it is the ratio of module output power to area and is given by ( 2).

(2)

Where

is the current at the maximum power point and is the voltage at the maximum power point. is area of module and is global irradiance which is considered 1000 W/m².

D. Parts of a PV Module

a. Front Surface It is basically a cover of glass. It should have low capability of reflection for concerned wavelength of sunlight. It should have high transmission. Iron glass is normally used due to some reasons such as low cost, stability, strength, high transparency as it is impermeable to gases and water [START_REF] Honsberg | pvcdreom[END_REF].

b. Encapsulant It provides strong bonds between the cells in module. It needs to be stable for different range of operating temperatures and must have low thermal resistance and highly transparent [START_REF] Honsberg | pvcdreom[END_REF]. A thin layer of EVA (ethylene vinyl acetate) is used at back and front surface of assembled cells.

c. Back Surface It is a back sheet for PV module which can be made from glass or thin polymer. Its thermal resistance should be low [START_REF] Honsberg | pvcdreom[END_REF]. Fig. 5 shows the layers in the PV module. a. Crystalline Solar cell technology Crystalline solar cells are the most efficient solar cells in the market. The PV cells based on silicon are generally long lasting and more efficient than any other cells [START_REF]Solar facts and advice[END_REF]. However, higher temperature for operation results in reduction of efficiency.

1. Monocrystalline solar cell Technology They are the most efficient solar cells made from wafers of silicon after a complex process of fabrication [START_REF]Solar facts and advice[END_REF]. Their efficiency is 15-18%. They are available in many shapes such as round, square or semi-round. Their thickness lie between 0.2-0.3mm. Square or semi-round cells are a bit expensive than round ones as more material is wasted during their production but the round ones are still used rarely because they don't properly utilize the space of module. They are available in different colors such as black, dark blue (with anti-reflection coating) and grey (without anti-reflection coating). Fig. 6 shows a layered structure of monocrystalline solar cell.

Polycrystalline solar cell technology

Their cost per unit area is less than monocrystalline solar cells but the structure of module for polycrystalline module and monocrystalline module are similar. Use of large square cells increase the efficiency of the module. Their efficiency is 13-16%. They are available in square shapes with 0.24-0.3mm thickness. They are available in different colors such as blue (with anti-reflection coating), grey, gold, brown, green (without anti-reflection coating) and silver. Fig . 7 shows a polycrystalline module and cell. They have high ability to absorb light, so a low thickness such as 0.001mm would be sufficient to convert the incident irradiation [START_REF]Planning and installing photovoltaic systems[END_REF]. They generally do not have a standard size of wafer like crystalline solar cells. The substrate of thin film technology can be of any size coated with a material of semiconductor. Despite the fact that they their efficiency is low per unit area, they have many advantages in comparison to the crystalline technology. They are not very sensitive to high temperature for operation. They are not very sensitive to shading due to narrow and long strip design whereas shaded cell in crystalline module will have effect on the performance of whole module. Their energy yield is higher than crystalline under certain condition. They can utilize the low and diffuse intensity of light in a better way. Fig. 8 shows a thin film solar module.

Amorphous Silicon solar cell technology

In this technology, a very thin layer of silicon is deposited on glass, metal or plastic. It is not preferred to install this technology on roof as its efficiency is low per unit area. Another drawback of this technology is degradation with induced light which decreases the efficiency of module during the starting 6-12 months before the nominal power output is levelled at a stable point or value. This effect is called Staebler-Wronski effect [START_REF]Planning and installing photovoltaic systems[END_REF]. They have relatively lower efficiency 5-7%. The thickness of substrate is 1-3mm while the coating thickness is 0.001mm with about 0.0003mm amorphous silicon. They can be in blue, reddish-brown and blue-violet colors.

Vol. 4, Issue 10, PP. 184-189, October 2017 ISSN: 2409-2770 Fig. 9 shows the amorphous silicon module and layered structure of its cell.

Copper indium Selenium technology

It has the highest efficiency among the thin film solar cell technology. The di-selenide is also sometimes alloyed with Sulphur or gallium. Unlike amorphous silicon, it is not degraded with induced light. The cost for its processing method is low and it uses very less active material [START_REF]Planning and installing photovoltaic systems[END_REF]. Its spectrum response is wide for sunlight. Their efficiency is 9-11%. The thickness of substrate is 2-4mm with coating thickness of 3-4μm. They can be in black or dark grey colors.

Micromorphous Technology

This technology combines amorphous silicon and microcrystalline in a tandem structure which improves the module performance. They are capable of responding to wide band of wavelength of solar spectrum, so their efficiencies can reach to 9.1%. Their degradation with induced light is less than amorphous cells [START_REF]Planning and installing photovoltaic systems[END_REF]. Fig. 12 shows a layered structure of the micromorphous technology along with its spectrum response.

III. RESULTS AND DISCUSSION

If the PV system was not used, grid will provide all power to the load as shown in Fig. 13. The grid connected PV system used in this paper is shown in Fig. 14. It is composed of PV array that is connected to the grid. Without the use of solar system, all the power to the load will be provided by the grid but with the use of the hybrid system, the power taken from the grid decreases as shown in Fig. 17. No power is taken from the grid from 9am-6pm because the PV output power during this time is greater than the load demand. If the generated PV output power is greater than the power required by the load, the excess PV power is sold to the grid as shown in Fig. 18. The daily cost of electricity without the PV system is 295.53PKR (Pakistani Rupees) and after the installation of this solar PV system it becomes 130.83PKR. The amount of electricity that is being sold to the grid generates a revenue of 51.9PKR on daily basis. So we will be charged 78.93PKR for electricity usage on daily basis which means that with the introduction of PV system in the home, we can save 216.6PKR daily.

Figure 1 .

 1 Figure 1. Effects of irradiance on current and voltage of PV module

Figure 2 .

 2 Figure 2. Effect of ambient temperature on current and voltage of module

Figure 3 .

 3 Figure 3. Maximum power point

Figure 4 .

 4 Figure 4. Fill factor

Figure 5 .

 5 Figure 5. Layers in PV module

Figure 6 .

 6 Figure 6. Monocrystalline PV module

Figure 7 .

 7 Figure 7. Polycrystalline module and cell b. Thin film solar cell Technology Thin film represents second generation of PV. Because of their less consumption of energy and less material required for production, they are cheaper as compared to crystalline solar cells. The semiconductor materials used include cadmium telluride, copper indium selenium and amorphous silicon.They have high ability to absorb light, so a low thickness such as 0.001mm would be sufficient to convert the incident irradiation[START_REF]Planning and installing photovoltaic systems[END_REF]. They generally do not have a standard size of wafer like crystalline solar cells. The substrate of thin film technology can be of any size coated with a material of semiconductor. Despite the fact that they their efficiency is low per unit area, they have many advantages in comparison to the crystalline technology. They are not very sensitive to high temperature for operation. They are not very sensitive to shading due to narrow and long strip design whereas shaded cell in crystalline module will have effect on the performance of whole module. Their energy yield is higher than crystalline under certain condition. They can utilize the low and diffuse intensity of light in a better way. Fig.8shows a thin film solar module.

Figure 8 .

 8 Figure 8. Thin film solar module

Figure 9 .

 9 Figure 9. Amorphous modules and amorphous cell layered structure

  Figure 10. CIS module and CIS cell layered structure3. Cadmium Telluride solar cell technologyThis technology has the lowest cost for production as compared to any other thin film technology. They are not degraded with induced sunlight[START_REF]Planning and installing photovoltaic systems[END_REF]. They have 7-8.5% efficiency. The substrate is 3mm thick with a coating thickness of 0.005mm. They are available in dark green or black colors. Fig.11shows the Cadmium telluride module and layered structure of its cell

Figure 12 .

 12 Figure 12. Layered structured of micromorphous technology with its spectrum response

Figure 13 .

 13 Figure 13. Bloack diagram without PV system

Figure 15 .

 15 Figure 15. Load demand of home Fig. 16 shows the PV output power for a 2KW array used in this simuation. The technology used for its manufacturing is the polycrystalline silicon.

Figure 16 .

 16 Figure 16. 2KW PV array output power

Figure 17 .

 17 Figure 17. Power taken from the grid

Figure 18 .

 18 Figure 18. Power sold to the gridThe cost of electricity decreases after the use of PV system. The cost of the home with and without the solar system is shown in Fig.19.

Figure 19 .

 19 Figure 19. Cost of energy with and without PV system

Vol. 4, Issue 10, PP. 184-189, October 2017 ISSN: 2409-2770 IV. CONCLUSION

This paper has shown the advantages of PV integration with the main power grid. The system becomes reliable and stable and can handle the changes in solar system. The grid connectivity balances the intermittent nature of solar PV and helps to provide power to the load continuously. Moreover, with the use of renewable energy , the power usage from the main grid is minimized.


Mubasher Hussain

Naeem Arbab

Adeel Khan

High Frequency Modeling of Transformer Using Black Box Frequency Response Analysis

Keywords: Power transformer, Black box analysis, Frequency response analysis, resonance frequency. I

Lightning Surges consequently induce high frequency overvoltage to transformers. Therefore, it is alluring to study the transfer voltage of lightning surges from primary to secondary side of transformer. Keeping in mind the end goal to do lead this review, high frequency Siemens power transformer of rating 25KVA, 25KV/400V is to be examined. In this paper, an advanced and modified high frequency model of transformer is presented for protection of load against lightning transients. Suggested model is modified form of N.A Sabiha model and based on black box, two ports four terminal network theory. For No load condition, Transformer parameters are calculated on two resonance frequencies of 450 KHz &1 MHz using fast Fourier transform. Matlab/Simulink is used for simulation analysis. Both Time and frequency domain validate the accuracy. Resemblance between the measured and calculated results confirms the precision of proposed model when an impulse of 1.2/50µs is applied to transformer terminals.

INTRODUCTION

Lightning surges shifted to consumer low voltage side through the interwinding capacitance of transformer. So, it is paramount importance to design the transformer to withstand the high voltages and high frequency stresses during lightning. Lightning impulse test devised to proof the high frequency behavior of transformer [START_REF] Liang | Modeling of transformer windings under very fast transient overvoltages‖[END_REF]- [START_REF] Samarawickrama | Impulse Generator Optimum Setup for Transient Testing of Transformers Using Frequency-Response Analysis and Genetic Algorithm[END_REF]. Different high frequency protected modelsform lightning are Proposed to study the transient behavior of transformers for both loaded [START_REF] Sabiha | Lightning-Induced Overvoltages Transmitted Over Distribution Transformer With MV Spark-Gap Operation-Part I: High-Frequency Transformer Model[END_REF]- [START_REF] Silva | Power transformer modeling based on wide band impedance and admittance measurements[END_REF] and unloaded conditions [START_REF] Vaessen | Transformer model for high frequencies[END_REF]- [START_REF] Piantini | Voltage surges transferred to the secondary of distribution transformers[END_REF]. The high-frequency behavior can be model by different ways.

In mechanical way computing a lumped electrical system in the light of geometry, winding stresses during transients and material properties, yet requires mechanical detailed data like winding material, gap between windings, physical structure, material properties including dimension and core size, about the transformer [START_REF] Abed | Physics-based high-frequency transformer modeling by finite elements[END_REF]- [START_REF] Abeywickrama | High-frequency modeling of power transformers for use in frequency response analysis (fra),‖ Power Delivery[END_REF]. Winding deformation can be calculated by FRA using finite element method. Conditional monitoring of large power transformer using SFRA is presented in [START_REF] Sood | Condition monitoring of power transformer using sweep frequency response analysis[END_REF]. Using frequency analyzer, the values for L, R and C were calculated for equivalent model of transformer. When inductance increases then disk deformation and local breakdown occur. Value of resistance is dependent on resonance frequency. The desirable mechanical data is hardly ever provided by the transformer manufacturer.

When detailed data and internal structure of transformer is not present, black box model is appropriate to acquire high frequency behavior of transformer [START_REF] Sabiha | Lightning-Induced Overvoltages Transmitted Over Distribution Transformer With MV Spark-Gap Operation-Part I: High-Frequency Transformer Model[END_REF]- [START_REF] Sabiha | Experimental verification of distribution transformer model under lightning strokes[END_REF], [START_REF] Piantini | Modeling of Three-Phase Distribution Transformers for Calculating Lightning Induced Voltages Transferred to the Secondary‖[END_REF]- [START_REF] Piantini | Voltage surges transferred to the secondary of distribution transformers[END_REF], [START_REF] Holdyk | Wideband Modeling of Power Transformers Using Commercial sFRA Equipment[END_REF]- [START_REF] Zhongyuan | A high-frequency circuit model of a potential transformer for the very fast transient simulation in gis,‖ Power Delivery[END_REF]. In [START_REF] Heindl | Transformer modeling based on standard frequency response measurements[END_REF] M Heindl compares white box, gray box and black box model. In White box model complexity is high; bandwidth is low as compared to gray and black box. But it allows deeper system view. While gray box lies between white and black box model. In [START_REF] Aghmasheh | Gray Box Modeling of Power Transformer Windings for Transient Studies[END_REF] artificial method is used for gray box analysis of transformer parameter calculations. The numbers of unknown parameter are reduced using both Weibull Distribution function and exponential function. In [START_REF] Jurišić | Transformer model for calculation of high frequency, transmitted overvoltages[END_REF] capabilities of a black box model were analyzing to characterize the transformer at high frequency. Measured values and simulated values in EMTP-RV for transmitted over-voltages are compared.

Black Box model has several terminals and based on terminal measurements. While in [START_REF] Sabiha | Lightning-Induced Overvoltages Transmitted Over Distribution Transformer With MV Spark-Gap Operation-Part I: High-Frequency Transformer Model[END_REF] N.A Sabiha presented transformer model for dual resonance frequencies which is based on two ports four terminals network theory. Aforesaid model is modified form of [START_REF] Piantini | Modeling of Three-Phase Distribution Transformers for Calculating Lightning Induced Voltages Transferred to the Secondary‖[END_REF] Piantini model, based on single resonance frequency. In aforesaid model resonance frequency is calculated by transfer function. The high frequency behavior of power transformer based on several resonance points in a wide frequency band is due to inductive and capacitive behavior of transformer [START_REF] Gustavsen | Wide band modeling of power transformers,‖ PowerDelivery[END_REF]. Frequency band is divided into low, medium and high frequencies but these levels do not have distinct boundaries. The transformer equivalent T or Pi model is based on lumped parameters.

In [START_REF] Noda | Accurate modeling of coretype distribution transformers for electromagnetic transient studies[END_REF] transfer of surges form primary to secondary side and effect of internal capacitance on winding skin effect of transformer were determines. In proposed model, the parameters were determined by frequency-characteristic measurement using an impedance analyzer. Hysteresis and saturation effect not discussed because CIGRE WG suggested that the hysteresis effect can be neglected in lightning surges. Similarly, Penetration of magnetic flux can be neglected at 1MHz frequency or for higher frequencies. In his model, various surges calculation is carried out by EMTP. In [START_REF] Župan | Transformer Windings Parameters Calculation and Lightning Impulse Voltage Distribution Simulation[END_REF] transformer winding parameters like R, L and C matrices were found using numerical method for lightning test. In [START_REF] Bigdeli | Optimized Modeling of Transformer in Transient State with Genetic Algorithm[END_REF] the transformer modeling is optimized using genetic algorithm and one of important application in fault detection is discussed. Some recent models of transformers have been developed in [START_REF] Chong | A Novel Algorithm to detect Internal Transformer Faults‖[END_REF]- [START_REF] Paulraj | Modeling and location of faults in power transformer using Transfer Function and Frequency Response Analysis[END_REF] to study the transferred surgeries and fault diagnostic techniques by using a program of transition electromagnetic called EMTP/ATP and Orcad-Pspice. However, all these models need some sophistication. So, modified model of power transformer at two resonance frequencies is presented here. Similarly, when impulse applied on secondary side of test transformer and primary side was kept open circuit then calculate the value of Vs, Is, and Vp as shown in figure 3. Apply the Fast Fourier transform to the discrete data of these currents and voltages wave foams and using transfer function method find out the values of parameters. Amplitude of Z11 and phase angle is given in figure 5. Similarly, the value of Z12 is shown in figure 6. The value of Z22 is shown in figure 7.

CONCLUSION

The modified model for transformer load protection from lightning surges of specific frequency is tested at 4KHz and 1 MHz.The resonance frequency for both Z12 and Z21 is same which validate the proposed model. Therefore, it can be very easily realized that the transformer model for protection of load from transients or lightening signal is very efficient. The proposed technique can be used for online conditional monitoring of transformer in future.

  Mubasher Hussain: Research Scholar at University of Engineering and Technology, Peshawar, Pakistan. Email:mubasher_hussain921@yahoo.com Cell: 0092-3455606120. Naeem Arbab: Professor at University of Engineering and Technology, Peshawar, Pakistan. Adeel Khan: Research Scholar at University of Engineering and Technology, Peshawar, Pakistan. Email: adeel_khan182@yahoo.com Cell: 0092-.3347354535 ISSN: 2409-2770In[START_REF] Vaessen | Transformer model for high frequencies[END_REF] PTM Vaessen proposed transformer model for high frequencies. The Proposed model was for no load condition and based on black box analysis. For value of inductance (L) the value for frequency ω will be minimum or nearly equal to zero and for the value of capacitance (C) the value of frequency ω is very high nearly equal to infinity. The value of L and C determined from imaginary part of Z(ĩω).

  II. TWO PORTS NETWORK THEORY Two ports network theory is used to determine Z-Parameters (impedances), Y-parameter (Admittance), Hparameters (Hybrid) and T-Parameters (Transmission). In this theory two ports have four terminals and Network is represented by black box. The driven source may be voltage or current. But here the driven source is impulse voltage. Based on connection of impedances, two ports network can be classified into T or π (Pi) network. Using open circuit test, we determine the different parameters of T-model transformer. The linear T-network equations are (1) (2) When impulse is applied on primary side of transformer and secondary remain open then we find the following quantities. Similarly, when impulse is applied on secondary and primary remain open then we obtain III. EXPERIMENTAL SETUP The experiment was executed in High Voltage Lab ESEIAAT_TRI of Universitat Politècnica de Catalunya (UPC)-Technological University of Catalonia, Barcelona Spain. The rating of test transformer was 25KVA, 25KV/400V DYn5 (Delta start connected with earth neutral).

Fig1.

  Fig1. Experimental setup for transformer testing

Fig2.

  Fig2. Impulse Voltage on HV side, Impulse current On HV Side, and Secondary voltage.

Fig3.

  Fig3. Impulse Voltage on LV side, Impulse current on LV side, and Primary voltage When impulse applied on secondary, transfer function for secondary side was calculated. T(S) verifies that Z12 and Z21 both resonate on same frequency as shown in fig 4. From this T(S), find the two resonance frequencies for which have inductive behavior.

Fig 4 :

 4 Fig 4: Transfer Function for Z12 and Z21

Fig 5 :

 5 Fig 5: Magnitude and Phase angle for Z11

Fig6:

  Fig6: Magnitude and Phase angle of Z12

Fig7:

  Fig7: Magnitude and Phase angle for Z22

Fig8:

  Fig8: Magnitude and Phase angle for Z21

Fig9.

  Fig9. Proposed T-Model for Transformer

TABLE I .

 I THE VALUES OF PARAMETER OF PROPOSED TRANSFORMER

	Elements	Values
	R1	3.013 Ω
	C1	0.20657315e-7
	R2	0.624 Ω
	L2	3.70244e-9H
	R3	558.5405 Ω
	L3	0.058967e-3H
	C3	0.00043e-6F
	R4	500 Ω
	L4	0.067305e-3H
	C4	0.001859e-6F
	R5	12.54 Ω
	L5	0.3748e-5H
	R6	53.213 Ω
	L6	1.123656e-5H
	IV.	PROPOSED MODEL

The proposed T-Model of the power transformer is shown in figure

9

.