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Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018), pp. 1–8

R. Bujack and K. Rink and D. Zeckzer (Editors)

Visualizing Electrical Power Systems as Flow Fields

Samantha Molnar1,2and Kenny Gruchalla1

1National Renewable Energy Lab, Golden, CO USA

2University of Colorado Boulder, Boulder, CO USA

Abstract

We describe a method for visualizing data ﬂows on large networks. We transform data ﬂow on ﬁxed networks into a vector

ﬁeld, which can be directly visualized using scientiﬁc ﬂow visualization techniques. We evaluated the method on power ﬂowing

through two transmission power networks: a small, regional, IEEE test system (RTS-96) and a large national-scale system (the

Eastern Interconnection). For the larger and more complex transmission system, the method illustrates features of the power

ﬂow that are not accessible when visualizing the power transmission with traditional network visualization techniques.

1. Introduction

We introduce a new technique to visualize ﬂow on networks by

constructing vector ﬁelds based on the network’s ﬂow of infor-

mation. Our method allows us to use standard ﬂow visualization

techniques on large networks to highlight salient features without

removing important topological information. We explore applica-

tions of our method to electric power systems, but emphasize that

this technique is applicable to any complex network with informa-

tion exchange.

An electrical power system is a network of components con-

nected by lines that deliver electricity from generators to con-

sumers. Power systems around the world are undergoing signiﬁ-

cant changes in recent years, with the addition of renewable gen-

eration sources such as wind and solar. It is crucial to understand

how renewable generation sources inﬂuence power ﬂow on the ex-

isting system so we can determine critical junctions and interac-

tions in the system. Visualization of power ﬂow can help maintain

and increase the reliability of power systems. There are generally

two categories of electric power grids: transmission systems which

span long distances carrying high-voltage power from generators to

electrical substations, and distribution systems which deliver power

from electrical substations to individual consumers. Distribution

systems typically have a radial topology forming acyclic connected

graphs, while transmission systems tend to have much more com-

plex interconnected networks forming cyclic non-planar graphs. At

a given instant in time, these networks can be considered directed

graphs with some magnitude of power ﬂowing from one vertex

along an edge to another vertex. The magnitude and direction of

power ﬂowing on these graphs is dynamic, varying in time. The

topology of these graphs can change with the operation of circuit

breakers or switches; however, these are relatively rare events. For

our investigation, we are only concerned with the dynamics of the

ﬂow not the topology.

We evaluate our visualization approach on power ﬂowing

through two transmission power networks: a small IEEE test sys-

tem (RTS-96) and a large system (the Eastern Interconnection). The

small test system is simple enough to use traditional visualization

techniques, allowing us to determine if the vector ﬁeld represents

the power ﬂow accurately. The Eastern Interconnection provides a

real-world test case that is too complex for existing visualization

techniques to convey the ﬂow trends (see Figure 1).

There are documented needs to visualize general trends of ﬂow

on complex network structures; for example, the ﬂow of infor-

mation on biological networks [AKK∗10] and neural networks

[LSKK16], and the ﬂow of power on electrical networks [CK15]

are compelling examples. Most techniques involve representing the

ﬂow explicitly on line graphs with glyphs (e.g., arrows) and line

attributes (e.g., color and width) [OW00b]. These techniques are

often sufﬁcient for small networks but are inefﬁcient or unreadable

for larger networks.

To visualize large-scale networks some form of aggregation be-

comes neccessary. Two common approaches are chord diagrams

and edge bundling. A chord diagram represents directed inter-

relationships between groups of objects, where these objects are ar-

ranged along a circle and connected with Be´zier curves [KSB∗09].

Chord diagrams can be used to represent data ﬂow on a directed

graph [GNB16] by grouping vertices into regions and representing

the net ﬂow between regions in the chord diagram. The chord di-

agram provides a quantitative view of net ﬂow, but abstracts that

ﬂow away from the toplogy of the network.

Edge bundling is a technique to simplify a graph by spatailly

grouping similar graph edges, decluttering and revealing under-

lying structure in large networks [LHT17]. However, large trans-

mission networks, like the Eastern Inconnection, have dense multi-

scale mesh-like topologies that provide few similar graph edges for

edge bundling to a group. For example, we applied geometry based

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

2S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields

Figure 1: A model of the Eastern Interconnection transmission network presents a visualization challenge with ﬂows across 70,000 trans-

mission lines. The labeled areas indicate independent system operators controlling power in those regions. We would like to understand how

and where these areas exchange power.

edge bundling (GBEB) [CZQ∗08] to the Eastern Interconnection.

GBEB bundles edges based on a kernel density estimation of the

primary edge directions. As seen in Figure 2, this method does well

with edges in the North East coming from Canada to the United

States, but it fails to declutter the majority of the Eastern Intercon-

nection. This is because the power system is mainly composed of

star-like motifs that come to a single node from various directions.

We also applied kernel density edge bundling (KDEB) [HET12],

which is the base algorithm for other edge bundling techniques that

use edge attributes as an indicator for bundling [PHT15]. KDEB

determines the density of edges, computes the gradient of this den-

sity, and iteratively moves edges in the direction of the gradient

to bundle edges. Figure 3 shows KDEB applied to the Eastern In-

terconnection, and while more edges are bundled than the GBEB

approach, it does little to alleviate the clutter of the network. Al-

though the network is dense, this is generally uniformly distributed

over space. Furthermore, many of the edges that are close together

in the topology do not have a uniform direction, which makes it un-

clear how and in what direction they should be bundled. The con-

clusion we can draw from applying these edge bundling techniques

is that due to the mesh-like structure of the Eastern Interconnec-

tion, there is little advantage to bundling the edges of the network

to try and reduce visual clutter and obtain useful information about

power ﬂow across the network. An alternate approach is required

to understand power ﬂow across the network.

Our technique does not visualize aggregated edges, but instead

visualizes aggregated ﬂow across the network. We can ﬁnd no de-

scription in the literature of transforming the ﬂow across a com-

plex network into a vector ﬁeld for the visualization of ﬂow trends.

Lobov et al. [LSKK16] appear to use a similar technique to un-

derstand properties of neural networks in response to a stimulus;

however, their technique is not well documented, and it is unclear

how they form their vector ﬁeld. Nevertheless, this provides an ex-

cellent example of the need to further analyze parameter choices for

this type of visualization in order to understand how these choices

affect analysis results.

Our method allows us to use standard ﬂow visualization tech-

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 3

Figure 2: The result of geometry based edge clustering on the East-

ern Interconnection [CZQ∗08]. Long transmission lines coming

from Canada are bundled using this technique, but small star-like

motifs in the South Eastern United States are mostly untouched.

Figure 3: The result of kernel density edge bundling on the Eastern

Interconnection [HET12]. Longer transmission lines are moved to

odd positions which do not help lessen clutter in the network.

niques on large networks to highlight salient features without re-

moving important topological information. Furthermore, we will

show that insightful knowledge of power exchange on the system

can be obtained using our visualization technique.

2. Methods

We analyze general ﬂow patterns across a network by creating a

vector ﬁeld based on the network dynamics and topology. Let the

network in question be represented by Gwith Vvertices and E

edges. We take each directed edge ei,j∈Ewhich connects nodes

i,j∈Vand form a vector, ˆei,j, based on its location in space. Each

ˆei,jhas an associated ﬂow, fi,j. To begin constructing the vector

ﬁeld, we place the network in an n×nsquare mesh M. Each cell in

the mesh, ml,k∈M, where ldeﬁnes the row and kdeﬁnes the col-

umn, will contain a vector. The function that computes this vector

is shown in Equation 1.

Vtotal (l,k,p,σ) = Vf l ow(l,k) + Vgrad(l,k,σ) + Vkern(l,k,p)(1)

Where Vf low is the vector sum of all edge ﬂows through a cell,Vgrad

is the gradient of a Gaussian convolution to attract the ﬂow towards

the topology, andVkern is a Gaussian weighting of nearby cells.

2.1. Calculating Vf l ow

Vf low represents the ﬂow of information on the network. For each

cell in the grid that contains ˆe, we compute the sum of the vector

ﬂows, deﬁned by Equation 2.

Vf low (l,k) =

s=|E|

∑

s=0

δ(ˆes,ml,k)∗(fsˆes)(2)

δ(ˆes,ml,k) = ˆes∈ml,k1

otherwise 0

A depiction of Vf low of the RTS-96 system is shown in Figure 4a.

Notice that very few of the cells in the mesh are ﬁlled with non-zero

vectors. This will cause problems for traditional ﬂow techniques

that seed streamlines or path lines at random positions. There is

also little guarantee that ﬂow lines seeded at node positions will

remain in the ﬂow ﬁeld without other forces pushing it back into

this area. For these reasons, we add a gradient force to the vector

ﬁeld.

2.2. Calculating Vgrad

The purpose of Vgrad is to ﬁll cells that do not directly intersect the

network with vectors that point towards the topology of the net-

work. This would allow streamlines seeded away from the network

topology to converge toward the network topology. If we imagine

placing the network on a rubber sheet, we would see the sheet sink

down where the lines are, and balls rolling on the sheet would tend

to fall into the valleys, or the edges of the network (see Figure 4b).

To achieve this effect we ﬁrst calculate the weighted sum of each

cell in the network, where the weight represents how much of each

edge in Ethe cell contains. We traverse each edge, ei,j, in unit

vector intervals, and for each step we compute which cell ml,kis

closest to the current position and add 1 to that cell’s weight wl,kin

the matrix W. We then compute Vgrad based on Equation 3.

Vgrad(l,k,σ) = ∇1

2πσ2e−wl,k

2σ2(3)

The only tunable parameter in Vgrad that we explore in this work is

σwhich, in our metaphor, represents how fast ﬂow moves towards

the edges of G. As σincreases, the number of cells with a non-zero

vector decreases.

2.3. Calculating Vkern(l,k,p)

Finally, in order to make the vector ﬁeld less abrupt at the bound-

ary of cells that contain network edges and cells that do not, we

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

4S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields

Figure 4: Vector ﬁeld representation of the RTS-96 test system with

a25 ×25 grid. a) directly represents the ﬂow on the network by

encoding Vf l ow.b) Vgrad directs ﬂow back towards the topology of

the network.

add a small vector to each cell that is the average of its neighbors.

For example, if σ=1, then Vgrad points cells without ﬂow directly

towards the topology, exhibited in Figure 5a. Vkern (l,k,p)smooths

this transition and slightly points cells without edges towards the

majority direction of its neighbors, shown in Figure 5b. In Equa-

tion 4, prepresents the size of the neighborhood we average over;

p=1 represents the surrounding ring of neighbors around a cell

(of which there are at most 8), then p=2 represents the next sur-

rounding ring and so on. The function g(p)is a gaussian function

that determines how much weight each neighbor cell contributes

to the average. In this paper, we focus on varying pto understand

how the size of a neighborhood changes the vector ﬁeld rather than

changing the weight of the contribution from the neighborhood.

Vkern(l,k,p) =

l+p

∑

s=l−p

k+p

∑

r=k−p

g(p)(Vfl ow(s,r) + Vgrad(s,r)) (4)

The parameter space for Vtot al (l,k,p,σ)is large and various in-

Figure 5: A graph edge represented as a vector ﬁeld. Black vectors

indicate cells that contain an edge with ﬂow and red vectors indi-

cate cells that do not. a) is constructed without Vkern (l,k,p)and b)

is constructed with Vkern (l,k,p)and p =1.

puts can change the types of conclusions one could draw about the

network ﬂow. In the following section we will explore how various

inputs of Vtotal change the observed ﬂow dynamics on two power

system test cases.

3. Experimental Results - Electric Power Systems

There are few techniques for the visualization of power systems

data [MJC12]. The simple one-line diagram, a static diagram with

single lines and schematic symbols to indicate the path and com-

ponents on an electric circuit [vM06], is a standard representation

for the static structure of the system. Most visualization research

has focused on augmenting the one-line diagram with ﬂow data.

Various researchers [MC93,LQ98] have proposed encoding elec-

trical ﬂow as line width and color and including arrows for ﬂow

direction. Two-dimensional contours have been used to visualize

network data [WO00,OW00b], market data [OW00a], and system

stability margins [OKW99,KW02]. However, these techniques do

not scale to large transmission systems with tens of thousands of

lines. The visualization of large transmission networks more typi-

cally aggregate the line ﬂows to the net ﬂow or interchange between

regions of the network. These aggregated ﬂows have been repre-

sented as ribbons in a chord diagram [GNB16] (see Figure 6). Net

interchange visualizations provide critical information on power

transfer between large regions in aggregate but fail to show how

that interchange occurs. For example, in Figure 6 we see ≈3 GW

of power ﬂowing from SERC to TVA, and ≈3 GW of power ﬂow-

ing from TVA to PJM. But we do not know where along that in-

terconnection this exchange occurs. Furthermore, we can speculate

that some of the ﬂow from SERC to TVA is transferred to PJM, but

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 5

we cannot decisively know this based on the chord diagram. Ad-

ditional locational information could be useful to determine where

additional transmission lines should go, which transmission lines

are under stress at various times, and potentially change market ex-

change calculations.

Figure 6: A chord diagram of a time step of the Eastern Renew-

able Generation Integration Study (ERGIS). Large transmission

networks have tens of thousands of lines, directly visualizing these

lines is impractical. Transmission studies such as ERGIS have re-

sorted to only visualizing aggregated ﬂow between regions. There

is signiﬁcant ﬂow between SERC and TVA, and TVA to PJM, but it

is unclear if the ﬂow from SERC to TVA is then going to PJM.

3.1. RTS-96 System – Validation Case

IEEE RTS-96 [GWA∗99] is a test system commonly used in trans-

mission power system reliability studies. The system is a cyclic

non-planar graph with 73 vertices connected by 120 edges com-

prised of three subnetworks. RTS-96 is small enough to directly

visualize ﬂows on the network with arrow glyphs indicating direc-

tion and line properties (e.g., color, width) representing ﬂow magni-

tude (see Figure 7). Clearly, a system this small is best represented

with these traditional techniques. We apply our method to RTS to

evaluate if our vector ﬁeld construction represents the power ﬂow

consistently.

We depict these vector ﬁelds using streamlines, but other vec-

tor ﬁeld visualization techniques (i.e., path lines and line interval

convolution) are directly applicable. To evaluate the correctness of

the constructed vector ﬁeld, we systematically seeded stream lines

near sources (i.e., buses with electrical generation) on RTS-96 and

followed those stream lines to the sinks (i.e., nodes with electrical

loads). The streamlines do travel from sources to sinks on paths

that generally follow the network (see Figure 8a and b). However,

how closely they follow the network depends on the parameters of

the vector ﬁeld construction. A signiﬁcant ﬂow difference can be

observed between Figure 8a and Figure 8b at the location of the

bottom orange seed area. The ﬁnal destination of the streamlines

changes from one node to another. Since Figure 8b has a larger

neighborhood and based on the direction the streamlines originate

from, they go towards the bottom right nodes instead of the farthest

right node. We explore the vector ﬁeld construction parameters n,

Figure 7: RTS-96 topology with line widths representing power

magnitude and arrows indicating direction. Generator buses (i.e.,

sources) are colored green, load buses (i.e. sinks) are colored or-

ange, and the swing bus is colored red.

p, and σon the RTS-96 system (a subset of that investigation is

shown in Figure 9). First, as the grid size nbecomes larger (and the

cells smaller), the network topology becomes more deﬁned in the

vector ﬁeld, as there is less aggregation per cell. For example, com-

pare n=100 in Figure 9a and n=1000 in Figure 9d. When there

are fewer cells, the larger ﬂows dominate the vector space and show

more of the cumulative dynamics on the system. When there are

more cells, many of the smaller ﬂows are captured. In areas where

a cell contains an intersection, the constructed ﬁeld will push ﬂow

in the aggregate direction of all edges in that cell weighted by their

magnitude. Second, as we increase σrelative to the grid size, the

number of cells with non-zero vectors increases and stream lines

can be seeded farther away from the network. Although, the stream

lines converge to the network, the visual features created as σin-

creases may be difﬁcult to interpret. For example, in Figure 9c a

saddle has formed in the middle of the network that could be mis-

construed as a pathway in the network. This feature occurs because

σhas become large enough compared to the grid size to ﬁll cells in

the very center of the network. Based on Vgrad cells to the right of

the saddle point towards the right part of the network and cells to

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

6S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields

Figure 8: a and b) Streamlines seeded near generators gener-

ally follow the network path to the system’s loads center. How the

stream lines follow the network path is impacted by the construc-

tion parameters. Colors are based on which area the streamlines

originate from, area 1 is purple, area 2 is orange and area 3 is

green. The images were both constructed with n =1000 and σ=1

and vary in kernel width a) p =10 b) p =20.

the left of the saddle point towards the left of the network. Under-

standing how domain experts might interpret these ﬂow patterns

will be necessary and is a topic for future work. But with proper

parameter selection, many of these artifacts are avoidable. Finally,

the number of neighbors considered in Vkern changes how much in-

ﬂuence larger ﬂow magnitudes have on the direction of the overall

ﬂow. We ﬁnd that a smaller neighborhood stays closer to the true

topology of the system, while large neighborhoods provide clear

illustrations of the cumulative ﬂow. The choice for this parameter

is dependent on how much aggregation one wishes to show in the

visualization.

There is one notable artifact visible in RTS-96 for all of the

streamlines shown in Figure 9, a spiral sink occurs in the lower

right hand area. This feature forms in a non-planar area of the net-

work, where lines in the network cross but do not intersect in the

actual physical topology (i.e., they are not connected by a node).

While there is a sink located in the vicinity, the intersection of an

edge close to this sink causes a spiraling effect where there in fact

is none. Our treatment of edge crossings is beneﬁcial if one wishes

to observe the general ﬂow trend in the network, but it may not give

an accurate depiction of small-scale ﬂow features. As the mesh grid

becomes ﬁner, this effect is less obvious because the cell containing

the intersection plays a much smaller role in the overall vector ﬁeld

which provides a more detailed view of the ﬂow.

Figure 9: Streamlines of various inputs for RTS-96 vector ﬁeld

where the left column has n =100 and the right column has

n=1000.a) p=1,σ=1b) p=1,σ=2c) p=4,σ=4d) p=10,

σ=1e) p=10,σ=2f) p=40,σ=4. We scale the neighbor-

hoods for the different grid sizes so that they represent the same

amount of space, and all of the vector ﬁelds use g(p) = .05e−p−1

200 .

Animated version here.

3.2. ERGIS System – Motivating Case

The Eastern Interconnection is the largest power system in North

America and one of the most complex power systems in the world.

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 7

We applied our visualization technique to data from the Eastern

Renewable Generation Integration Study (ERGIS) [BTP∗16] that

models the Eastern Interconnection with over 70,000 transmission

lines. This network provides a real-world test case for our visualiza-

tion technique, where directly rendering the line ﬂows is impracti-

cal. We compared the aggregated ﬂow visualization from the study,

which shows us how much net power ﬂows between the regions, to

our results (see Figure 10). In general, the path of the stream lines

are conﬁrmed by the chord diagram. For example, there is signif-

icant ﬂow between SERC and TVA and between SPP and MISO

(reference Figure 1 for region labeling). While the chord diagram

provides a quantitative measure of the net interchange, the stream-

lines provide more qualitative detail to where along the interface

that interchange is occurring. We can gain more insight into the

Figure 10: Streamlines, distributed proportionally to the vector

magnitude, showing detailed ﬂow of power for the Eastern Inter-

connection (n =1000,σ=1, p =10). Large amounts of power is

ﬂowing in to the mid-west region.

ﬂow characteristics by isolating the streamline seeding in partic-

ular regions and analyzing their path. Figure 11 shows ﬂow be-

tween different areas in the Eastern Interconnection. Based on the

aggregate ﬂow in the chord diagram (Figure 6), we know that a

signiﬁcant amount of power ﬂows from SERC to TVA, and from

TVA to PJM. However, what we can now see from Figure 11 is

that a signiﬁcant amount of power that ﬂows from SERC to TVA

then ﬂows to PJM. Most of that SERC-TVA ﬂow originates in Al-

abama and not elsewhere in the region. Additionally, we can see

bi-directional ﬂow between MISO and TVA; although, in Figure 6

we are only privy to relatively small net interchange coming in to

TVA from MISO. This is new directional information that we could

only speculate based on the original visualizations. Preliminary ex-

amination of these results with domain scientists indicate that this

novel vector ﬁeld construction provides useful locational informa-

tion that can be used during development, testing, and analysis of

power ﬂow models and simulations.

Figure 11: Streamlines colored based on which region they orig-

inated from. Flows originating from SERC traveling through TVA

into PJM are clearly visible. Likewise bi-directional ﬂow can be

seen between TVA and MISO.

The vector ﬁeld construction parameters have a similar effect on

the large-scale system as they did on the small test system. Denser

meshes better resolve individual transmission lines, while larger

kernel sizes provide more aggregated versions of the ﬂow. A spi-

raling sink artifact, like the one apparent in RTS-96, can also be

found in our Eastern Interconnection construction in eastern Ne-

braska (see Figure 10). Like RTS-96, there appears to be a non-

planar cluster of transmission lines in this area of Nebraska. In all

likelihood, this is an artifact of the construction, but with alternate

parameters this artifact does disappear.

4. Conclusions and Future Work

We introduced a method for visualizing data ﬂows on large net-

works by transforming those ﬂows into a vector ﬁeld. The method

clearly shows promise. On our small-scale test case we veriﬁed

that the technique represents the predominate ﬂow features. On

our large-scale test case we were able to identify complex ﬂow

exchanges between different areas of the Eastern Interconnection

transmission system. We have a better understanding of how power

ﬂows between regions with our visualization than an aggregate

view can present and a traditional line glyph rendering cannot han-

dle.

However, the method has some limitations. The construction

captures the most dominate ﬂows at the expense of averaging out

the details of the ﬂows. More concerning, non-planar areas of the

graph can lead to spiraling sink artifacts. A more sophisticated vec-

tor ﬁeld allowing for probabilistic and/or conditional cells could

improve these situations. Developing these multi-state cells and the

visualization algorithms to support them is left for future work.

Our parameter investigation indicates that larger grid sizes yields

more accurate ﬂow information while larger neighborhoods causes

submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)

8S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields

larger ﬂow magnitudes to dominate more of the vector space. Fu-

ture work will extend the investigation to time-varying vector ﬁeld

visualization methods so we can understand how ﬂow evolves on

a network over time. User studies will be necessary to understand

how domain experts interpret the ﬂow visualizations in relation to

the ﬂow on the underlying network, which may provide a mecha-

nism to better tune the parameters of the construction.

5. Acknowledgements

This work was supported by the U.S. Department of Energy under

Contract No. DE-AC36-08GO28308 with Alliance for Sustainable

Energy, LLC, the Manager and Operator of the National Renew-

able Energy Laboratory. Funding provided by U.S. Department of

Energy Ofﬁce of Energy Efﬁciency and Renewable Energy. NREL

is a national laboratory of the U.S. Department of Energy, Ofﬁce

of Energy Efﬁciency and Renewable Energy, operated by the Al-

liance for Sustainable Energy, LLC. The U.S. Government retains

and the publisher, by accepting the article for publication, acknowl-

edges that the U.S. Government retains a nonexclusive, paid-up, ir-

revocable, worldwide license to publish or reproduce the published

form of this work, or allow others to do so, for U.S. Government

purposes. The authors would like to thank Nicholas Brunhart-Lupo

and Kristi Potter for useful conversations and insight on this work.

We would also like to thank Aaron Bloom and Clayton Barrows for

preliminary examination of the results.

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