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CONFERENCE PAPER - This is the version accepted by the conference. Final version uploaded to IEEE Xplore.
doi: 10.1109/AI4G50087.2020.9311085 https://ieeexplore.ieee.org/document/9311085
Citation: J. C. F. Li, D. Zimmerle and P. Young, "Optimizing Networked Rural Electrification Design using Adaptive Multiplier-
Accelerated A* Algorithm," 2020 IEEE / ITU International Conference on Artificial Intelligence for Good (AI4G), Geneva,
Switzerland, 2020, pp. 164-169.
Optimizing Networked Rural Electrification Design using
Adaptive Multiplier-Accelerated A* Algorithm
Jerry C.F. Li
Department of Systems Engineering
Colorado State University
Fort Collins, CO, USA
jerry.li@colostate.edu
Daniel Zimmerle
Energy Institute
Colorado State University
Fort Collins, CO, USA
dan.zimmerle@colostate.edu
Peter Young
Department of Electrical & Computer
Engineering
Colorado State University
Fort Collins, CO, USA
peter.young@colostate.edu
Abstract— Networked rural electrification can potentially
improve energy resources utilization, reduce cost and
enhance supply reliability. Identifying optimal connection
paths is critical for proper network design. To overcome the
inefficiency of applying standard A* path-finding method to
complex topography, multiplier-accelerated A* (MAA*)
algorithm, which utilizes a modified heuristic, has been
developed in previous research. While MAA* can generally
reduce computation time by ~90% at the cost of ~10%
optimality, the computation burden can still be remarkable
for some areas with intricate topological variations. This
paper proposes an adaptive version of MAA*. By introducing
intermediate nodes in MAA*, the new algorithm significantly
simplifies computations in complex regions. This greatly
facilitates the analysis and design of optimal network for
cost-effective electricity supply to users in remote, difficult-
to-reach areas.
Index Terms-- rural electrification; SDG7; A* algorithm;
path finding.
I. INTRODUCTION
Enacted by United Nation [1], the seventh Sustainable
Development Goal (SDG7) envisions affordable, reliable,
sustainable and modern energy will be accessible to all by
2030. As reported by recent studies [2,3], the progress of
electrification is remarkable. Population without electricity
access decreased from 1.2 billion to 840 million during
2010-2017. On the other hand, the reports also pointed out
the problem of rural-urban divide. Among the unserved
population, 732 million people are living in rural areas.
Electrification of these areas requires extra effort and could
be costly due to increased complexity. For example, owing
to uncertainties in resource and demand forecasts for small
geographical areas, correct planning for renewable energy
generation for individual village or small town is
challenging. Longer-term, urbanization policy may
unexpectedly impact population and hence energy demand
[4-6]. In addition, location of the village or town may not
be optimal for efficient generation [6-8].
To facilitate electrification for these difficult areas, a
networked rural electrification scheme has been proposed
[9]. The scheme uses a cost-optimized network to connect
villages in a wider area together, and each village is
supplied by (i) centralized generation sites with good
resources such as strong solar radiation and/or wind for
efficient generation, and (ii) supplementary local
generation modules that can be added or removed based
upon local demand (Figure 1).
Figure 1 Networked Rural Electrification
Viability of this approach depends on the cost of
building the network, and hence depends on correctly
identifying an optimal, or near-optimal, connection
topology that accounts for the topography of the area. To
properly design the network, the two steps are (i) finding
(near) optimal paths for connecting any two villages, and
also optimal paths connecting each village to the
centralized generation sites, and (ii) synthesizing the
lowest-cost connection topology based the optimal paths
identified. While this approach is conceptually easy to
understand, designing such a network is difficult in practice
because of high computational complexity. This paper
proposes an efficient algorithm that could significantly
reduce the complexity and facilitate the design process.
To perform the two mentioned tasks, standard methods
are available. For example, the widely used A* search
algorithm appears to be a reasonable solution for finding
the paths, and minimum spanning tree (MST) can be used
to obtain the minimum cost network. However, as
explained in Section II, standard A* algorithm is very
inefficient for this application and common acceleration
techniques are also not applicable due to significant
topological (cost) variations. Since hundreds of
combinations may have to be evaluated, a fast method
suitable for identifying optimal interconnection paths is
highly desirable. This will become even important if there
are considerable uncertainties in input map and many
Monte Carlo simulations will be required to ensure
robustness of the solution. Multiplier-accelerated A*
(MAA*) algorithm [9] has hence been developed to
improve the computational efficiency of pathfinding. While
MAA* can generally reduce computation time by ~90% at
the cost of ~10% optimality, the computational burden may
still be large for areas with intricate topological variations.
This paper proposes an adaptive version of MAA*. By
introducing intermediate nodes in MAA*, the new
algorithm significantly simplifies computations in complex
regions. This greatly facilitates the analysis and design of
optimal network for cost-effective electricity supply to
users in remote, difficult-to-reach areas.
II. BRIEF REVIEW OF MULTIPLIER-ACCELERATED A*
ALGORITHM AND IT’S PERFORMANCE
Although A* is a proven algorithm for optimal path
finding, computation complexity is problematic when
applied to a large search space. Complexity of the
algorithm is described by O(bϵd) [10], where b is the
branching factor, ϵ is the error in heuristic estimate defined
as (h* - h)/h* (where h* and h are the actual and estimated
cost from the node to final goal), and d is the solution depth
– i.e. the path length in this application. Figure 2 illustrates
the effect of ϵ and d under a branching factor of, for
example, 2.13 with constant step cost.
Figure 2 Time complexity vs search depth
As seen, time complexity increases exponentially when
the heuristic error ϵ and the solution depth d increase even
under constant step cost assumption. To improve
computation efficiency, there are two approaches: (i)
reducing d by simplifying the search space, and (ii)
reducing ϵ by designing better heuristic estimates.
For some applications (e.g. video games), simplifying
search space is quite straight-forward. Methods such as
Quadtree, NavMesh etc. essentially divide space into
simple geometric shapes and use a significantly fewer
waypoints to represent these shapes and hence resulting a
much smaller d [11, 12]. Standard A* algorithm can then
be applied to these waypoints. Many other acceleration
techniques [13, 14] are also based on different geometric
simplifications. Viability of geometric simplification
depends heavily on isotopicity of the search space. For
networked rural electrification, these techniques are not
applicable because varying topography leads to a high
degree of cost anisotropicity. The search space, therefore,
cannot be effectively reduced.
Reducing error in the heuristic estimate is even more
challenging since ϵ is highly dependent on the properties of
the search space. Using a simple heuristic estimate to
closely track the actual cost throughout the entire space is
not usually achievable, and the high degree of
anisotropicity in this application will lead to even larger ϵ.
Researchers have developed more sophisticate heuristic
estimates in order to reduce complexity [15-17], but most
of these methods are only applicable under some specific
conditions. For networked rural electrification, search space
may take any physical form and these methods are not
directly relevant.
Multiplier-accelerated A* (MAA*) Algorithm was
developed to tackle this problem. Essentially, MAA* is an
algebraic, instead of geometric, search space reduction
method that selectively ignores, instead of reduces, error in
heuristic estimate. The principle is illustrated in Figure 3.
Figure 3 Multiplier-accelerated A* algorithm
The heuristic cost (H) in MAA* is a modulated R2
Euclidean distance that decreases as the current search
point becomes closer to the end point.
H = U × ( 1 + m × U / L ) (1)
where U is the R2 distance between current and end
points
L is the R2 distance between start and end points
m is the heuristic multiplier
In a canonical A* algorithm, when ϵ is large, the search
jumps back to earlier paths in the open list and restarts
frequently due to wrong cost estimation. If degree of
anisotropicity is high, the search may even return to early
search nodes from late stage search nodes because the
range of estimated cost of the early nodes can vary widely
under these topologies. However, restarting from very early
nodes will unlikely lead to a better path due, again, to large
ϵ. Therefore, the rationale of Equation (1) is to exaggerate
the heuristic cost for nodes far away from the destination,
and hence reduce the chance of jumping back to very early
nodes during the later stages of the search. Effectively,
MAA* reduces the search space by ignoring some error-
driven search incentive when those incentives would be
unlikely to lead to better paths. However, it is important to
realize that MAA*, unlike A*, cannot guarantee optimality
since the modulated heuristic cost H may over-estimate
actual cost and hence may violate admissibility criterion of
A*[18]. To trade-off between optimality and computation
complexity, user can choose different values of m and c,
which, respectively, define the level of exaggeration of the
heuristic cost of distant nodes and the region in which to
resume, if preferred, to standard A* algorithm.
Figure 3 and Table 1 illustrate the performance of
MAA* algorithm. Computation time and optimality are
used as metrics. Although computation time is not an
authoritative measure – it will vary with coding quality and
machine loading – the results can still clearly demonstrate
the advantage of the new algorithm.
Calculations are based on a 30 x 30 demonstrative map,
with a cost function composed of (i) path length, (ii)
incremental costs due to changing elevation in routing, and
(iii) accessibility of the locations along the path. Similar
results are observed with a larger and more realistic 300 x
300 map (Figure 4). Details are provided in [9].
When c is fixed at 10 and m is small (0.1, 0.2 or 0.5),
the path identified by MAA* is very close to (or exactly the
same as) the global minimum found by standard A*
algorithm. Computation time has reduced but remains in
the same order of magnitude.
When larger m is used (1 or 2), location the path
identified deviates remarkably from the global minimum
but the cost is not very different. On the other hand,
computation time has been reduced very significantly. In
other words, it is a near-optimal path that requires much
less time to identify.
Figure 3 Optimal paths connecting (14,21) and (34,29)
For practical purposes, small cost over-estimation
should be acceptable since maps created from surveys or
aerial (satellite or unmanned aerial vehicle) photography
will also have intrinsic tolerances at comparable level.
Table 1 Optimal connection between (14,21) and (34,29)
Connection: (14,21) - (34,29)
Computed
Optimal Cost
Computation
Time (sec)
Normalized
Computation
Time
% Over-
estimate
Standard A* 47.9 198.6 100.0% 0.0%
Accelerated A*
m = 0.1 47.9 169.8 85.5% 0.0%
m = 0.2 47.9 146.9 74.0% 0.0%
m = 0.5 48.8 117.7 59.3% 1.9%
m = 1 50.7 20.3 10.2% 5.8%
m = 2 50.7 5.8 2.9% 5.8%
m = 5 58.6 3.9 2.0% 22.3%
In the above example, MAA* has significantly
accelerated (near) optimal path finding under anisotropic
search space, thereby accelerating networked rural
electrification routing.
III. LIMITATION OF MULTIPLIER-ACCELERATED A*
ALGORITHM
Generally, MAA* remains effective even for large,
anisotropic map. However, the search space reduction
strategy used by the algorithm may not work well for some
very complex topologies.
In Figure 4, there are two target connections A and B
on a 300x300 grid. Connection A is a typical connection
and MAA* can efficiently identify the (near) optimal path
for connecting the villages. Connection B, however, is very
different. Both start and end nodes, although not far away,
are lying within a region of low accessibility (i.e. the blue
rectangle). In addition to topographical variation, many
locations in this region are difficult to access due to natural
or human-related reasons. High cost nodes within the
region can lead to drastic path cost increase in each step of
movement inside the region; i.e. ϵ is large.
Figure 4 Performance of MAA* on a 300 x 300 map
If human intuition is used for routing, one would likely
exit the region as quickly as possible to skirt the difficult
area. This is consistent with results using MAA* (Figure 5,
for m=5 and 10). Unfortunately, the MAA* computation
time for this case is extremely long even for relatively large
m.
Figure 5 Searching costly zone
As in all cases, large ϵ will lead to exponentially
increasing computation burden as per O(bϵd). However, this
specific problem can be qualitatively understood as a high
frequency switching, or path oscillation, between search
paths inside and outside of the difficult region. High cost
paths within the region have made the long paths outside of
the region comparatively inexpensive. As the search within
the region begins, it soon encounters move so expensive
that the algorithm infers that some paths outside of the
region could be possibly be less expensive. The algorithm
then commences to explore these outside paths. The outside
search remains favorable for many steps because the step
cost increment, in general, will be less than the step cost
increment within the region. However, after many steps,
the outside exploration accumulates sufficient path cost that
it becomes more expensive than the previous search path
within the region. The algorithm then jumps back into the
region to continue the search. However, since the single
step cost in the region is large, cost of the inside search
path will soon become more expensive than outside paths
stored in the open list. The algorithm will again jump to the
outside paths and continue for quite some more steps, and
then jump back to inside paths. The process repeats
inefficiently in this mode. In fact, using MAA* (m=5) to
find the optimal path for Connection A only creates a
closed list (i.e. nodes being completely explored) of size
557, but the corresponding list size for Connection B is
22,555. This means that only 0.6% of the search space
needs to be explored in case of Connection A, but 25% in
case of Connection B despites it appears to be much
shorter.
Thus, even with MAA*, computation time is too long
to be practical unless m is very large. However, when m is
large, optimality of the solution is questionable. As shown
in Figure 5, MAA* can complete the computation in about
9 minutes when m=30, but the solution is obviously far
from optimal. When smaller m is used, optimality is
improved but computation time is too long (Table 2).
Table 2 MAA* computation time for Connection B (on i5 machine with
typical configuration)
IV. ADAPTIVE MULTIPLIER-ACCELERATED A*
ALOGRITHM
To tackle difficult regions, the adaptive multiplier-
accelerated A* (AMAA*) algorithm has been developed.
As illustrated in section III, the problem originates from
frequent switching between inside and outside search paths,
which is fundamentally rooted at the large ϵ in the difficult
region. Thus, instead of performing search solely based on
cost comparison, AMAA* uses a directed search approach
based on a projection of ϵ. This is done by adaptively
defining and inserting intermediate nodes (P1 and P2 in
Figure 6) according to the specific structure of the search
space, and then applying MAA* algorithm to the newly
constructed line segments.
Figure 6 Principle of AMAA*
Intermediate node(s) are defined based upon (i)
location desirability, and (ii) minimization of ϵ.
If one starts at (110, 140), an intermediate node on its
slight right is more desired than a node on slight left
because the former is closer to the destination (180, 180).
Similarly, a node slightly above is preferred over a node
slightly below. For minimal ϵ, direct calculation is possible
but computational quite heavy. Therefore, it is indirectly
measured by examining the coarse cost of movement in a
number of directions. Scores are assigned to potential
candidates based on the two criteria. The method is
illustrated in Figure 7.
Figure 7 Screening for intermediate node
(i) Place k evenly distributed concentric circles on the
start node such that the final circle passes through the
end node;
(ii) Draw a line passing through the start and end nodes.
Label the intersection closest the end node as “0”;
(iii) Draw r-1 lines passing through the start node so that
the concentric circles are evenly divided. Label each
radii sequentially.
This creates 2kr points formed by the intersections of
the lines and circles. These are potential candidates for
intermediate nodes. For convenient programming,
evaluation is done as follow.
(iv) For the 2r nodes on each circle, calculate the R2
distance to the end node, compared to the R2 distance
between start node and end node to create a
normalized score, L. A negative change indicates
movement towards the destination and will be assigned
with a low L. In this scheme, a low score indicates
high desirability. Normalization is necessary because
measured parameters will be different for different
circles;
(v) For the 2r nodes on each circle, costs for moving from
start node to these nodes are calculated and normalized
by corresponding R2 distance, and denoted as D. Thus,
by examining the change in D along the k nodes on
one of the 2r arrows, one can coarsely deduce the
difficulty for moving in that direction and hence
estimate whether ϵ is increasing or decreasing;
(vi) The final score for each point is obtained by
multiplying L and D.
Due to length limitation, computational details are not
discussed in this paper. In addition, alternative methods for
creating additional intersections that enhance optimality
are also skipped. However, Tables 3 and 4 illustrate how
the method works in practice.
With k=10 and r=5, there will be 100 potential
candidates. The analyses are tabulated in 10 tables based on
the k circles, and summarized in one additional table. Table
3 is the second of the 10 tables, and serves as an illustration
of the format and information contained.
Table 3 Analysis of circle 2
By combining the 10 tables, a summary of final score
for all 100 candidates is obtained (Table 4).
Table 4 Summary of final score
Inspecting all the directions, it has been found that the
scores in direction 3 decrease quickly and remain stable
after decreasing, even though the initial score at circle 1 is
high. This indicates that moving along direction 3 is (i) not
expensive (D score) and (ii) not moving too far away from
the destination (L score). In contrast, early scores for
direction 0, 1 and 2 are low, but increase markedly when
moving toward the outer circles; these directions are
expensive and should be avoided. It is to be noted that
direction 8 is also a possible choice, but further illustration
is skipped due to space constraints.
It is desirable to move to low score area as soon as
possible. Since direction 3 attains a low final score at circle
2, the intersection is chosen as an intermediate node.
Coordinates of this node, according to Table 4, are (98,
150). The method can also be applied to the end node
(180,180), obtaining the intermediate point (162,196).
MAA* algorithm is then applied to (110,140)-(98,150),
(98,150)-(162,196), and (180,180)-(162,196) to find the
final path (Figure 8)
Figure 8 Optimal path finding by AMAA*
Results obtained by using AMAA* algorithm are
summarized in Table 5.
Table 5 Computation results, AMAA*
Compared to Table 2, speed improvement is achieved
by AMAA*. When m=1, MAA* and AMAA* are
delivering the same optimality (932.43 vs 934.83), but the
former took 664h to compute while the later only took
1.7h. With larger m, the improvement is even more
significant. For example, AMAA* provides both better
optimality and computation efficiency when m=5. Table 6
summarizes the differences.
Table 6 Comparing AMAA* to MAA*
As qualitatively discussed in section III, frequent
switching between paths inside and outside of the high cost
region causes significant computational load. This section
quantitatively demonstrated this excess load, and provided
an effective method to avoid it. With AMAA*, it is
practically feasible to solve optimal path finding problems
under very complex topography. However, it should be
emphasized that AMAA*, like any other heuristic search
methods, cannot guarantee optimality.
V. CONCLUSION
Achieving SDG7 by providing affordable, reliable,
sustainable, and modern energy to all in 2030 is
challenging. Networked rural electrification can potentially
accelerate the process by reducing system cost, enhancing
reliability, and offering installation flexibility. However,
designing the required optimal network under some
complex topographies could be computationally
prohibitive. In this paper, it has been demonstrated that
AMAA* algorithm can resolve the computation issue for
some complex situations, and hence facilitate networked
rural electrification. Since AMMA* focuses solely on
optimal path finding, further works of developing an
integrated framework for complete network synthesis is
desirable.
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