Optimizing the Energy Efficiency of a Lighting Network using Graph
Theory
EKATERINA GOSPODINOVA
Faculty of Engineering and Pedagogy
Technical University of Sofia
Sliven 8800,
BULGARIA
ORCID: 0000-0001-9083-7135
Abstract: - In this paper, we discuss how to make electric street lighting systems more energy efficient by
creating an algorithm and mathematical model for optimizing parameters, minimizing active power losses, and
finding the best topology for the lighting network when it is being designed or updated. Scientific and
technological progress has led to an increase in the complexity of every human being's daily life. Companies,
institutions, and countries constantly need to find modern tools to help them make the best decisions. Graph
theory has numerous applications to many everyday problems. It can resolve and simplify them. An algorithm
was developed to determine the shortest length in the form of a modified Dijkstra graph, with nodes supporting
the street lights and ribs being the wires connecting them.
Key-Words: - graph theory, algorithm Dijkstra, optimization, energy efficiency, lighting, mathematical model,
topology.
Received: April 9, 2023. Revised: February 18, 2024. Accepted: March 8, 2024. Published: April 23, 2024.
1 Introduction
A priority direction for energy development is the
creation of a new generation of urban energy
systems based on modern technological means and
distributed intelligent management systems. One of
the main components of the city's energy system is
street lighting. In the context of this work, a power
supply set of street lighting systems, consisting of
power and lighting networks, as well as the
installations themselves, is considered. Their
operational life and the safety and comfort of city
dwellers depend on the reliability of their
functioning. These systems are energy-intensive
objects; electricity consumption for lighting needs
can reach 30–40% of the total energy consumption
in the city. Because saving energy is one of the
priorities of developed economies, ensuring energy
efficiency is one of the most important areas to
modernize the city's electricity system. By energy-
efficient operation, we mean the lowest electricity
consumption while providing the standard level of
road and sidewalk lighting, [1]. The analysis of
scientific publications and technical solutions in this
field showed that several approaches are currently
used to increase the energy efficiency of street
lighting, namely: replacing light sources with other
more efficient
ones; optimizing the lighting network; and
algorithms for energy-efficient management. It
should be noted that in the existing scientific
studies, the application of methods for optimizing
the parameters of network elements of street
lighting was carried out without formalizing
boundary conditions that reflect the requirements of
regulatory documents in the field of electricity.
Additionally, some solutions consider simulating
how all the parts of street lighting work together.
This lets you model different systems and pick the
best elements and parameters for controlling and
figuring out how energy-efficient each mode is
based on the results of optimizing the lighting
network's parameters. The calculation can be aimed
at reducing active power losses by reducing the
power of street lighting installations, increasing the
cross-section of power supply lines, and
compensating reactive power, [1], [2], [3]. The
results of the study show that these measures can
reduce active power losses by up to 45%. Noting the
undeniable practical importance of the listed
methods, it is worth mentioning that they analyze
specific ways to reduce losses in existing networks
but do not offer ways to optimize street lighting
parameters at the design stage. In the analyzed
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Ekaterina Gospodinova
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developments, the only criterion for choosing the
measures to reduce losses is maximization, but the
costs of introducing energy-saving measures and
their impact on illumination are not taken into
account, [4]. Mathematical models of the system are
also proposed to support decision-making for the
design and modernization of street lighting systems
that allow for maximizing energy efficiency. Even
though in the scientific community, the issue of
optimizing the parameters of distribution network
elements has been developed quite thoroughly, [5],
[6], the authors do not know of any works that offer
mathematical models and methods for optimizing
the parameters of electrical networks for street
lighting.
There are automated systems for controlling
lighting fixtures for open spaces, such as "Arman",
"Helios", "Zora", and "Modul-C." All of them were
found to use deterministic algorithms for operating
mode selection with a single input variable, either
readings of light sensors, geographic coordinates of
a point, or a predefined schedule for operating light
heating installations. The comparative analysis of
control systems showed that the most promising
should be the development of control systems with a
mode of operation according to sensor readings and
the use of intelligent algorithms. It is noted that the
lighting network with the highest energy efficiency
will have light sources that allow smooth regulation
of energy and light flow consumption. Papers, [2],
are devoted to the development of street lighting
control systems using artificial intelligence methods.
Currently, such management systems are being
implemented as pilot projects and are not widely
used, despite their considerable economy.
The purpose of this paper is to present a
theoretical approach to graph theory and a solution
to the routing problem for wiring in a street lighting
network. For a fixed location (the lighting node), the
algorithm will be designed to optimize a specific
wiring route. A changed version of Dijkstra's
algorithm is suggested for a brand-new network.
This algorithm can figure out a topology with the
fewest number of wires by using the known
locations of the power sources on streetlight poles,
[7], [8], [9].
At the same time, the relationship between the
number M of light poles and the distance V between
them will be studied, and the reduction of the
distance between them will be sought as a function
of the distance V, [10]. Finally, the total distance
between the nodes will be calculated. The most
difficult process—the mathematical formula for
node selection—will also be presented, and an
optimal topology will be found.
2 A Model for Optimizing the
Parameters of the Lighting
Network
We assume that the lighting network will be
represented as a graph in the process of developing
a methodology for determining the ideal cross-
section of the wires and the number of lighting
fixtures. The nodes (n) in the graph will be the street
lighting poles, and the ribs will be the wires
connecting them. In this instance, a matrix
connection C [n x n] can be used to depict the
topology of the lighting network. As an optimization
criterion, we provide:
1. Electrotechnical standard for reducing active
power losses (Figure 1):
Fig. 1: Network lighting area
(1)
where n is the total number of nodes and k and j are
the number of nodes; In the lighting network's
regular operating state, Ik,j is the current flowing
along the branch linking nodes k and j, and Rk,j is
the resistance of the branch that connects them, Om,
[11].
2. The financial standard for reducing operating and
capital expenses related to energy loss while
building the network is:
(2)
(3)
(4)
(5)
(6)
where Sp is the price of street lighting poles, and Sc
is the cost of cables. Sl is the price of lamps, k,j is
the distance in kilometers that a segment of the
network connects nodes j and k, Suc is a wire's unit
cost per kilometer. Ssp is the price of a single pole,
including installation. Ssl is the price of a single
lamp, installed. P[n] is the load vector's number of
non-zero entries, where n≠0.
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3. Light standard to optimize roadway lighting:
(7)
where A is the width of the illuminated road,
sidewalk, etc., and I is the distance between the
lamps, m. We get the goal function S'(x)→max for
the task of finding the best cross-section of wires
and the number of lamps in the lighting network by
using multiplicative convolution as a scalarization
method:
(8)
where x=(x1,x2) is a vector of desired values (wire
cross-section, number of devices), S'(x) is a super
criterion, S1,S2...,Sn are the values of optimization
criteria, and P1,P2...,Pn are the weights of
corresponding criteria. As a result, the optimization
problem's boundary conditions can be defined as
follows:
- Long-term wire current that is permitted:
(9)
where Ia is the allowed current for the chosen
conductor segment, Uk is the voltage at node k, and
Sk,j is the power of the branch linking nodes k and j,
VA.
– voltage dropouts inside the illumination system:
(10)
where U1 is the supply voltage, V;
An extra prerequisite for the border is that the
required quantities must be discrete.
We suggest determining the type of wire and its
cross-section, for which the maximum operating
current defines the resistance and its specific
pricing, as the standard series sets the cross-section
of the power supply wires of the lighting network.
By progressively adding installations and ensuring
that boundary conditions are met, the mode of the
lighting network is determined. The values of the
optimization criteria and the super criterion S'(x) are
computed for this variation if the boundary
requirements are met. The option with the highest
super criterion value among all the options is chosen
if no boundary criteria are satisfied.
The electrical part's generated simulation model
consists of:
1. A model of an electricity source Since the
lighting network is not the only burden on the
substation, I suggest that the supply voltage value
serve as a representation of the power source while
taking the external load into account.
2. A lighting network model. Given that insulated
wire fills the supply lines and that a 0.4 kV voltage
powers the lighting installations, the model advises
ignoring the section currents, which are negligible
due to the high insulation resistance, and the wire's
capacitance, which is suitably low due to the low
network voltage. Consequently, a series connection
of active and inductive resistance can be used to
mimic power line sections. It is suggested that a
matrix C of dimension nxn, where n is the number
of nodes in the network, be used to mathematically
depict the topology of the lighting network. The
lighting network branch, the power connection
point, and the SL connection point are all regarded
as nodes. C(j,k) = 1 if nodes k and j are directly
connected; otherwise, C(j,k) = 0. In the same way,
the resistances and lengths of each section of the
network can be set using the matrix as electric load
vectors S[n], P[n], and Q[n]. Each vector element
represents the load in the node that corresponds to
that number. The voltage vector at the network node
U [n] and the matrix branch currents I [n x n] define
the network's mode of operation.
3. The comparison of the load models showed that
when the constant power load model is used, the
voltages in the nodes and the power losses in their
branches are calculated with an error of no more
than 3%. Based on this, it was determined to carry
out additional research using the mathematical
constant power load model Sn=const.
By combining the mathematical models of the
power supply, lighting network, and load, it is
possible to compute the lighting's operational
parameters and simulate the ES SL with various
lighting network settings. The study's goal is to
determine the mode parameters of the currents in the
branches and the voltages in the network nodes
since lighting networks have a lot of nodes and their
current load varies between the first and last
sections. I suggest utilizing an iterative computation
approach, which has the advantages of being highly
accurate and simple to formulate. The lighting
network mode is iteratively calculated using a
function block from the Simulink simulation
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environment and an m-function in the MATLAB
environment.
To get both the natural and surface illumination
values for different operating modes, the Simulink
simulation environment uses a mathematical model
that is built as a functional block (Figure 2).
Fig. 2: Components of the functioning simulation
model
3 Dijkstra's Algorithm
Finding the best pathways is a crucial part of using
graphs to solve problems. The Dijkstra and
Bellman-Ford algorithms are the most commonly
utilized methods for resolving this issue. Both
directed and undirected coherent networks with
positive weights can be solved using Dijkstra's
technique. Bellman-Ford's technique should be
applied when there are negative weights because
Dijkstra's algorithm will yield inaccurate results,
[12]. It is a greedy algorithm because, up until it
reaches the terminal node, it chooses the locally
optimal solution at each step and then synthesizes
all of the earlier answers that have led to the best
result. The following needs to be defined before the
algorithm is executed, [12], [13], [14]:
• G(V, E) graph
• N(n): Nodes in the system.
• W (v, u) stands for the weights assigned to nodes v
and u.
• Launch: Launch node.
• A vector of size N that holds the node's distance
from the origin.
• A vector of size N that stores the shortest path to
each node that comes right before the preceding
one.
All of the vertices are split into two groups while the
algorithm runs:
• T = 0, which keeps track of every tested node as
well as the best route that led to their discovery.
• All of the network nodes that need to be checked
are contained in Set F = V.
A node is eliminated from the current set after
each test. The method stops when there are no more
nodes to test or when the set F goes to zero. To keep
the path as short as possible, a node v that is not part
of the set T is selected at each step. The vector is
then updated with a new value for each neighbor of
the node that, if it turns out to be the shortest path, is
not in the set T.
Before we can formulate a lighting network
routing problem mathematically, we need to
establish the following assumptions:
• One of a kind is the delivery node, or light pole.
•The cables that link the poles have the same
capacity and are similar.
•A single wire can be used to connect two poles.
• The number of pupils positioned atop the pillars is
unalterable.
• One network covers a single section of the city;
• Networks can be joined to one another.
The overall distance between the street light poles
will serve as the primary optimization criterion.
4 Mathematical Model for
Optimization
The following function reduces the total distance
between street light poles, [15], [16], [17]:
, (11)
One significant drawback is that lighting
fixtures require k poles, meaning that only one
wire may be placed at the separation between x
and y, [18], [19].
s.t. (12)
Route connectivity m, or the number of networks in
each area:
V, (13)
Meeting the restriction that node visits can only be
made once, except the city's collecting point for all
networks, [20], [21], [22]:
(14)
Making certain that every pillar was linked to the
one that was nearest to it, [23], [24], [25]:
(15)
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Applying the constraint that the capacity of a wire is
not exceeded:
(16)
Using the restriction that if the equivalent pole is not
present in this network, pole X should not be
connected to pole x:
, (17)
imposing the limitation that a pair of pillars can only
be joined once [26], [27].
(18)
The choice variables that follow have to be binary.
In other words, they have to choose between 1 and
0.
(19)
Below are the main criteria for selecting an
element and placing it in the corresponding m sets.
The main criterion for element selection is the
objective function. It is defined as the minimum
distance between the sums of the shortest distances
contained in Qi. In this order, we will analyze and
present the process of finding a minimal number of
subgraphs N containing Qx. Through an algorithmic
procedure, we will place suitable nodes in the
corresponding subsets Qx R. The elements of the
nodes will be denoted as follows:
(20)
Only in cases where two or more potential
nodes have the same value does the second
criterion come into play. The node that has the
least eccentricity in this instance will be chosen.
(21)
In recent years, the stability results of divergent
systems have had a strong effect on impulsive
profits. All impulse gains should be stabilizing
impulses. In the convergent behavior of some
divergent systems, most impulse controllers allow a
finite number of destabilizing impulses. It follows
from the obtained results that an infinite number of
destabilizing pulses can be allowed to reduce these
conservatives using the applied method.
In this instance, the element that is chosen will
be the node that has the most neighbors. The
following is computed in each step:
The objective function is used to determine the
potential elements at nodes qi and r.
The main component is:
Next, determine whether nodes satisfy the
maximum distance requirement;
Aligning them with the proper Qi group;
Locating the following lamppost;
Determining the shortest path between two
adjacent poles;
The addition of a node, or multiple nodes, to set
F; that is, their classification from single to commit.
To develop the model, specific economic
indicators related to the creation and implementation
of such a smart grid were identified. The models
chosen for the study are multifactorial and
regression. The lack of incentives to build is due to
technology versus transformation and energy
efficiency policies.
The advantages are: the ability to proactively
remove potential causes of accidents, prevent
unplanned power outages, and replace depreciated
assets before they actually fail. It is also possible to
perform energy monitoring, which gives you the
flexibility to determine the power supply according
to your needs. The energy community frequently
updates the Energy Efficiency Roadmap, which
grows in scope with the adoption of new legislation
in the field of electricity. The goal of the economic
design of a smart grid is to find the lowest costs for
the system. Overall, the problem is multifaceted,
with all components in the system and system
configuration having some impact on performance
and thus cost. Therefore, the optimal design must
find the optimal number of components and, at the
same time, provide the possibility for the system to
reduce the electrical load. Such a street lighting
network is usually characterized by high installation
costs and low operating costs. It enables limited
network losses and the forecasting of energy
consumption and production levels according to
climatic conditions.
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5 Application of the Algorithm to
Solve a Street Lighting Network's
Route
This is the task that this essay will go on to detail.
The issue is specifically with an integrated system
that delivers uniform objects (light fixtures) to
permanent sites in an urban environment called light
poles and electrical sources. We'll examine the path
taken by the pillars and the feeder, which serves as
the starting point. There will be a map available in
the form of a graph that includes all light poles, or
nodes, as well as the beginning position. All node
weights and distances will also be provided.
6 A Lighting Network’s Topology
and Its Shortest Wire Lenght
Determination
Description of the problem
The following factors are considered when
determining the routes that will connect the nodes:
• The primary goal is to shorten the wires' travel
distance.
• The maximum amount of space that can be
allowed between street lighting poles and the power
source
• Within a predetermined window of time, power
transmission must be finished.
• electricity with a fixed capacity.
A modified Dijkstra algorithm has been developed
as a methodology to solve the problem of
determining the optimal topology of a newly
designed street lighting network. This allows one to
determine the topology of the lighting network with
the least number of wires required, given the
coordinates of the power source and the street
lighting poles. The following steps are part of the
methodology:
Fig. 3: Summarizing the nodes
1. Building the lighting network schedule,
accounting for the lighting project's maximum
allowable wire sag and the spacing between street
lighting installations (Figure 3) and (Table 1).
Table 1. Adjacency list
2. Creation of the resultant graph's adjacency matrix
(Figure 4) and (Table 2).
3.Using a modified Dijkstra algorithm, determine
the shortest path between each light fixture and the
power source.
In contrast to Dijkstra's traditional algorithm,
the label value of each node in the proposed t
method is the entire length of the lighting network
(the graph's edge), including the lighting bodies,
rather than the distance from the power source to the
street lighting installation (a node of the graph).
Additionally, when a node's label is changed, the
adjacency matrix is altered. The updated label's
column is set to zero, and while searching for the
path from which the update came, the node's
connection is highlighted.
In one part, the updated adjacency matrix is
used to make a graph network that shows the
lighting network's topology from pole to pole with
the shortest wire length (Figure 4).
Fig. 4: Graph - topology of the lighting network
The designed techniques are put into practice in
the MATLAB environment and enable the
identification of the lighting network's ideal version
during the design stage.
After that, the adjacency matrix is computed using
the appropriate weights. The calculation for each
element Nxy N is as follows:
(22)
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Given the orientation of the graph, Nxy≠ Nyx
There are nodes in the urban network N.
The set S n expresses the nodes of urban
areas. There are a total of 7 regions: |S|=|N|
− 1 = m m².
The origin-destination node in a city
network N is taken to be: network node = 7.
There are k = 1 networks in each region.
Lighting pole capacity for the regional
node: C = 15.
The maximum separation between pillars
that is allowed is W =2.
Table 2. Adjacency Matrix with Weights for the
Network
7 Conclusion
A review of the available technical fixes and a
rationale for the indicator system were conducted. In
the process of designing or updating the street
lighting network, an ideal topology was looked
after, and a technique for optimizing the
characteristics of its parts with the least amount of
active power loss was created. The following
elements of the street lighting electrical system were
included in the simulation model that was
developed: power supply, lighting network, street
lighting installation, and a system for controlling the
system's operating mode. Unlike current methods,
the optimization process for the network elements is
based on a selection of parameters based on the least
capital cost, regulatory compliance, and maximum
lit area criteria.
Multifactorial problem-solving is required. To
determine the best solution, the problem's needs
must be completely specified from the beginning.
Finding the best answer will take less time if the
suggested approach is implemented using a
programming language like Python. Finding the
differences between the routing problems, analyzing
them, and then making any necessary adjustments to
the algorithm to enable its execution are the
objectives.
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Ekaterina Gospodinova
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