# A geographic information system-based modelling, analysing and visualising
of low voltage networks: The potential of demand time-shifting in the power
quality improvement

Tomislav Antić<EMAIL_ADDRESS>Tomislav Capuder
<EMAIL_ADDRESS>
###### Abstract

The challenges of power quality are an emerging topic for the past couple of
years due to massive changes occurring in low voltage distribution networks,
being even more emphasized in the years marked by the novel COVID-19 disease
affecting people’s behaviour and energy crisis increasing the awareness and
need of end-users energy independence. Both of these phenomena additionally
stress the need for changes in the planning and operation of distribution
networks as the traditional consumption patterns of the end-users are
significantly different. To overcome these challenges it is necessary to
develop tools and methods that will help Distribution System Operators (DSOs).
In this paper, we present a geographic information system (GIS)-based tool
that, by using open source technologies, identifies and removes errors both in
the GIS data, representing a distribution network, and in the consumption data
collected from the smart meters. After processing the initial data, a
mathematical model of the network is created, and the impact of
COVID-19-related scenarios on power quality (PQ) indicators voltage magnitude,
voltage unbalance factor (VUF), and total voltage harmonic distortion (THDu)
are calculated using the developed harmonic analysis extension of the
pandapower simulation tool. The analyses are run on a real-world low voltage
network and real consumption data for different periods reflecting different
COVID-19-related periods. The results of simulations are visualised using a
GIS tool, and based on the results, time periods that are most affected by the
change of end-users characteristial behaviour are detected. The potential of
the end-users in the PQ improvement is investigated and an algorithm that
shifts consumption to more adequate time periods is implemented. After
modifying the consumption curve, power quality analysis is made for newly
created scenarios. The results show that the pandemic negatively affect all
analysed PQ indicators since the change in the average value of PQ
disturbances increased both during the hard and post-lockdown period. The
time-shifting of consumption shows significant potential in how the end-users
can not only reduce their own energy costs but create power quality benefits
by reducing all relevant indicators.

###### keywords:

, demand time-shifting , geographic information system , low voltage networks
, power quality analysis

[*] [*]

organization=University of Zagreb Faculty of Electrical Engineering and
Computing, Department of Energy and Power Systems, city=Zagreb,
country=Croatia

Integrated tool for modelling, analysing, and visualising a network is
presented

Simulations are run to determine the impact of end-user pattern consumption
change on power quality (PQ)

The results show negative impact of passive end-user demand changes on PQ
indicators

Demand time-shifting is shown to be an efficient method for reducing PQ
disturbances

##  1 Introduction

###  1.1 Impact of COVID-19 on power systems

Besides the well-known technical challenges caused by the continuous increase
of LC technologies share and their uncoordinated installation and operation
[1, 2], in 2020, 2021, and part of 2022 the COVID-19 pandemic has created
numerous changes in the planning and operation of power systems and new
challenges for system operators. The analysis of the German and other European
power systems showed that the pandemic increased the existing power system’s
uncertainty despite the kept high level of security. The authors in [3]
investigated the impact of COVID-19 on the CO2 emissions and showed that in
2020 the emissions decreased in the industry, transportation, and construction
sector. The authors in [4, 5] analyse the impact of the pandemic on
electricity markets and market operations and show that both electricity
prices and power demand were seriously affected by the reaction to the virus.
Results of a detailed analysis of the impact of COVID-19 on the behaviour of
end-users show that electricity consumption of residential low voltage (LV)
end-users has significantly increased [6, 7]. The focus of already published
research on the COVID-19 impact on LV distribution networks is mostly on the
change in consumption and its correlation with environmental, technical, and
market aspects of planning and operation of distribution networks. However,
the existing literature does not address the impact of the pandemic on power
quality and the changes in people’s behaviour in the post-pandemic period and
if their potential in providing flexibility services changes. To overcome the
identified gap, we present a detailed analysis of power quality in the LV
distribution network over different pandemic-related periods and observe if it
is possible to improve deterioration only by shifting demand with the minimum
disruption of end-users’ comfort.

###  1.2 Open-source power system simulations

The use of power system simulation tools developed using different programming
languages is continuously increasing. The development of pandapwoer, the
Python-based open-source tool for different analyses of steady-state power
systems is presented in [8]. The mathematical model and the pandapower-based
tool for harmonic analysis in both balanced and unbalanced radial distribution
networks were presented in [9]. Another open-source platform for modelling,
control, and simulations of local energy systems, OPEN, was described in [10].
MATPOWER, an open-source Matlab-based power system simulation package that
provides a high-level set of power flow, optimal power flow (OPF), and other
tools targeted toward researchers, educators, and students are described in
detail by the authors in [11]. Major benefits of open source tools include the
possibility to modify and extend their functionalities and to integrate them
with other tools and external databases. Such integration with smart meter
measurements and geographic information system data is described in detail in
the rest of the paper.

###  1.3 The power of geographic information systems

Due to the increasing complexity of distribution networks, it becomes
insufficient to rely only on power system simulation tools. Such tools must be
integrated with other tools, systems, and databases. One of the systems being
more used in the planning and operation of smart distribution networks is the
geographic information system (GIS), a system that contains both technical and
geospatial attributes of distribution networks. The authors in [12] present an
integration of QGIS, an open source GIS tool, with OpenDSS, an open source
tool for power system simulations. Since GIS is characterized by georeferenced
data, other applications of GIS-based tools include determining the PV hosting
capacity or finding an optimal location for installing a PV power plant [13,
14], developing an optimal design of renewable-powered EV charging stations
[15], or visualising the results of power flow calculations [16]. A potential
disadvantage of using GIS data in its integration with the power system
analysis tool is the high probability of different errors in the initial data
set. Those errors often make the data unusable in its initial form and further
processing and editing are required [17]. Other than GIS data, data collected
from smart meters is often considered the key in different distribution
network analyses [18, 19] or forecasts [20]. A detailed analysis of the
potential use and advantages of smart meters in the planning and operation of
smart networks is presented in [21, 22]. Unlike the other papers that are
mostly focused on the single use of GIS in the planning and operation of power
systems, in this paper, GIS is applied in editing data representing LV network
elements and visualisation of the results of power system simulations. On top
of that, we present a detailed step-by-step guide for detecting and editing
specific geospatial errors that prevent the direct use of GIS data.

###  1.4 Mitigation of power quality issues

To prevent power quality disturbances and other technical problems in
distribution networks, it is becoming more important to investigate approaches
for mitigating these issues. Solutions can be oriented on the utilization of
physical devices such as inverters that help in the mitigation of harmonic
distortion [23, 24] or re-phasing switches that mitigate network unbalances
[25], the potential of end-users where they can help in the improvement of
power quality [26] and in avoiding harmonics [27] under multiple demand
response schemes created based on the profiles created in the laboratory
environment, or on their combination as described in [28]. Besides observing
only the potential of end-users’ increase or decrease in electricity demand,
the improvement of networks’ technical conditions can be achieved by the time-
shifting of demand. However, this approach is in most cases applied only on
the improvement of one value, e.g., voltage magnitude along the feeder [29].
As seen in the literature review, papers often focus on the improvement of the
limited set of technical quantities and solutions are often tested on
benchmark networks or in a controlled environment. We propose a demand time-
shifting algorithm aimed at the comprehensive improvement of PQ indicators.
The mathematical model of the algorithm is tested on real-world data - demand
measurements collected from smart meters and real-world LV network topology
information.

###  1.5 Contributions

In this paper, we propose an integrated open source tool used for processing
and editing GIS and smart meter data, power quality analyses, and visualising
the results of the simulations. QGIS, Python, and SQL are used for processing
and editing GIS data, smart meter data is processed using Python and SQL,
power quality is analysed using the pandapower tool and its developed harmonic
calculations extension, while the results are visualised using QGIS.
Additionally, Python is used for the detection of time periods that were most
affected by the pandemic-related changes. After the detection, a demand time-
shifting algorithm is implemented in order to investigate if it is possible to
reduce the PQ deterioration only by using end-users potential and without the
investment in the additional physical devices. Power quality analyses in this
paper are limited only to voltage magnitude, VUF, and THD since these are the
only disturbances that can be calculated using the presented model. Other
disturbances cannot be detected by quasi-static time series simulations using
pandapower or any other tool with the same set of functionalities and their
analysis is out of the scope.

To summarise, based on the literature review and identified research gaps, the
proposed contributions of this paper are:

  * 1.

An integrated open source tool used for editing and processing GIS and smart
meters data, creating LV network models, performing power quality analyses,
and the visualisation of the simulations results.

  * 2.

A comprehensive analysis of PQ indicators in the real-world LV network during
periods related to realistic end-users consumption pattern changes, i.e.
COVID-19-related pre-pandemic, pandemic, and post-pandemic periods. Calculated
values are compared to limitations in European standards and the impact of the
pandemic on the distortion of power quality is investigated.

  * 3.

Implementation of the end-users demand time-shifting scheme based on the
detection of PQ distorted intervals. The developed methodology reduces values
of PQ indicators and in some cases even improves PQ and, unlike the papers
that rely only on the possibility of end-users to increase or decrease their
consumption when needed, results in the effect of less discomforting and more
acceptable time-shifting of end-users’ consumption on the PQ improvement.

The rest of the paper is organized as follows: Section 2 gives an explanation
of detected errors in GIS and smart meters data and methods used for removing
the errors, while the results of analyses defined for a real-world Croatian
case study are defined in Section 3. Section 4 presents the methodology and
the results after the implementation of an algorithm used for the improvement
of PQ. Finally, conclusion is presented in Section 5.

##  2 Tool description

Using both GIS and smart meters data presents great potential in the
improvement of the planning and operation of smart distribution networks and
the increase of the network’s observability. The problem with this data is
that in a high number of cases it cannot be used in its initial form, which
presents an obstacle for Distribution System Operators (DSOs). Also, not all
data about distribution networks is stored in one system or database.
Therefore it is necessary to develop an integrated tool that enables the
communication between several systems and databases, that has the possibility
of removing errors and preparing data for further analyses.

In the first step, the open source-based tool presented in this paper is used
for editing the data and removing the errors in the initial data set. Features
of open source technologies QGIS, Python programming language, and PostgreSQL
with the postgis extension, are applied for processing the initial GIS and
smart meters data. GIS data contains technical and geospatial information
about the Croatian real-world LV network, while smart meters data contains
information about the end-users’ consumption. The focus in this paper is put
on data representing a LV network since the main focus was to investigate the
potential of residential end-users in the improvement of technical conditions
in distribution networks. In case of observing MV networks, the presented tool
can be used for processing the GIS data and in case of identifying new errors
it can be updated in order to successfully solve all detected problems.

###  2.1 GIS data

In the process of identifying the errors in the initial GIS data set, several
errors were detected, and afterward, the data was edited in order to enable
using it in the creation of the network’s mathematical model:

  * 1.

Continuity of a polyline

  * 2.

Unknown beginning and ending node of an LV cable/line

  * 3.

Disconnection of an LV cable/line and an MV/LV substation

  * 4.

Disconnection of an LV cable/line and an LV switch cabinet

  * 5.

Unknown technical attributes of an LV cable/line

  * 6.

LV cable/line without known ending node

  * 7.

Redundance of point objects

Each polyline that represent the main line of the observed distribution
network feeds additional branches of a distribution network. Main lines and
branches are connected at a point object called connection point. Even though
main line is topologically connected with branches, without breaking a
polyline into smaller parts, i.e., segments, determining the beginning and the
ending node of each line will not be possible. Figure 1 shows an example of
dividing a line object into segments, where each colour represents an
individual segment created by using the developed tool.

Figure 1: Dividing a line object into segments

Some point objects, e.g., connection points or points that represent end-
users, are located at the beginning or the ending node of a line object, i.e.,
some point objects are topologically connected with line objects representing
an LV line/cable. In those cases, based on the equality of coordinates, the
beginning and ending nodes of each line object are determined. This change
cannot be visualised since only attributes of a line object are changed in a
database so they could be used in the creation of a mathematical model of a
network based on the edited GIS data.

Some point objects are not located at the beginning or the end of a line, but
still, need to be connected with lines. An example of those objects are points
representing MV/LV substations or LV switch cabinets. Depending on which node,
the beginning or the ending, of a line is unknown, the distance between the
first or last coordinate of a line and a point object representing a
substation or an electric switch cabinet is determined, and if it is within
the radius defined by a user, the point object is attached as a beginning or
ending node of a line. Figure 2 shows an example of a topological connection
of a point object representing a substation with line objects representing
lines and cables.

Figure 2: Connection of a point with line objects

After all point objects are processed there are still some lines without
attached beginning or/and ending nodes. In those cases the missing node is
replaced with a point object called a virtual node, which is an element
without a connected end-user, i.e., the value of the load at that node is
equal to zero. An example of creating a point object representing the virtual
node element is shown in Figure 3.

Figure 3: Creating a point object

Some point objects cannot be connected with LV cables/lines since they do not
belong to the observed network, i.e., these point objects are not a part of
the network and they were input by an accident. Those objects must be deleted
from the database and the GIS data set, since they will cause errors in
further calculations, e.g., an unbalanced power flow calculation becomes
impossible. Figure 4 shows an example of deletion of an extra node, i.e., of a
node representing an end-user that is not connected with the rest of a
network.

Figure 4: Deletion of topologically unconnected node

In the last step of GIS data editing, unknown technical attributes of LV
lines/cables are replaced with values for the standard type of a line or a
cable that is most common in a database.

###  2.2 Smart meters data

Similar to the error detection in the initial GIS data set, errors are
identified in the collected smart meters data:

  * 1.

outlier values

  * 2.

zero values

  * 3.

missing values

Outlier values are defined as the values that significantly differ from the
other consumption data, e.g., the consumption of an end-user at one time
period is 1000 times larger than in the previous one. In some time periods,
the value of consumption is equal to zero. Even though it is possible that
consumption lowers at certain periods, there are still several devices in each
household that consume energy for basic functions. In some periods, due to a
loss of connection between the smart meter and a database or any other
obstacle, values of consumption are not stored in the database. Those values
are defined as missing ones. All detected errors are resolved in a way that an
outlier, zero, or a missing value in a certain period is replaced with the
value of consumption in a previous time period. If there are multiple
consecutive time periods with some of the detected errors, there is a
significant number of intervals in which the initial data cannot be used, or
detected errors could not be corrected, the whole consumption data collected
from a certain smart meter is neglected. For the purpose of the analysis in
this paper, such data is replaced with consumption collected from other smart
meters, modified with a randomly determined factor, since it is not expected
that two end-users have identical consumption in each time period. An example
of the smart meter data editing is shown in Table 1.

Table 1: Example of the smart meter data editing

| Power (W)
---|---
|  | Outlier
---
value
| Zero
---
value
| Missing
---
value
$\Rightarrow$ |  | Outlier
---
value
| Zero
---
value
| Missing
---
value
Time | Initial data | Edited data
00:00 | 575 | 575 | 575 | 575 | 575 | 575
00:15 | 583 | 583 | 583 | 583 | 583 | 583
00:30 | 50000 | 0 | - | 583 | 583 | 583
00:45 | 580 | 580 | 580 | 580 | 580 | 580

###  2.3 Power quality analysis and visualisation of the results

After all the data is processed and edited and all errors are removed from the
initial data sets, prior to calculations it is necessary to merge the data
related to the end-users consumption with the GIS data. Since end-users
consumption and GIS data are stored in two unrelated databases, the
communication between the two systems is established as a part of the
developed tool. After the merging of the data, all prerequisites for analyses
of LV distribution networks are satisfied.

The second step of the developed tool includes the creation of a mathematical
model of an LV network using pandapower, an open source Python-based library
for power systems calculation [8]. Additionally, one of the latest pandapower
extensions presented in [9] is used for the unbalanced harmonic analysis. LV
network elements such as nodes, lines, transformers, and loads (determined by
locations of end-users) are created from the edited GIS data, while the value
of each load’s active power is defined from the smart meter data. Since only
active consumption is measured, reactive power of loads is calculated with
power factor, randomly chosen from the $[0.95-1]$ interval. Once the network
with all its elements is created, pandapower is used for time-series
simulations and calculation of voltage magnitude, voltage unbalance factor
(VUF), and total voltage harmonic distortion (THDu). Consumption measurements
are collected in the 15-minute interval over one week and they represent the
average value of consumption for every end-user in every 15-minute interval.
Because of that, calculated values of PQ indicators voltage magnitude, VUF,
and THD are also average 15-minute values and not values captured at a certain
moment. Therefore, the focus of analyses is on the quasi-static time series
three-phase harmonic power flow and PQ indicators that are meaningful in the
defined time interval. Other indicators such as transients require the
installation of additional monitoring devices and cannot be calculated with
the used approach and are out of the scope of the paper.

Once the simulations are completed, the results are stored in an external
database and used for visualisation using QGIS. In this paper, visualisation
is intended to show the change of PQ indicators during the pandemic with
respect to the pre-pandemic period, i.e., we investigate and visualise how
much did PQ deteriorate or improve due to the changes caused by differences in
end-users behaviour during different pandemic-defined scenarios. Results of
the scenario that considers the implementation of the consumption time-
shifting are also presented, which enables a simple and intuitive assessment
of the end-users potential in the general improvement of power quality
disturbances.

Tools used in each part of the analyses and the visualisation in this paper,
and the developed integrated tool as a whole are presented in Figure 5.

Figure 5: Integrated tool description

##  3 Case Study and Results

###  3.1 Case Study

An LV network used for the analysis of the impact of different periods related
to COVID-19 on the power system is a part of the real-world Croatian
distribution network. An urban network consists of an MV/LV transformer, 10
feeders, 179 nodes, from which 139 represent end-users, and 177 overhead lines
and underground cables. However, due to the visibility and clarity of the
representation of the results, only one LV feeder, with 66 nodes, 43 three-
phase and single-phase connected users, and 64 lines, is modelled. The
modelled LV feeder is presented in Figure 6.

Figure 6: Modelled LV feeder

Three weeks that represent pre-lockdown, hard lockdown, and post-lockdown
periods are chosen to minimise the season’s impact on electricity consumption.
The first period is set in late winter when the temperatures are not so low,
and end-users heat less than in December or January. The second period is set
in spring and is characterized by conditions that do not require heating or
cooling. Finally, the third period is set in late summer, when the
temperatures do not go as high as in July and August, and therefore end-users
do not use air conditioners as often as in other summer months. Therefore,
consumption and consequentially PQ indicators should not differ so much in the
chosen representative weeks. Even more important, consumption during each
representative week has captured specific patterns and end-users’ behaviour
that characterized different periods during the pandemic. Moreover,
observation of these specific weeks allows the analysis of the impact of the
pandemic on technical conditions in a residential LV network and also presents
the opportunity to investigate if some periods were more favourable in terms
of the end-users’ flexibility provision.

The data from representing weeks presents the cumulative active consumption of
end-users connected to an urban LV network, which is converted to the more
suitable active power data. The active consumption data is collected in
15-minute intervals. Since the smart meters installed in the network defined
in the case study do not collect the reactive consumption data, reactive power
is calculated from active power and power factor whose value is chosen with
uniform randomization from the $[0.95-1.0]$ interval. Together with the data
defining the elements of an LV network, this data is used as input into
pandapower and its harmonic calculation extension.

###  3.2 Results

The results of simulations are visualised using the QGIS open source tool.
Results show the average change of a PQ indicator during hard (Figure 7) and
post-lockdown (Figure 8) periods, compared to the pre-lockdown period. The
change is calculated with eq. (1) where $PQ_{indicator}$ represents the value
of voltage magnitude, VUF, or THDu during each observed COVID-19-related
period. Finally, the average change for each PQ indicator in every node is
calculated as a mean value of calculated $\Delta_{indicator}$ values in all
periods.

$\Delta_{indicator}=\frac{PQ_{indicator,hard/post}-PQ_{indicator,pre}}{PQ_{indicator,pre}}$
(1)

The defined method of calculating the change considers the pre-lockdown as a
referent period that observed end-users’ habits before the pandemic. Despite
the opening of several business sectors and the relaxation of measures in most
of countries during the post-lockdown period, the habits of end-users were
changed, e.g., people are working from home more than in the pre-lockdown
period. Therefore, the post-lockdown period is also characterized by increased
electricity consumption in LV networks. Hence, the change in the value of each
PQ indicator is observed with respect to the referent pre-lockdown period. It
is important to mention that the change of the value of a certain PQ indicator
does not mean that it increased or decreased for the calculated amount, e.g.,
if the value of THDu during the hard lockdown period increased to 3% from 2%
in the pre-lockdown period, the increase of 50% and not 1% will be visualised
using the developed tool. The reason why we focused only on these three PQ
indicators in our analyses is the interval in which their values can be
calculated. While there are other indicators such as transients, voltage
swells or sags, these disturbances are characterized by a short duration or a
milliseconds time frame in which they can be detected. There are two main
reasons why it is not possible to consider them in analyses - the first being
the need for additional measuring equipment since the only available data are
the consumption measurements. The second reason is the limitation in terms of
possible analyses conducted by the used power system simulation tool. However,
other tools used in quasi-static power system simulations share this
limitation and make the analyses of dynamic disturbances impossible and not
realistic.

####  3.2.1 Pre-lockdown period

To better evaluate the change of values of PQ indicators during hard and post-
lockdown periods, voltage magnitude, VUF, and THDu are also analysed for the
pre-lockdown period in order to calculate values of PQ indicators in the base
scenario and determine if there are any violations of defined PQ limitations
or if values of observed PQ indicators come close to the threshold values. The
minimum value of voltage magnitude is 0.933 p.u., while the maximum is 1.001
p.u. Since most of the European standards and national grid codes define an
allowed interval of [0.9-1.1 p.u.] for the value of voltage magnitude,
calculated values are not outside the allowed interval in any period during
the observed week. However, the minimum value during the pre-lockdown period
comes close to the lower bound of the allowed voltage magnitude interval,
meaning that any further increase in consumption caused by the pandemic or
electrification of heating and transport could cause a violation of defined
limitations. The largest VUF value during the pre-lockdown period is equal to
0.539%, which is below the 2% threshold (3% in networks with predominant
single-phase connected loads). Even though most of the end-users’ devices are
single-phase connected to a network none of the devices have power that would
create unbalance problems in a network. Single-phase connected devices lead to
increased unbalance, but the value of VUF still does not violate the defined
limitations. However, installation of any load with large nominal power or a
single-phase connected PV will additionally increase the difference between
phase consumption and consequentially phase voltages, i.e., voltage unbalance.
A similar situation is with THDu. Even without the integration of LC units
that are power electronics interfaced to a network, end-users in LV networks
possess a high number of non-linear loads that increase harmonic pollution.
The largest value of THDu in the observed network is 0.886% which is
significantly lower than the maximum allowed value which is equal to 8%. Same
as in the case of voltage magnitude and VUF, the increase in consumption and
further integration of LC units will only contribute to the harmonic
distortion in a network.

####  3.2.2 Hard lockdown period

Figure 7(a) presents the average change of voltage magnitude in each node
during the hard lockdown period compared to the referent, pre-lockdown period.
The results show that the average value of voltage magnitude decreased in all
nodes. The decrease is larger than 0.3% at the nodes further from the referent
node representing a substation. Voltage magnitude at first several nodes
decreased for less than 0.1%, while at all other nodes the decrease is in the
range between 0.1% and 0.3%. Even though the value of voltage magnitude is
close to the lowest value in the pre-lockdown period, change during the hard-
lockdown period does not significantly decrease voltage magnitude. The lowest
value is equal to 0.928 p.u., which is negligibly lower compared to the pre-
lockdown period. Even though the results show that values of voltage magnitude
do not violate the limitations even in the case of enlarged consumption, the
decrease presents a concerning trend for DSOs. Since end-users’ habits changed
during the pandemic, higher consumption is expected in future scenarios and it
must be considered in the planning and operation of distribution networks.
Also, further and more rapid integration of LC units including heat pumps and
EVs will continue the decrease of voltage magnitude and if not prevented,
violations of limitations will almost certainly happen.

(a) Change of voltage magnitude during hard lockdown

(b) Change of VUF during hard lockdown

(c) Change of THD during hard lockdown

Figure 7: Change of PQ indicators during hard lockdown

The change in average VUF value during the hard lockdown period, compared to
the pre-lockdown periods is shown in Figure 7(b). Unlike the average change of
voltage magnitude, which was in the range of -0.327% and -0.004%, the change
of VUF is significantly larger and is between 64.62% and 106.39%. In most
nodes, the average change is lower than 70% or between 70% and 80%, while in
several nodes it is larger than 80%. Even though the average change in each
node seems very high and concerning, the maximum value of VUF in the hard
lockdown period is equal to 0.659%. The results of the analysis show that
voltage unbalance increased due to enlarged consumption but it still did not
lead to the violation of the threshold value. However, the distribution of
consumption among phases is still close to symmetrical, which means that
voltage unbalance still does not present a problem for DSOs. Same as with
voltage magnitude, an increase of VUF presents a concerning trend which will
in the future be more emphasized with the integration of additional single-
phase consumption and production units.

Figure 7(c) shows the impact of increased consumption during the hard lockdown
period on harmonic distortion. Even though it is not the only factor in the
calculation of THDu, increased consumption causes higher current at the
fundamental frequency and also higher current at higher frequencies since it
is defined as the percentage of fundamental frequency current. Such increased
current directly increases higher-order harmonic voltages, which together with
already shown decreased voltage magnitude at fundamental frequency leads to
higher values of THDu in the observed LV network. Compared to the pre-lockdown
period, voltage distortion increased between 12.7% and 14.22% in all nodes,
and in more than half of the nodes, the change is larger than 13.5%. However,
the largest value of THDu is 1.027%, which is significantly lower than the 8%
threshold value. Despite the maximum value being lower than the limitation,
any increase in harmonic distortion caused by higher consumption could present
a potential problem for DSOs, since it is expected for consumption to continue
to increase in the future, together with the share of power electronics.

####  3.2.3 Post-lockdown period

In the post-lockdown period analysed in this paper, most of the government’s
restrictions were abandoned, however, a significant number of end-users still
remained working from home which continued the trend of increased consumption
and changes related to PQ in residential LV networks.

Figure 8(a) shows the change of voltage magnitude during the post-lockdown
period, with respect to pre-lockdown. Compared to the hard-lockdown period,
the average change of voltage magnitude in this scenario is lower and it even
increased in some nodes, with the largest average increase of 0.124%. The
largest average decrease of voltage magnitude is -0.048%, meaning that the
average value of voltage magnitude is similar to the pre-lockdown period.
Also, the minimum voltage magnitude is 0.939 p.u., which is negligibly larger
than in the pre-lockdown period. Despite the fact that the behaviour of end-
users changed, leading to more often working from home, the post-lockdown
period is located in the late summer, almost fall, when consumption is
traditionally smaller compared to winter periods, in which the pre-lockdown
period is set. The pre-lockdown period is set in late winter when heating is
still necessary, especially in continental, more cold parts, where the
observed LV network is located. Therefore, the impact of seasonality is more
emphasised than the post-pandemic-related changes in end-users’ behaviour and
the results are more similar to the pre-lockdown than the hard lockdown
period.

(a) Change of voltage magnitude during post lockdown

(b) Change of VUF during post lockdown

(c) Change of THDu during post lockdown

Figure 8: Change of PQ indicators during post lockdown

The change of VUF during post-lockdown is shown in Figure 8(b). As mentioned
before, the connection of large single-phase loads or single-phase connected
generation units is more influential on voltage unbalance than seasonality or
a scenario related to the pandemic. However, similar to during the hard
lockdown period, the results show that there is a higher difference between
phase consumption in the post-lockdown period. Compared to the pre-lockdown
period, VUF increased in all nodes, but the change that occurs is generally
smaller than during the hard lockdown period. The lowest average value of
change is equal to 52.64% and the largest to 116.36%. Despite the large
change, values of VUF do not violate the 2% limitation in any node and time
period, with the largest value of 0.56%.

As mentioned before, the value of THDu is largely affected by the situation at
the fundamental frequency. Even though the results of the voltage magnitude
analysis show that there are some nodes in which the average value of voltage
magnitude increased, a more detailed analysis of changes in the post-lockdown
period shows that there are more time periods with the decrease of voltage
magnitude than with increase. Also, the value of THDu is dependent on more
parameters than only on consumption and voltage magnitude at the fundamental
frequency. The results in Figure 8(c) show that the average harmonic
distortion increased in all nodes in the observed network, with all changes in
the range between 1.51% and 7.21%, which are smaller values compared to the
changes during the hard lockdown. Also, the maximum value of THDu is equal to
0.819%, which is lower than the maximum that occurred during the hard lockdown
but also during the pre-lockdown period. It is interesting to mention that the
trend of the change of THDu in this scenario is almost identical to the trend
of the voltage magnitude change. Nodes with the largest average increase of
THDu are also the nodes with the average decrease of voltage magnitude and
those with the lowest increase of THDu are those with the largest increase of
voltage magnitude. The perceived trend additionally brings the value of the
average harmonic distortion change into correlation with the average change of
current and voltage magnitude at the fundamental frequency.

##  4 Improvement of the PQ indicators

###  4.1 Methodology

Even though changes in end-users’ habits that arose as a consequence of
restrictions during the pandemic have created numerous challenges for DSOs,
they also enable potential for a larger exploitation of their flexibility in
the planning and operation of distribution networks. In most cases, end-users
can help a DSO by increasing or decreasing their consumption or by time-
shifting of electricity consumption. Since there are papers that implement a
demand-side management scheme and investigate if it is possible to improve the
values of different PQ indicators only by increasing and decreasing end-users
demand or test the time-shifting approach in a controlled laboratory
environment, the focus in this paper is put on the comprehensive improvement
of PQ in a real-world LV network only by shifting consumption to other time
periods.

In order to avoid unnecessary time-shifting of consumption, happening too
often, which will disrupt the comfort of end-users and discourage them from
providing such and other similar flexibility services in the future, we
developed an algorithm that detects time periods most suitable for time-
shifting of consumption. The developed algorithm is a part of an integrated
framework presented in this paper. For each day, an algorithm finds 10 time
periods in which the change of electricity demand during the hard lockdown
period was the largest. The change of electricity demand during hard and post-
lockdown is defined with eq. (2). Even though the trigger for activating the
time-shifting service could be set in a different way, e.g., the change of
demand was 30% larger compared to pre-lockdown, such definition would lead to
too often activation of the time-shifting of consumption service. Therefore,
the limitation was set to 10 intervals with the highest value of change for
each day.

$\Delta_{hard/post-pre}=\frac{P_{hard/post,t}-P_{pre,t}}{P_{pre,t}}$ (2)

After all such time periods are detected, the surplus of demand is shifted in
eight consecutive time periods (2 hours) within 32 periods (8 hours) before or
after the detected surplus. Time periods in which demand will be shifted are
determined as those with the smallest total electricity demand since they will
be most adequate for the increase of electricity demand. The step in which the
minimum sum is found is defined with eq. (3).

$\begin{gathered}\min\sum_{i=0}^{7}P_{t+i}\\\ \forall t\in[T-32,T-8]\ or\
[T+1,T+24]\end{gathered}$ (3)

After the detection of the eight most suitable time periods $[t,t+7]$, the
value of the demand increase in each time period $t+i$ is calculated using eq.
(4). Such an approach ensures fair distribution of the total increase of
demand among consecutive time periods. An algorithm continuously updates the
values of demand, i.e., if there are two consecutive periods detected as the
most suitable for time-shifting of consumption, an algorithm will find
adequate periods according to updated and not initial demand values.

$\begin{gathered}\Delta_{t+i}=\frac{P_{t+j}}{\sum_{j=0}^{7}P_{t+i}}\cdot\frac{(P_{hard/post,t}-P_{pre,t})}{P_{pre,t}},\quad\forall
i\in[0,7]\end{gathered}$ (4)

It is important to mention that only active power was observed in the
described approach. Reactive power was calculated with the same power factor
as in the initial case, before the improvement of the hard lockdown and post-
lockdown consumption curves.

###  4.2 Results

After the implementation of the proposed methodology, the same analyses were
made as in Section 3.2 with the modified, time-shifted consumption curve,
which is compared with the results obtained using the pre-lockdown period. The
results of that comparison will show if the time-shifting of consumption
helped in the PQ improvement or if it did not have any effect and the values
remained unchanged.

After the verification of an algorithm used for the detection of time
intervals most adequate for time-shifting, more detailed analyses were
conducted in order to investigate the effect of time-shifting of consumption
on PQ indicators in LV distribution networks. The change of value of the PQ
indicators is calculated with respect to the initial value, before the
implementation of the demand time-shifting algorithm, using eq. (5).

$\Delta_{indicator}=\frac{PQ_{indicator,improved}-PQ_{indicator,initial}}{PQ_{indicator,initial}}$
(5)

Table 2 shows the result of the change of voltage magnitude, VUF, and THDu
after the implementation of the demand time-shifting algorithm for the hard
lockdown period. The results present maximum increase, maximum decrease,
average change, and median change for all defined PQ indicators. The maximum
increase of voltage magnitude is around 4.5% and the maximum value of voltage
magnitude decrease is more than 3%. Even though values of both median and
average change are larger than zero, such change of the values that are in the
interval between 90% and 100% are not so significant. The time-shifting of
demand decreases peak demand and voltage magnitude in some intervals but
consequentially increases both demand and voltage magnitude in other periods,
causing a so called rebound effect. This effect does not present problems for
DSOs since it cannot cause overvoltage or undervoltage, despite the increase
in average and median voltage magnitude change. A more detailed analysis of
results shows that there are not values close to the upper voltage limitation.

Both VUF and THDu values are decreased in most periods, which can be seen from
the negative median changes of the mentioned values. Same as in the case of
voltage magnitude, due to the rebound effect, values of PQ indicators increase
in some periods but the value of the increase is around ten times smaller than
the increase in case of voltage magnitude. The maximum value of VUF is
detected as an outlier value that does not adequately present the potential of
the used methodology. Due to this and other similar outliers, the average
value of the VUF change is larger than zero. Therefore, using a median value
is a better indicatior of the improvement of PQ distrubances.

Table 2: Change of PQ indicators during hard lockdown - time-shifted
consumption

|  | Maximum
---
increase (%)
| Maximum
---
decrease (%)
| Average
---
change (%)
| Median
---
change (%)
| Voltage
---
magnitude
4.4458 | -3.1886 | 0.3715 | 0.2501
VUF | 26446.6319 | -99.156 | 16.9229 | -9.5514
THDu | 91.2088 | -48.3703 | -11.6702 | -12.3757

Since the measures related to the COVID-19 pandemic in most countries are
seriously relaxed or even completely abandoned, the results related to the
improvement of PQ indicators due to demand time-shifting in the hard lockdown
period are no longer as relevant as they were a few months or years ago.
However, some end-users’ habits remain changed even after the hard lockdown
and in this paper we test if these habits affect consumption in a way that the
improvement of PQ after the implementation of the demand time-shifting
algorithm is still possible.

As it can be seen in Table 3, despite relaxing and abandoning the
COVID-19-related measures, end-users can still participate in the improvement
of PQ indicators. Even though the values of change are smaller when compared
to the potential of demand time-shifting in the hard lockdown period, trends
of the voltage increase caused by the demand rebound effect, the median
decrease VUF and THDu and outlier maximum value of VUF remain the same as in
the hard lockdown period.

Table 3: Change of PQ indicators during post-lockdown - time-shifted
consumption

|  | Maximum
---
increase (%)
| Maximum
---
decrease (%)
| Average
---
change (%)
| Median
---
change (%)
| Voltage
---
magnitude
2.8399 | -2.4235 | 0.2728 | 0.1811
VUF | 17230.1585 | -99.8613 | 7.7298 | -8.8347
THDu | 76.0736 | -46.7923 | -9.9796 | -10.1633

Improvement of PQ by implementing demand-shifting algorithm is achieved both
in the hard lockdown period but also in more recent post-lockdown period.
Although the improvement is less significant for the consumption curve in the
post-lockdown period, values of the observed changes show that the
implementation of the presented algorithm was not useful only during hard
lockdown but even in the case of relaxed and abandoned measures.

There are many other proposed approaches for the improvement of values of PQ
indicators, and many of them include the implementation of physical devices
such as voltage regulators, phase switching devices, or active and passive
harmonic filters. Even though their efficiency is high and they provide the
possibility to improve certain technical conditions without compromising the
comfort of end-users, their integration in a network often requires investment
creating additional and potentially unwanted costs for end-users. Based on the
results and the efficiency of the tested approach, the demand time-shifting
solution proposed in this paper can be considered as an alternative method for
mitigating disturbances in LV distribution networks. However, the proposed
method also requires the replacement of the metering equipment, the
development of control algorithms, and the installation of equipment needed
for receiving and answering signals for decreasing or increasing electricity
demand.

##  5 Conclusion

Together with the rapid integration of LC units, the changes in end-users’
habits, such as those exhibited during the pandemic lockdowns, present the
biggest challenge in the planning and operation of end-users that the DSOs
faced recently. To overcome these challenges, now more than ever, there is an
emerging need for the development of tools that can be used both in the short-
term and long-term planning but also in the everyday operation of distribution
networks affected by the penetration of LC units and long-term changes caused
by the pandemic.

One of such tools is presented in this paper, where we rely on functionalities
of open source technologies Python, SQL with the postgis extension, and QGIS,
and develop a GIS-based framework that is used for processing GIS and smart
meters data, creation of an LV network’s mathematical model, power quality
analyses, and finally, the visualisation of the simulation results. In the
first step, several typical errors in GIS and smart meters data are identified
and removed using Python and SQL with postgis extension so they can be used in
the second step, creating a mathematical model of a network. A mathematical
model of a network is created with GIS data containing both technical and
geospatial attributes and representing the elements of distribution networks
such as substations, lines, or electric switch cabinets. A mathematical model
is created with pandapower, the Python-based simulation tool. Pandapower is
also used in the third step, where it is used together with the harmonic
analysis extension, developed by the authors of this paper, for power quality
simulations. PQ indicators voltage magnitude, VUF, and THDu are calculated and
the average change of their value during the hard and post-lockdown periods
are compared to the values in the initial, pre-lockdown periods. The final
step of the developed tool includes a GIS visualisation with the QIGS open
source tool. The results show that on average, all PQ indicators deteriorated
during both pandemic periods, with larger disturbances in the hard lockdown
period.

According to calculated values, it is possible to conclude that increased
consumption during the pandemic presents a potential problem for the DSOs,
especially in the networks that are expected to integrate new LC units. To
overcome the potential issue, an algorithm that detects time periods most
suitable for the shifting of end-users’ consumption is implemented as an
extension of the presented framework. An algorithm creates a new consumption
curve, with minimally discomforting end-users. An implementation of time-
shifting of consumption successfully decreased PQ disturbances, with even
smaller deterioration of PQ indicators than in the pre-lockdown period.

The results and conclusions presented in the paper are valid for only one
analysed LV feeder and they are not relevant to the analysis of all LV
distribution networks in the area. However, the main contribution of the
developed GIS-based open source tool is the possibility given to a user who
can analyse a variety of different feeders or a whole network in order to draw
more general conclusions. The feeder in this paper is used as a proof of
concept and a test case used for verification of the developed tool. Moreover,
the conclusions related to the potential of end-users’ flexibility in the PQ
improvement are valid no matter the type of network, the only thing that
changes is the extent to which end-users can help in mitigating PQ
deterioration.

Future work includes further investigation of the potential of using both end-
users’ flexibility and utilization of physical devices, how they affect each
other’s operation, and how they affect the operation costs of DSOs caused by
PQ disturbances but also on the finances of end-users considering the
incentives for providing different services. The emphasis of future work is
planned to be on investigating the economic feasibility of integrating
different solutions for the improvement of PQ indicators, including a detailed
economic analysis of the ways through which end-users can be financially
encouraged to provide services that will lead to a safer and more reliable
operation of LV networks.

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# Scale-invariance underlying the logistic equation and its social
applications

A. Hernando1, A. Plastino2, 3 1 Laboratoire Collisions, Agrégats, Réactivité,
IRSAMC, Université Paul Sabatier 118 Route de Narbonne 31062 - Toulouse CEDEX
09, France
2National University La Plata, IFLP-CCT-CONICET, C.C. 727, 1900 La Plata,
Argentina
3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,
Spain

###### Abstract

On the basis of dynamical principles we derive the Logistic Equation (LE),
widely employed (among multiple applications) in the simulation of population
growth, and demonstrate that scale-invariance and a mean-value constraint are
sufficient and necessary conditions for obtaining it. We also generalize the
LE to multi-component systems and show that the above dynamical mechanisms
underlie large number of scale-free processes. Examples are presented
regarding city-populations, diffusion in complex networks, and popularity of
technological products, all of them obeying the multi-component logistic
equation in an either stochastic or deterministic way. So as to assess the
predictability-power of our present formalism, we advance a prediction,
regarding the next 60 months, for the number of users of the three main web
browsers (Explorer, Firefox and Chrome) popularly referred as “Browser Wars”.

###### pacs:

89.70.Cf, 05.90.+m, 89.75.Da, 89.75.-k

##  I Introduction

It is well-known that the logistic equation (LE) (sometimes called the
Verhulst model or logistic growth curve) is a phenomenological model of
population growth first published by Pierre Verhulst in the 1840’s. The model
is continuous in time, but a modification of the pertinent equation to a
discrete quadratic recurrence equation known as the logistic map is also
widely used. The continuous version of the logistic model for the evolution of
the population $x(t)$ is described by the differential equation

$\dot{x}(t)=kx(t)\left(1-\frac{x(t)}{N}\right),$ (1)

where $k$ is the Malthusian parameter (rate of maximum population growth) and
$N$ is the so-called carrying capacity (i.e., the maximum sustainable
population). The LE has as a solution

$x(t)=\frac{N}{1+(N/x(0)-1)e^{-kt}},$ (2)

i.e., the sigmoid function. The discrete version of the LE is the celebrated
logistic map. A typical application of the logistic equation refers to a
1838-model of population growth, originally due to Verhulst, in which the rate
of reproduction is proportional to both the existing population and the amount
of available resources, all else being equal. The Verhulst equation was
published after he had read Thomas Malthus’ An Essay on the Principle of
Population. Verhulst derived his logistic equation to describe the self-
limiting growth of a biological population. Today, proper referencing to the
logistic equation’s variegated applications to multiple fields of endeavor
would require pages and pages of citations. Of this immense body we just
mention, as a tiny sample, 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 .

Some ad-hoc LE-deductions have been previously published in a case-by-case
basis. One such demonstration is that provided by A. D. Zimm for companies or
firms zimm , that grow according to their commercial successfulness with a
classic linear Marshallian price-volume relationship. Other analytical
derivations are also found with some ad-hoc assumptions6 ; toston . Out
present goal is to describe an universal and generic dynamical mechanism that
leads in natural fashion to the logistic equation. The above cited derivations
do not have a dynamical character, as ours does. Our procedure is based on

  * •

_scale symmetry_ and

  * •

a _mean-value constraint_.

We will show that these are necessary an sufficient conditions for a LE-
derivation. The two items above are empirically known to be related to the LE
zimm ; 6 ; toston  but they are used here for the first time as its pure
mathematical basis.

##  II Derivation from dynamical principles

Consider an $n-$components system, each of them characterized by a population
$x_{i}$. Let us further assume that a multiplicative, time-evolution of
population takes place via free proportional growth, i.e.,

$\dot{x}_{i}(t)=k_{i}(t)x_{i}(t),$

where $k_{i}$ is the growth-ratio per unit-time for the $i$-th component.
Scale-symmetry is here apparent so that it proves convenient to transform
coordinates to $u_{i}=\log(x_{i})$ as in Refs. prl, ; epjb, . Thereby one is
led to the linear equation

$\dot{u}_{i}(t)=k_{i}(t),$

where the scale-invariance of $x$ is now a translational invariance in $u$. We
assume that the total population is finite, namely

$\sum_{i=1}^{n}x_{i}(t)=\sum_{i=1}^{n}e^{u_{i}(t)}=N,$

and that also $n$ remains constant, so that $\langle x\rangle=N/n$. For each
arbitrarily small time-interval $\Delta t$, the $u-$population grows freely
via

$u^{\prime}_{i}(t+\Delta t)=u(t)+\Delta tk_{i}(t),$

but conservation of the mean value $\langle e^{u}\rangle$ demands a global
“self-correction” process that should respect the original symmetry of the
system (translational for $u$). Accordingly,

$u_{i}(t+\Delta t)=u_{i}^{\prime}(t+\Delta t)+A,$

where $A$ is a value that guarantees fulfillment of
$\sum_{i=1}^{n}e^{u_{i}(t+\Delta t)}=N$. This is achieved if

$A=-\ln\left[\frac{1}{N}\sum_{j=1}^{n}\,e^{u_{j}^{\prime}(t+\Delta
t)}\right]=-\frac{\Delta t}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)},$

where a Taylor-expansion to first order is justified since $\Delta t$ is
arbitrarily small. One is then led to

$u_{i}(t+\Delta t)=u_{i}(t)+\Delta
t\left[k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)}\right].$

In the continuum-limit one finds

$\dot{u}_{i}(t)=k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)e^{u_{j}(t)},$

that written in $x-$parlance leads to what we call the _multi-component
logistic equation_ (MCLE)

$\dot{x}_{i}(t)=x_{i}(t)\left(k_{i}(t)-\frac{1}{N}\sum_{j=1}^{n}k_{j}(t)x_{j}(t)\right).$
(3)

A matrix version of this equation is presented in the Appendix, together with
with some applicability perspectives. It is easy to check that the MCLE
retains the original scale-symmetry of $x$, and also exhibits translational
symmetry in $k$. The latter allows for some arbitrariness in the definition of
the $k_{i}$ rates in the fashion $k^{\prime}_{i}=k_{i}-k_{0}$. The same
results obtain for primed or unprimed $k$’s. If the $k_{i}$ are constant, or
exhibit just a slow dependence on $t$ (quasi-statics), the solution to the
MCLE is

$x_{i}(t)=\frac{Nx_{i}(0)e^{k_{i}t}}{\sum_{j=1}^{n}x_{j}(0)e^{k_{j}t}},$ (4)

where $x_{i}(0)$ are the initial conditions of the evolutive-process. The
logistic equation is directly derived from the MCLE Eq. (3) in a
straightforward fashion. We recover Eq. (1) by i) considering a bi-component
system ($n=2$), ii) defining $x(t)\equiv x_{1}(t)$, $k\equiv k_{1}-k_{2}$, and
iii) taking into account that $x_{2}(t)=N-x_{1}(t)$. Similarly, the sigmoid
function Eq. (2) is recovered from Eq. (4) with the same assumptions. The
second component acts here as a population-reservoir and the first component
becomes the only evolutive degree of freedom.

##  III Possible physical regimes

According to the nature of the growth-ratios $k_{i}$, variegated kinds of
processes can be described by the MCLE.

  1. 1.

Constant values or deterministic time-dependencies lead to mechanical systems
exhibiting deterministic evolution while

  2. 2.

adding noise to the pertinent mean values gives rise to stochastic systems,
with interesting behaviors and applications.

Without aiming to be exhaustive, we consider here three different tableaus for
MCLE-applicability, according to the amount of ‘noise’ in the system: totally
stochastic (or thermodynamic regime), an intermediate level of randomness
(involving diffusive processes), and totally deterministic dynamics.

###  III.1 Thermodynamic regime

Consider a multi-component case with dozens or hundreds of elements, and a
very high level of noise. Assume that each $k_{i}$ describes the derivative of
a Wiener process. We write $k_{i}(t)=\overline{k}_{i}+\sigma_{i}\xi(t)$, where
$\overline{k}_{i}$ is the time-average of $k_{i}$, $\sigma_{i}$ the standard
deviation measured in a certain interval $\Delta t$, and $\xi(t)$ an
independent normal-distributed random number. Defining
$\langle\sigma^{2}\rangle=\sum_{i=1}^{n}\sigma_{i}^{2}/n$, one asserts that we
thermalize the system if i)
$|\sigma_{i}-\sigma_{j}|^{2}/\langle\sigma^{2}\rangle\ll 1$ and ii)
$|\overline{k}_{i}-\overline{k}_{j}|^{2}/(\langle\sigma^{2}\rangle\Delta t)\ll
1$, $\forall i,j$, i.e., if all elements exhibit similar deviations and the
differences between mean values are much smaller than the noise. If $n$ is
large enough (as stated above, in the hundreds), dynamical equilibrium is
encountered after some finite time, meaning that the system is well-described
by the MaxEnt approach. The MaxEnt solution for scale-free systems describes
an equilibrium density $p_{X}(x)$ that follows the general form prl

$p_{X}(x)dx=\exp\left[-\sum_{a}f_{a}(x)\right]\frac{dx}{x},$

where $f_{a}(x)$ is the $a$-th constraint of the system. For a constraint in
the normalization ($n$ is invariant) we have $f_{\langle 1\rangle}(x)=\mu$,
and for one in the mean value we write $f_{\langle x\rangle}(x)=\lambda x$,
where $\mu$ and $\lambda$ are constants, univocally determined by the
fulfillment of each constraint. The density-distribution obeys in this case
the relation epjb

$p_{X}(x)dx=\frac{1}{\Gamma(0,\lambda x_{0})}\frac{e^{-\lambda
x}}{x}dx;\leavevmode\nobreak\ \leavevmode\nobreak\ x_{0}\leq x<\infty,$

where $\Gamma$ is the incomplete gamma function and $x_{0}$ is the smallest
allowed population for the elements (that can be 1). The rank-distribution is
then written as

$x=\frac{1}{\lambda}\Gamma^{-1}\left[0,\Gamma(0,\lambda x_{0})r/n\right],$ (5)

where $r$ is the (continuous) rank from 0 to $n$, and $\Gamma^{-1}(z)$ denotes
the inverse function of $\Gamma:$ $\Gamma(\Gamma^{-1}(z))=z$. The value of
$\lambda$ is obtained from the mean value

$\frac{e^{-\lambda x_{0}}}{\lambda\Gamma(0,\lambda x_{0})}=\frac{N}{n}.$ (6)

In order to test the above relationships we have carried out a calculation
that simulates human population dynamics. We also compare the result with
empirical data regarding city and place-populations, using as example data
from Ohio State (United States) ohio . We consider $n=1000$ random walkers
that mimic the population of cities, and set the total population to
$N=6000000$. We fix the minimum allowed size at the Dunbar’s number epjb
$x_{0}=150$ people (empirically known to be the usual size of small human
communities, and related to the maximum of social relationships/links that a
human can comfortably handle). We make the walkers to stochastically obey the
MCLE Eq. (3) so as to simulate migration patters between cities, with a
constant total population. We evolve the walkers in small intervals $\Delta
t=0.03$ and generate gaussian-distributed random numbers at each interval for
each $k_{i}$, using $\sigma_{i}=1$ for all $i$. Due to the translational
symmetry in $k$, we can set $\overline{k}_{i}=0$ for all $i$. To respect the
minimum size, a walker’ ‘move’ is not accepted if it leads to a value lower
than the Dunbar’s one. We start all walkers at $x_{i}=N/n$. After some
iterations we get the equilibrium distribution of Fig. 1, which perfectly fits
the MaxEnt prediction ($\lambda=0.00533$ humans-1). The available empirical
data covers years 2010, 2000 and 1990 with $n=1204$, $1015$, and $928$ cities
and places, respectively. We discard places with populations of under 150
people (67, 48 and 39 centers respectively). We also discard very large cities
(their potential economic correlation with the rest of the country compromises
the hypothesis of isolated systems). Excluding the four largest cities, the
total population is $N=6318170$, $6019960$, and $5477830$, respectively. We
have checked the proportional growth condition comparing $\log(x_{i})$ vs.
$|\dot{x}_{i}/x_{i}|^{2}$ (or equivalently, $u_{i}$ vs. $|\dot{u}_{i}|^{2}$).
No correlation between these two observables is expected for scale-invariance.
We have found a correlation coefficient of $0.0018$, as shown in Fig. 1, thus
confirming the proportional growth hypothesis (the same correlation
coefficient in the precedent simulation is $0.0027$). According to Eq. (6),
the predicted values of $\lambda$ are $0.00585$, $0.00507$, and $0.00513$,
respectively. A direct fit of the data to the form Eq. (5) yields
$\lambda=0.00636(2)$, $0.00502(3)$, and $0.00522(3)$, respectively, close
enough to the former values so as to confirm the MaxEnt prediction.

Figure 1: Top panel: behavior of random walkers obeying the stochastic MCLE
(blue dots) compared with the associated MaxEnt prediction (black line).
Inset: walkers’ squared-relative-growth $|\dot{u}_{i}|^{2}$ vs. logarithmic
size $u$. There is no correlation in view of the linear regression (red line).
Bottom panel: Rank-distributions for Ohio (red: year 2010; green: year 2000;
blue: year 1990) and rank-distribution of random walkers obeying the
stochastic MCLE (yellow) compared with the corresponding MaxEnt predictions
(black lines). Inset: same as upper panel with the empirical data. In view of
the regression line, no correlation is detected.

###  III.2 Intermediate regime

Let us pass now to consideration of an intermediate stochastic regime in the
bi-component case. One party acts as a population-reservoir while the other
obeys Eq. (1). Defining the new variable

$y(t)=-\log(N/x(t)-1),$ (7)

the LE linearizes itself for $\dot{y}(t)=k$. The rate $k$ is here again the
derivative of a Wiener process $k(t)=\overline{k}+\sigma\xi$, but we now
include a drift obeying $|\overline{k}|^{2}>\sigma^{2}\Delta t$. Working with
an ensemble of independent walkers following this equation is equivalent to
handling Brownian motion in $y$-space. Consequently, the usual diffusion
equation for the density of walkers for $p_{Y}(y,t)$ ensues BM

$\partial_{t}p_{Y}(y,t)=-\overline{k}\partial_{y}p_{Y}(y,t)+D\partial_{y}^{2}p_{Y}(y,t),$

where $D$ is the diffusion-coefficient, related to $\sigma_{k}$ via
$\sigma_{k}=\sqrt{2D/\Delta t}$, and $\Delta t$ stands for the time-interval
used in the random-walk numerical simulation. The diffusion-equation’s kernel
is a Gaussian

$p_{Y}(y,t)dy=\frac{1}{\sqrt{4\pi
Dt}}e^{-\frac{(y-y_{0}-\overline{k}t)^{2}}{4Dt}}dy,$

with $y_{0}$ a reference-value. If at $t=0$ all walkers are located in
$x$-space at $x_{0}=N/(1+e^{-y_{0}})$, they will later evolve via

$p_{X}(x,t)dx=\frac{1}{\sqrt{4\pi
Dt}x(1-x/N)}e^{-\frac{(\log(N/x-1)-y_{0}+\overline{k}t)^{2}}{4Dt}}dx,$ (8)

since $p_{X}(x,t)dx=p_{Y}(y,t)dy$.

We have verified this prediction with a diffusion process taking place inside
a scale-free ideal network (SFIN) ournets , a random network with a degree-
distribution following the scale-free ideal gas one $p(c)\sim c^{-1}$, where
$c\leq c_{M}$. We have generated a SFIN of $N=20000$ nodes with a maximum
degree of $c_{M}=100$-connections and carried out a multitude of cluster-
growth processes ournets ; clusters . Diffusion in networks generally starts
i) by using a randomly chosen node as a seed, ii) with its first neighbors
being added to the cluster in the first iteration, iii) and the neighbors of
those “first" neighbors, afterwards (and so on). The process ends when all
nodes of the network belong to the cluster. The size of the cluster at each
iteration depends on the particular node selected as seed, via its position
inside the network. We depict in Fig. 2 the result of a large number of these
processes, indicating the size of the cluster at each iteration. All of them
start with $x=1$ and end up with $x=N$, but processes exhibit deviations at
intermediate steps. The associated median clearly follows a logistic
evolution, as shown in Fig. 2. Changing to the variable $y$ of Eq. (7) we find
a straight line with slope $\overline{k}=3.09$. The deviations can be
described by $y$-random walkers with $\sigma=0.7$ ($\Delta t=1$). The
statistics of the processes can be nicely described with Eq. (8) via
$x_{0}=1$, $D=0.245$, and with the above value of $\overline{k}$, as
illustrated by the comparison depicted in Fig. 2.

Figure 2: Diffusion inside a scale-free ideal network (dots) compared with the
analytical one provided by Eq. (8), derived from the logistic equation.

###  III.3 Deterministic regime

We study now the deterministic evolution of a multi-component system with few
elements and very low level of (or without) noise. Assume that the growth-
ratios $k_{i}$ are now constants or represent a quasi-static evolution in
time. Assume further that we have data about the temporal population-evolution
but do not known the explicit values (or the tendency) of the $k_{i}$ rates.
These can be easily obtained from the solution of MCLE Eq. (3) taking
advantage of its property of translational symmetry in $k$. By arbitrarily
setting $k_{1}=0$, all the remaining values are obtained from the population
data thanks to the functions

$h_{i}(t)=\log\left[\frac{x_{i}(0)}{x_{i}(t)}\frac{x_{1}(t)}{x_{1}(0)}\right].$
(9)

If the growth-ratios are constants, $h_{i}(t)=k_{i}t$, the entire evolution-
path can be predicted (for any arbitrary time). If our functions $h_{i}(t)$
exhibit small time-fluctuations we can parameterize them, via a fitting
procedure, to any given analytical form. The desired solution is obtained by
substituting the arguments in the exponentials of Eq. (3) by these functions
$k_{i}t\rightarrow h_{i}(t)$. We have tested this last statement using data
regarding web-browsers’ statistics so as to study the past and future of the
(popularly called) _Browser Wars_ wikiBW . We considered the $n=3$ system
composed of Microsoft Explorer (E), Mozilla Firefox (F), and Google Chrome
(C). Our analysis of the popularity of each uses data from _w3schools_ w3  and
_statcounter_ sc  (depicted in Fig. 3).

We take $N=100$% and choose the origin $t=0$ at March 2012 (as this
communication was being written). Setting $k_{E}=0$ we have applied Eq. (9) to
the data, finding a small dependence on time in both $k_{F}$ and $k_{C}$. We
show in Fig. 3 that the functions $h_{i}(t)$ can be nicely fitted to a simple
exponential form $h(t)=ae^{-bt}t+c$ (that can be regarded as a pure
exponential time-dependence of $k$ plus a correction on the initial value
$x(0)$ via $e^{c}$). Note that the small fluctuations of the data become more
apparent as we approach the reference point at $t=0$. However, our accuracy
remains sufficiently high for our purposes. We obtain

$\begin{array}[]{rl}h_{F}(t)=&0.0074(5)\exp[-0.021(1)t]t-0.008(14)\leavevmode\nobreak\
\mathrm{and}\\\ h_{C}(t)=&0.0579(24)\exp[-0.0097(10)t]t+0.104(26),\end{array}$

for _w3schools_ , and

$\begin{array}[]{rl}h_{F}(t)=&0.0022(5)\exp[-0.043(6)t]t+0.026(13)\leavevmode\nobreak\
\mathrm{and}\\\ h_{C}(t)=&0.055(2)\exp[-0.015(1)t]t+0.015(27),\end{array}$

for _statcounter_ , that are compared in the top panels of Fig. 3 to empirical
data. Our predictions for the popularity evolution are evaluated using Eq. (4)
and substituting $k_{E}t$, $k_{F}t$, and $k_{C}t$ by the above described
extrapolations of $h_{E}$, $h_{F}$, and $h_{C}$. A proper correction is
finally added in the later case to improve the fitting by using
$N^{\prime}=1.03N$. We depict in Fig 3 our monthly prediction for the next 5
years regarding browsers’ usage. In the two reported instances, Google Chrome
grows till coming ahead in the competition, saturating effects being
noticeable at 80% and 60%, respectively. Thus, according to our prediction,
Google Chrome will win the competition but it will not acquire such a dominant
position as the MS Explorer attained in the past.

Figure 3:  Left panels: data from _w3schools_. Right panels: data from
_statcounter_. Top panels: increment rate of M Firefox (red squares) and G
Chrome (yellow triangles) defined as $(h(t)-c)/t$ relative to MS Explorer (see
text), compared with the analytical fit (solid lines). Bottom panels: relative
users of MS Explorer (blue circles), M Firefox (red squares) and G Chrome
(yellow triangles), compared with our prediction (lines).

##  IV Conclusions

We have been able here to demonstrated that scale-invariance and a mean-value
constraint are sufficient and necessary conditions for obtaining the LE from
first dynamical principles. Then, the LE was generalized to multi-component
systems. This allowed us to show that these dynamical mechanisms underlie
interesting scale-free processes, which was illustrated with reference to
city-populations phenomena, diffusion in complex networks, and popularity of
Net Browsers.

##  Appendix A Generalization of the multi-component logistic equation to a
matrix equation

We generalize here the formalism discussed above. If we define a new set of
variable $\chi_{i}=\sqrt{x_{i}/N}$, the total-population’s constraint can be
recast in the fashion $\sum_{i=1}^{n}\chi_{i}^{2}=1$. This condition becomes
formally equivalent to the conservation of the modulus of a vector
$\mathbf{\chi}=\\{\chi_{i}\\}_{i=1}^{n}$ in a $n$-dimensional space. We can
also generalize the definition of the growth-ratios $k_{i}$, promoting them to
a matrix $K_{ij}=k_{i}\delta_{ij}$, and write the MCLE as a matrix equation.
Using bra-ket notation $\mathbf{\chi}=|\chi\rangle$ one has

$\partial_{t}|\chi\rangle=\frac{1}{2}\left\\{K-\frac{\langle\chi|K|\chi\rangle}{\langle\chi|\chi\rangle}\right\\}|\chi\rangle.$

This equation is formally identical to that used in quantum physics to find
the ground state wave-function of a Hamiltonian (here, $K$). We are speaking
of the Imaginary Time Method (ITM) widely used in the literature ITM . The
mean-value term is also used to guarantee the conservation of the
normalization of $|\chi\rangle$ during the process. All our examples can be
regarded as particular applications of this formalism, calling attention to
the “functional" definition of our effective ‘Hamiltonians’ $K$. In our
examples $K$ has a diagonal form, which only indicates that we were working in
the eigenbasis of the dynamics defined by $K$. A generalized definition can
include off-diagonal terms as well, indicative of some kind of interaction
between populations. Density functional theories (DFT) also use the above
equation for many-body quantum systems DFT . A phenomenological Hamiltonian is
defined by means of a parametric functional form, than can also be a
functional of the own state-vector $\chi$. The associated parameters are
chosen so as to reproduce well-known empirical facts regarding the system of
interest. We expect that the bridge we have here built up between the MCLE and
the DFT can open new vistas with respect to the possibility of studying scale-
free systems. Such approach would take advantage of the huge experience
accumulated regarding DFT methods.

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