![(Normalized) total power and electric water heater demand in a representative workweek (11 November to 15 November 2019) in Uruguay. Data from the ECD-UY dataset [6].](https://www.researchgate.net/publication/355709535/figure/fig1/AS:11431281349736770@1743751948191/Normalized-total-power-and-electric-water-heater-demand-in-a-representative-workweek_Q320.jpg)
Access to this full-text is provided by MDPI.
Content available from Applied Sciences
This content is subject to copyright.
applied
sciences
Article
A Thermal Discomfort Index for Demand Response Control in
Residential Water Heaters
Rodrigo Porteiro 1,*,† , Juan Chavat 2,† and Sergio Nesmachnow 2, *,†
Citation: Porteiro, R.; Chavat, J.;
Nesmachnow, S. A Thermal
Discomfort Index for Demand
Response Control in Residential
Water Heaters. Appl. Sci. 2021,11,
10048. https://doi.org/10.3390/
app112110048
Academic Editor: Amjad
Anvari-Moghaddam
Received: 30 June 2021
Accepted: 6 September 2021
Published: 27 October 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Administración Nacional de Usinas y Transmisiones Eléctricas, Montevideo 11300, Uruguay
2
Computer Science Institute, Engineering Faculty, Universidad de la República, Montevideo 11300, Uruguay;
juan.pablo.chavat@fing.edu.uy
*Correspondence: rporteiro@ute.com.uy (R.P.); sergion@fing.edu.uy (S.N.)
† These authors contributed equally to this work.
Featured Application: The methodology described in this article is applicable to design proper
management strategies for demand response in smart electricity grids to fairly select water heaters
to intervene while guaranteeing the lower discomfort of users.
Abstract:
Demand-response techniques are crucial for providing a proper quality of service under
the paradigm of smart electricity grids. However, control strategies may perturb and cause discomfort
to clients. This article proposes a methodology for defining an index to estimate the discomfort
associated with an active demand management consisting of the interruption of domestic electric
water heaters. Methods are applied to build the index include pattern detection for estimating
the water utilization using an Extra Trees ensemble learning method and a linear model for water
temperature, both based on analysis of real data. In turn, Monte Carlo simulations are applied
to calculate the defined index. The proposed approach is evaluated over one real scenario and
two simulated scenarios to validate that the thermal discomfort index correctly models the impact
on temperature. The simulated scenarios consider a number of households using water heaters
to analyze and compare the thermal discomfort index for different interruptions and the effect of
using different penalty terms for deviations of the comfort temperature. The obtained results allow
designing a proper management strategy to fairly decide which water heaters should be interrupted
to guarantee the lower discomfort of users.
Keywords:
demand response; smart grid; discomfort index; water heaters; thermal model;
computational
intelligence
1. Introduction
Energy demand management is a crucial idea for the modern paradigm of smart
cities. The concept of smart electricity networks, or smart grids, refers to electrical grids
enhanced by including operation and management features to improve the controlling of
production and distribution of energy [
1
]. Smart grids are mainly oriented to maintain
a reliable and secure infrastructure to allow properly satisfying the demand growth, the
integration of distributed energy resources, smart storage, and other features related to
smart devices and real-time information provided to clients [
2
]. Information and Commu-
nication Technologies (ICT) are very closely connected to smart grids, as they provide the
basis for communicating and processing information that is very useful at different levels
to implement the aforementioned services [3].
The daily consumption pattern of electricity involves periods with higher-than-
average electricity consumption (peaks) and other periods with lower-than-average electric-
ity consumption (valleys). A common situation in the daily operation of electric grids is
that electricity generation and transmission systems may not always meet peak demand
Appl. Sci. 2021,11, 10048. https://doi.org/10.3390/app112110048 https://www.mdpi.com/journal/applsci
Appl. Sci. 2021,11, 10048 2 of 21
requirements. In these situations, power demand management strategies are helpful tools
for operation of electric grids.
Power demand management refers to the proper administration of power consump-
tion for end consumers in a smart grid in order to promote better energy utilization. Two
of the most widely applied actions for power demand management are load management,
whose main goal is modifying, reducing, or shifting the demand, and energy conservation,
which is mainly focused on reducing the demand, e.g., via technological improvements.
In turn, several other actions have been applied for demand management, including fuel
substitution and load building [
4
]. Among load management techniques, the most used are
peak reduction, oriented to reduce power consumption in periods of maximum demand;
valley filling, whose main goal is to promote energy utilization in off-peak periods,; and
load shifting from peak to off-peak periods.
Several demand response and demand management tools can be applied to mitigate
overloads in the electrical system. One of the simplest yet most effective methods for direct
load control is allowing the electric company to remotely control user devices. This method
is properly applied to control those devices with a thermostat, especially those that have
important thermal inertia. On the one hand, remote control is a very effective technique
to achieve peak reduction and load shifting at critical periods. On the other hand, the
benefits of the achieved reduction of energy consumption in the overall operating cost of
the electrical system must be weighted against the loss of comfort that the users of the
controlled devices may have. In order to define an economic value to the loss of comfort
associated with an intervention, to be taken into account in the business model of the
electric company, the discomfort of users must be properly evaluated with quantifiable
metrics in advance.
In this line of work, this article proposes and evaluates a methodology to calculate
a Thermal Discomfort Index (TDI) associated with a remote intervention for load man-
agement from the electric company. The computed index evaluates the discomfort for
users, generated by the intervention of electric water heater appliances. The main goal of
the index is to be a valuable tool for defining a method to manage a set of water heaters
to defer electrical demand by properly identifying those appliances that have the lower
impact on user comfort to be interrupted first. This way, the values of the computed TDI
make it possible to decide in which order water heaters should be interrupted to minimize
the overall total discomfort of a set of users. The key aspect of the proposed strategy is to
know in advance the value of TDI in order to decide if it is economically profitable to carry
out an intervention.
The proposed methodology for computing TDI is based on real data, a linear tem-
perature model, and a forecasting model for water utilization applying ensemble learning
and Monte Carlo simulation. The proposed TDI was developed and evaluated with data
from the electrical system in Montevideo, Uruguay. This is a relevant case study since
electricity is the main resource used for heating and other domestic activities in Uruguay,
far superior to natural gas and other sources. In Montevideo, as in other main Uruguayan
cities, more than 90% of households have a thermostat-controlled electric water heater
(according to the 2019 household survey by National Statistics Institute [
5
]). Since the
electric water heater is one of the most energy-intensive household appliance (accounting
for 34% of residential energy consumption, on average), it is an ideal candidate for remote
load control as a demand management technique.
The experimental analysis was performed both on real and simulated scenarios built
using data from real electric water heaters in Uruguay, gathered in the ECD-UY dataset [
6
].
ECD-UY includes utilization of time series data about power consumption of several appli-
ances, including water heaters, and aggregated consumption for representative households
in the main Uruguayan cities.
Results reported for the considered case studies demonstrate that the proposed index
managed to capture the impact of thermal discomfort, fulfilling the goal of sorting electric
water heaters to be properly managed by applying a direct control strategy, and allows spe-
Appl. Sci. 2021,11, 10048 3 of 21
cial cases to be defined where a particular water heater is required to never be interrupted
for security reasons.
This article extends our previous conference article “Demand response control in elec-
tric waterheaters: evaluation of impact on thermal comfort” [
7
], presented at III Iberoamer-
ican Congress on Smart Cities. The main contributions beyond those of the previous
conference article include: (i) an extension of the developed temperature model for electric
water heaters, only measuring the electrical state of the device; (ii) an improved procedure
to predict water utilization by applying data analysis to real electricity consumption data
from the ECD-UY dataset; (iii) the definition of an index to approximate the discomfort
associated with an active demand management interruption of the water heater; and (iv)
an extended evaluation of the proposed methodology for several cases of active demand
management interruption of water heaters over different realistic scenarios.
The article is organized as follows. Section 2presents the formulation of the demand
management problem through direct control of electric appliances. Section 3reviews
related works. Section 4describes the proposed approach to define a TDI for direct
control of electric water heaters. Section 5reports the experimental validation of the water
utilization forecasting, the temperature model, and the proposed TDI for realistic case
studies. Finally, Section 7presents the main conclusions and lines for future work.
2. Demand Management and Direct Control of Electric Water Heaters
This section presents the main concepts of demand management strategies. In particu-
lar, direct load control strategies applied to load shifting are described, and the problem of
affecting comfort of the end-user is discussed.
2.1. Demand Management and Direct Load Control
The traditional model of an electrical system supplies electricity to end consumers
through a unidirectional flow of energy, which is delivered by centrally controlled genera-
tors. However, in the last thirty years, power grids all over the world became decentralized
systems, thus resulting in distributed energy resources that have fostered new business
models and specific transformations of energy markets. The concept of energy demand
management emerged within these new business models.
Energy demand management involves a set of techniques oriented to modify the
energy demand of consumers of an electric grid to fulfill specific goals [
8
]. The subset of
techniques oriented to reduce the energy demand of consumers in the short term are known
as demand response methods. A specific technique within demand response methods is
direct load control. The main idea of direct load control is to provide the energy company the
permission to control (i.e., switch off) the devices of end-users, which is usually obtained
via specific agreements that grant users a monetary incentive. Load control is considered
an effective technique to achieve immediate power reduction in a very short time and is
very useful to deal with peak reduction needs [
9
] in order to obtain a more stable grid
operation. Another situation where load control methods are very useful is to provide
frequency regulation services [
10
], with the main goal of maintaining the system frequency
close to the utility frequency (i.e., the nominal oscillation frequency of alternating current,
60 Hz in the Americas and Asia, and 50 Hz in other sites), preventing deviations that affect
generators and also make the grid unstable. This article addresses some aspects related
to the direct load control of electric water heaters, mainly focusing on the impact of using
direct load control in the thermal comfort of end-users.
2.2. Load Shifting by Direct Control of Electric Water Heaters
In most countries, the profile of total electricity consumption shows two pronounced
peaks: early in the morning, before starting the working day, and near two hours after
people return to their homes from work. Usually, the power consumption of electric water
heaters has a high correlation with the total consumption: both present coincident peaks,
as shown in Figure 1, for a representative analysis using data from the electric water heater
Appl. Sci. 2021,11, 10048 4 of 21
consumption subset, part of the ECD-UY dataset. This correlation is explained by the high
impact of water heaters compared with other domestic appliances. For example, in the
case study analyzed in this article (Uruguay), water heaters account for 27% of the total
residential electricity consumption, on average, peaking at 35%, according to the subset
data. A specific feature of water heaters is that they have the ability to accumulate energy
in the form of heat inside the water tank. This thermal inertia allows proper planning of
switch on/switch off periods to help the grid operation while trying to affect the thermal
comfort of users as little as possible.
According to the correlation between total energy consumption and water heaters’
energy consumption, the amount of energy associated with electric water heaters within
a demand peak can be deferred by switching off the devices in a proper moment and
switching them on in the future. This strategy allows implementing a load shifting on the
demand curve, since the total amount of energy consumed in the period remains equal
but the load profile is modified. Several studies have addressed the load shifting problem
using direct control of devices [
11
,
12
], but few articles have focused on quantifying the
thermal discomfort generated by the real application of this strategy.
Figure 1.
(Normalized) total power and electric water heater demand in a representative workweek
(11 November to 15 November 2019) in Uruguay. Data from the ECD-UY dataset [6].
2.3. Problem Formulation
The problem proposes determining an index for the quantitative evaluation of the
discomfort generated by the application of load shifting using a direct control of electric
water heaters. Two main challenges must be addressed to solve the problem: determining
proper variables for the analysis and, since no-deterministic behavior is involved, proposing
an effective methodology to estimate their values.
The first step is evaluating relevant variables for the problem. In the case of electric
water heaters, the main variables affecting the comfort of the users is the temperature of the
water in the tank and variables that determine the utilization pattern of water heaters. Re-
garding user comfort, installing a remote device to the water heater that allows measuring
its power and switching it off or on is reasonable, since it can be achieved via specific Inter-
net of Things controllers without modifying the water heater
structure [13,14]
. However,
installing a remote thermometer to measure the temperature of the tank requires modifying
the structure of the water heater, which implies a significant monetary investment. Thus,
non-intrusive methods [15] are preferred for the analysis.
The main goal of defining a proper direct control strategy is to determine the dis-
comfort introduced by the same controlling action among several electric water heaters to
decide which of them intervene and minimize the probability of generating discomfort.
Therefore, it is not crucial to compute the temperature exactly, but a good approximation
is enough for analyzing differences. Another relevant aspect for the analysis is that the
user perceives thermal discomfort only when using water and the temperature is below a
given threshold. At any other time, or if the water temperature is still above the threshold
despite the intervention, users are not aware of the switching-off actions.
Two variables must be estimated to define a discomfort index in the event of an
intervention: the water utilization and the temperature of the water in the tank. By
estimating these two variables, the proposed TDI takes into account the period of time
Appl. Sci. 2021,11, 10048 5 of 21
that the water temperature is below the comfort temperature for a person using the water.
This way, the proposed TDI properly models the human discomfort associated with an
intervention.
3. Related Works
This section reviews relevant related works about load reduction techniques and
thermal discomfort evaluation.
Several works in the literature have described the main concepts of peak load reduction
strategies and specific implementations in smart grids [
9
,
16
]. In particular, direct load
control allows utilities to remotely manage electricity demand by modifying the operation
of end-use devices to perform load shifting [
11
,
17
]. A special type of devices to perform
load shifting are thermostat-controlled appliances (TCA). The main feature of TCA is that
they provide flexibility to select the desired temperature, i.e., the thermostat set point [
18
,
19
].
In addition, some TCAs can also store energy in the form of heat, providing a great
advantage over non-storage appliances when performing load shifting strategies. This is
the case of domestic electric water heaters, which are the focus of this article.
Nehrir et al. [
20
] proposed and analyzed an interactive demand-side management
strategy for electric water heaters, but the research did not elaborate on specific aspects
of thermal discomfort of users. A recent article by Xiang et al. [
21
] introduced a complex
strategy to minimize thermal discomfort related with electric water heater control. The
proposed strategy requires information to build a time-varied weight matrix based on
detected utilization patterns of domestic electric water heaters. In turn, the weight matrix
and other information are used to generate a customer satisfaction prediction index. Due
to the large amount of information required to build the model, the approach is difficult to
apply in practice.
Demand response control strategies for TCA can be effectively implemented provided
that thermal comfort is not compromised. Thus, quantifying the impact on user comfort is
a crucial aspect, and it has been the focus of several works. Kampelis et al. [
22
] evaluated
the thermal discomfort in demand response control of several appliances used for heating,
ventilation, and air conditioning in a university building. A daily discomfort score was
proposed for demand response events to reduce the cost of energy, but the proposed
strategy requires knowing the real temperature. Regarding electric water heaters, the study
of the hot water utilization profile by end-users is a key aspect for estimating discomfort.
Tabatabaei and Klein [
23
] studied whether a smart heating system can benefit from good
predictions of the user behavior. Pirow et al. [
24
] proposed an algorithm for estimating
domestic hot water utilization, but the technique requires the installation of temperature
and vibration sensors.
The main factor that defines comfort is the water temperature. Thus, a model to
estimate water temperature from measured information is crucial for the effectiveness
of the comfort evaluation. Paull et al. [
25
] proposed a water heater model to estimate
the temperature of the water in the tank as a function of time and the related variables,
including the thermal losses and the water utilization. Data from smart meters, recorded
at 15 min intervals, were used for validation. The model was proposed to be applied in a
multiobjective demand-side management program. The study by Lutz et al. [
26
] provided
a comprehensive empirical analysis of a simplified energy consumption model for water
heaters considering the variation of the temperature of the water in the tank. Results of the
proposed Water Heater Analysis Model (WHAM) model were compared with data from
water heaters simulation programs (TANK, WATSIM, and WATSMPL). WHAM obtained
very accurate prediction results, while being significantly faster. In addition, WHAM
requires less detailed engineering information about the water heaters. Finally, comfort
evaluation must be considered in the problem of controlling a subset of electric water
heaters, e.g., by building a ranking to sort all interruptible devices according to appropriate
criteria. Yin et al. [
27
] proposed a scheduling strategy based on a temperature state priority
Appl. Sci. 2021,11, 10048 6 of 21
list. In turn, Al-Jabery et al. [
28
] analyzed a scheduling strategy for electric water heaters
based on approximate dynamic programming techniques and q-learning.
The analysis of related works allows concluding that few articles have studied the
thermal comfort effect when applying direct load control of electric water heaters. Existing
approaches are based on installing specific sensors or devices to monitor water temperature
or on building sophisticated algebraic formulations for modeling utilization patterns. This
article contributes in this line of works by proposing an approach to evaluate the ther-
mal discomfort of an intervention on electric water heaters, without requiring installing
additional devices (e.g., a thermometer to measure the water temperature). Instead, the pro-
posed model follows a non-intrusive approach, applying data analysis and computational
intelligence and demonstrates its effectiveness on realistic problem instances.
4. The Proposed Approach for Defining a Discomfort Index
This section describes the proposed approach for defining a discomfort index for
demand response via direct control of water heaters. The approach applies ideas described
in the previous sections and follows a data analysis approach [
29
,
30
] considering inforation
from a group of remotely controlled electric water heaters located in Uruguay.
4.1. Data Preparation
The data used in this article were provided by the Uruguayan National Electricity
Company (UTE). Data are available in the “Electric water heater consumption” dataset,
one of the three subsets included in EDC-UY [
6
], an effort to build a national database
of energy consumption by gathering data from several households located in the main
Uruguayan cities.
In this article, only the electric water heater consumption records were used. These
records have a sample period of one minute and cover a date range from 2 July 2019 to 26
October 2020. Customer records were filtered by the recording length, keeping only those
with more than 5 months of recording (i.e., at least 216,000 records).
The disaggregated electric water heater data have several gaps caused by different
problems that arose during the data collection process (e.g., malfunctioning of the data
transmission network, power failures, etc.). The gaps were filled using two techniques:
resampling and refilling. The resampling technique normalizes the sample period to an
exact minute. First, the records are grouped by customers to build one-minute record
containers, starting from the date and time of the first record. Then, records whose date
and time match with the date range of the container are assigned to it. In case one or more
records match the same container, the minimum consumption value is set; otherwise, a
null value is set (i.e., it corresponds to a missing record). The resulting data are taken as
the input of the refilling technique. First, refilling detects the data gaps (i.e., consecutive
missed records) and refills each one of them according to the following criteria. Starting
from both extremes of the gap up to seven minutes forward/backwards, the missing data
is recreated by a linear interpolation method. Finally, if missing values are still present
at the gap (i.e., the gap is larger than 14 min), the null value is assigned to all missing
values. The described process results in normalized time series of consumption values
without gaps. For data preparation purposes, a Jupyter notebook was implemented, and
the basis of the scripts was provided by the ECD-UY dataset. The Jupyter notebook uses
Python (version 3) programming language and libraries Pandas and Numpy. The resulting
notebook is available to download from https://bit.ly/3h133qu (accessed on 23 October
2021).
An example of the effects of the refilling procedure on electric water heater activations
is presented in Figure 2(missing records) and Figure 3(after processing). Both figures
show the same water heater activations in the same date and time range. The first graphic
shows the dataset before refilling the gaps and include the missing records. The second
graphic was captured after the data were processed.
Appl. Sci. 2021,11, 10048 7 of 21
Figure 2.
A day (14 November) of the electricity consumption of an electric water heater (customer
id. 115747 from dataset ECD-UY) before refilling the data gaps.
Figure 3.
A day (14 November) of the electricity consumption of an electric water heater (customer
id. 115747 from dataset ECD-UY) after refilling the data gaps.
4.2. Water Utilization Forecasting Model
4.2.1. Overall Description
The proposed model for water utilization forecasting is based on a specific charac-
teristic of electric water heaters: when this appliance is on, its power consumption can
be considered as constant, since it just has slight variations. Therefore, the load curve of
an electric water heater can be represented in a binary format (0 when the appliance is
off and a given value
C
when the appliance is on). A pattern similarity approach [
15
] is
proposed to estimate water utilization. To properly characterize the utilization patterns,
let us define an on block as the time interval in which the electric water heater is switched
on continuously. From the analysis of the power consumption time series, some of these
on blocks are associated with water utilization by the user and other (shorter) on blocks
correspond to thermal recoveries to maintain the target temperature of the water.
4.2.2. Methodology
The proposed forecasting model is based on identifying the on blocks associated with
water utilization and discarding those corresponding to thermal recoveries. In this regard,
a threshold duration is defined, and any on block shorter than the threshold duration is
considered to be a thermal-recovery on block and not considered for the analysis. The
proposed approach is robust, since discarding short blocks is not relevant for the main goal
of identifying long-term utilization blocks, which are generally associated with showers.
The analysis considers as a baseline an electric water heater with a capacity of 60 L and
an average water outlet flow rate, which is representative of a highly efficient appliance.
In fact, these electric water heaters are the main target of a campaign to promote energy
efficiency in residential buildings in Uruguay. For an electric water heater with a capacity
of 60 L, the empirical duration of an on block is on average eight times the duration of the
utilization period. The proposed approximation is applied to convert the information about
on blocks into an estimation of the water utilization by users. An example of the identified
on blocks and the inferred water utilization patterns obtained with the aforementioned
procedure is presented in Figure 4. In the analysis, the proposed model is applied to
forecast the water utilization each minute in a period of two hours. Thus, the output of the
model is a vector of 120 Boolean values, indicating the water utilization (or lack thereof) in
each minute.
Appl. Sci. 2021,11, 10048 8 of 21
Recoveries One utilization, one turn on Many utilizations, one turn on
ON ON with utilization
Figure 4. On blocks and water utilization patterns in the electric water heater consumption.
4.2.3. Formulation
The forecasting model for water utilization is based on an extremely randomized
trees (ExtraTrees) regression model. ExtraTrees is an ensemble learning method for solving
classification and regression problems that operates as a Random Forest (RF) technique.
ExtraTrees defines a set of decision trees that is trained with training data, and the resulting
output class (for classification problems) or prediction (for regression problems) is defined
as the mode (for classification) or mean/average prediction (for regression) of the con-
sidered decision trees [
31
]. Using a large number of trees allows properly dealing with
overfitting problems that arise when using few trees. ExtraTrees has two main differences
with the standard procedure defined by RF: (i) each tree is trained using all the training
data (and not using just a bootstrap sample, as the standard RF method), and (ii) a further
step of randomization is included in the top-down splitting in the tree learner by selecting
a random cut point for each considered feature in the problem (according to a uniform
empirical distribution to select between the values for each feature in the training set). The
split that computes the best result is then used to split the considered node of the tree [
32
].
ExtraTrees have proven to be an accurate predictor for electricity-related problems (e.g., for
demand forecasting in industrial and residential facilities [7]).
Input features considered for the proposed ExtraTrees regression method include:
•
Use (
~
u
, 120 Boolean values): indicating whether a water utilization occurs in the past
120 min.
•Month (m, integer): indicating the month of the horizon to forecast.
•Day (d, integer): indicating the day of the horizon to forecast.
•Hour (h, integer): indicating the hour of the horizon to forecast.
•Dayofweek (dw, integer): indicating the day of the horizon to forecast.
•
Workingday (
wd
, Boolean): indicating if the horizon to forecast is a working day or not.
Fine-tuning of the proposed ExtraTrees regression method was performed using grid
search techniques for hyperparameter settings. Hyperparameters are external parameters,
inherent to the learning model, whose values cannot be set or estimated from training data.
Hyperparameter values affect the quality of the resulting model, and appropriate values
must be set before launching the learning process. Grid search techniques define a search
space as a grid of different combinations of candidate hyperparameter values and proceed
to evaluate every combination in the grid. In this article, the GridSearchCV method from
scikit-learn
was used. GridSearchCV (the CV stands for cross validation) was applied
with varying parameters: number of estimators and max tree depth, considering 10-folds
cross-validation and the predetermined evaluation metrics over the model. After the model
is trained, a vector with 120 Boolean values (
~
F
), representing the water utilization forecast
for the next two hours, is obtained according to the Equation: Pred(~
u,m,d,h,dw,wd) = ~
F.
Appl. Sci. 2021,11, 10048 9 of 21
4.3. Water Temperature Model
4.3.1. Overall Description
The general formulation of equations for heating and cooling water in thermal tank
devices, such as water heaters for domestic use, include a large number of variables.
Existing models, even when they simplify the formulations, depend on several variables,
including the insulation factor, the ambient temperature, the flow of water used, the time
of use, and the tank volume, among other factors [
26
]. To overcome the difficulties of
knowing beforehand all these variables, which are often difficult to determine in practice,
the pattern similarity approach proposed for estimating water utilization is applied to
define a linear temperature model, which provides a good approximation in order to
estimate the TDI. The linear model provides a reasonable approximation, since the cooling
and heating curves in this model are straight lines.
Five parameters are defined to build the temperature model from gathered data about
on blocks and water utilization:
1. Tmin
is the temperature at which the electric water heater is turned on by the action of
the thermostat when the water is cooling.
2. Tmax
is the temperature at which the electric water heater is turned off by the action
of the thermostat when the water is heating.
3. cheat is the slope of the line when the electric water heater is turned on.
4. ccool
is the slope of the line when the electric water heater is turned off and no water
is being used.
5. cuse is the slope of the line when using water, whether or not water is being used.
The considered parameters depend on several factors. In this article, a specific ap-
proach is proposed to compute a robust approximation for each parameter value, i.e., to
guarantee that all possible temperature approximation errors always produce underes-
timated temperature values. Thus, the proposed model is conservative about comfort
estimation, in order to not affect the quality of service provided to the user.
The model assumes that the user sets the thermostat at a temperature value of 60
◦
,
which is the suggested temperature not only to achieve energy efficiency in households, but
also due to health concerns, such as avoiding the proliferation of Legionella bacteria [
33
,
34
].
Without loss of generality, the variation range defined by
Tmin =
55
◦
and
Tmax =
65
◦
is
considered. In any case, the proposed model is fully extensible to work with other values of
Tmin
and
Tmax
. The use of this model in an industrial context should consider: (i) variations
in the set point of the thermostat (e.g., values extracted from statistics or provided by the
users via survey or web/mobile application), and (ii) variations in the temperature of the
room, in the average water utilization, and in the temperature of supplied water. Variations
are captured by recomputing the coefficients after variations occur. Since the model is
estimated from on-and-off data of the water heater in real time, the variations of
Tmin
and
Tmax
should be updated frequently. After that, coefficients
cheat
,
ccool
, and
cuse
should be
recomputed using the updated values of
Tmin
and
Tmax
. These dynamics allow properly
modeling different climate conditions and user preferences.
Figure 5presents a schema of the proposed model definition from on blocks and water
utilization. The black line on the upper graphic is the water temperature. On the bottom,
grey on blocks represents thermal recoveries, and on blocks caused by water utilization are
marked in orange. The analysis of the water temperature curve indicates that the water
cools with a slope
ccool
until it reaches the value
Tmin
; at that moment, the electric water
heater turns on. The heating phase starts with a slope
cheat
until the temperature reaches
the value
Tmax
. Then, another cooling phase occurs until a water utilization causes a much
faster cooling with a slope
cuse
. During the water utilization, the electric water heater turns
on almost immediately after opening the water stream. When the water utilization ends,
the electric water heater remains on because the water temperature is below
Tmin
, so it
heats the water with a slope
cheat
until temperature
Tmax
is reached, where the gray on block
ends. Assuming the described behavior, an algebraic approach can be applied to compute
the three slopes.
Appl. Sci. 2021,11, 10048 10 of 21
Tmax
Tmin
ON
ON with
utilization
Ccool
Cheat
Cuse
Figure 5. Linear temperature model.
The described procedure allows computing an approximation of the water tempera-
ture in a given interval from a set of on blocks and water utilization data in that interval.
Therefore, a temperature forecast can be obtained from a set of on blocks predicted for a
future time interval. The forecasting method can be applied to other situations, e.g., to
simulate a remote switch off of the electric water heater, associated with a demand response
action, and obtain the water temperature forecast for this event.
4.3.2. Formulation
The linear temperature model is applied to determine the values of coefficients
ccool
,
cheat
, and
cuse
. The input data for the temperature model are the values of
Tmin
and
Tmax
and the information about on blocks and water utilization.
The coefficients of the model are calculated as follows:
•
Coefficient
ccool
.First, two consecutive temperature recoveries are identified. Then,
∆rec
is calculated as the time between the the end of the first recovery and the beginning
of the next recovery. Finally,
ccool
is calculated by Equation (1). The procedure is
described in the left box of Figure 6. From the graphic, ccool <0.
ccool = (Tmin −Tmax)/∆rec (1)
•
Coefficient
cheat
.First, a recovery is identified and
durrec
is defined as its duration.
Then,
cheat
is computed by Equation (2). The procedure is described in the middle box
of Figure 6. From the graphic, it is clear that ccheat >0.
cheat = (Tmax −Tmin)/durrec (2)
•
Coefficient
cuse
.The first step is identifying a temperature recovery followed by a water
utilization event. Then,
Tuse
ini
and
Tuse
end
are defined as the temperature at the beginning
and end of the utilization, respectively. To compute
Tuse
ini
, the value of
ccool
(already
computed) is used as the slope to draw the line that passes through
Tmax
at the end of
the recovery, and intersects with the start of the utilization. Similarly, to compute
Tuse
end
,
the value of
cheat
(already computed) is used as the slope to draw the line that passes
through the end of the on block associated with the utilization and intersects with the
end of the utilization.
duruse
is defined as the duration of the utilization. Finally,
cuse
is calculated by Equation (3). The procedure is described in the right box of Figure 6.
cuse = (Tuse
end −Tuse
ini )/duruse (3)
Appl. Sci. 2021,11, 10048 11 of 21
Tmax
Tmin
-ΔT
Δrec
ON ON with utilization
ΔT
durrec
Tini
use
Tend
use
duruse
-ΔT
Figure 6. Graphic representation of ccoo l (left box), cheat (middle box), and cuse (right box).
4.4. Defining the Thermal Discomfort Index
The proposed index is conceived to capture the thermal impact that a user suffers
due to an intervention by the electrical company in the electric water heater. The TDI is
defined using the defined water use forecasting model and the temperature model. Since
every forecast has uncertainty, the index is defined in terms of the expected value of the
difference of the aforementioned temperatures, as expressed by Equations (4) and (5).
TDI(I,u,w)=
tend(u)
Z
tini(u)
(Tn(t,w)−Tint(t,w))dt +ρZ
t∈τ
(Tcomf −Tint(t,w))dt (4)
In Equation (4),
TDI(I
,
u
,
w)
represents the discomfort index of an interruption
I
, a
water utilization
u
, and a realization
w
that defines a single scenario of water use and
temperature evolution. The values
tini(u)
and
tend(u)
are the starting and finishing time of
utilization
u
. For every realization
w
,
Tn(t
,
w)
is the temperature curve without interruption
and
Tint(t
,
w)
is the temperature curve with interruption. Both curves are obtained using
the temperature model described in Section 4.3 for the realization w.
Finally,
Tcomf
is the lowest water temperature that does not produce discomfort to
the users, and
τ
represents the time interval in which
Tint(t
,
w)≤Tcomf
. The value
ρ
is a
parameter that acts as a penalty over the area below the comfort temperature.
TDI(I)= Ew
∑
u∈U(w)
TDI(I,u,w)
(5)
In Equation (5),
I
denotes the interruption of the electric water heater by the electric
company. In turn,
Ew[•]
represents the expected value of the expression inside parenthesis
with respect to the random variable
w
. Each realization of the random variable
w
generates
a different forecast of water use. Therefore, a forecast of the temperature obtained using
the model described in Section 4.3 is also generated.
U(w)
is the set of water utilization
intervals in the analyzed time horizon associated with the forecast generated by w.
Figure 7presents a visual representation of
TDI(I
,
u
,
w)
. Two temperature curves are
represented on the upper graphic. The full black line represents the water temperature
when no interruption occurs. In turn, the dotted black line represents the water temperature
when an interruption occurs at time
t∗
. The green area between the curve of water tempera-
ture without interruption and the curve of water temperature with interruption (defined by
the polygon PQRTU) is computed by the integral
Rtend(u)
tini(u)(Tn(t
,
w)−Tint(t
,
w))dt
. This area
represents the heat loss due to the interruption. In turn, the red area below the comfort tem-
perature (defined by the polygon RST) is computed by the integral
Rt∈τ(Tcomf −Tint(t
,
w))dt
Appl. Sci. 2021,11, 10048 12 of 21
and weighted by the penalty
ρ
. Introducing the penalty term
ρ
is interesting to provide the
model the flexibility for adjusting the weight of the red area below the comfort temperature
with respect to the whole area of temperature reduction (PQSU).
Tmax
Tmin
Tconf
P
Load shifting
U
Q
R
T
S
ON
ON with
utilization
Shifted
load
Interruption
temp. with
interruption
temp. without
interruption
Figure 7. Graphical representation of TDI.
The expected value of TDI, as defined in Equation (4), is computed by applying a
Monte Carlo (MC) simulation method. First, the MC method samples 100 realizations of
the value
w
, with a normal distribution
N(
0, 1
)
. Then, for each value of
w
, the following
procedure is applied:
•
The next 12 h are forecasted using the model described in Section 4.2. Six iterations
are applied, considering that the Extra Trees regressor is defined for a period of two
hours (120 observations).
•
Using the temperature model described in Section 4.3 and the water utilization forecast,
the water temperature is obtained for the next 12 h.
•
An interruption of
k
minutes is simulated and the temperature for the next 12 h is
obtained using the proposed temperature model.
•
Since
Tmax
,
Tmin
,
Tcomf
are known, Equation (5) is applied to compute
TDI(I
,
u
,
w)
for
the current realization wand all uses.
• An auxiliary variable Suses(w)is defined by Equation (6).
Suses(w) = ∑
u∈U(w)
TDI(I,u,w)(6)
Finally, after computing
Suses(w)
for each realization, TDI is computed as the empirical
expected value defined in Equation (7).
TDI(I) =
w=100
∑
w=1
Suses(w)/100 (7)
Appl. Sci. 2021,11, 10048 13 of 21
5. Experimental Validation
This section presents the experimental validation of the proposed approach for defin-
ing a TDI.
5.1. Methodology
The methodology for the experimental evaluation includes two steps: validation of
forecasting models and validation of the proposed index.
5.1.1. Validation of Forecasting Models
The first step of the experimental evaluation consists of validating the two models
required to calculate the TDI (water utilization forecasting and water temperature). For
the validation of the aforementioned models, the standard mean absolute percentage
error (MAPE) metric is used to evaluate the forecasting capabilities. MAPE is defined in
Equation (8), where
actuali
represents the measured value for
t=i
,
predi
represents the
predicted value, and nis the predicted horizon length.
MAPE =100 ×1
n
n
∑
i=1
actuali−predi
actuali
(8)
5.1.2. Validation of the Proposed TDI
After determining the forecasting accuracy of the proposed models, the second step of
the experimental evaluation consists of validating the TDI calculation and utilization in
realistic scenarios in order to properly evaluate the thermal discomfort of users. For this
purpose, three experiments were designed based on scenarios that adequately represent
the real operation of water heaters. A water heater dynamic simulator was developed
and used to generate real scenarios. The description of the simulator and the experiments
performed are presented in Section 6.
5.2. Development and Execution Platforms
Data processing algorithms and the proposed models to build the TDI were imple-
mented using Python and well-known open source libraries such as Pandas, Numpy and
Tensorflow. Data processing and the experimental analysis were performed on the high
performance platform of National Supercomputing Center (Cluster-UY), Uruguay [35].
5.3. Evaluation of the Water Utilization Forecasting
Metrics defined in Section 4.2 were applied to evaluate the implementation of the
water utilization forecasting model. A subset of ECD-UY was used, consisting of ten
electric water heaters with more than five months of measurements. The grid search was
performed on a two-dimensional grid to determine the best values for the number of trees
in the forest and the maximum depth of the tree. The best parameter setting found applying
the grid search configuration was n_estimators = 50, max_depth = 200.
Using the best parameter configuration, the ExtraTrees regressor achieved a MAPE
value of 11.79 in just 4.09 s of execution time. This accuracy is adequate for estimation
purposes to compute TDI, considering the high variance of individual water utilization. The
method is useful for generating scenarios to apply the Monte Carlo simulation approach in
order to estimate the empirical probability distribution of water utilization.
5.4. Evaluation of the Water Temperature Model
The linear model described in Section 4.3 was evaluated for a real case study corre-
sponding to an electric water heater with a thermometer to measure the temperature of
the water in the tank. The defined setting of the thermostat allowed knowing in advance
the values of parameters
Tmin
and
Tmax
. The corresponding values are
Tmin =55 °C
and
Tmax =65 °C
. Then, the other parameters of the model were calculated as described in
Section 5.3. Finally, data of twelve hours on blocks of the electric water heater were used
Appl. Sci. 2021,11, 10048 14 of 21
to estimate the temperature and compared with the real temperature measured. Table 1
reports the comparison of the real and the estimated temperature, and the largest difference
in the three long utilization periods in the twelve hours analyzed.
Table 1. Accuracy of the water temperature model.
1st Utilization 2st Utilization 3st Utilization
measured temperature 59.09 °C 53.03 °C 58.34 °C
linear temperature 59.88 °C 55.12 °C 59.41 °C
difference 0.79 °C 2.09 ◦C 1.07 °C
The second utilization had the largest temperature difference (
2.7 °C
, marked in light
blue in Table 1), which represents a percentage error of 4.5% in the worst case. The other
utilizations had a significantly lower error. The accuracy of the temperature model is
adequate for the purpose of estimating TDI.
6. Application of the Proposed Model: TDI Calculation
This section describes the application of the proposed model for TDI calculation over
relevant sample scenarios.
6.1. Overall Description
The main challenge when designing the TDI was to properly capture the differences
between demand response strategies in order to fairly select water heaters to interrupt
while minimizing the discomfort of users. Therefore, evaluating the TDI calculation is
important to have real scenarios that capture the utilization profile of the users of electric
water heaters. Evaluating the proposed methodology over a real scenario is not an easy task.
The scenario must include a set of real water heaters large enough to carry out experiments
on real water uses, and real interventions must be set up. To overcome these difficulties, a
common approach in the related literature [
22
,
26
] consists of using simulations of thermal
appliances.
A simulator was implemented to perform the experimental evaluation in those scenar-
ios where real data is not available. The consists of two modules: the individual water heater
module simulates the energy dynamics of a water heater, based on the work by Lutz [
26
]
and the household utilization module generates scenarios of water utilization for a group of
households, using a vector of hourly probabilities of water usage as input.
The proposed TDI is evaluated in three different scenarios accounting for different
number of water heaters, households, and priorities. Scenario #1 considers two water
heaters and real data for both temperature and the electrical state (ON/OFF) of the water
heaters. Scenarios #2 and #3 considers a large number of water heaters, for which real
data about the electrical state are available. In turn, the developed temperature and water
utilization models, implemented in the simulator, were applied to validate the proposed
TDI. The main details of the evaluation are reported in the following subsections.
6.2. Scenario #1: Evaluation of a Simple Case Study with Two Real Water Heaters
One of the main challenges related to the definition of TDI is modeling the differences
of temperature (
∆T
, a quantitative factor) between performing an interruption in different
moments. A relevant case is analyzing the
∆T
values situations in the interruption affects
the most to comfort.
As a relevant sample study, the comparison of the TDI for two different values of
ρ
and two particular electric water heaters (
EWH1
and
EWH2
) is presented. The considered
electric water heaters model two different utilization patterns from two different users. On
weekdays, EWH1has two consecutive utilizations, and EWH2is only used once. The TDI
associated with a 20 min interruption between 20:10 and 20:30 was considered for this case.
Figure 8presents the empirical distribution of uses (
P(u)
) from 19:00 to 22:00 for
EWH1
Appl. Sci. 2021,11, 10048 15 of 21
(left) and
EWH2
(right). For each case, the interruption period is represented by a green
band.
19 20 21 22
hour
P(u)
19 20 21 22
hour
P(u)
(a) (b)
Figure 8.
Probability distribution for the water utilization of two electric water heaters. (
a
)
EWH1
;
(b)EWH2.
For the presented example, it is expected for the TDI value to be higher for
EWH2
than
for
EWH1
, because in the hours immediately after the interruption analyzed, the average
historical utilization is higher for
EWH2
. On the other hand, as the value of
ρ
increases,
it is expected that the gap between the TDI of both electric water heaters becomes larger.
Table 2
reports the TDI values computed for each electric water heater, considering two
different values of ρ(ρ=1, and ρ=2).
Table 2. TDI applied for the interruption for EWH1and EWH2.
Appliance ρ=1ρ=2
EWH13362.3 °C s 4108.2 °C s
EWH28109.6 °C s 11,041.7 ◦C s
The results in Table 2confirm that the proposed index correctly models discomfort.
The TDI value is higher for
EWH2
, and the difference widens when considering larger
penalty values (ρ). Results show that TDI properly models the differences of temperature
between a scenario with an interruption and a scenario without interruption, as expected.
6.3. Scenario #2: Evaluation on a Group of Households with Water Heaters
Another challenge related to the definition of TDI is to prioritize which group of water
heaters should be interrupted to reduce electricity demand while minimizing discomfort
(e.g., for demand response management in a peak situation). In this experiment, a group
of twenty households
H
with electric water heaters was studied using the developed
simulator. Ten households (
H1
,
. . .
,
H10
) have a water use profile concentrated in the
morning hours and the other ten households (
H11
,
. . .
,
H20
) in the evening hours. Figure 9
presents the average water use profiles for each type of household.
A simulation of a complete day for the twenty defined households was performed. In
turn, the TDI associated with a 20 min interruption between 7:10 and 7:30 (
TD Im
) and the
TDI associated with a 20 min interruption between 20:10 and 20:30 (
TD Ia
) were computed
for the twenty households using the method presented in Section 4.4.
Table 3reports the computed TDI values for each household. Table 4presents the
two rankings defined: Ranking
m
sorts households according to the values of
TD Im
and
Ranking
a
sorts households according to the values of
TD Ia
, both from lowest to higher
values. H1, . . . , H10 households are highlighted in green and H11, . . . , H20 in blue.
Appl. Sci. 2021,11, 10048 16 of 21
0246 8 10 12 14 16 18 20 22
0
0.1
0.2
0.3
0.4
hour
P(u)
Morning
Afternoon
Figure 9. Water use profiles for a day.
Table 3. T DImand TDIafor each water heater.
T D Im(°C s)TD Ia(°C s)
H13140.2 64.1
H23170.4 55.9
H33131.0 67.2
H43158.6 77.1
H53171.4 79.8
H63176.7 77.4
H73161.5 31.2
H83110.8 74.7
H93100.5 66.8
H10 3146.9 77.2
H11 71.6 1962.8
H12 87.3 1996.9
H13 97.8 2017.7
H14 15.4 2019.5
H15 55.8 1993.2
H16 96.6 1986.6
H17 29.4 2018.9
H18 38.1 1954.8
H19 82.6 1977.3
H20 72.6 1956.2
Table 4. Rankings of water heaters sorted by TDI in ascending order.
Order
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ranking mH14 H17 H18 H15 H11 H20 H19 H12 H16 H13 H9H8H3H1H10 H4H7H2H5H6
RankingaH7H2H1H9H3H8H4H10 H6H5H18 H20 H11 H19 H16 H15 H12 H13 H17 H14
For the presented case study, and considering that the probability of affecting comfort
when interrupting water heaters in the morning is higher for households
H1
,
. . .
,
H10
, these
ten households are expected to be in the last ten positions of Ranking
m
. Analogously,
H11
,
. . .
,
H20
are expected to be in the last ten positions of Ranking
a
. Results in Table 4
confirm that the TDI properly sorts the water heaters according to the real discomfort
produced for users, as expected. These results allow for very precise information to decide
about specific demand-management interventions.
Appl. Sci. 2021,11, 10048 17 of 21
6.4. Scenario #3: Setting ρto Tune the Interruption Priority of a Special Water Heater
In several circumstances, an electric water heater is installed with specific objectives
and should not be considered to be interrupted in a demand-management event. An
example of this situation is a nursing home, where hot water is used for sanitary purposes.
A simple approach to deal with this scenario is excluding the electric water heater from
the list of appliances to be interrupted by the energy utility. This approach is not flexible
because it only allows inserting or removing a device from the interruptible list and does
not allow prioritizing special appliances. However, using the proposed TDI definition, the
parameter
ρ
is used to independently weight those special electric water heaters. In this
experiment, the case presented in
Section 6.3
is extended by adding a household (
HE
) with
an special water heater. The hypothesis of the experiment is that the special water heater in
household
HE
has a water use profile almost constant throughout the day, and it cannot be
interrupted in demand-response events. Figure 10 shows in orange the hourly probability
vector of this special water heater. Green and blue dotted lines correspond to the profiles
of households with water use concentrated in the morning and in the evening, respectively.
The experiment consists of adding the device with the same value of the parameter
ρ=
1
used in the base experiment and calculating Ranking
m
and Ranking
a
. Then, the value of the
parameter
ρ
for the special device is set to a very large value (
ρ=
1000), and differences
are analyzed.
0246 8 10 12 14 16 18 20 22
0
0.1
0.2
0.3
0.4
hour
P(u)
Special
Morning
Afternoon
Figure 10. Water use profiles for a day in the special household.
The proposed scenario was studied via simulations. Two special household values
were included,
H(1)
E
using
ρ=
1 and
H(1000)
E
using
ρ=
1000. Table 5reports the TDI values
obtained in the simulations, considering the two special households
H(1)
E
(highlighted in
red) and
H(1000)
E
(highlighted in orange). Table 6reports the order defined by
Rankingm
and
Rankinga
considering
H(1)
E
, and Table 7reports the order defined by
Rankingm
and
Rankingaconsidering H(1)
EH(1000)
E.
Appl. Sci. 2021,11, 10048 18 of 21
Table 5. T DIm
and
TD Ia
for each water heater with
ρ=
1, adding a special household using
ρ=
1
(red), and another special household ρ=1000 (orange).
T D Im(°C s)TD Ia(°C s)
H13140.2 64.1
H23170.4 55.9
H33131.0 67.2
H43158.6 77.1
H53171.4 79.8
H63176.7 77.4
H73161.5 31.2
H83110.8 74.7
H93100.5 66.8
H10 3146.9 77.2
H11 71.6 1962.8
H12 87.3 1996.9
H13 97.8 2017.7
H14 15.4 2019.5
H15 55.8 1993.2
H16 96.6 1986.6
H17 29.4 2018.9
H18 38.1 1954.8
H19 82.6 1977.3
H20 72.6 1956.2
H(1)
E354.4 309.5
H(1000)
E72,305.4 71,588.1
Results reported in Table 6demonstrate that the special electric water heater
H(1)
E
is
not ranked at the end of either Ranking
m
or Ranking
a
. A different situation is observed in
the results reported in Table 7, which indicate that using the value
ρ=
1000 only for the
special electric water heater
H(1000)
E
, this appliance appears last in both rankings Ranking
m
and Rankinga, as expected.
Table 6.
Rankings of water heaters sorted by TDI using
ρ=
1 in ascending order, considering the
special household.
Order
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
RankingmH14 H17 H18 H15 H11 H20 H19 H12 H16 H13 H(1)
EH9H8H3H1H10 H4H7H2H5H6
RankingaH7H2H1H9H3H8H4H10 H6H5H(1)
EH18 H20 H11 H19 H16 H15 H12 H13 H17 H14
Table 7.
Rankings of water heaters sorted by TDI in ascending order using
ρ=
1000 for the special
household and ρ=1 for the rest of the households.
Order
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
RankingmH14 H17 H18 H15 H11 H20 H19 H12 H16 H13 H9H8H3H1H10 H4H7H2H5H6H(1000)
E
RankingaH7H2H1H9H3H8H4H10 H6H5H18 H20 H11 H19 H16 H15 H12 H13 H17 H14 H(1000)
E
Overall, results obtained for scenario #3 validate the proposed approach and con-
firm the effect of modifying the
ρ
parameter for some water heaters to allow flexible
demand-response strategies to be designed. For instance, a possible strategy is to define a
classification of the complete set of households in three classes and associate each class with
a different
ρ
value to define water heaters that are not interruptible,interruptible if necessary,
and no restrictions for interruption. The flexibility of the proposed TDI using different values
Appl. Sci. 2021,11, 10048 19 of 21
of
ρ
facilitates, in a real scenario that is continually changing in structure, performing active
demand management to quickly adapt to the needs of the electrical system.
7. Conclusions and Future Work
This article presented an approach to evaluate the impact on the thermal comfort of
direct demand response control using electric water heaters. An index associated with
the thermal discomfort was defined according to the following procedure. First, a water
utilization forecasting model was built using real power data from a set of electric water
heaters and applying an ensemble learning technique. Then, a linear model was developed
to estimate the temperature of the water in the electric water heater tank. Finally, the TDI
associated with an intervention on the electric water heater was defined stochastically, and
calculated via Monte Carlo simulation. A specific water heater simulator was developed
for the evaluation of the proposed TDI.
The computational models and the reliability of the proposed index were evaluated in
three real case studies. The first case considered two electric water heaters from different
users with different average historical utilization. The TDI values were analyzed for
both electric water heaters for different penalization factors
ρ
. Results confirmed that the
proposed index correctly models discomfort since higher TDI values were computed for
the electric water heater with the higher average historical utilization. The difference in
TDI values increased when considering larger penalty values. The second case analyzed
the use of TDI for sorting water heaters according to the discomfort caused by their
intervention. Two sets of ten households were generated and simulated, considering
two different utilization patterns and two interruptions (in the morning and the evening).
Results confirmed that in the ranking generated with the TDI computed for the morning
interruption, households with high probability of water utilization in the morning were
in the last ten positions of the ranking. A similar result was obtained for the afternoon
interruption, as expected. The third case study explored the use of parameter
ρ
to tune
the interruption priority, considering an additional special household and two values of
ρ
. Results demonstrated that the new special household was in eleventh position in both
rankings when
ρ=
1 and changed to the last position in both rankings when
ρ=
1000,
properly modeling a non-interruptible appliance (e.g., for sanitary reasons). The main lines
of future work are related to estimating an economic value of the TDI index (in USD/MWh),
useful to characterize the profit of reducing the energy demanded by a set of electric water
heaters by applying the interruption action in order to compare this strategy with other
demand response techniques (e.g., using fuel generators or batteries). Expanding the
developed simulator to consider all generators in the system is another line for future
work to fairly compare demand response strategies by using TDI values and the economic
impact of an interruption.
Author Contributions:
R.P.: research conceptualization, data analysis, code development, manuscript
writing, manuscript revision and correction. J.C.: data analysis, data cleansing, manuscript writing,
manuscript revision and correction. S.N.: research conceptualization, data analysis, manuscript
writing, manuscript revision and correction. All authors have read and agreed to the published
version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Acknowledgments:
The work of S. Nesmachnow was partly supported by ANII and PEDECIBA,
Uruguay.
Conflicts of Interest: The authors declare no conflicts of interest.
Appl. Sci. 2021,11, 10048 20 of 21
References
1. Momoh, J. Smart Grid: Fundamentals of Design and Analysis; Wiley-IEEE Press: Hoboken, NJ, USA, 2012.
2.
Assessment of Demand Response & Advanced Metering; Technical Report AD-06-2-00; Federal Energy Regulatory Commission:
Washington, DC, USA, 2006.
3.
European Automotive Research Partner Association. Smart Grids European Technology Platform. Available online: https:
//www.etip-snet.eu/ (accessed on 21 January 2021).
4. Bhattacharyya, S. Energy Demand Management. In Energy Economics; Springer: London, UK, 2011; pp. 135–160.
5.
Instituto Nacional de Estadística, Uruguay. Microdatos de la Encuesta Continua de Hogares. 2019. Available online: http:
//www.ine.gub.uy/microdatos (accessed on 17 August 2020).
6.
Chavat, J.; Nesmachnow, S.; Graneri, J.; Alvez, G. ECD-UY: Detailed household electricity consumption dataset of Uruguay. Sci.
Data 2020, submitted.
7. Porteiro, R.; Nesmachnow, S.; Hernández-Callejo, L. Electricity demand forecasting in industrial and residential facilities using
ensemble machine learning. Rev. Fac. Ing. Univ. Antioq. 2020,102, 9–25.
8.
Deng, R.; Yang, Z.; Chow, M.; Chen, J. A Survey on Demand Response in Smart Grids: Mathematical Models and Approaches.
IEEE Trans. Ind. Inform. 2015,11, 570–582. [CrossRef]
9.
Hassan, N.U.; Khalid, Y.I.; Yuen, C.; Tushar, W. Customer engagement plans for peak load reduction in residential smart grids.
IEEE Trans. Smart Grid 2015,6, 3029–3041. [CrossRef]
10.
Tang, R.; Wang, S.; Yan, C. A direct load control strategy of centralized air-conditioning systems for building fast demand
response to urgent requests of smart grids. Autom. Constr. 2018,87, 74–83. [CrossRef]
11. Ozoh, P.; Apperley, M.; Olayiwola, M. Modelling of household power consumption and its effects on load shifting strategies. J.
Sci. Arts 2018,18, 1067–1072.
12.
Witherden, M.; Rayudu, R.; Tyler, C.; Seah, W. Managing peak demand using direct load monitoring and control. In Proceedings
of the Australasian Universities Power Engineering Conference, Hobart, TAS, Australia, 29 September–3 October 2013; pp. 1–6.
13.
Orsi, E.; Nesmachnow, S. Smart home energy planning using IoT and the cloud. In Proceedings of the IEEE URUCON,
Montevideo, Uruguay, 23–25 October 2017.
14.
Orsi, E.; Nesmachnow, S. IoT for smart home energy planning. In Proceedings of the XXIII Congreso Argentino de Ciencias de la
Computación, La Plata, Argentina, 9–13 October 2017; pp. 1091–1100.
15.
Chavat, J.P.; Nesmachnow, S.; Graneri, J. Non-intrusive energy disaggregation by detecting similarities in consumption patterns.
Rev. Fac. Ing. Univ. Antioq. 2020,98, 27–46. [CrossRef]
16.
Yoon, A.; Kang, H.; Moon, S. Optimal Price Based Demand Response of HVAC Systems in Commercial Buildings Considering
Peak Load Reduction. Energies 2020,13, 862. [CrossRef]
17.
Huang, D.; Billinton, R. Effects of load sector demand side management applications in generating capacity adequacy assessment.
IEEE Trans. Power Syst. 2011,27, 335–343. [CrossRef]
18.
Lu, N.; Katipamula, S. Control strategies of thermostatically controlled appliances in a competitive electricity market. In
Proceedings of the IEEE Power Engineering Society General Meeting, San Francisco, CA, USA, 16 June 2005; pp. 202–207.
19.
Perfumo, C.; Braslavsky, J.H.; Ward, J. Model-based estimation of energy savings in load control events for thermostatically
controlled loads. IEEE Trans. Smart Grid 2014,5, 1410–1420. [CrossRef]
20.
Nehrir, M.; LaMeres, B.; Gerez, V. A customer-interactive electric water heater demand-side management strategy using fuzzy
logic. In Proceedings of the Winter Meeting IEEE Power Engineering Society, New York, NY, USA, 31 January–4 February 1999;
Volume 1, pp. 433–436.
21.
Xiang, S.; Chang, L.; Cao, B.; He, Y.; Zhang, C. A Novel Domestic Electric Water Heater Control Method. IEEE Trans. Smart Grid
2019,11, 3246–3256. [CrossRef]
22.
Kampelis, N.; Ferrante, A.; Kolokotsa, D.; Gobakis, K.; Standardi, L.; Cristalli, C. Thermal comfort evaluation in HVAC Demand
Response control. Energy Procedia 2017,134, 675–682. [CrossRef]
23.
Tabatabaei, S.; Klein, M. The role of knowledge about user behaviour in demand response management of domestic hot water
usage. Energy Effic. 2018,11, 1797–1809. [CrossRef]
24.
Pirow, N.; Louw, T.; Booysen, M. Non-invasive estimation of domestic hot water usage with temperature and vibration sensors.
Flow Meas. Instrum. 2018,63, 1–7. [CrossRef]
25.
Paull, L.; MacKay, D.; Li, H.; Chang, L. Awater heater model for increased power system efficiency. In Proceedings of the
Canadian Conference on Electrical and Computer Engineering, St. John’s, NL, Canada, 3–6 May 2009.
26.
Lutz, J.; Whitehead, C.; Lekov, A.; Winiarski, D.; Rosenquist, G. WHAM: A Simplified Energy Consumption Equation for Water
Heaters; American Council for and Energy-Efficient Economy Summer Study on Energy Efficiency in Buildings: Washington, DC,
USA, 1998
27.
Yin, Z.; Che, Y.; Li, D.; Liu, H.; Yu, D. Optimal scheduling strategy for domestic electric water heaters based on the temperature
state priority list. Energies 2017,10, 1425. [CrossRef]
28.
Al-Jabery, K.; Xu, Z.; Yu, W.; Wunsch, D.C.; Xiong, J.; Shi, Y. Demand-side management of domestic electric water heaters using
approximate dynamic programming. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 2016,36, 775–788. [CrossRef]
29.
Massobrio, R.; Nesmachnow, S.; Tchernykh, A.; Avetisyan, A.; Radchenko, G. Towards a cloud computing paradigm for big data
analysis in smart cities. Program. Comput. Softw. 2018,44, 181–189. [CrossRef]
Appl. Sci. 2021,11, 10048 21 of 21
30.
Luján, E.; Otero, A.; Valenzuela, S.; Mocskos, E.; Steffenel, L.; Nesmachnow, S. An integrated platform for smart energy
management: the CC-SEM project. Rev. Fac. Ing. Univ. Antioq. 2019,97, 41–55. [CrossRef]
31. De Ville, B. Decision trees. Wiley Interdiscip. Rev. Comput. Stat. 2013,5, 448–455. [CrossRef]
32. Zhang, C.; Ma, Y. (Eds.) Ensemble Machine Learning; Springer: New York, NY, USA, 2012.
33.
Lévesque, B.; Lavoie, M.; Joly, J. Residential Water Heater Temperature: 49 or 60 Degrees Celsius? Can. J. Infect. Dis.
2004
,
15, 11–12. [CrossRef]
34.
Sangoi, J.; Ghisi, E. Energy Efficiency of Water Heating Systems in Single-Family Dwellings in Brazil. Water
2019
,11, 1068.
[CrossRef]
35.
Nesmachnow, S.; Iturriaga, S. Cluster-UY: Collaborative Scientific High Performance Computing in Uruguay. In High Performance
Computing; Springer International Publishing: Berlin/Heidelberg, Germany, 2019; pp. 188–202.
Available via license: CC BY
Content may be subject to copyright.
