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Citation: Pardiñas, Á.Á.; Durán
Gómez, P.; Echevarría Camarero, F.;
Carrasco Ortega, P. Demand–
Response Control of Electric Storage
Water Heaters Based on Dynamic
Electricity Pricing and Comfort
Optimization. Energies 2023,16, 4104.
https://doi.org/10.3390/en16104104
Academic Editor: Surender
Reddy Salkuti
Received: 25 April 2023
Revised: 8 May 2023
Accepted: 13 May 2023
Published: 15 May 2023
Copyright: © 2023 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/).
energies
Article
Demand–Response Control of Electric Storage Water Heaters
Based on Dynamic Electricity Pricing and Comfort Optimization
Ángel Á. Pardiñas * , Pablo Durán Gómez, Fernando Echevarría Camarero and Pablo Carrasco Ortega
Energy Division, Galicia Institute of Technology, 15003 A Coruña, Spain; pduran@itg.es (P.D.G.);
fechevarria@itg.es (F.E.C.); pcarrasco@itg.es (P.C.O.)
*Correspondence: aalvarez@itg.es; Tel.: +34-652-77-40-76
Abstract:
Electric Storage Water Heaters (ESWH) are a widespread solution to supply domestic hot
water (DHW) to dwellings and other applications. The working principle of these units makes them
a great resource for peak shaving, which is particularly important due to the level of penetration
renewable energies are achieving and their intermittent nature. Renewable energy deployment
in the electricity market translates into large electricity price fluctuations throughout the day for
individual users. The purpose of this study was to find a demand–response strategy for the activation
of the heating element based on a multiobjective minimization of electricity cost and user discomfort,
assuming a known DHW consumption profile. An experimentally validated numerical model was
used to perform an evaluation of the potential savings with the demand–response optimized strategy
compared to a thermostat-based approach. Results showed that cost savings of approximately 12%
can be achieved on a yearly basis, while even improving user thermal comfort. Moreover, increasing
the ESWH volume would allow (i) more aggressive demand–response strategies in terms of cost
savings, and (ii) higher level of uncertainty in the DHW consumption profile, without detriment
to discomfort.
Keywords:
Electric Storage Water Heater; demand response; optimization; thermal comfort; domestic
hot water
1. Introduction
The residential building sector accounts for a very significant percentage of total
energy consumption. Already in 2004, this percentage was 16% worldwide, and even
higher for developed countries and regions (e.g., 22% in the USA, 28% in the UK and 26%
in the EU) [
1
]. These figures are even higher today, according to a more recent study [
2
]
based on data from the International Energy Agency for 2020, and this increase is driven
by non-OECD (Organisation for Economic Co-operation and Development) countries, in
contrast with the slight decrease of energy consumption in the residential building sector
for OECD countries. These numbers are also supported by the statistics by Eurostat [
3
] for
the EU, with 27% of final energy consumption in 2020 for the residential building sector.
All in all, 30% of global CO
2
emissions are linked to the building sector [
2
], which explains
why targeting this sector is crucial in the fight against climate change.
Within the building sector, domestic hot water (DHW) is responsible for 13% of energy
consumption in the world [
2
] and approximately 15% in EU households [
3
]. Currently,
water heating production is still dominated by fossil fuels in the EU, with approximately
half of its final energy consumption proceeding from natural gas or oil and petroleum
products [
3
]. Electric Storage Water Heaters (ESWH) are an inexpensive alternative to shift
these numbers towards decarbonization and electrification of the building sector. An ESWH
is an insulated water tank, with capacity ranging from 5 L to several hundreds of liters,
with an electric resistance element immersed in the water to heat it up to temperatures
in the range of 60
◦
C to 80
◦
C [
4
]. Even if they could be outperformed in efficiency by,
Energies 2023,16, 4104. https://doi.org/10.3390/en16104104 https://www.mdpi.com/journal/energies
Energies 2023,16, 4104 2 of 25
for example, heat pumps to produce hot water, ESWHs are very simple and represent an
opportunity to increase the renewable share of the building sector as countries and users
move forward on the deployment of renewable sources to produce electricity. Moreover,
the storage capacity inherent to ESWHs is key to peak shaving capabilities and demand
response (also known as demand-side management).
Demand response refers to the practice of adjusting electricity usage in response to
price signals, which helps to balance the energy system and support Distribution System
Operators (DSOs). Essentially, when energy prices are high, demand response encour-
ages consumers to reduce their consumption, while when prices are low, consumers are
incentivized to increase their usage. By responding to these price signals, consumers can
help to balance the overall demand for energy and prevent overload on the system, which
ultimately benefits both the consumers themselves and the DSOs responsible for managing
the energy grid. Furthermore, consumers may opt to additional benefits by their direct par-
ticipation in balance and local flexibility markets, which are currently under development.
Different processes and services can operate according to demand response, among which
are ESWHs, which simply can decouple DHW production and consumption considering
the intraday electricity price or the availability of locally produced renewable electricity.
Demand response is one of three main flexibility sources according to the Universal Smart
Energy Framework [5].
Any demand–response strategy linked to ESWHs is highly dependent on the DHW
consumption profile for the specific user [
6
–
8
], and on the variability of this profile from
one day to the next (due to climate conditions, day of the week, season, etc.). Thus, before
delving further into demand–response strategies, control algorithms, etc., the topic of DHW
consumption profiles should be defined. Fuentes et al. [
9
] reviewed different technical
standards and studies on DHW consumption profiles, identifying patterns and parameters
that define the occurrence and duration of water draw-offs, the objective being to assist
researchers and designers in the correct dimensioning of systems and on developing
improved control strategies. The authors of this review stated that these profiles combine
random and patterned features, which happen at household-specific times. This was also
pointed out in Edwards et al. [
10
], which, based on measurements for 73 dwellings in
Canada, concluded that DHW consumption profiles change importantly from house to
house, but as soon as the pattern is decided, it remains rather constant. Lomet et al. [
11
]
observed in the data from eight dwellings that weekly periodicity, random fluctuations,
and profiles differ depending on the dwelling, season and day of the week. The review
by Fuentes et al. [
9
] indicates the importance of having updated data for the generation of
draw-off profiles, since DHW consumption has decreased importantly in the last decades
due to the installation of metering devices in buildings, increased cost of water and heat, etc.
Stochastic models to generate DHW usage profiles were developed by Hendron et al. [
12
],
Ritchie et al. [
13
] and Jordan and Vajen [
14
,
15
], based on data recorded in the USA, South
Africa and Europe (Switzerland and Germany), respectively. Najafi and Fripp [
16
] indicated
that a weakness with stochastic models is that they do not account for previous water draw-
offs during the day to redefine the profile, and they introduced joint probability to solve
this. Time-series forecast of hot water consumption for a specific household was suggested
by Gelazankas and Gamage [
7
] as a suitable alternative to stochastic models. Exponential
smoothing, seasonal decomposition or ARIMA were seen as positive techniques for 24-h-
ahead forecasting.
Another important aspect related to demand–response strategies and ESWHs is com-
fort. A general definition of the term “hot water comfort” is given in the VDI 6003 [
17
]: “A
high comfort level for hot water is given if the required volume of hot water and mass flow
is available at each draw-off point, at any time and at the desired temperature”. This could
be translated as the ability to meet any given load with stable temperature. According
to [
18
], this can either be achieved by a large hot water volume or a high heater switch-off
temperature. Thus, comfort is opposed to energy efficiency and operation cost minimiza-
tion. Leaving aside the well-known issues about Legionella, a deadly hazard to humans,
Energies 2023,16, 4104 3 of 25
thriving at typical hot water use temperatures around 35–40
◦
C [
19
], any user-focused
demand–response strategy for ESWHs should have thermal comfort as a priority. Moreover,
most DHW consumption events are more critical in terms of thermal comfort than those
for HVAC [
6
], and slight increases in the energy bill are preferred to cold showers [
20
].
Table 1gathers different works from the literature defining comfort indicators, in some
cases utilized in demand–response strategies and optimizations.
Table 1. Hot water comfort indicators from the literature.
Reference Indicator Name Observations
Porteiro et al. [21]
Thermal Discomfort Index
Indicates the impact that users overcome due to the remote
intervention of the electricity company on the ESWH, which may cause
the water temperature to fall below the comfort temperature.
Non-normalized indicator, in [
◦
C
·
s], the higher its value, the higher the
priority given to a particular ESWH. Penalization factor may be
applied to identify which users should not suffer interruptions.
Wu et al. [22] Comfort index value
Amount of time there is no mixing of cold water and water from the
tank since the temperature in the tank is below the comfort/expected
temperature. Non-normalized indicator, with positive values in [◦C]
Najafi and Fripp [16] Discomfort function Based on temperature and volume of water. Nonlinear: no discomfort
if temperature above a threshold, but discomfort if below.
Belov et al. [23] Thermal discomfort
Thermal discomfort is seen as individual-dependent and depending on
the difference between tap water temperature and user-desired
temperature. Non-normalized indicator, dimensionless.
Kapsalis and Hadellis [24]
Kapsalis et al. [25]Normalized comfort cost
Difference of actual bath water temperature from the preferred
temperature at a time slot, multiplied by mass flow rate at the same
time slot, normalized with the maximum mass flow rate during a day
and temperature difference between preferred temperature and
minimum acceptable temperature. Values ranging between 0 and 1.
Shen et al. (2021) [26] Comfort fulfilment
Time at which the temperature of water draw-offs is above a certain
value, from the total time with draw-offs, in percentage (from
0 to 100%).
Shi et al. [27] Comfort index
Comfort treated as a constraint to be maximized, and ranging between
0 and 1.
Tejero-Gómez and
Bayod-Rújula [28]Discomfort coefficient Calculated as ratio of tank temperature to maximum temperature
achievable, and ranging between 0 and 1.
Barja-Martínez et al. [29] Discomfort cost Savings due to shifting the load multiplied by the time delay.
Non-normalized, and values in [€].
Demand response applied to ESWHs has been developed for close to 30 years, to the
best of our knowledge. One of the first approximations to a demand–response control for a
system of residential water heaters was described in 1996 by Dolan et al. [
30
]. Since then,
demand–response works have evolved, incorporating, e.g., learning-based techniques or
genetic algorithm optimizations [
6
]. There are mainly two approaches to demand response
and ESWHs: either a power grid-oriented approximation, focused on clustering heaters
and regulating them remotely to accommodate demand and electricity production [
8
,
30
,
31
],
or an end-user-oriented approximation, with objectives such as energy cost reduction,
thermal comfort or maximized use of electricity generated onsite, e.g., [
25
,
32
–
34
]. Even
some studies that belong to the first group, with grid focus, claim to keep user comfort
and cost optimization under consideration [
27
,
35
], since the lack of comfort control in the
approach aggregating ESWHs for grid balance was seen as a concern [
32
]. Table 2collects
different works on the subject, summarizing their main findings and conclusions.
The present study could be classified into the user-oriented approach, being the
objective to evaluate the potential in terms of cost savings by applying a demand–response
strategy in the current context in Spain with day-ahead, dynamic electricity prices, and
with user comfort in the multiobjective optimization. Considering the state of the art
discussed in previous paragraphs, one of the main novelties in the present article is that the
Energies 2023,16, 4104 4 of 25
optimization is based on an experimentally validated two-volume model that represents
more accurately the temperature of water draw-offs from the ESWH if compared to the
single-volume approach. Moreover, it considers yearly operation to evaluate the actual
potential, which also represents better different situations in the electricity market and on
user behavior (intraweek variations, seasonal effects, holidays). Different analyses were
performed to understand how different aspects and parameters affect the demand–response
strategy and the benefits that could be achieved.
Table 2.
Demand response with ESWHs in the literature. “U” stands for user-oriented approach, and
“G” for grid-oriented approach.
Reference U vs. G ESWH Model Observations
Dolan et al. [30] G Single-volume
Disabling of different percentages of water heaters for specific time
intervals, aiming at reducing the peak power. Peak power
reductions up to 25%, unless off-periods are too long.
Nehrir et al. [31] G Single-volume
Shift peak power demand to off-peak periods. Effective strategy to
leveling power demand profiles. Participation of customers is
needed, more effective with financial incentives, real-time
pricing, etc.
Paull et al. [32] U Single-volume
Water usage modelling based on past electricity consumption
household data. Possible to distinguish between heat losses and
draw-offs.
Diao et al. [8] G Two-volume
Central controller can modify temperature setpoint, which was
found as very effective, and ON/OFF signal for emergency support
and with random delay reconnection.
Passenberg et al. [33] U Single-volume
Cost minimization linked to PV generation. Legionella considered
a constraint. Algorithm guarantees sufficient hot water (comfort)
even with uncertain hot water draw-off profiles. Effective use of
forecast on weather, PV-generation, water and electricity demands
and electricity prices allow energy cost reductions.
Lin et al. [6] U Single-volume
Day-ahead, dynamic electricity price framework. ARIMA selected
to forecast customer water demand pattern. Genetic
algorithm-based optimization, with cost minimization constrained
by heating element maximum output and the ESWH capability
(linked to comfort). Balanced energy consumption throughout the
day, cost savings reaching 49% and almost negligible unavailability
of hot water.
Kapsalis and Hadellis [24]
Kapsalis et al. [25]U Single-volume
Heuristic algorithm scheduling ESWH operation with dynamic
electricity pricing. Optimization which can be steered towards
either cost or comfort by user preference. [25] adds condition to
optimization, namely maximum number of low-temperature time
slots allowed. Positive results were reported.
Tabatabei and Klein [36] U Single-volume
Demand–response control based on overheating the ESWH right
before peak price hours, to have available energy for consumption.
Not positive from energy-efficiency perspective due to increased
energy losses [4], but financially beneficial during cold months.
Results depend on user profile (shower vs. nonshower).
Barja-Martínez et al. [29] U Single-volume
Unknown temperature in tank and draw-off profile, decisions
based on predictions, conservative approach to guarantee hot water
availability and user comfort. Multiobjective minimization based
on electricity cost and discomfort, and proposes end user selectable
parameters concerning comfort. 7% savings were attained.
Booysen et al. [34] U Single-volume
Known water draw-off patterns, evaluation of different optimal
control strategies alternative to thermostat control, in some case
considering Legionella sterilization. Focus is on energy
optimization (not cost), and they ranged between 8% to 18%,
depending on the method chosen.
Wu et al. [22] G & U Single-volume
Multiobjective optimization, with weighting factors for cost and
comfort. Very different results depending on these factors. Users
follow grid requirements by price incentive during off-peak hours.
Energies 2023,16, 4104 5 of 25
Table 2. Cont.
Reference U vs. G ESWH Model Observations
Najafi and Fripp [16] U Single-volume
Multiobjective optimization for the control of ESWHs based on
electricity cost and discomfort. Stochastic definition of water
draw-offs, joint probability random distribution.
Shen et al. (2021) [26] U Single-volume
Forecast of DHW draw-off profiles based on data from South
African dwellings. To account for the uncertainties on future
draw-offs, a range of probable DHW consumption rates is
calculated. Objective to minimize electricity cost, two electricity
tariff levels (off peak, on peak), thermal comfort to be maintained.
30% cost reductions were achieved with comfort fulfilment rate of
about 99%.
Tejero-Gómez and
Bayod-Rújula (2021) [28]U Single-volume
Low-cost energy management system for ESWHs. Dynamic pricing
tariff, DHW consumption probability based on previous data.
Maximum temperature reached during off-peak hours for
sterilization against Legionella. Recalculations on the optimized
heating patter applied if unexpected draw-offs occur or deviations
in the expected temperature happen. Savings over 30% annually.
Porteiro et al. (2021) [21] G & U Single-volume
Grid balancing with aggregated ESWHs is the focus (90%
penetration in Uruguay for DHW supply, thus great potential).
Comfort is considered in the optimization, and special consumers
are not to be interrupted.
Shi et al. (2022) [27] G & U Single-volume
Electric water heaters respond to power grid company
requirements, but in the meantime optimizing the electricity cost
and maximizing comfort index for a single heater.
Clift et al. (2023) [35] G & U N-nodes
Aggregated control of ESWHs to balance grid while looking into
price optimization for consumers. Tank with two heating elements,
top (emergency heating) and bottom. Authors indicate that this
configuration is beneficial. Relevant individual cost savings, with
large potential for grid balancing at national level.
2. Materials and Methods
2.1. ESWH Models
The definition of a demand–response strategy applied to ESWHs requires a sufficiently
accurate thermodynamic model that captures the behaviors occurring in it, but simple
enough to reduce the computation time of the optimization problem. As detailed in Table 2,
most previous works have utilized a single-volume (also known as single-node or fully-
mixed) approximation, with very few exceptions. The first impression when evaluating
data from an actual ESWH would be that there is a certain level of stratification, even
with rather small volumes, that cannot be appropriately modelled with the single-volume
approach. In this line, Clift et al. (2023) [
35
] pointed out that ignoring stratification reduces
the demand–response capacity by 34%, increasing modelled energy costs by 21%.
To confirm this, three models were prepared, and evaluated against experimental data
with a setup described below and shown in Figure 1.
•
Single-volume model. Its main advantage is simplicity, considering fully mixing of the
whole water volume, but misrepresents the real behavior of water storage, neglecting
stratification and underestimating the temperature of water draw-offs. In this case,
implemented as indicated by Paull et al. [37].
•
Partial differential equation model. It considers spatial (vertical) discretization of the
ESWH in n volumes, each of these fully mixed. Thus, it accounts for stratification,
buoyancy effects, heat transfer by conduction between volumes, etc. In this analysis,
the model used was as suggested by Lago et al. [
38
], considering slow buoyancy effects
via max function, which the authors claim to be acceptable for short enough time
discretization (time step length
∆
t). A disadvantage with this model is that it is more
computationally intensive, which could be an issue for optimization problems.
Energies 2023,16, 4104 6 of 25
•
An intermediate approach between the two previous options would be the two-
volume (two-node) model. To the best of our knowledge, it was first suggested by Diao
et al. [
8
], and it assumes a hot volume, at a temperature close to the ESWH temperature
setpoint, and a cold volume at approximately the mains water temperature. The hot
part increases or decreases in volume (height) depending on the relation between the
heat input and the water draw-offs at the given time step. Correspondingly, the cold
section decreases or increases in volume, respectively, filling the remaining part of
the tank. When the tank is full, i.e., the hot volume equals the total volume, then it is
evaluated as in the single-volume model.
Energies 2023, 16, x FOR PEER REVIEW 6 of 26
the relation between the heat input and the water draw-os at the given time step.
Correspondingly, the cold section decreases or increases in volume, respectively,
lling the remaining part of the tank. When the tank is full, i.e., the hot volume equals
the total volume, then it is evaluated as in the single-volume model.
(a)
(b)
Figure 1. (a) Simplified sketch of the ESWH utilized for the validation tests, also indicating parameters
required for the definition of the numerical models. (b) Picture of the ESWH used for the validation
test campaign.
The two-volume approach has been seen by the authors of the present article as a
positive compromise between simplicity and rather accurate representation of the process
occurring in the ESWH. However, some areas for improvement were identied in the
formulation by Diao et al. [8]:
• The heating process described in the article, particularly after discharge of the tank,
represents quite precisely the process existing in tanks charged with a hot water
stream through the top port and from an external source, but not as much in the
ESWH. In an ESWH, when there is a draw-o, cold water enters the boom of the
tank and pushes the hot volume to the top. The heating element, typically located at
the boom, will heat up mainly the cold volume until it reaches the temperature of
the hot volume, and then all the water in the tank will continue with the heating
process (if needed). An example of this is shown in the Appendix A, as Figure A1.
• There is no conduction between hot and cold volumes.
• Two problem formulations (single-volume and two-volume) need to be considered,
switching between them depending on the tank charging level.
We tried to address these challenges in the modied two-volume model presented in
the following paragraphs. As represented in Figure 1a, the ESWH is still divided into two
volumes, hot volume (V1), with a height equal to h, and cold volume (V2), with a height
equal to L − h, L being the total length of the tank. In a discretized formulation, the changes
in temperature for each volume from the current time step, t, to the next, t + 1, with a time
step length equal to Δt, are as represented in Equations (1) and (2).
FM
di
L
hV1
V2
TO
TB
TT
MAINS
WATER
W
HOT
WATER
HEATER
Figure 1.
(
a
) Simplified sketch of the ESWH utilized for the validation tests, also indicating parameters
required for the definition of the numerical models. (
b
) Picture of the ESWH used for the validation
test campaign.
The two-volume approach has been seen by the authors of the present article as a
positive compromise between simplicity and rather accurate representation of the process
occurring in the ESWH. However, some areas for improvement were identified in the
formulation by Diao et al. [8]:
•
The heating process described in the article, particularly after discharge of the tank,
represents quite precisely the process existing in tanks charged with a hot water stream
through the top port and from an external source, but not as much in the ESWH. In an
ESWH, when there is a draw-off, cold water enters the bottom of the tank and pushes
the hot volume to the top. The heating element, typically located at the bottom, will
heat up mainly the cold volume until it reaches the temperature of the hot volume,
and then all the water in the tank will continue with the heating process (if needed).
An example of this is shown in the Appendix A, as Figure A1.
•There is no conduction between hot and cold volumes.
•
Two problem formulations (single-volume and two-volume) need to be considered,
switching between them depending on the tank charging level.
Energies 2023,16, 4104 7 of 25
We tried to address these challenges in the modified two-volume model presented in
the following paragraphs. As represented in Figure 1a, the ESWH is still divided into two
volumes, hot volume (V
1
), with a height equal to h, and cold volume (V
2
), with a height
equal to L
−
h,Lbeing the total length of the tank. In a discretized formulation, the changes
in temperature for each volume from the current time step, t, to the next, t+ 1, with a time
step length equal to ∆t, are as represented in Equations (1) and (2).
TV1,t+1=TV1,t+∆t·
Pheater,V1+U·AV1·Tamb −TV1,t+kV1·Ac·TV2,t−TV1,t
h
cpV1·Ac·h·ρV1
(1)
TV2,t+1=TV2,t+∆t·
Pheater,V2+U·AV2·(Tamb−TV2,t)−.
m·cp,m·(TV2,t−Tm)+kV2·Ac·TV1,t−TV2,t
L−h
cpV2·Ac·(L−h)·ρV2!(2)
The different symbols used in these equations are explained in the list below:
•
C
p
[J kg
−1
K
−1
], water specific heat capacity (isobaric), evaluated at V
1
,V
2
, and the
mains (m) water temperatures.
•
U[W m
−2
K
−1
], overall heat transfer coefficient for the tank, representing heat transfer
between the tank and surroundings.
•
A
V1
and A
V2
[m
2
], heat transfer area with the surroundings for the hot and cold
volumes, respectively. The ESWH was approximated as a perfect cylinder with inner
diameter di.
•Tamb [◦C], ambient (surroundings to the ESWH) temperature.
•.
m[kg s−1], water draw-off mass flow rate.
•Tm[◦C], mains water temperature.
•k[W m−1K−1], water thermal conductivity, evaluated at V1and V2temperatures.
•Ac[m2], tank cross sectional area, evaluated with inner diameter diin Figure 1a.
•ρ[kg m−3], water density, evaluated at V1and V2temperatures.
•
The heating element power (P
heater
), if activated during the given time step, would be
distributed between the hot (P
heater,V1
) and cold (P
heater,V2
) volumes, depending on
the value of h, the heating element length (L
heater
) and its position within the ESWH,
i.e., it will be very much dependent on the tank characteristics and even on the relative
orientation heating element/tank. As seen in Equations (3) and (4), it was assumed
that the heating element capacity is linearly distributed along its length (L
heater
), and
it is positioned parallel to the tank axis, and as close to the bottom as possible.
Pheater,V1=max(0, Pheater·(h−L+Lheater)/Lheater )(3)
Pheater,V2=Pheater −Pheater,V1(4)
The change of height of the hot volume, h, after the time step and due to water
draw-offs, would be calculated as in Equation (5).
ht+1=ht−.
m
ρV1·Ac·∆t(5)
After each time step, the model checks if the temperature of the cold volume V
2
, due
to the effect of the heater input, has become higher or equal to the temperature of the hot
volume V
1
, i.e., T
V1,t+1 ≤
T
V2,t+1
. If that is the case, h=L, the whole tank is considered to
be at equal temperature until there are new water draw-offs.
As observed in previous paragraphs, the model formulation indicated above assumes
perfect stratification, being the only interaction between the hot and cold volumes due
to conduction through the interface. However, the validation tests detailed in Section 2.2
Energies 2023,16, 4104 8 of 25
suggest a higher degree of interaction between the volumes, and a mixing factor is defined
in that section, calibrating the numerical model so as to represent reality more accurately.
The control of the heating element (ON/OFF) in the base case (thermostat control)
would be according to the temperature at the thermostat position, activating the heating ele-
ment when this temperature falls below the setpoint (T
set
) minus a certain offset (
∆
T
ON/OFF
)
and deactivating when the temperature rises above the setpoint. From the perspective of
the two-volume model described here, this would imply that T
V2,t+1
should be considered
when the hot volume has not reached the position of the thermostat, i.e.,
h< (L−htherm)
,
while T
V1,t+1
would be the value for comparison otherwise, i.e.,
h≥(L−htherm)
. In a
demand–response approach as suggested in the present article, the heating element sta-
tus will be decided from a day-ahead optimization considering the final conditions from
the previous day, the daily profile assumed as known and according to dynamic electric-
ity pricing.
Water properties utilized in the models (density, thermal conductivity, specific heat
capacity) were initially taken from the database CoolProp [
39
] through the wrapper for
Python. However, several function calls to CoolProp per time step have a strong impact on
the computation time. Considering the temperature range needed for the current study,
roughly from 10
◦
C to 80
◦
C, it was more efficient computationally to implement specifically
developed correlations for density (Equation (6)) and thermal conductivity (Equation (7))
in the form of quadratic equation functions of the water temperature (in Kelvin), in both
cases with an R
2
> 0.999. In the case of the specific heat capacity, a constant value of
4186 J/kg
−1
K
−1
was assumed due to the very slight fluctuation of this property within the
given temperature range (within ±0.25%).
ρ=748.925 +1.921·T−3.563·10−3·T2(6)
k=−7.475·10−1+7.442·10−3·T−9.734·10−6·T2(7)
2.2. Experimental Setup and Validation Tests
An experimental campaign was performed with two objectives in mind: (i) validation
of the two-volume numerical model developed within this study, and (ii) selection of
the most suitable alternative from the models described at the beginning of the section,
i.e., single-volume, two-volume, or partial differential equation, that could be further
used to evaluate the potential of an optimized demand–response strategy for ESWHs.
Figure 1b shows a picture of the setup, being the ESWH the main component, modified
to accommodate two temperature sensor pockets with an approximate length of 100 mm
and located at around 100 mm from the top and 200 mm from the bottom. A digital
temperature sensor was located in each temperature pocket (TT and TB in Figure 1a). The
ESWH was connected to the mains water (cold water supply), and the hot water that would
be delivered to a user (water draw-off) was measured by a dedicated flow meter (FM in
Figure 1a). The power input to the heating element was measured through an active power
meter. The most important features about the setup and its components and sensors are
included in Table 3.
The characteristics of the ESWH available for the validation experiments were used
for the definition of the numerical model and for experimental validation. A dedicated test
was performed to evaluate the heat losses from storage to the ambient, in which the water
in the tank was heated from cold conditions to approximately 67
◦
C, and then it was left
to ambient conditions (T
amb ≈
20
◦
C) with the heating element disabled. The water in the
tank decreased approximately 4
◦
C in almost 6 h, i.e., the resulting overall heat transfer
coefficient was 1.36 W m−2K−1.
Several tests were performed to define the suitability of the three ESWH models
indicated in the previous subsection. A detailed account of these tests and comparison
with each individual model is included in Appendix A, but for the sake of concision, only
some aspects are summarized in the core of this article.
Energies 2023,16, 4104 9 of 25
Table 3. Main characteristics of the components and sensors from the experimental setup.
Device Type Value/Range Observations
ESWH, volume 76 L According to the user manual.
ESWH, di370 mm
ESWH, L695 mm
Thermostat height, htherm 50 mm From the bottom.
Heating element length, Lheater 140 mm From the bottom.
Heating element power, Pheater, max 1.95 kW Measured. Rated value is 2 kW.
Temperature sensors, TT & TB 1-wire digital thermometer −10 ◦C–85 ◦C Accuracy ±0.5 ◦C
Flow meter, FM Vortex 1–30 L/min Accuracy ±3%
Active power meter, W Single-phase, with clamp 230 Vac/50 A Accuracy ±1%
Temperature sensor in FM, TO NTC 50K −40 ◦C–85 ◦C Accuracy ±1◦C
•
Single-volume model. It provides a positive representation of the heating process, but
fails to simulate the temperature of water draw-offs from the top of the tank due to
the full-mixing consideration.
•
Partial differential equation model (10 nodes). If the time step length is sufficiently
short (
∆
t= 1 s), there is a very accurate representation of heating and draw-off pro-
cesses. However, such short time step lengths are unpractical for daily optimizations,
and with longer periods, the slow-buoyancy approach followed would not appropri-
ately represent the heating process.
•
Two-volume model. The first approximation, as described in the previous subsection,
was by considering that the two volumes, hot and cold, were perfectly separated,
with perfect stratification. However, the results from a validation test with several
1 min water draw-offs of approximately 6 L/min each from a fully-charged tank and
with the heating element deactivated (Figure 2a) showed that, after the eighth cycle,
water from the tank was no longer at 60
◦
C. This is described by the purple dots in the
figure (T_TO_Test), corresponding to temperature measurements in the flow meter
downstream of the tank (please keep in mind that the temperature measurements in
the flow meter are only representative when there is flow). In other words, a 76 L
tank cannot deliver many more than 50 L at the setpoint conditions or above (in this
case 60
◦
C). In terms of energy draw-off (mains water temperature at around 14
◦
C),
a value of 3.59 kWh was calculated from the test, while 4.14 kWh (+15.4%) came
out from the numerical model. Moreover, the cold volume temperature measured
(T_TB_Test) was significantly higher than the mains water temperature (represented in
the model T_TB_Model). Both the lower effective capacity of the tank and the higher
cold-volume temperature could be explained by (i) a certain level of mixing between
hot and cold volumes due to disturbed stratification, and (ii) heat transfer between
the cold volume and the water delivered from the top of the tank, since the pipe is
arranged from top to bottom and the connections are at the bottom (Figure 1b). This
second effect should increase in time as the cold volume becomes larger (deactivated
heating element in this case).
To account for these effects, a mixing factor MF is suggested, which would recalculate
the temperatures after each time step for the hot and cold volumes as represented in
Equations (8) and (9), respectively.
TV1,t+1,MF =Ac·h·ρV1−MF·.
m·∆t·TV1,t+1+MF·.
m·∆t·TV2,t+1
Ac·h·ρV1
(8)
TV2,t+1,MF =MF·.
m·∆t·TV1,t+1+Ac·(L−h)·ρV2−MF·.
m·∆t·TV2,t+1
Ac·(L−h)·ρV2
(9)
Energies 2023,16, 4104 10 of 25
Energies 2023, 16, x FOR PEER REVIEW 10 of 26
(a)
(b)
Figure 2. Comparison between experimental data and the two-volume model for a test with several
water draw-os starting from hot conditions and with the heating element disabled. (a) Numerical
model considered perfect stratication. (b) Model with a mixing factor equal to 0.2. Solid lines
represent results from the model, while dots are datapoints from the tests. T_TT, T_TB and T_TO
represent temperatures at the upper and lower locations in the tank, and at the ow meter
downstream of the tank, respectively, while FR is the water draw-o ow rate.
To account for these eects, a mixing factor MF is suggested, which would recalculate
the temperatures after each time step for the hot and cold volumes as represented in
Equations (8) and (9), respectively.
(8)
(9)
As shown in Figure 2b, a mixing factor equal to 0.2 would beer approximate the
model to the actual behavior of the ESWH available for validation. This is even clearer in
terms of energy aained from the water draw-os. According to the corrected model (with
mixing factor = 0.2), the energy adds up to 3.6 kWh, only 0.3% higher than in the validation
test. Moreover, the “mixing” eect also leads to a closer approach to the actual
temperature measured in the cold part of the tank (T_TB_Model vs. T_TB_Test).
So far in the validation, the time step length considered in the models was equal to
1 s (period between measurements in the validation test campaign 10 s). However, daily
optimizations with 1 s time step length would be unpractical due to the high computation
time and eort. Thus, a sensitivity analysis of this parameter Δt was performed, showing
that, up to 30 s, dierences between numerical and experimental data in terms of energy
delivered from the tank were contained (at +2.5%), but increasing to one minute would
involve discrepancies above 10%.
In conclusion, the two-volume model with a mixing factor equal to 0.2 and a time
step length of 30 s was considered in the evaluations within this study. More information
about the validation of the models and the criteria used for the selection of the right model
and of the mixing factor are included in Appendix A.
2.3. Denition of Base Case and Optimized Case with Demand–Response Strategy
The main objective of this study is to evaluate the potential of implementing a
demand–response strategy for an ESWH, considering dynamic electricity pricing and
minimization of user discomfort. Thus, this subsection describes several crucial aspects in
Figure 2.
Comparison between experimental data and the two-volume model for a test with several
water draw-offs starting from hot conditions and with the heating element disabled. (
a
) Numerical
model considered perfect stratification. (
b
) Model with a mixing factor equal to 0.2. Solid lines
represent results from the model, while dots are datapoints from the tests. T_TT, T_TB and T_TO
represent temperatures at the upper and lower locations in the tank, and at the flow meter downstream
of the tank, respectively, while FR is the water draw-off flow rate.
As shown in Figure 2b, a mixing factor equal to 0.2 would better approximate the
model to the actual behavior of the ESWH available for validation. This is even clearer in
terms of energy attained from the water draw-offs. According to the corrected model (with
mixing factor = 0.2), the energy adds up to 3.6 kWh, only 0.3% higher than in the validation
test. Moreover, the “mixing” effect also leads to a closer approach to the actual temperature
measured in the cold part of the tank (T_TB_Model vs. T_TB_Test).
So far in the validation, the time step length considered in the models was equal to
1 s (period between measurements in the validation test campaign 10 s). However, daily
optimizations with 1 s time step length would be unpractical due to the high computation
time and effort. Thus, a sensitivity analysis of this parameter
∆
twas performed, showing
that, up to 30 s, differences between numerical and experimental data in terms of energy
delivered from the tank were contained (at +2.5%), but increasing to one minute would
involve discrepancies above 10%.
In conclusion, the two-volume model with a mixing factor equal to 0.2 and a time step
length of 30 s was considered in the evaluations within this study. More information about
the validation of the models and the criteria used for the selection of the right model and of
the mixing factor are included in Appendix A.
2.3. Definition of Base Case and Optimized Case with Demand–Response Strategy
The main objective of this study is to evaluate the potential of implementing a demand–
response strategy for an ESWH, considering dynamic electricity pricing and minimization
of user discomfort. Thus, this subsection describes several crucial aspects in this analysis,
such as the dynamic electricity pricing scenario chosen, the DHW consumption profile
definition and constraints and setpoints in the evaluation or the optimization problem.
2.3.1. Dynamic Electricity Pricing
Dynamic electricity pricing is an approach to transfer part of the wholesale market
volatility to end users, and is becoming a reality for households and small commercial
customers in an increasing number of countries due to the more generalized availability
of smart meter data and a more efficient market. Thus, users are penalized with high
electricity prices when there is larger demand and low availability of renewable (cheap)
energy, and they can benefit when demand is low and there is high electricity production
from renewables. This is key for demand response and demand flexibilization at the user
Energies 2023,16, 4104 11 of 25
side. In the present study, the case of Spain was considered, based on hourly tariffs for small
consumers with contracted power below 10 kW. The information for the year 2022 was
considered, obtained from a public database [
40
] and applying the general electricity tax
(5.1127%) and VAT (21%). Currently, these taxes are exceptionally low, namely 0.5% and 5%,
respectively, due to the current situation in Europe, but it is expected that they will return to
general values in the near future. Figure 3a illustrates the daily mean electricity cost (taxes
included) throughout 2022 and the standard deviation for each day, thus representing the
volatility of the market. Figure 3b represents, on the other hand, the hourly tariffs for four
days used as examples.
Energies 2023, 16, x FOR PEER REVIEW 11 of 26
this analysis, such as the dynamic electricity pricing scenario chosen, the DHW
consumption prole denition and constraints and setpoints in the evaluation or the
optimization problem.
2.3.1. Dynamic Electricity Pricing
Dynamic electricity pricing is an approach to transfer part of the wholesale market
volatility to end users, and is becoming a reality for households and small commercial
customers in an increasing number of countries due to the more generalized availability
of smart meter data and a more ecient market. Thus, users are penalized with high
electricity prices when there is larger demand and low availability of renewable (cheap)
energy, and they can benet when demand is low and there is high electricity production
from renewables. This is key for demand response and demand exibilization at the user
side. In the present study, the case of Spain was considered, based on hourly taris for
small consumers with contracted power below 10 kW. The information for the year 2022
was considered, obtained from a public database [40] and applying the general electricity
tax (5.1127%) and VAT (21%). Currently, these taxes are exceptionally low, namely 0.5%
and 5%, respectively, due to the current situation in Europe, but it is expected that they
will return to general values in the near future. Figure 3a illustrates the daily mean
electricity cost (taxes included) throughout 2022 and the standard deviation for each day,
thus representing the volatility of the market. Figure 3b represents, on the other hand, the
hourly taris for four days used as examples.
(a)
(b)
Figure 3. (a) Daily mean and standard deviation of the electricity cost in Spain throughout 2022. (b)
Hourly uctuations of the electricity cost for four days in 2022.
2.3.2. DHW Consumption Prole
As suggested by dierent studies [6–8], any demand–response strategy linked to an
ESWH is highly dependent on the DHW consumption proles for the specic user.
Among the dierent alternatives for the denition of these proles, that resulting from
the works by Jordan and Vajen [14,15] was used as the baseline in the current study. These
works consisted of measurements of DHW consumption paerns in Germany and
Swierland obtained within the scope of the Solar Heating and Cooling Program of the
International Energy Agency (IEA SHC), Task 26. Based on these data, Jordan and Vajen
used a stochastic approach to determine 1 min proles for which the probability of ow
rate and time of occurrence would depend on the type of load (bath, shower, medium and
short), the intraday probability, the intraweek probability, the seasonal probability and
the existence of holiday periods. According to Jordan and Vajen [14,15], in their default
case, the average daily DHW consumption at 45 °C throughout the year was equal to 200
Figure 3.
(
a
) Daily mean and standard deviation of the electricity cost in Spain throughout 2022.
(b) Hourly fluctuations of the electricity cost for four days in 2022.
2.3.2. DHW Consumption Profile
As suggested by different studies [
6
–
8
], any demand–response strategy linked to an
ESWH is highly dependent on the DHW consumption profiles for the specific user. Among
the different alternatives for the definition of these profiles, that resulting from the works
by Jordan and Vajen [
14
,
15
] was used as the baseline in the current study. These works
consisted of measurements of DHW consumption patterns in Germany and Switzerland
obtained within the scope of the Solar Heating and Cooling Program of the International
Energy Agency (IEA SHC), Task 26. Based on these data, Jordan and Vajen used a stochastic
approach to determine 1 min profiles for which the probability of flow rate and time of
occurrence would depend on the type of load (bath, shower, medium and short), the
intraday probability, the intraweek probability, the seasonal probability and the existence
of holiday periods. According to Jordan and Vajen [
14
,
15
], in their default case, the average
daily DHW consumption at 45
◦
C throughout the year was equal to 200 L/day. This
value properly matches the consumption profile L [
41
] for which the EWSH used in the
experimental validation was defined by its manufacturer. According to [
4
], an L tapping
pattern corresponds to 4–5 person family, with shower and some baths and peak tapping
at 60 ◦C equal to 201 L.
A different approach was established on the holiday definition between the current
study and the profile definition from Jordan and Vajen [
14
,
15
]. The latter sets two
14-day
periods for which the probability of consumption would sink to approximately 100 L/day.
However, this seems to be an arbitrary choice, which could depend on many factors
intrinsic to the household. A different approach was considered in the current study, which
the authors consider more up to date and appropriate for the case of Spain, with zero
consumption during 14 days in the year. The holiday period selected started 8 August 2022
in this study.
All this considered, a random profile of DHW consumption was generated for 2022.
Figure 4a represents the daily water draw-off (volume) throughout the year, while Figure 4b,
depicts the 1 min profile generated for 9 January 2022. It is important to remember that the
Energies 2023,16, 4104 12 of 25
DHW consumption according to these profiles is at 45
◦
C,
.
V45 ◦C
, meaning that there will
be a mixing downstream of the tank with the mains water. Thus, the actual draw-off from
the tank at the specific time,
.
VESWH
, will depend on the water temperature at the specific
time, T
V1,t
, and the mains water temperature, T
mains
, calculated according to Equation (10).
In case the temperature in the tank is below 45
◦
C, then the volumetric flow rate from the
tank will be equal to that indicated by the profile, i.e.,
.
VESWH
=
.
V45 ◦C
. Concerning the
mains water temperature, the case of A Coruña (Spain) was considered, with values that
range from 10 ◦C in January or February to 16 ◦C in July/August [42].
.
VESWH,t=.
V45 ◦C,t
45 −Tmains,t
TV1,t−Tmains,t
(10)
Energies 2023, 16, x FOR PEER REVIEW 12 of 26
L/day. This value properly matches the consumption prole L [41] for which the EWSH
used in the experimental validation was dened by its manufacturer. According to [4], an
L tapping paern corresponds to 4–5 person family, with shower and some baths and
peak tapping at 60 °C equal to 201 L.
A dierent approach was established on the holiday denition between the current
study and the prole denition from Jordan and Vajen [14,15]. The laer sets two 14-day
periods for which the probability of consumption would sink to approximately 100 L/day.
However, this seems to be an arbitrary choice, which could depend on many factors
intrinsic to the household. A dierent approach was considered in the current study,
which the authors consider more up to date and appropriate for the case of Spain, with
zero consumption during 14 days in the year. The holiday period selected started 8 August
2022 in this study.
All this considered, a random prole of DHW consumption was generated for 2022.
Figure 4a represents the daily water draw-o (volume) throughout the year, while Figure
4b, depicts the 1 min prole generated for 9 January 2022. It is important to remember that
the DHW consumption according to these proles is at 45 °C, 45 ℃, meaning that there
will be a mixing downstream of the tank with the mains water. Thus, the actual draw-o
from the tank at the specic time, ESWH, will depend on the water temperature at the
specic time, TV1,t, and the mains water temperature, Tmains, calculated according to
Equation (10). In case the temperature in the tank is below 45 °C, then the volumetric ow
rate from the tank will be equal to that indicated by the prole, i.e., ESWH = 45 ℃ .
Concerning the mains water temperature, the case of A Coruña (Spain) was considered,
with values that range from 10 °C in January or February to 16 °C in July/August [42].
ESWH,t45 ℃,tmains
mains
(10)
(a)
(b)
Figure 4. (a) Daily water draw-os generated for the study (at 45 °C). (b) 1 min DHW consumption
prole (at 45 °C) for an example day.
2.3.3. Demand–Response Strategy Based on Optimization
With the scenario of dynamic electricity pricing and DHW consumption proles
dened, the potential for a demand–response strategy was evaluated by comparing it with
a conventional control with a thermostat (base case) and considering the features of the
ESWH used for the validation of the numerical models (Table 3). For the control by
thermostat, a temperature setpoint, Tset, and oset for reactivation, ΔTON/OFF, were selected
as shown in Table 4, suciently high to meet Legionella sterilization. For the optimized
demand–response strategy, the procedure will be stated in the following paragraphs.
Figure 4.
(
a
) Daily water draw-offs generated for the study (at 45
◦
C). (
b
) 1 min DHW consumption
profile (at 45 ◦C) for an example day.
2.3.3. Demand–Response Strategy Based on Optimization
With the scenario of dynamic electricity pricing and DHW consumption profiles
defined, the potential for a demand–response strategy was evaluated by comparing it with a
conventional control with a thermostat (base case) and considering the features of the ESWH
used for the validation of the numerical models (Table 3). For the control by thermostat, a
temperature setpoint, T
set
, and offset for reactivation,
∆
T
ON/OFF
, were selected as shown
in Table 4, sufficiently high to meet Legionella sterilization. For the optimized demand–
response strategy, the procedure will be stated in the following paragraphs.
Table 4. Values given to different parameters in the evaluation and optimizations.
Parameter Value Observations
Setpoint thermostat control, Tset 65 ◦C
Offset reactivation, ∆TON/OFF 5◦C
Daily sterilization condition [34], TLeg 60 ◦C
At least once a day, during 11 min.
Maximum water temperature, Tmax 80 ◦C Avoid scaling.
Comfort temperature, Tcomf 45 ◦C
Time step length for optimization,
∆
t
opt 1 h Electricity price changes by hour.
Time step length model calculation, ∆t30 s
Two indexes were defined for comparison and optimization purposes:
•
Cost Index, CI, for a specific day and as formulated in Equation (11). It is linked to
the electricity cost at a given hour of the day, c
el,i
, and the heating element factor of
utilization for the same hour, FU
i
. This factor of utilization is defined by the control
strategy in each case (thermostat or optimized for demand response) and understood
Energies 2023,16, 4104 13 of 25
as the fraction of the rated heating element power used during that hour. The CI is
normalized with the cost of running the heating element continuously during that day.
CIday =∑24
i=1cel,i·Pheater,max·FUi
∑24
i=1cel,i·Pheater,max
(11)
•
Discomfort Index, DI, for a specific day and as described in Equation (12). It is
calculated as the sum of the volumetric flow rate in a given time step (at 45
◦
C)
multiplied by the temperature deficit (if any) between the water in the tank and the
comfort temperature, T
comf
, for the same time step. The DI is normalized through the
total water volume to be delivered during that day (also at 45
◦
C) and the temperature
difference between the comfort temperature and the mains water temperature that day.
The number of time steps equals 2880 each day, since the
∆
t= 30 s was considered.
Concerning the comfort temperature, it was assumed as equal to 45
◦
C, according to
references [23,35], even if other authors indicate lower comfort temperatures [22,28].
DIday =
∑steps
i=1
.
V45 ◦C,i·∆t·max0, Tcomf −TV1,i·∆t[◦C]
∑steps
i=1
.
V45 ◦C,i·∆t·Tcomf −Tmains,day (12)
In the case of the optimization, these two indicators are related through a savings
index, SI, which is a weight factor to be selected by the end user and with values between 0
and 1. The savings index defined in the present article is used as weight to CI and DI in
the optimization function f(function to be minimized), as shown in Equation (13). The
lower the savings index, the higher the priority the user gives to discomfort minimization,
and vice versa. A very similar definition could be found in Wu et al. [
22
], with the main
difference that the authors also defined a scale factor
µ
to the cost index. This weighting
approach also resembles that suggested by Kapsalis and coworkers [
24
,
25
], but in the
present study the algorithm does not sweep different weights until it finds the optimum,
i.e., a sufficiently high weight factor with minimum impact on comfort.
f=SI ·C I +(1−S I)·DI (13)
The variable used in the optimization was the heating element utilization factor, FU
i
,
for the optimization time step length,
∆
t
opt
= 1 h. As the reader may point out, different
time steps lengths were considered for the optimization,
∆
t
opt
, and the numerical model
calculation,
∆
t. The reason for this was to reduce the computation time, and it is also
justified by the fact that electricity prices change on an hourly basis. An alternative to
reduce computation time would have been to increase the numerical model time step
length,
∆
t, but this was disregarded to avoid losing accuracy with the model. It is also
worth pointing out that the heating element typically operates as an ON–OFF device. From
a practical point of view, the utilization factor could be implemented in two ways: (i) with
a power regulator, so that the heater output would be equal to the rated power multiplied
by the factor of utilization during the optimization time step length, or (ii) through the
intermittent operation of the heating element, activated for a total time equal FU
i·∆
t
opt
,
and deactivated during (1 −FUi)·∆topt.
The temperature of the tank in the optimization problem should be constrained due to
two aspects. On the one hand, Legionella sterilization, which was defined as in [
34
], by
reaching a water temperature of 60
◦
C and keeping it for at least 11 min. Other studies claim
slightly different times or temperatures, e.g., 60
◦
C for 32 min (on a weekly basis) [
35
], or
65
◦
C on a daily basis [
43
]. On the other hand, the upper limit for the water temperature in
the ESWH was set to 80
◦
C, for two reasons. First, because scale formation is linked to high
temperatures, and scale and biofilms help on the survival of Legionella [
4
]. Second, because
higher water temperatures promote heat losses to the ambient air [
36
]. These constraints
Energies 2023,16, 4104 14 of 25
were implemented as additional weights to the optimization function (Equation (13)), so
that the optimization solutions not fulfilling them would be automatically discarded.
To summarize, the day-ahead optimization could be now performed considering
that: (i) the water draw-off profile was assumed as known, (ii) the hourly electricity
prices were available the previous day around 20:15–20:30 in the evening, (iii) the function
to be minimized was defined with a cost index, a discomfort index and both weighted
through a savings index, (iv) temperature constraints to the optimization were introduced
in the problem as strong penalizations to the optimization function and (v) the heating
element utilization factor was used as the optimization variable. The differential evolution
optimization method in Python was utilized for this optimization (package scipy.optimize),
with a maximum number of iterations equal to 200 (never needed), population size equal to
5, tol (relative tolerance) equal to 0.01 (default value) and atol (absolute tolerance) equal to
0.01. The final conditions of the tank for one day, namely hot and cold volume temperatures,
and hot volume height h, were considered as initial conditions for the optimization of the
following day.
3. Results and Discussion
3.1. Savings Index and Multiobjective Optimization
Figure 5represents the impact on the optimized objective function components, cost
index (CI) and discomfort index (DI), when sweeping the savings index, SI, from 0 to
0.9. The purpose is to evaluate quantitatively how a focus on savings, with an increase
in SI, penalizes comfort, and vice versa. As had been represented in Equation (13), CI is
multiplied in the objective function directly by the SI, while DI is multiplied by (1
−
SI).
Electricity cost data and DHW consumption profiles for the two first weeks in the year
(1–14 January) were considered in this evaluation. The discomfort indexes, DI, and cost
indexes, CI, indicated in Figure 5correspond to their mean daily values for the 14-day
period and resulting from the optimization for each savings index, SI (value next to each
dot in the graph). Unsurprisingly, the higher the priority given to comfort, the closer SI is
to 0, the lower the DI and the higher the operational costs, represented by the CI. On the
other hand, the acceptance of slight discomfort may lead to significant cost savings. As
an example, increasing the SI from 0 to 0.5 would involve a reduction of the CI close to
a 7%, with a DI (expressed in percentage) that rises from 0.7% to 1.5%. This had already
been observed by Wu et al. [
22
], who had a similar definition of the weighting factors (our
savings index), and found that the electricity cost could be reduced by a factor of 3.3 when
the optimization changed from a comfort focus to a cost focus.
Energies 2023, 16, x FOR PEER REVIEW 15 of 26
Figure 5. Discomfort index DI vs. cost index CI obtained from the multiobjective optimization and
as a function of the savings index SI, represented as the value linked to each dot.
Due to the temperature constraints imposed on the optimization problem, i.e.,
Legionella sterilization aspect of reaching 60 °C on a daily basis during at least 11 min,
and maximum of 80 °C to avoid scaling and heat losses, the optimization problem and the
results of CI or DI are not purely dependent of the SI. At the lowest end of the SI range
(from 0 to 0.5), there is a very limited eect on the DI, the reason being that the Legionella
constraint also maintains discomfort under a relative control. All this considered, 0.5 was
selected as the savings index for the yearly simulation and comparison with the
thermostat control discussed in Section 3.2.
3.2. Thermostat Control vs. Optimized Heater Control
Figure 6 gathers multiple graphs to compare the rather dierent behavior of the
ESWH with a conventional thermostat control strategy vs. an optimized demand–
response operation (SI = 0.5) and corresponding to an example day (9 January). It is an
interesting day since there is a bath, and thus the DHW consumption is approximately
double of the average daily value, namely 403 L at 45 °C during that day.
(a)
(b)
Figure 5.
Discomfort index DI vs. cost index CI obtained from the multiobjective optimization and as
a function of the savings index SI, represented as the value linked to each dot.
Due to the temperature constraints imposed on the optimization problem, i.e., Le-
gionella sterilization aspect of reaching 60
◦
C on a daily basis during at least 11 min, and
Energies 2023,16, 4104 15 of 25
maximum of 80
◦
C to avoid scaling and heat losses, the optimization problem and the
results of CI or DI are not purely dependent of the SI. At the lowest end of the SI range
(from 0 to 0.5), there is a very limited effect on the DI, the reason being that the Legionella
constraint also maintains discomfort under a relative control. All this considered, 0.5 was
selected as the savings index for the yearly simulation and comparison with the thermostat
control discussed in Section 3.2.
3.2. Thermostat Control vs. Optimized Heater Control
Figure 6gathers multiple graphs to compare the rather different behavior of the
ESWH with a conventional thermostat control strategy vs. an optimized demand–response
operation (SI = 0.5) and corresponding to an example day (9 January). It is an interesting
day since there is a bath, and thus the DHW consumption is approximately double of the
average daily value, namely 403 L at 45 ◦C during that day.
Energies 2023, 16, x FOR PEER REVIEW 15 of 26
Figure 5. Discomfort index DI vs. cost index CI obtained from the multiobjective optimization and
as a function of the savings index SI, represented as the value linked to each dot.
Due to the temperature constraints imposed on the optimization problem, i.e.,
Legionella sterilization aspect of reaching 60 °C on a daily basis during at least 11 min,
and maximum of 80 °C to avoid scaling and heat losses, the optimization problem and the
results of CI or DI are not purely dependent of the SI. At the lowest end of the SI range
(from 0 to 0.5), there is a very limited eect on the DI, the reason being that the Legionella
constraint also maintains discomfort under a relative control. All this considered, 0.5 was
selected as the savings index for the yearly simulation and comparison with the
thermostat control discussed in Section 3.2.
3.2. Thermostat Control vs. Optimized Heater Control
Figure 6 gathers multiple graphs to compare the rather dierent behavior of the
ESWH with a conventional thermostat control strategy vs. an optimized demand–
response operation (SI = 0.5) and corresponding to an example day (9 January). It is an
interesting day since there is a bath, and thus the DHW consumption is approximately
double of the average daily value, namely 403 L at 45 °C during that day.
(a)
(b)
Energies 2023, 16, x FOR PEER REVIEW 16 of 26
(c)
(d)
Figure 6. Graphs corresponding to an example day (9 January) and comparing the base case
(Thermostat) and the optimized demand–response strategy (Optimized). (a) Temperature in the
tank upper volume. (b) Height of the hot volume. (c) Hourly electricity cost. (d) Draw-o ow rates
for a portion of the day at which there is a bath.
Starting with the temperature prole throughout the day (Figure 6a), there are very
clear dierences between the thermostat control and the optimized demand–response
strategy. First, the starting temperatures (and also the level of charge of the tank according
to Figure 6b) are completely dierent. For the thermostat control, the temperature at the
top of the tank (hot volume) will only fall below the setpoint minus the temperature oset
in case of a very high consumption of DHW. On the other hand, the optimized demand–
response strategy from the previous day decided not to heat the water in the tank at the
end of that day, meaning that the tank starts rather cold the “current” day. This could be
seen as a point for improvement of the optimizing technique, since it may be actually more
cost ecient to heat up the tank the previous day, depending on the hourly electricity
prices of the previous and current days. In further works, the optimization could be re-
evaluated as soon as the following day electricity prices are available, typically around
20:15–20:30 in the evening.
If the thermostat-control temperature prole is analyzed in detail, it can be observed
that the temperature according to the model at the top (hot volume) is higher than the
actual setpoint for the thermostat (Tset = 65 °C). This is due to the fact that, after a short
water draw-o, the thermostat could be indicating that heating is needed (temperature
below Tset − ΔTON/OFF), even if the tank is almost fully charged (h not far from 0.7 m). As
explained in Section 2.1, the heating element capacity was assumed as linear through its
length, meaning that part of the heat input from the heating element would be heating up
the cold volume, to fulll the thermostat condition, while the other part would be heating
up the hot volume above the temperature setpoint.
The temperature prole for the optimized demand–response strategy for the
evaluated day starts at a very low temperature and charge level, and the tank is heated up
importantly in the rst hour, when there is a valley in electricity price (Figure 6c). Only
very small DHW consumptions would be expected in those rst hours, having a very low
impact on the DI. The large water draw-o happening around hour 204 (9 January, 12:00)
and corresponding to a bath is anticipated by the optimized strategy by increasing the
water temperature close to 80 °C to reduce the impact on discomfort, even if the cost of
electricity may not be as convenient at that time of the day. Figure 6d also shows that, due
to this higher temperature if compared to the thermostat control, a lower water ow rate
from the tank is needed to achieve the DHW ow rate requested at 45 °C. Another similar
example of anticipation to a relevant DHW consumption happens around hour 212 (9
January, 20:00), but in this case corresponding to a shower.
Figure 6.
Graphs corresponding to an example day (9 January) and comparing the base case (Thermo-
stat) and the optimized demand–response strategy (Optimized). (
a
) Temperature in the tank upper
volume. (
b
) Height of the hot volume. (
c
) Hourly electricity cost. (
d
) Draw-off flow rates for a portion
of the day at which there is a bath.
Starting with the temperature profile throughout the day (Figure 6a), there are very
clear differences between the thermostat control and the optimized demand–response
strategy. First, the starting temperatures (and also the level of charge of the tank according
to Figure 6b) are completely different. For the thermostat control, the temperature at the
top of the tank (hot volume) will only fall below the setpoint minus the temperature offset
in case of a very high consumption of DHW. On the other hand, the optimized demand–
response strategy from the previous day decided not to heat the water in the tank at the end
of that day, meaning that the tank starts rather cold the “current” day. This could be seen
Energies 2023,16, 4104 16 of 25
as a point for improvement of the optimizing technique, since it may be actually more cost
efficient to heat up the tank the previous day, depending on the hourly electricity prices of
the previous and current days. In further works, the optimization could be re-evaluated as
soon as the following day electricity prices are available, typically around 20:15–20:30 in
the evening.
If the thermostat-control temperature profile is analyzed in detail, it can be observed
that the temperature according to the model at the top (hot volume) is higher than the
actual setpoint for the thermostat (T
set
= 65
◦
C). This is due to the fact that, after a short
water draw-off, the thermostat could be indicating that heating is needed (temperature
below T
set −∆
T
ON/OFF
), even if the tank is almost fully charged (hnot far from 0.7 m). As
explained in Section 2.1, the heating element capacity was assumed as linear through its
length, meaning that part of the heat input from the heating element would be heating up
the cold volume, to fulfill the thermostat condition, while the other part would be heating
up the hot volume above the temperature setpoint.
The temperature profile for the optimized demand–response strategy for the evaluated
day starts at a very low temperature and charge level, and the tank is heated up importantly
in the first hour, when there is a valley in electricity price (Figure 6c). Only very small DHW
consumptions would be expected in those first hours, having a very low impact on the DI.
The large water draw-off happening around hour 204 (9 January, 12:00) and corresponding
to a bath is anticipated by the optimized strategy by increasing the water temperature close
to 80
◦
C to reduce the impact on discomfort, even if the cost of electricity may not be as
convenient at that time of the day. Figure 6d also shows that, due to this higher temperature
if compared to the thermostat control, a lower water flow rate from the tank is needed to
achieve the DHW flow rate requested at 45
◦
C. Another similar example of anticipation to
a relevant DHW consumption happens around hour 212 (9 January, 20:00), but in this case
corresponding to a shower.
The CI for the example day with thermostat control equals 0.38, which corresponds
to approximately 3.78
€
in electricity cost, while with the optimized demand–response
strategy it adds up to 0.31, i.e., 3.09
€
for the day, thus 18.3% savings. All this happens in
combination with a slight reduction of the discomfort for the user, being the DI equal to
0.13 with thermostat control and 0.11 with the optimized demand–response strategy. The
reason is that the storage is allowed to go as high as 80
◦
C, meaning that more effective
energy is stored with the same volume and when the electricity is cheaper. This was also
found by Kapsalis et al. [25].
A yearly analysis was performed to evaluate the potential savings due to implementing
an optimized demand–response strategy compared to the thermostat control in the ESWH.
As shown in Figure 7, the monthly savings are more relevant in those months with higher
electricity cost fluctuations (represented by the monthly standard deviation) and larger
DHW needs (colder months, also in terms of mains water temperature). March is the
month with the largest monthly standard deviation and also the largest monthly savings.
August also shows very important electricity cost volatility, but it does not translate into
savings. The main reason for this is that the DHW consumption profile was defined with
14 days of holidays in August, meaning that savings are half of what could be expected for
a regular month with that electricity cost pattern. On the other end are May, June or July,
with lower fluctuations in electricity costs, leading to lower savings. Yearly savings add up
to 146
€
, which is approximately 11.9% of the cost with the thermostat control. To be able to
put these savings into a context, a cost estimation of implementing the electronics capable
of retrieving information from the sensors in the tank and performing the day-ahead
optimization based on the electricity tariffs was conducted. Depending on the capabilities
of the electronics and on the economy of scale, a cost in the range of 25
€
to 35
€
could
be expected, i.e., approximately 20% of the expected annual savings and payback well
below one year. This percentage could decrease further in a future scenario in which even
larger intraday price fluctuations could be expected due to the penetration of renewables.
Moreover, having such a low-cost solution for ESWHs is crucial, since users buying an
Energies 2023,16, 4104 17 of 25
ESWH, which cost in the range of 170
€
to 200
€
in the capacity evaluated here, may not be
willing to purchase additional equipment unless it pays off in a very short period.
Energies 2023, 16, x FOR PEER REVIEW 17 of 26
The CI for the example day with thermostat control equals 0.38, which corresponds
to approximately 3.78 € in electricity cost, while with the optimized demand–response
strategy it adds up to 0.31, i.e., 3.09 € for the day, thus 18.3% savings. All this happens in
combination with a slight reduction of the discomfort for the user, being the DI equal to
0.13 with thermostat control and 0.11 with the optimized demand–response strategy. The
reason is that the storage is allowed to go as high as 80 °C, meaning that more eective
energy is stored with the same volume and when the electricity is cheaper. This was also
found by Kapsalis et al. [25].
A yearly analysis was performed to evaluate the potential savings due to
implementing an optimized demand–response strategy compared to the thermostat
control in the ESWH. As shown in Figure 7, the monthly savings are more relevant in
those months with higher electricity cost uctuations (represented by the monthly
standard deviation) and larger DHW needs (colder months, also in terms of mains water
temperature). March is the month with the largest monthly standard deviation and also
the largest monthly savings. August also shows very important electricity cost volatility,
but it does not translate into savings. The main reason for this is that the DHW
consumption prole was dened with 14 days of holidays in August, meaning that
savings are half of what could be expected for a regular month with that electricity cost
paern. On the other end are May, June or July, with lower uctuations in electricity costs,
leading to lower savings. Yearly savings add up to 146 €, which is approximately 11.9% of
the cost with the thermostat control. To be able to put these savings into a context, a cost
estimation of implementing the electronics capable of retrieving information from the
sensors in the tank and performing the day-ahead optimization based on the electricity
taris was conducted. Depending on the capabilities of the electronics and on the
economy of scale, a cost in the range of 25 € to 35 € could be expected, i.e., approximately
20% of the expected annual savings and payback well below one year. This percentage
could decrease further in a future scenario in which even larger intraday price uctuations
could be expected due to the penetration of renewables. Moreover, having such a low-cost
solution for ESWHs is crucial, since users buying an ESWH, which cost in the range of
170 € to 200 € in the capacity evaluated here, may not be willing to purchase additional
equipment unless it pays o in a very short period.
Figure 7. Monthly savings with the optimized demand–response strategy compared to thermostat
control. Also monthly average (mean) cost of electricity and standard deviation to illustrate
electricity cost uctuations during that month.
The savings aained through this evaluation can be compared to other studies from
the literature working in a similar line of research. Kapsalis and Hadellis [24] calculated
cost savings around 30% compared to thermostat control and for one day of evaluation.
A similar percentage of cost reduction was determined by Naja and Fripp [16], compared
Figure 7.
Monthly savings with the optimized demand–response strategy compared to thermostat
control. Also monthly average (mean) cost of electricity and standard deviation to illustrate electricity
cost fluctuations during that month.
The savings attained through this evaluation can be compared to other studies from
the literature working in a similar line of research. Kapsalis and Hadellis [
24
] calculated
cost savings around 30% compared to thermostat control and for one day of evaluation. A
similar percentage of cost reduction was determined by Najafi and Fripp [
16
], compared to
thermostat control (setpoint at 65
◦
C), or Shen et al. [
26
]. In the case of Lin et al. [
6
], savings
were even higher, reaching 49% compared to the conventional thermostat control. On the
other hand, other studies such as Booysen et al. [
34
] or Barja et al. [
29
] indicated more
contained savings. In the former, 13.1% median (energy) savings would be expected from a
demand–response strategy based on energy matching and with the Legionella sterilization
condition, compared to thermostat control. The latter explains the rather low savings
obtained, 7%, due to the conservative demand–response strategy followed by the study,
which focused on cases with very limited information on the tank status or demand profiles.
The conclusion after this benchmarking exercise would be that relevant savings are always
expected, but the extent of savings is very closely related to the width of electricity cost
fluctuations during the day. While in the current study, it was very uncommon that the
ratio of highest to lowest hourly electricity cost was above 3 for a day, in the case by Shen
et al. [
26
], the ratio of on-peak to off-peak electricity cost was at least 9.5 and even 26 during
the summer season.
Concerning user discomfort, the yearly evaluation was used to compare the DI due
to the demand–response strategy with that from the conventional thermostat control.
Independently of the strategy, it is clear that the occurrence or not of baths determines the
level of discomfort. On average, days with baths showed DI (expressed in percentage) of
9.7% and 14.3% for the optimized demand–response and thermostat strategies, respectively.
Without baths, these values were as low as 0.4% and 0.1%, respectively. It could be argued,
based on this, that the ESWH considered would be rather short in capacity to meet the
DHW demands defined by the profile, particularly when baths are considered.
3.3. ESWH Volume and Optimization
The concluding finding of the previous subsection motivated an additional evaluation
to delve into the optimization of demand response, focusing on increased capacity (volume)
of the ESWH and maintained other features such as heating element rated capacity, tank
diameter, etc. To avoid excessive computation time, the evaluation was limited to one
month, January. January should be rather representative, since the mean cost of electricity
Energies 2023,16, 4104 18 of 25
this month is almost equal to the mean cost of electricity during 2022 (364
€
/MWh vs.
365 €/MWh), even if the standard deviations differ (88 €/MWh vs. 135 €/MWh).
As shown in Figure 8, three volumes, 100 L, 125 L and 150 L, were evaluated in
addition to the reference volume of 76 L. The demand–response optimized strategies for
each of these volumes were obtained with three savings indexes, namely 0.5, 0.7 and 0.9.
Results have shown that increasing the ESWH volume has a negative effect on the CI, i.e., a
higher operational cost (electricity cost) is needed in the optimized scenario. This could
be expected, since more mass needs to be heated above the requested temperatures e.g.,
for daily Legionella sterilization, and in addition, a larger tank has a higher heat exchange
surface with the environment, elevating heat losses. A similar conclusion was already
indicated by Kapsalis et al. [
25
]. In return, the DI is lower as the storage volume rises. As an
example, a sufficiently large volume of 150 L leads to relatively low DI even if the savings
index is 0.7 or even 0.9, which would represent a very aggressive strategy towards cost
savings. In parallel, this could reduce the CI due to the higher volume, even below what
would correspond to the reference case, 76 L, with a savings index of 0.5. In conclusion,
more aggressive demand–response strategies in terms of savings could be expected with
higher ESWH volumes (even keeping the heating element power).
Energies 2023, 16, x FOR PEER REVIEW 19 of 26
Figure 8. Inuence of ESWH volume on the CI (dots connected by solid lines) and DI (dots
connected by doed lines) as a function of the savings index. CI and DI are monthly mean indexes,
considering consumption proles, conditions and electricity costs for January.
3.4. Sensitivity Analysis of the DHW Consumption Prole
As mentioned above, so far in the current study the DHW consumption prole has
been considered as known, generated according to the stochastic approach and data from
the studies by Jordan and Vajen [14,15]. However, experience tells that this is not the case,
and even if there are certain paerns, DHW consumptions occur with a certain level of
randomness (in time of the day and volume drew-o). The aim of this subsection is to
evaluate the impact that operating according to the demand–response strategy (heater
activation/deactivation or factor of utilization) optimized for the “known” DHW
consumption prole, prole “a”, may have on several randomly generated proles. Two
types of randomly generated proles were included in this analysis. On the one hand,
proles “a1” and “a2” assume that the bath day is xed, i.e., the bath day coincides with
the bath day in prole “a”. On the other hand, proles “b” and “c” do not x the bath day
but follow the original approach by Jordan and Vajen [14,15], seing only one bath day
per week, which is more likely to happen on the weekend. This distinction was made
because the bath has the largest DHW consumption, both in duration and in ow rate. It
was assumed that the uncertainty on the bath day could have a huge impact on discomfort
if the ESWH is operated with an optimized demand–response strategy generated with
another bath day in mind. An additional condition needed to be implemented in the
model for this analysis, namely a heater deactivation when the temperature in the tank
reached 80 °C, even if the optimized heater activation strategy would indicate that the
heater should be on.
The results of this evaluation, which was performed considering the proles,
conditions and electricity costs for January, are included in Table 5. The optimization of
the demand–response strategy was performed according to the DHW consumption
prole “a”, establishing one reference case for each ESWH volume considered. As
expected, the DI for this prole “a” was the lowest, since the heating element activation
strategy was optimized accordingly. Changing the DHW consumption prole involved a
signicant increase of the DI, but the impact on user discomfort would be milder in case
of having a larger tank (in this case represented by 125 L compared to the reference case
of 76 L). Surprisingly enough, there was not much benet on knowing the bath day
beforehand, i.e., even if the DI with proles “a1” and “a2” was lower than with proles
“b” and “c”, the dierence was not as high as expected. Finally, it could be seen that the
CI in the optimized case (prole “a”) was the largest of each series, the reason being the
Figure 8.
Influence of ESWH volume on the CI (dots connected by solid lines) and DI (dots connected
by dotted lines) as a function of the savings index. CI and DI are monthly mean indexes, considering
consumption profiles, conditions and electricity costs for January.
3.4. Sensitivity Analysis of the DHW Consumption Profile
As mentioned above, so far in the current study the DHW consumption profile has
been considered as known, generated according to the stochastic approach and data from
the studies by Jordan and Vajen [14,15]. However, experience tells that this is not the case,
and even if there are certain patterns, DHW consumptions occur with a certain level of
randomness (in time of the day and volume drew-off). The aim of this subsection is to
evaluate the impact that operating according to the demand–response strategy (heater
activation/deactivation or factor of utilization) optimized for the “known” DHW consump-
tion profile, profile “a”, may have on several randomly generated profiles. Two types of
randomly generated profiles were included in this analysis. On the one hand, profiles “a1”
and “a2” assume that the bath day is fixed, i.e., the bath day coincides with the bath day in
profile “a”. On the other hand, profiles “b” and “c” do not fix the bath day but follow the
original approach by Jordan and Vajen [14,15], setting only one bath day per week, which
is more likely to happen on the weekend. This distinction was made because the bath has
the largest DHW consumption, both in duration and in flow rate. It was assumed that
the uncertainty on the bath day could have a huge impact on discomfort if the ESWH is
Energies 2023,16, 4104 19 of 25
operated with an optimized demand–response strategy generated with another bath day
in mind. An additional condition needed to be implemented in the model for this analysis,
namely a heater deactivation when the temperature in the tank reached 80
◦
C, even if the
optimized heater activation strategy would indicate that the heater should be on.
The results of this evaluation, which was performed considering the profiles, con-
ditions and electricity costs for January, are included in Table 5. The optimization of the
demand–response strategy was performed according to the DHW consumption profile “a”,
establishing one reference case for each ESWH volume considered. As expected, the DI for
this profile “a” was the lowest, since the heating element activation strategy was optimized
accordingly. Changing the DHW consumption profile involved a significant increase of the
DI, but the impact on user discomfort would be milder in case of having a larger tank (in
this case represented by 125 L compared to the reference case of 76 L). Surprisingly enough,
there was not much benefit on knowing the bath day beforehand, i.e., even if the DI with
profiles “a1” and “a2” was lower than with profiles “b” and “c”, the difference was not as
high as expected. Finally, it could be seen that the CI in the optimized case (profile “a”) was
the largest of each series, the reason being the additional restriction to the heater so as to
limit the temperature to 80 ◦C with the other consumption profiles.
Table 5.
Sensitivity evaluation of the DHW consumption profile on the CI and DI (expressed
in percentage), and assumed savings index equal to 0.5. Two ESWH volumes were considered,
76 L (default) and 125 L. CI and DI are monthly mean indexes, considering consumption profiles,
conditions and electricity costs for January.
Profile CI [%] DI [%]
76 L
a (Initial profile) 20.6 1.9
a1 (new profile, bath days fixed) 19.7 10.4
a2 (new profile, bath days fixed) 19.7 9.9
b (new profile, bath days random) 19.9 11.6
c (new profile, bath days random) 19.7 10.4
125 L
a (Initial profile) 21.4 0.2
a1 (new profile, bath days fixed) 21.3 4.2
a2 (new profile, bath days fixed) 21.4 3.6
b (new profile, bath days random) 21.2 6.7
c (new profile, bath days random) 21.4 5.4
4. Conclusions
In the current context of renewable energy penetration and the fight against climate
change, Electric Storage Water Heaters (ESWH) may become a resource to decarbonize do-
mestic hot water production, now dominated by fossil fuel boilers. ESWHs are inexpensive
in investment, but not in operation costs, particularly if compared with more efficient tech-
nologies such as heat pumps. Thus, optimizing electricity costs with a demand–response
strategy in a dynamic electricity price scenario may become very positive for end users
owning ESWHs. The present work delved into this subject, by defining a demand–response
strategy for ESWHs based on a multiobjective optimization focusing on cost savings and
user comfort. For this purpose, a two-volume numerical model was proposed and experi-
mentally validated, and it was used to evaluate the potential savings due to this optimized
demand–response strategy compared to a conventional thermostat control of the heating
element in the ESWH. The yearly analysis performed indicated that electricity cost savings
around 12% could be achieved, considering the hourly electricity costs in Spain for 2022
and assuming as known the DHW consumption profile, which was generated following
a stochastic approach from the literature. In general, it was observed that the higher the
fluctuations in electricity price within a certain day or month, the larger the potential to
Energies 2023,16, 4104 20 of 25
achieve savings. Moreover, user discomfort was not penalized by the demand–response
strategy compared to the thermostat control, mainly since the water temperature was
allowed to go above the thermostat setpoint in the optimized demand–response strategy.
Another interesting finding was that increasing the volume of ESWHs can be very beneficial.
Even if, in principle, the cost of heating a larger tank is higher, more volume means more
aggressive approaches in terms of cost savings can be adopted in the optimization, without
impact on user comfort. Lastly, a user with a larger tank will also be less affected by DHW
consumption that deviates from those used to optimize the demand–response strategy.
Some areas of improvement have been identified and will be addressed in future
works. The first aspect is the implementation of artificial intelligence techniques to predict
water draw-offs based on past usage information and to enable replanning capabilities in
response to nonpredicted DHW consumptions. The optimization can then automatically
focus on either comfort or savings depending not only on user preferences but also on
the predictability of the consumption profile. Nonintrusive and inexpensive techniques
for accurately evaluating DHW consumption are key to the success of this approach,
and there are actors already working in this area. Additionally, the optimized demand–
response strategy should be open to other factors that are very relevant in the current
electricity market landscape, such as on-site electricity generation (PV), storage or even
more fluctuating electricity prices, which may sometimes reach negative values.
Author Contributions:
Conceptualization, Á.Á.P., F.E.C. and P.C.O.; methodology, Á.Á.P. and P.D.G.;
software, Á.Á.P. and P.D.G.; validation, Á.Á.P.; formal analysis, Á.Á.P.; writing—original draft
preparation, Á.Á.P.; writing—review and editing, Á.Á.P., P.D.G., F.E.C. and P.C.O.; visualization,
Á.Á.P.; project administration, F.E.C. and P.C.O.; funding acquisition, F.E.C. and P.C.O. All authors
have read and agreed to the published version of the manuscript.
Funding:
This research was supported by CERVERA Research Program of CDTI, the Industrial and
Technological Development Centre of Spain, through the Research Projects HySGrid+ (grant number
CER-20191019).
Data Availability Statement:
The data supporting reported results can be found in the public
repository Zenodo under the https://doi.org/10.5281/zenodo.7861588 (accessed on 12 May 2023).
Acknowledgments:
The authors would like to thank the great contribution by Julio César Mérida
Sánchez on the preparation of the experimental setup and data acquisition system.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. Experimental Validation of Numerical Models
As indicated in Section 2, three numerical models of different complexity were con-
sidered to emulate the performance of an ESWH for domestic uses: the single-volume
model [
37
], a two-volume model produced by the authors of this article and building on
that suggested by [
8
], and the partial differential equation model [
38
]. In the main text
of the present article, the selection of the model for the optimization process, namely the
modified two-volume model, was justified in a summarized form. This appendix delves
into this subject, by presenting the validation process in detail and discussing why the
different models were disregarded in favor of the two-volume approach.
Starting with the single-volume model, it is clearly the simplest approach and the
solution leading to a sufficiently good representation of the heating processof the tank, both
undisturbed by draw-offs from cold conditions (Figure A1a) and with a single draw-off
(3 min) during the heating process (Figure A1b).
Energies 2023,16, 4104 21 of 25
Energies 2023, 16, x FOR PEER REVIEW 21 of 26
accurately evaluating DHW consumption are key to the success of this approach, and
there are actors already working in this area. Additionally, the optimized demand–
response strategy should be open to other factors that are very relevant in the current
electricity market landscape, such as on-site electricity generation (PV), storage or even
more uctuating electricity prices, which may sometimes reach negative values.
Author Contributions: Conceptualization, Á.Á.P., F.E.C. and P.C.O.; methodology, Á.Á.P. and
P.D.G.; software, Á.Á.P. and P.D.G.; validation, Á.Á.P.; formal analysis, Á.Á.P.; writing—original
draft preparation, Á.Á.P.; writing—review and editing, Á.Á.P., P.D.G., F.E.C. and P.C.O.;
visualization, Á.Á.P.; project administration, F.E.C. and P.C.O.; funding acquisition, F.E.C. and
P.C.O. All authors have read and agreed to the published version of the manuscript.
Funding: This research was supported by CERVERA Research Program of CDTI, the Industrial and
Technological Development Centre of Spain, through the Research Projects HySGrid+ (grant
number CER-20191019).
Data Availability Statement: The data supporting reported results can be found in the public
repository Zenodo under the hps://doi.org/10.5281/zenodo.7861588 (accessed on 15 May 2023).
Acknowledgments: The authors would like to thank the great contribution by Julio César Mérida
Sánchez on the preparation of the experimental setup and data acquisition system.
Conicts of Interest: The authors declare no conict of interest.
Appendix A. Experimental Validation of Numerical Models
As indicated in Section 2, three numerical models of dierent complexity were
considered to emulate the performance of an ESWH for domestic uses: the single-volume
model [37], a two-volume model produced by the authors of this article and building on
that suggested by [8], and the partial dierential equation model [38]. In the main text of
the present article, the selection of the model for the optimization process, namely the
modied two-volume model, was justied in a summarized form. This appendix delves
into this subject, by presenting the validation process in detail and discussing why the
dierent models were disregarded in favor of the two-volume approach.
Starting with the single-volume model, it is clearly the simplest approach and the
solution leading to a suciently good representation of the heating process of the tank,
both undisturbed by draw-os from cold conditions (Figure A1a) and with a single draw-
o (3 min) during the heating process (Figure A1b).
(a)
(b)
Energies 2023, 16, x FOR PEER REVIEW 22 of 26
(c)
(d)
Figure A1. Single-volume results vs. validation test data. (a) Heating without water consumptions.
(b) Heating with a water draw-o in the middle. (c) Continuous draw-o at 6 L/min, heating
element disabled. (d) Several 1 min draw-os at 6 L/min, heating element disabled.
However, the single-volume model starts to fail with the representation of the
temperature of the water drawn from the tank very early in the process. In Figure A1c, the
validation test consisted of a continuous draw-o with an average ow rate close to
6 L/min, from hot conditions (initial temperature around 60 °C) and with the electric
heater disabled. The single-volume model indicates rather early in the process that the
temperature is below any comfort temperature, e.g., lower than 40 °C around 8 min, while
in the test, it happened 4 min later. Similar behavior occurs with several consecutive water
draw-os with the same ow rate as above, disabled heating element, and with duration
of 1 min each (Figure A1d).
Concerning the partial dierential equation model (10 volumes), the same validation
analysis against experimental data was performed. As observed in the graphs in Figure
A2, there is a rather good approach in most cases between test data and model results.
There are still discrepancies that could be explained by some other eects such as the time
delay for the temperature measurements (also applicable to the two-volume model). As
observed for example in Figure A2b, the temperature measured at the top of the tank by
the sensor located in the sleeve/pocket (red dots) started falling in temperature earlier than
that sensor located in the ow meter downstream of the tank (purple dots), as expected,
but then the laer fell more steeply due to its construction (in-ow sensor), while the
sensor in the pocket took longer. On the other hand, buoyancy occurring during the
heating of the tank is not that well represented if the time step length is increased from
1 s (Figure A2e) to 30 s (Figure A2f). The lowest part of the tank is heated by the heating
element (represented by the black solid line T_TB) but with a time step length equal to
30 s it did not translate in the actual heating that occurs through natural convection for
the top parts of the tank (red solid line T_TT). The model could be tuned with a buoyancy
factor (F) on the buoyancy part of the discretized dierential equation from Lago et al. [38]
(see Figure A2f) in case the time step length needed to be increased for reduced
computation time.
Figure A1.
Single-volume results vs. validation test data. (
a
) Heating without water consumptions.
(
b
) Heating with a water draw-off in the middle. (
c
) Continuous draw-off at 6 L/min, heating element
disabled. (d) Several 1 min draw-offs at 6 L/min, heating element disabled.
However, the single-volume model starts to fail with the representation of the tempera-
ture of the water drawn from the tank very early in the process. In Figure A1c, the validation
test consisted of a continuous draw-off with an average flow rate close to 6 L/min, from
hot conditions (initial temperature around 60
◦
C) and with the electric heater disabled. The
single-volume model indicates rather early in the process that the temperature is below any
comfort temperature, e.g., lower than 40
◦
C around 8 min, while in the test, it happened
4 min later. Similar behavior occurs with several consecutive water draw-offs with the same
flow rate as above, disabled heating element, and with duration of 1 min each (Figure A1d).
Concerning the partial differential equation model (10 volumes), the same validation
analysis against experimental data was performed. As observed in the graphs in Figure A2,
there is a rather good approach in most cases between test data and model results. There
are still discrepancies that could be explained by some other effects such as the time delay
for the temperature measurements (also applicable to the two-volume model). As observed
for example in Figure A2b, the temperature measured at the top of the tank by the sensor
located in the sleeve/pocket (red dots) started falling in temperature earlier than that sensor
located in the flow meter downstream of the tank (purple dots), as expected, but then the
latter fell more steeply due to its construction (in-flow sensor), while the sensor in the
pocket took longer. On the other hand, buoyancy occurring during the heating of the tank
is not that well represented if the time step length is increased from 1 s (Figure A2e) to 30 s
(Figure A2f). The lowest part of the tank is heated by the heating element (represented by
the black solid line T_TB) but with a time step length equal to 30 s it did not translate in
the actual heating that occurs through natural convection for the top parts of the tank (red
solid line T_TT). The model could be tuned with a buoyancy factor (F) on the buoyancy
Energies 2023,16, 4104 22 of 25
part of the discretized differential equation from Lago et al. [
38
] (see Figure A2f) in case the
time step length needed to be increased for reduced computation time.
Energies 2023, 16, x FOR PEER REVIEW 23 of 26
(a)
(b)
(c)
(d)
(e)
(f)
Figure A2. Partial dierential equation model vs. validation test data. (a) Continuous draw-o at 6
L/min, heating element disabled. (b) Continuous draw-o at 17 L/min, heating element disabled. (c)
Several 1 min draw-os at 6 L/min, heating element disabled. (d) Several 1 min draw-os at 6 L/min,
heating element enabled if requested by thermostat. (e) Heating without water consumptions,
model time step length Δt = 1 s. (f) Heating without water consumptions, model time step length
Δt = 30 s.
Concerning the modified two-volume model, the first approach was by considering
that the two volumes, hot and cold, were perfectly separated, with perfect stratification. As
mentioned in Section 2.1, the results from the test with several 1 min water draw-offs of
approximately 6 L/min each and no heating element activation showed that after the eighth
Figure A2.
Partial differential equation model vs. validation test data. (
a
) Continuous draw-off
at 6 L/min, heating element disabled. (
b
) Continuous draw-off at 17 L/min, heating element
disabled. (
c
) Several 1 min draw-offs at 6 L/min, heating element disabled. (
d
) Several 1 min
draw-offs at 6 L/min, heating element enabled if requested by thermostat. (
e
) Heating without water
consumptions, model time step length
∆
t= 1 s. (
f
) Heating without water consumptions, model time
step length ∆t= 30 s.
Concerning the modified two-volume model, the first approach was by considering
that the two volumes, hot and cold, were perfectly separated, with perfect stratification. As
Energies 2023,16, 4104 23 of 25
mentioned in Section 2.1, the results from the test with several 1 min water draw-offs of
approximately 6 L/min each and no heating element activation showed that after the eighth
cycle, water from the tank is not at 60
◦
C anymore (Figure 2a), being the effective capacity
of the tank, from the tests, lower than expected from the model. The implementation of
the mixing factor solved the challenge positively enough, as detailed in that same section
(Figure 2b).
At this point, the selection of model was between two: the two-volume model with
mixing factor or the partial differential equation model with buoyancy factor if increased
the time step length. The MAPE (mean absolute percentage error) was calculated for each
case to elaborate on the decision between both models. The results comparing the value
measured at the upper sensor in the tank (TT) and that from the model at that same position
for different tests are included in Table A1. The partial differential equation model stands
out under most tests, followed by the two-model with mixing factor = 0.2.
Table A1.
Mean average percentage error between tests and the different models evaluated. Tem-
peratures evaluated in Kelvin. Test a, heating from cold tank. Test b, with intermediate draw-off.
Test c, continuous draw-off at 6 L/min from hot tank and disabled heating element. Test d, continu-
ous draw-off at 17 L/min from hot tank and disabled heating element. Test e, several one-minute
draw-offs (6 L/min) with heating element disabled.
MAPE [%] 1-Vol 2-Vol_MF = 0 2-Vol_MF 0.1 2-Vol_MF 0.2 10-Vol
Test a 0.16 0.12 0.12 0.12 0.22
Test b 0.39 0.28 0.30 0.32 0.32
Test c 3.68 1.29 1.05 1.21 1.15
Test d - 3.46 3.08 2.95 2.90
Test e 3.36 3.01 2.54 2.37 1.66
A final aspect on the decision was computation time. The two-volume model ended
up being 20% faster for one day simulation (not even optimization of demand–response
strategy) than the partial differential equation model. This, combined with rather good
MAPE and a very positive match between tests and the model of the energy draw-off
(Section 2.1), led to the decision for the two-volume model with mixing factor = 0.2 as the
modelling approach in the current article.
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