Demand–Response Control of Electric Storage Water Heaters Based on Dynamic Electricity Pricing and Comfort Optimization

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.

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 CO₂ 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, 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 encourages 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 participation 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 [6,7,8], 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 comfort. 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 minimization. Leaving aside the well-known issues about Legionella, a deadly hazard to humans, 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 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 [€].

gathers different works from the literature defining comfort indicators, in some cases utilized in demand–response strategies and optimizations.

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., [25,32,33,34]. 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 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.
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.

collects 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 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.

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 Δt). A disadvantage with this model is that it is more computationally intensive, which could be an issue for optimization problems.
• 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.

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.

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.

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 (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).

$$T_{V_1,t+1}=T_{V_1,t}+\Delta t·\left(\frac{P_{\mathrm{heater},V_1}+U·A_{V_1}·\left(T_{\mathrm{amb}}-T_{V_1,t}\right)+k_{V_1}·A_c·\frac{T_{V_2,t}-T_{V_1,t}}{h}}{c_{p{V}_1}·A_c·h·{\rho }_{V_1}}\right)$$

(1)

$$\begin{array}{c}{T}_{V_2,t+1}=T_{V_2,t}+\Delta t·\hfill \\ \left(\frac{P_{\mathrm{heater},V_2}+U·A_{V_2}·\left(T_{\mathrm{amb}}-T_{V_2,t}\right)-\dot{m}·c_{p,\mathrm{m}}·\left(T_{V_2,t}-T_{\mathrm{m}}\right)+k_{V_2}·A_c·\frac{T_{V_1,t}-T_{V_2,t}}{L-h}}{c_{p{V}_2}·A_c·\left(L-h\right)·{\rho }_{V_2}}\right)\hfill \end{array}$$

(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 V₁, V₂, 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 [m²], heat transfer area with the surroundings for the hot and cold volumes, respectively. The ESWH was approximated as a perfect cylinder with inner diameter d_i.
• T_amb [°C], ambient (surroundings to the ESWH) temperature.
• $\dot{m}$ [kg s⁻¹], water draw-off mass flow rate.
• T_m [°C], mains water temperature.
• k [W m⁻¹ K⁻¹], water thermal conductivity, evaluated at V₁ and V₂ temperatures.
• A_c [m²], tank cross sectional area, evaluated with inner diameter d_i in Figure 1a.
• ρ [kg m⁻³], water density, evaluated at V₁ and V₂ temperatures.
• 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.

$$P_{\mathrm{heater},V_1}=\max \left(0,P_{\mathrm{heater}}·\left(h-L+L_{\mathrm{heater}}\right) /L_{\mathrm{heater}}\right)$$

(3)

$$P_{\mathrm{heater},V_2}=P_{\mathrm{heater}}-P_{\mathrm{heater},V_1}$$

(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).

$$h_{t+1}=h_t-\left(\frac{\dot{m}}{{\rho }_{V_1}·A_{\mathrm{c}}}\right)·\Delta t$$

(5)

After each time step, the model checks if the temperature of the cold volume V₂, due to the effect of the heater input, has become higher or equal to the temperature of the hot volume V₁, 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 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 element when this temperature falls below the setpoint (T_set) minus a certain offset (ΔT_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 − h_therm), while T_V1,t+1 would be the value for comparison otherwise, i.e., h ≥ (L − h_therm). In a demand–response approach as suggested in the present article, the heating element status 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 electricity 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 R² > 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%).

$$\rho =748.925+1.921·T-3.563·{10}^{-3}·T^2$$

(6)

$$k=-7.475·{10}^{-1}+7.442·{10}^{-3}·T-9.734·{10}^{-6}·T^2$$

(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. 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, di		370 mm
ESWH, L		695 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

.

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⁻² K⁻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.

• 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 (Δt = 1 s), there is a very accurate representation of heating and draw-off processes. However, such short time step lengths are unpractical for daily optimizations, and with longer periods, the slow-buoyancy approach followed would not appropriately 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.

$$T_{V_1,t+1,MF}=\frac{\left(A_c·h·{\rho }_{V_1}-MF·\dot{m}·\Delta t\right)·T_{V_1,t+1}+MF·\dot{m}·\Delta t·T_{V_2,t+1} }{A_c·h·{\rho }_{V_1}}$$

(8)

$$T_{V_2,t+1,MF}=\frac{MF·\dot{m}·\Delta t·T_{V_1,t+1}+\left(A_c·\left(L-h\right)·{\rho }_{V_2}-MF·\dot{m}·\Delta t\right)·T_{V_2,t+1}}{A_c·\left(L-h\right)·{\rho }_{V_2}}$$

(9)

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 Δt was 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 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.

2.3.2. DHW Consumption Profile

As suggested by different studies [6,7,8], 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 DHW consumption according to these profiles is at 45 °C, ${\dot{V}}_{45 °\mathrm{C}}$, 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, ${\dot{V}}_{\mathrm{ESWH}}$, 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., ${\dot{V}}_{\mathrm{ESWH}}$ = ${\dot{V}}_{45 °\mathrm{C}}$. 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].

$${\dot{V}}_{\mathrm{ESWH},t}={\dot{V}}_{45 °\mathrm{C},t} \frac{45-T_{\mathrm{mains},t} }{T_{V_1,t}-T_{\mathrm{mains},t} }$$

(10)

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, ΔT_ON/OFF, were selected as shown in

Table 4. Values given to different parameters in the evaluation and optimizations.

Parameter	Value	Observations
Setpoint thermostat control, Tset	65 °C
Offset reactivation, ΔTON/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, Δtopt	1 h	Electricity price changes by hour.
Time step length model calculation, Δt	30 s

, sufficiently high to meet Legionella sterilization. For the optimized demand–response strategy, the procedure will be stated in the following paragraphs.

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 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.

$$C{I}_{\mathrm{day}}=\frac{{{\displaystyle \sum }}_{i=1}^{24}{c}_{\mathrm{el},i}·P_{\mathrm{heater},\max }·F{U}_i}{{{\displaystyle \sum }}_{i=1}^{24}{c}_{\mathrm{el},i}·P_{\mathrm{heater},\max }}$$

(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 Δt = 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].

$$D{I}_{\mathrm{day}}=\frac{{{\displaystyle \sum }}_{i=1}^{\mathrm{steps}}{\dot{V}}_{45 °\mathrm{C},\mathrm{i}·\Delta \mathrm{t}} ·\underset{}{\max }\left(0,T_{\mathrm{comf}}-T_{V_1,\mathrm{i}·\Delta \mathrm{t}} [°\mathrm{C}] \right)}{{{\displaystyle \sum }}_{i=1}^{steps}{\dot{V}}_{45 °\mathrm{C},\mathrm{i}·\Delta \mathrm{t}} ·\left(T_{\mathrm{comf}}-T_{\mathrm{mains},\mathrm{day}}\right)}$$

(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.

$$f=SI·CI+\left(1-SI\right)·DI$$

(13)

The variable used in the optimization was the heating element utilization factor, FU_i, for the optimization time step length, Δt_opt = 1 h. As the reader may point out, different time steps lengths were considered for the optimization, Δt_opt, and the numerical model calculation, Δt. 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, Δt, 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 · Δt_opt, and deactivated during (1 − FU_i) · Δt_opt.

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 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 5 represents 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 5 correspond 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.

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 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 6 gathers 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.

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 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 − ΔT_ON/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 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 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.

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 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).

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 consumption 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 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, conditions and electricity costs for January, are included in

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

. 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.

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 domestic 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 technologies 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 experimentally 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 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.