Model Predictive Control of Electric Water Heaters in Individual Dwellings Equipped with Grid-Connected Photovoltaic Systems

Abstract

The residential sector is energy-consuming and one of the biggest contributors to climate change. In France, the adoption of photovoltaics (PV) in that sector is accelerating, which contributes to both increasing energy efficiency and reducing greenhouse gas (GHG) emissions, even though the technology faces several issues. One issue that slows down the adoption of the technology is the “duck curve” effect, which is defined as the daily variation of net load derived from a mismatch between power consumption and PV power generation periods. As a possible solution for addressing this issue, electric water heaters (EWHs) can be used in residential building as a means of storing the PV power generation surplus in the form of heat in a context where users’ comfort—the availability of domestic hot water (DHW)—has to be guaranteed. Thus, the present work deals with developing model-based predictive control (MPC) strategies—nonlinear/linear MPC (MPC/LMPC) strategies are proposed—to the management of EWHs in individual dwellings equipped with grid-connected PV systems. The aim behind developing such strategies is to improve both the PV power generation self-consumption rate and the economic gain, in comparison with rule-based (RB) control strategies. Inasmuch as DHW and power demand profiles are needed, data were collected from a panel of users, allowing the development of profiles based on a quantile regression (QR) approach. The simulation results (over 6 days) highlight that the MPC/LMPC strategies outperform the RB strategies, while guaranteeing users’ comfort (i.e., the availability of DHW). The MPC/LMPC strategies allow for a significant increase in both the economic gain (up to 2.70 EUR) and the PV power generation self-consumption rate (up to 14.30%ps), which in turn allows the ${\mathrm{CO}}_2$ emissions to be reduced (up to 3.92 $\mathrm{k}\mathrm{g}{\mathrm{CO}}_{2.\mathrm{eq}})$. In addition, these results clearly demonstrate the benefits of using EWHs to store the PV power generation surplus, in the context of producing DHW in residential buildings.

1. Introduction

1.1. Context

In France, the residential sector is the largest energy consumer. In 2019, electricity demand accounted for 36.0% of the total energy demand, with 12.1% of this demand pertaining to domestic hot water (DHW) production, which is the second largest electricity-demand item [1]. Electricity is the primary source of DHW production, constituting 46.5% [2]. EWHs equipped with storage tanks are the most common EWH production system in France, with 11 million units and approximately 1 million units replaced annually. France stands out in Europe due to its stock of EWHs, largely influenced by electricity pricing featuring advantageous off-peak hours, offering significant financial benefits to owners of systems producing and storing DHW overnight for daytime use. However, this production method results in higher energy losses compared to instantaneous production, as indicated by the French Agency for Ecological Transition (ADEME). Consequently, in real-world conditions, such EWHs exhibit a relatively modest efficiency of around 70% in final energy, equivalent to 28% in primary energy [2]. Meanwhile, solar PV panels have the potential to help reducing the amount of power extracted from the distribution grid in the residential sector, in particular for DHW production. These past few years, the number of solar PV panels has clearly experienced an important growth in France, which is directly attributable to government incentives. However, misalignment is experienced with demand peaks in the residential sector resulting in a negative net load, as explained above.

In the residential sector, power demand, DHW demand, and PV power generation patterns are intricately correlated with control strategies. Demand patterns exhibit a cyclic nature, typically recurring every 24 h, modulated by seasonal variations and socioeconomic, demographic, and cultural factors. Conversely, PV power generation patterns exhibit a stochastic behavior, heavily influenced by seasonal dynamics. Full consideration of these factors is paramount when it comes to develop control strategies capable of handling real-world conditions and implement them in real systems. In the context of producing DHW, the challenge behind developing active demand-side management (ADSM) strategies is to minimize the impact on hot water availability while shaving peak demand during critical periods by taking advantage of the PV power generation surplus [3,4,5,6]. The success of control strategies highly depends on correctly identifying the power and DHW demand behavior [7,8]. Thus, achieving a detailed characterization of users’ power and DHW demand is of great relevance, as it allows for a more reliable assessment of the ADSM strategies efficiency. A deeper knowledge of power and DHW demand profile features allows for the design of innovative control strategies based on demand patterns [9,10].

In this context, the present work aimed first at identifying the factors that influence most of power demand and DHW demand in individual dwellings of southwest France. To this end, a survey was conducted by PROMES-CNRS and ART-Dev. Using the data collected, typical scenarios for DHW demand, non-DHW power demand, and PV power generation were developed and used to implement EWH control strategies. Those strategies aim at improving the PV power generation self-consumption rate in the context of individual dwellings of southwest France by using EWHs as a means of storing the unused PV power generation during the day, without compromising users’ comfort. In addition, the economic and environmental benefits related with managing EWHs using MPC are assessed, taking as a reference rule-based (RB) control approaches, and the computational complexity associated with implementing the MPC strategies is controlled, without compromising performance.

The state of the art of both factors influencing DHW demand and power demand in the residential sector and EWH control is presented in the next subsection of the paper.

1.2. State of the Art

1.2.1. Factors Influencing DHW Demand and Power Demand in the Residential Sector

Several factors are identified as influencing DHW and power demand profiles worldwide. These factors are primarily linked to demand habits [10,11], climatic conditions, the timing of activity onset and cessation throughout the day [10,12], the family structure [10,13,14,15,16], the social status [10], the education level [15], the dwelling type and characteristics (e.g., the year of construction, the building size, whether the building is an apartment or a house, whether it is a rental or ownership home, etc.), and the appliances and equipment used [17,18]. In this context, several research studies have dealt with the development of demand profiles, and several approaches and methods are proposed for forecasting DHW demand and power demand, such as technical guidelines [19,20] or statistical methods [13,18,21,22,23].

1.2.2. EWH Control

As the present work deals with the control of EWHs in individual dwellings, a literature review on control strategies for advanced demand-side management (ADSM) has been conducted. ADSM approaches allow for actively managing and dynamically adjusting power demand and supply in response to specific patterns, thus contributing to saving energy and improving grid reliability and stability. In [24], the impact of schedule control on the energy consumed by electrical water heaters is evaluated. The results show that a significant reduction in energy consumption can be achieved through scheduling without noticeable impact on users. In [25], the authors proposed an optimal water heater control approach based on dynamic programming (DP) with the aim of optimizing heating schedules. The results demonstrate the ability of the system in reducing the overall energy cost while preserving users’ comfort. In [26], an EWH control strategy based on DP and the classification of power consumption profiles was proposed. The k-means clustering algorithm was used for cluster analysis. The results highlight that the control strategy allows reducing peak demand and meeting the hot water demand.

In [27], advanced strategies for load shifting of groups of EWHs were proposed and evaluated. For small-size groups, the problem was formulated as a mixed-integer linear programming/mixed-integer quadratic programming (MILP/MIQP) problem, yielding relatively low computational efforts. As stated by the authors, such an approach is strongly limited by the dimensionality of the problem. For medium-size groups, the problem of optimal scheduling can be efficiently solved using a well-designed heuristic. When the dimension gets sufficiently large, the Fokker–Planck approach is used. In [28], the authors introduced a strategy for controlling EWHs using MILP. The two major contributions of the work are as follows: (1) discomfort is modeled as undelivered energy and (2) energy consumption is considered for scheduling. The results highlight the effectiveness of the proposed approach by improving both sides of the cost vs. discomfort trade-off. In [29], a demand-side management system, which was built on the centralized MILP-based control of EWHs, was designed for areas with high penetration of renewable energy sources. The results show that the proposed load control allows for the optimal dispatch of the EWHs so that the peak imports and exports in the area are minimized and reduces voltage fluctuations. In [30],, an approach based on mixed-integer nonlinear programming (MINLP) was proposed for enabling residential water heating to be converted from an inelastic to an elastic demand, which would respond to pricing incentives. The results demonstrate that the proposed approach could not only reduce the electrical water heating costs significantly, but improve customer experience as well, as thermal storage capability would enable more hot water to be delivered at times of peak consumption.

In [31], an optimal operation scheduling algorithm was proposed and evaluated under both a day-ahead real-time pricing tariff and a time varying bound on power consumption. The Dijkstra’s algorithm is applied to the operation scheduling of EWHs, which can be used as thermal energy buffers. The simulation results are presented under various realistic scenarios. The algorithm may be used for EHW control by optimally adjusting the temperature set point at each time step. In [32], a low-complexity heuristic algorithm, which schedules the operation of EWHs under dynamic electricity pricing, was proposed. It takes into account cost and comfort preferences and constraints and minimizes the energy cost without compromising comfort, operating either in cost- or comfort-oriented mode. In [33], a binary particle swarm optimization (BPSO) algorithm—the time-of-use (TOU) electricity rate is used as DR incentive–allowed optimizing the power usage of EWHs, considering both the customers’ electricity cost and the hot water demand. The authors highlighted that aggregating thousands of EWHs may have a positive effect on power system reliability, considering their operating reserve capacities as interruptible loads.

In [34], an approach was presented to automatically adapt DHW heaters to individual human behavior based on real IoT data. The human behavior was learned to create an optimal hot water schedule that adapts to each user, using either artificial neural networks or Gaussian processes with periodic kernels. In [35], two techniques based on Q-learning and action-dependent heuristic dynamic programming (ADHDP) were demonstrated for the demand-side management of domestic EWHs. The problem was modeled as a DP problem. According to simulations, these techniques reduce the cost of energy consumption significantly. The results also indicate that energy consumption is reduced during on-peak periods. In [36], a distributed model-free strategy was proposed to manage the EWH demand. Distributed reinforcement learning (RL) was adopted to independently control several EWHs based on a virtual tariff. Two strategies were proposed: the first strategy was based on measuring the battery state of charge (SOC) in real time while the second strategy was built on predicting the SOC 24 h in advance thanks to an artificial neural network. The results show that energy consumption is reduced. In [37], an RL-based approach was proposed for optimizing hot water production. As stated by the authors, the proposed approach was completely generalizable and did not require an offline step or human domain knowledge to build a model of the hot water vessel or the heating element. The occupant preferences were learnt on the fly. The results show that energy consumption is reduced. In [38], a fuzzy logic-based control strategy for shifting the average power demand of residential EHWs was proposed. A minimum temperature for hot water, which deals with users’ comfort, was used as a control variable. The control strategy has the ability to shift the average power demand of residential EWHs from on-peak to off-peak periods.

In [39], the challenge of scheduling a large population of EWHs was addressed by incorporating a physically-based aggregate model of the population into the day-ahead unit commitment. Numerical simulations illustrated the applicability and benefits of the approach. In [40], an optimal EWH scheduling algorithm was proposed with the aim of optimizing the self-consumption of a residential PV installation. In addition, a novel approach allowing a set of realistic PV production scenarios to be generated was presented. This study has shown the influence on self-consumption performance of an efficient management system. In [41], a smart scheduling and control system for EWHs—data-driven disturbance forecasts were used in a robust MPC scheme—was proposed to accomplish various demand side management objectives. Simulations are performed on a central EWH supplying DHW for a multi-unit apartment building with quantified prediction uncertainty. The results show that the proposed system is capable of anticipating DHW demand and reducing electricity cost without affecting users’ comfort. In [42], an MPC-based approach was proposed for the optimal dispatch of thermostatically controlled loads, such as air conditioners and EHWs. Uncertainty was handled by considering demand scenarios. The results show that the total cost arising from both energy acquisition and schedule deviations is minimized. In [43], an MPC strategy was proposed for maximizing the PV power generation self-consumption rate in a household by exploiting the flexible demand of an EWH. The results demonstrate that the proposed approach outperforms a traditional thermostatic controller. In [44], a framework allowing implementing MPC controllers on residential water heaters was proposed for load shifting applications. Novel methods for estimating water draw patterns without using a flow meter are proposed as well. The MPC strategies were compared through simulations, under two different time-varying price signals. The results provide practical insight into effective MPC design for EWHs in home energy management systems.

1.3. Purpose and Organization of the Paper

In this context, ADSM strategies for the control of EWHs in individual dwellings equipped with PV panels are proposed and assessed using simulations, as part of the SmartECS project in which PROMES-CNRS and ART-Dev are involved. The first two strategies are rule-based. In the first strategy—SRBS, for standard rule-based strategy—which is the reference strategy, DHW production exclusively relies on the power extracted from the distribution grid during off-peak hours. The second strategy—ERBS, for enhanced rule-based strategy—takes advantage of the unused PV power generation for DHW production both during on-peak and off-peak hours. The third and fourth strategies rely on economic MPC—the optimization problem, which is of MINLP type, is solved using a genetic algorithm—and aim at optimizing DHW production by appropriately identifying the heating times, based on the available and unused PV power generation, as well as the times to sell the PV power generation surplus, while satisfying users’ comfort constraints. In the third strategy—MPC, for model-based predictive control strategy—the EWH model and the constraints are nonlinear. In the fourth strategy—LMPC, for linear model-based predictive control strategy—the EWH model and the constraints are linear. The four control strategies are discussed and assessed using simulations.

The paper is organized as follows. Section 2 deals with the development of DHW demand and power demand models, taking advantage of the data collected from the survey conducted by the PROMES-CNRS and ART-Dev laboratories. A model of the daily power demand is proposed as well. The case study on which the simulations are based is presented in this section. The EWH model, the rule-based control strategies, and the MPC and LMPC strategies are detailed in Section 3, along with the associated optimization problems and the linearization process (LMPC strategy). The results are discussed in Section 4. The paper ends with a conclusion and an outlook to future work (Section 5).

2. Human and Social Science (HSS) Study

2.1. Context and Objectives

As mentioned above, the HSS study aimed first to collect data related to the demand habits of residents in the southern French context, as well as their socioeconomic characteristics, their demographic characteristics, and their dwellings and equipment specificities. A survey was launched in 2021 based on a review of the existing literature. The survey managed to collect information about energy consumption habits and identify the key variables influencing both DHW demand and power demand. To the best of the authors’ knowledge, this is the first comprehensive study, in a French context, dealing with the development of DHW demand models for both the summer and winter seasons using a bottom-up approach grounded in real data, i.e., quantile regression.

2.2. Dataset

The survey was conducted by PROMES-CNRS and ART-Dev between mid-April and mid-June 2021, during the COVID-19 pandemic, which was a public health emergency characterized by barrier gestures and rules of social distancing, lockdown measures, and a change in demand habits. In these conditions, collecting data was a challenging task, and 236 responses were obtained.

Table 1. Descriptive statistics of quantitative variable samples, where ADDDW is the approximated daily demand volume of DHW during the winter season, ADDDS is the approximated daily demand volume of DHW during the summer season, AEC is the annual electricity consumption, NUWM is the number of times per week the washing machine is used, NUCD the number of times per week the clothes dryer is used, TSFD is the total square footage of the dwelling, AgeHRP is the age of the household reference person, HHS is the household size, NYCH is the number of children under the age of 18 in the household, NOCH is the number of children over the age of 18 in the household, and NWH is the number of women over the age of 18 in the household.

Variable	Min	Mean	Median	Max
ADDDW	10	289.7	268.6	840
ADDDS	30	348.7	300	1060.7
AEC	1011	6001	5387	17,084
NUWM	0	3.261	3	10
NUCD	0	0.8592	0	7
TSFD *	15	97.72	95	250
AgeHRP *	21	44.74	44	82
HHS *	1	2.516	2	6
NCH *	0	0.8216	0	4
NYCH	0	0.108	0	2
NOCH	0	0.615	0	3
NWH *	0	1.005	1	3

* Variable included in the final QR models, as explained in Section 2.3.2, Section 2.3.3 and Section 2.3.4.

provides samples of quantitative variables and a summary statistics of these variables, where ADDDW is the approximated daily demand volume of DHW during the winter season, ADDDS is the approximated daily demand volume of DHW during the summer season, AEC is the annual electricity consumption, NUWM is the number of times per week the washing machine is used, NUCD is the number of times per week the clothes dryer is used, TSFD is the total square footage of the dwelling, AgeHRP is the age of the household reference person, HHS is the household size, NYCH is the number of children under the age of 18 in the household, NOCH is the number of children over the age of 18 in the household, and NWH is the number of women over the age of 18 in the household.

Table 2. Descriptive statistics of qualitative variable samples, where HS is the type of heating system, WM is the presence of a washing machine, CD is the presence of a clothes dryer, DOWD is the average daily dwelling occupancy during weekdays from 8 a.m. to 8 p.m., DOWE is the average daily dwelling occupancy during weekend days from 8 a.m. to 8 p.m., HRHBD is the habit of reducing the heating temperature in the bedrooms during the day, and HRHBN is the habit of reducing the heating temperature in the bedrooms during the night.

Variable	Category	Frequency
HS	Individual (1)	9.74%
	Collective (2)	88.14%
	I do not know (3)	2.12%
WM	Yes (1)	92.80%
	No (0)	7.20%
CD *	Yes (1)	37.29%
	No (0)	62.71%
DOWD *	Less than 4 h (1)	47.29%
	Between 4 and 8 h (2)	33.90%
	More than 8 h (2)	18.64%
DOWE *	Less than 4 h (1)	4.24%
	Between 4 and 8 h (2)	47.46%
	More than 8 h (2)	48.31%
HRHBD	Yes (1)	67.37%
	No (0)	32.63%
HRHBN *	Yes (1)	72.03%
	No (0)	27.97%

* Variable included in the final QR models, as explained in Section 2.3.2, Section 2.3.3 and Section 2.3.4.

provides an overview of some qualitative variables and a summary statistics of these variables, where HS is the type of heating system, WM is the presence of a washing machine, CD is the presence of a clothes dryer, DOWD is the average daily dwelling occupancy during weekdays from 8 a.m. to 8 p.m., DOWE is the average daily dwelling occupancy during weekend days from 8 a.m. to 8 p.m., HRHBD is the habit of reducing the heating temperature in the bedrooms during the day, and HRHBN is the habit of reducing the heating temperature in the bedrooms during the night.

2.3. DHW Demand and Power Demand Modeling

2.3.1. Quantile Regression

DHW demand and power demand vary among households, and the diversity of household behaviors yields heterogeneous effects. Consequently, using the ordinary least squares (OLS) method—a common technique for estimating coefficients of linear regression equations describing the relationship between one or more independent quantitative variables and a dependent variable—for modeling both DHW demand and power demand is not adequate, as it does not allow discerning the effects of variables across the entire distribution range of the dependent variable, i.e., DHW demand or power demand. That is why we have decided to use quantile regression (QR). QR, introduced by Koenker and Bassett in 1978 [45], allows the individual impact of independent variables on specific quantiles of the outcome variable to be estimated. The method expands upon the conventional mean regression model by estimating conditional quantiles of the response variable, such as the median. A key advantage of QR over OLS lies in its limited dependence on bootstrapped standard errors and heteroskedastic errors, as well as its lack of necessity for a Gaussian error structure. Specifically, QR offers increased robustness against nonnormal errors and outliers compared to standard OLS. Consequently, it provides a robust data characterization, allowing for the consideration of explanatory variables effects across the entire distribution of the dependent variable, rather than solely focusing on its conditional mean [46]. Equation (1) describes the standard form of QR, where $(Y_i,X_i)$, $i=1,...,n$ is a sample from some population, $X_i$ is a $k\times 1$ vector of the regressors, $Q_{\theta }\left(X_i\right|Y_i)$ is a quantile of $Y_i$ given $X_i$, and ${ϵ}_{\theta }$ is supposed to satisfy the quantile restriction $Q_{\theta \left({ϵ}_i\right|X_i)}=0$:

$$Y_i=X_i^{\prime }{\beta }_{\theta }+{ϵ}_{\theta }\mathrm{with}{Q}_{\theta }\left(Y_i\right|X_i)=X_i^{\prime }{\beta }_{\theta },0<\theta <100\%$$

(1)

In the context of this research work, the standard log-linear equation is chosen to estimate the significant determinants of DHW demand and power demand during the summer and winter seasons, as outlined in Equation (2), with $x\in \{\mathrm{S},\mathrm{W},\mathrm{E}\}$:

$$Y_{i,x}=X_{i,x}^{\prime }{\beta }_{\theta ,x}+{ϵ}_{\theta ,x}$$

(2)

where W deals with estimating the daily DHW demand during the winter season in liters per day at a temperature of 40 °C, S deals with estimating the daily DHW demand during the summer season in liters per day at a temperature of 40 °C, and E deals with estimating the daily power demand in kWh. Moreover, ${\mathrm{Y}}_{i,x}$ is the vector of demand values (for DHW in winter or summer, or for energy) in logarithmic form, ${\mathrm{X}}_{i,x}$ is the vector of explanatory variables, ${\beta }_{\theta ,x}$ is the vector of parameters to be estimated, ${ϵ}_{\theta ,x}$ is the vector of residuals, and $\theta$ is the quantile under consideration.

The stepwise method is used to identify factors influencing both DHW demand and power demand in winter and summer. The method aims at minimizing the Akaike’s entropy-based information criterion (AIC) [47,48], which serves for selecting the most suitable model among a set of alternatives by identifying the one that gives optimal estimates while preserving simplicity. In other words, the method aims at determining the best trade-off between model accuracy and complexity. Since we did not obtain the same selection for both variables—DHW demand during the winter and summer seasons and power demand—the decision is based on the selection that maximizes the number of explanatory variables, while minimizing the AIC criterion.

The modeling results are presented in Section 2.3.2, Section 2.3.3 and Section 2.3.4. The daily DHW demand parameters are summarized in

Table 3. Model parameters for the daily DHW demand in summer, where ${\beta }_{\mathrm{x}}$ is the estimated vector of parameters, NCH is the number of children in the household, NWH is the number of women over the age of 18 in the household, AgeHRP is the age of the household reference person, TSFD is the total square footage of the dwelling, and DOWD and DOWE are the average daily dwelling occupancy during weekdays and weekend days, from 8 a.m. to 8 p.m., respectively. ${\mathrm{R}}^1$ allows assessing the model accuracy.

Parameter	10%	25%	50%	75%	90%
${\beta }_{\mathrm{NCH}}$	0.290	0.296	0.279	0.216	0.282
${\beta }_{\mathrm{NWH}}$	0.181	0.194	0.251	0.254	0.245
${\beta }_{\mathrm{AgeHPR}}$	−0.0010	−0.006	0.002	−0.002	−0.004
${\beta }_{\mathrm{TSFD}}$	0.002	0.0003	−0.002	−0.002	−0.002
${\beta }_{\mathrm{DOWD}}$	−0.060	−0.048	−0.115	−0.102	0.076
${\beta }_{\mathrm{DOWE}}$	0.034	−0.022	0.044	0.106	0.102
${\beta }_0$	4.704	5.153	5.356	5.877	6.233
${\mathrm{R}}^1$	0.178	0.149	0.183	0.149	0.116

and

Table 4. Model parameters for the daily DHW demand in winter, where ${\beta }_{\mathrm{x}}$ is the estimated vector of parameters, NCH is the number of children in the household, NWH is the number of women over the age of 18 in the household, AgeHRP is the age of the household reference person, TSFD is the total square footage of the dwelling, and DOWD and DOWE are the average daily dwelling occupancy during weekdays and weekend days, from 8 a.m. to 8 p.m., respectively. ${\mathrm{R}}^1$ allows assessing the model accuracy.

Parameter	10%	25%	50%	75%	90%
${\beta }_{\mathrm{NCH}}$	0.195	0.272	0.290	0.251	0.177
${\beta }_{\mathrm{NWH}}$	0.219	0.076	0.230	0.271	0.233
${\beta }_{\mathrm{AgeHPR}}$	0.001	−0.003	−0.004	−0.005	−0.010
${\beta }_{\mathrm{TSFD}}$	0.003	0.001	0.0002	−0.0001	−0.001
${\beta }_{\mathrm{DOWD}}$	−0.037	−0.103	−0.073	−0.132	−0.109
${\beta }_{\mathrm{DOWE}}$	−0.237	−0.066	0.038	0.037	−0.082
${\beta }_0$	4.224	5.006	5.244	5.691	6.400
${\mathrm{R}}^1$	0.185	0.154	0.169	0.173	0.138

for the summer season and the winter season, respectively. The daily power demand parameters are summarized in

Table 5. Model parameters for the daily power demand, where HHS is the household size, AgeHPR is the age of household reference person, EWH is the electric water heater, TWH is the thermal water heater, SWH is the solar water heater, OWH is the other water heater, HP is the heat pump, EH is the electric heater, FH is the fire heater, RAC is the reversible air conditioning system, OH is the other heating system, CD is the clothes dryer, and HRHBN is the habit of reducing the heating temperature in the bedrooms during the night.

Parameter	10%	25%	50%	75%	90%
${\beta }_{\mathrm{HHS}}$	0.207	0.188	0.127	0.160	0.201
${\beta }_{\mathrm{AgeHPR}}$	0.008	0.011	0.010	0.009	0.008
${\beta }_{\mathrm{EWH}}$	0.162	0.346	0.374	0.183	0.268
${\beta }_{\mathrm{TWH}}$	0.072	0.343	0.324	0.133	0.215
${\beta }_{\mathrm{SWH}}$	0.266	0.212	0.284	−0.030	−0.115
${\beta }_{\mathrm{OWH}}$	−0.333	−0.263	−0.452	−0.179	−0.221
${\beta }_{\mathrm{HP}}$	0.261	0.267	0.380	0.346	−0.037
${\beta }_{\mathrm{EH}}$	0.324	−0.029	0.051	0.218	−0.041
${\beta }_{\mathrm{FH}}$	−0.090	−0.274	−0.154	−0.100	−0.382
${\beta }_{\mathrm{RAC}}$	0.592	0.292	0.088	0.201	−0.170
${\beta }_{\mathrm{OH}}$	0.163	0.062	0.064	−0.007	−0.371
${\beta }_{\mathrm{CD}}$	0.235	0.271	0.210	0.197	0.094
${\beta }_{\mathrm{HRHBN}}$	−0.216	−0.144	−0.169	−0.184	−0.145
${\beta }_0$	4.224	5.006	5.244	5.691	6.400
${\mathrm{R}}^1$	0.305	0.267	0.271	0.247	0.246

. In these tables, $\theta$ is the $10\mathrm{th}$, $25\mathrm{th}$, $50\mathrm{th}$, $75\mathrm{th}$, or $90\mathrm{th}$ quantile. The outcomes shed light on how variations in the predictors influence DHW demand and power demand at different points of the distribution, revealing households that exhibit lower or higher demand levels. Accuracy of the model is assessed using the ${\mathrm{R}}^1$ criterion, by minimizing the sum of absolute quantities of residuals. The aim is to find the model coefficients that minimize the sum of the absolute discrepancies between observed values and predicted values for each quantile.

2.3.2. Parametric Model of the Daily DHW Demand in Summer

The findings suggest that NCH (the number of children in the household), NWH (the number of women over the age of 18 in the household), TSFD (the total square footage of the dwelling), AgeHPR (the age of the household reference person), DOWD (the average daily dwelling occupancy during weekdays, from 8 a.m. to 8 p.m.), and DOWE (the average daily dwelling occupancy during weekend days, from 8 a.m. to 8 p.m.) are the most important factors influencing the daily DHW demand during the summer season in the residential park of southwest France. The QR model ($V_s^{\mathrm{QR}}$) is described by Equation (3), with its parameters summarized in Table 3:

$$\begin{array}{cc}\hfill V_s^{\mathrm{QR}}={\beta }_0& +\left({\beta }_{\mathrm{NCH}}\mathrm{NCH}\right)+\left({\beta }_{\mathrm{NWH}}\mathrm{NWH}\right)+\left({\beta }_{\mathrm{AgeHPR}}\mathrm{AgeHPR}\right)\hfill \\ & +\left({\beta }_{\mathrm{TSFD}}\mathrm{TSFD}\right)+\left({\beta }_{\mathrm{DOWD}}\mathrm{DOWD}\right)+\left({\beta }_{\mathrm{DOWE}}\mathrm{DOWE}\right)\hfill \end{array}$$

(3)

The next paragraphs in this subsection deal with analyzing the effect of the different variables on the daily DHW demand during the summer season.

Socioeconomic Variables: NCH, NWH, and AgeHRP

The first socioeconomic variable under consideration is NCH, described by estimated vector ${\beta }_{\mathrm{NCH}}$. This variable demonstrates a positive correlation with various quantiles, as illustrated in Table 3. As NCH increases by one individual, DHW demand rises between 21.6% and 29.6% during the summer season, depending on the quantile. This positive effect could be attributed to the presence of children in the dwelling during the summer vacation period.

The second socioeconomic variable under consideration is NWH, described by estimated vector ${\beta }_{\mathrm{NWH}}$. This variable has a significant effect over all quantiles (Table 3). As NWH increases by one individual, DHW demand rises between 18.1% and 26.8% during the summer season, depending on the quantile. This can be explained by the observed tendency for women to typically consume a greater volume of hot water during showering, as evidenced in the existing literature [10]. This indicates that women aged over 18 exhibit higher DHW demand patterns, which are often associated with hygiene practices.

The third socioeconomic variable is AgeHRP, described by estimated vector ${\beta }_{\mathrm{AgeHRP}}$. This variable demonstrates a negative effect on the 10th, 25th, 75th, and 90th quantiles; however, it has a positive effect on the 50th quantile. This negative effect can be explained by the fact that young and unemployed individuals prefer to reduce their DHW demand in order to lower their energy bill. Furthermore, employed individuals spend the majority of their time outside of the dwelling, which has a negative effect on DHW demand. However, the positive impact can be explained by the fact that older households tend to use more water per capita than younger households due to spending more time at the dwelling and using water-dependent appliances, as evidenced in the literature [18].

Technical Dwelling Variable: TSFD

The technical dwelling variable under consideration is TSFD, described by estimated vector ${\beta }_{\mathrm{TSFD}}$. This variable has various effects on the quantiles. For the 10th quantile, the daily DHW consumption is expected to rise by 0.2% with each additional square meter of dwelling area. In addition, the daily DHW consumption is expected to decrease by 0.2% for the 50th and 75th quantiles, respectively, and by 0.3% for the 90th quantile with each additional square meter of dwelling area. Such a variation can be explained by the fact that young occupants may not necessarily employ energy-efficient devices. However, the negative effect observed is explained by the fact that homeowner occupants are more sensitive to reducing their energy consumption by using energy-efficient devices. This sensitivity arises due to their status as property owners and larger households in some cases, making it more appealing for them to utilize energy-efficient devices. This is due to an economy-of-scale effect as well, whereby resource utilization becomes more efficient with increased occupancy, or to user behavior adaptations that emerge in response to high-occupancy conditions, as shown in the literature [10,18].

Household Behavior Variables: DOWD and DOWE

The first household behavior variable under consideration is DOWD, described by estimated vector ${\beta }_{\mathrm{DOWD}}$. The results highlight the negative effect of this variable on the daily DHW demand during the summer season for the 10th, 25th, 50th, and 75th quantiles. However, this variable has a positive effect on the 90th quantile. In each quantile, the negative effect is linked to occupancy duration, also known as dwell time, and DHW demand habits. Young individuals, employed individuals, students, and elderly do not consume a large quantity of DHW due to their limited financial resources and because of their absence from the dwelling during week days. However, the positive impact can be explained by the number of individuals present in the dwelling during the summer period, consuming hot water simultaneously, as shown in the literature [18].

The second household behavior variable under consideration is DOWE, described by estimated vector ${\beta }_{\mathrm{DOWE}}$. The results presented in Table 3 highlight the positive impact of this variable on the 10th, 50th, 75th and 90th quantiles. This can be explained by the fact that more persons are present in the dwelling at the same time during weekend days than during weekdays [18]. In addition, DOWE has a negative effect on the 25th quantile. This can be due to the fact that young individuals spend more time out of the dwelling during weekend days than the other age groups.

2.3.3. Parametric Model of the Daily DHW Demand in Winter

The findings suggest that NCH, NWH, TSFD, AgeHPR, DOWD, and DOWE are the most important factors influencing the daily DHW demand during the winter season in the residential park of southwest France. The QR model ($V_w^{\mathrm{QR}}$) is described by Equation (4), with its parameters summarized in Table 4:

$$\begin{array}{cc}\hfill V_w^{\mathrm{QR}}={\beta }_0& +\left({\beta }_{\mathrm{NCH}}\mathrm{NCH}\right)+\left({\beta }_{\mathrm{NWH}}\mathrm{NWH}\right)+\left({\beta }_{\mathrm{AgeHPR}}\mathrm{AgeHPR}\right)\hfill \\ & +\left({\beta }_{\mathrm{TSFD}}\mathrm{TSFD}\right)+\left({\beta }_{\mathrm{DOWD}}\mathrm{DOWD}\right)+\left({\beta }_{\mathrm{DOWE}}\mathrm{DOWE}\right)\hfill \end{array}$$

(4)

The next paragraphs in this subsection deal with analyzing the effect of the different variables on the daily DHW demand during the winter season.

Socioeconomic Variables: NCH, NWH, and AgeHRP

The first socioeconomic variable under consideration is NCH, described by estimated vector ${\beta }_{\mathrm{NCH}}$. The results demonstrate a positive correlation with all the quantiles of DHW demand in the winter season, as illustrated in Table 4. The estimates for NCH range from 17.7% to 29.0%, depending on the quantile.

The second socioeconomic variable under consideration is NWH, described by estimated vector ${\beta }_{\mathrm{NWH}}$. NWH has a significant and positive impact on all the quantiles of DHW demand during both the winter and summer seasons. This can be explained by the fact that women’s preferences and routines in terms of personal hygiene, such as showering and hand-washing, may influence DHW demand. These findings align with those of Marszal-Pomianowska et al. [49].

The third socioeconomic variable, described by estimated vector ${\beta }_{\mathrm{AgeHPR}}$, is AgeHRP. This variable exhibits heterogeneous effects across quantiles. Specifically, it has (1) a negative effect on the 25th, 50th, 75th, and 90th quantiles of DHW demand during the winter season and (2) a positive effect on the $10\mathrm{th}$ quantile. The negative effect can be attributed to several factors among which employed individuals tend to spend more time outside for work- and school-related reasons during the winter season [18].

Technical Dwelling Variable: TSFD

The technical dwelling variable under consideration is TSFD, described by estimated vector ${\beta }_{\mathrm{TSFD}}$. This variable has various effects on the quantiles of DHW demand during the winter season. For each additional square meter of dwelling area, the daily DHW demand is expected to rise by 0.3%, 0.1%, and 0.02% for the 10th, 25th, and 50th quantiles, respectively. This can be explained by the lack of energy-efficient devices used in these dwellings. However, for each additional square meter of dwelling area, the daily DHW demand is expected to decrease by 0.01% and 0.1% for the 75th and 90th quantiles, respectively, which is due in this case by the energy-efficient devices used in these dwellings [10,18].

Household Behavior Variables: DOWD and DOWE

The first household behavior variable under consideration is DOWD, described by estimated vector ${\beta }_{\mathrm{DOWD}}$. The results highlight the negative effect of DOWD on the daily DHW demand during the winter season for all quantiles, which is linked to the time spent at work for adults and the time spent at school for primary schoolchild or at university for students.

The second household behavior variable under consideration is DOWE, described by the estimated vector ${\beta }_{\mathrm{DOWE}}$. The results presented in Table 4 highlight the negative impact of DOWE on the 10th, 25th, and 90th quantiles, which can be explained by the tendency of young individuals and young couples to engage in outdoor activities during weekend days.

2.3.4. Parametric Model of the Daily Power Demand

The findings suggest the most important factors influencing the daily power demand in the residential park of southwest France are HHS (the household size), AgeHPR (the age of household reference person), HWS—which can be an EWH, a thermodynamic water heater (TWH), a solar water heater (SWH), or another type of water heater (OWH)—HSC—which can be a heat pump (HP), an electric heater (EH), a fireplace heater (FH), a reversible air conditioning (RAC) or another type of heater (OH)—CD (clothes dryer), and HRHBN (the habit of reducing the heating temperature in the bedrooms during the night). The QR model ($E^{\mathrm{QR}}$) is described by Equation (5), with its parameters summarized in Table 5:

$$\begin{array}{cc}\hfill E^{\mathrm{QR}}={\beta }_0& +\left({\beta }_{\mathrm{HHS}}\mathrm{HHS}\right)+\left({\beta }_{\mathrm{AgeHPR}}\mathrm{AgeHPR}\right)+\left({\beta }_{\mathrm{HWS}}\mathrm{HWS}\right)\hfill \\ & +\left({\beta }_{\mathrm{HSC}}\mathrm{HSC}\right)+\left({\beta }_{\mathrm{CD}}\mathrm{CD}\right)+\left({\beta }_{\mathrm{HRHBN}}\mathrm{HRHBN}\right)\hfill \end{array}$$

(5)

The next paragraphs deal with analyzing the effect of the different variables on the daily power demand.

Socioeconomic Variables: HHS and AgeHRP

The first socioeconomic variable under consideration is HHS, described by estimated vector ${\beta }_{\mathrm{HHS}}$, which demonstrates a positive correlation with all the quantiles of power demand, as illustrated in Table 5. This implies that as the household size increases, power demand also increases, which is coherent with the literature studies [46,50,51].

The second socioeconomic variable is AgeHRP, described by estimated vector ${\beta }_{\mathrm{AgeHRP}}$, which has a positive impact on all quantiles. When the age of the household reference person increases by one year, power demand increases by 0.8% to 1.1%, depending on the considered quantile. The results are considered significant and align with literature results [46,50,51].

Technical Dwelling Variables: HWS, HSC, and CD

The first technical dwelling variable under consideration is HWS, described by estimated vectors ${\beta }_{\mathrm{EWH}}$, ${\beta }_{\mathrm{TWH}}$, ${\beta }_{\mathrm{SWH}}$, and ${\beta }_{\mathrm{OWH}}$. The results indicate that mainly all HWSs have a positive effect on power demand, with the exception of water heating systems categorized as OWH systems whose effect on power demand is negative. However, the extent of this positive effect varies depending on the primary energy source, which confirm findings from studies on the fact that power demand patterns may vary based on the efficiency and usage of the HWS, highlighting the importance of considering such systems in energy policy and conservation efforts [46,50,51]. The next parts of this paper only focus on dwellings equipped with EWHs used as HWSs. Thus, the first objective of the SmartECS project is to investigate the relevance of storing the unused PV power generation during the day. The second reason is related to the fact that the EWH is the most commonly used instrument for DHW production in France, representing 46% of the water heaters used in the residential sector in France.

The second technical dwelling variable under consideration is HSC, described by estimated vectors ${\beta }_{\mathrm{HP}}$, ${\beta }_{\mathrm{EH}}$, ${\beta }_{\mathrm{FH}}$, ${\beta }_{\mathrm{RAC}}$, and ${\beta }_{\mathrm{OH}}$. In this research work, the focus is only put on dwellings equipped with electric heating (EH) systems due to the fact that such systems are the most used heating systems in southwest France, where the study is conducted, according to the statistics provided by INSEE [52].

The third technical dwelling variable under consideration is CD, described by estimated vector ${\beta }_{\mathrm{CD}}$, which has a significant effect on all considered quantiles. Households owning a clothes dryer consume between 9.4% and 27.1% more electricity than households not equipped with such equipment, which is consistent with literature findings [46].

Household Behavior Variable: HRHBN

The household behavior variable under consideration is HRHBN, described by estimated vector ${\beta }_{\mathrm{HRHBN}}$, which has a negative effect on power demand. The estimates range between −14.4% and −21.6%. Of course, households reporting a reduction in heating temperature in the bedrooms during the night consume less energy than others. The results are consistent with literature findings [46].

2.4. Case Study

This section deals with the dwellings considered in this study and the typical scenarios. Subsequently, an outline of the methods used to develop DHW and non-DHW power demand scenarios, as well as PV power generation scenarios, is provided.

2.4.1. Dwelling Description

The individual dwellings are all equipped with PV panels and domestic appliances such as heating systems, washing machines, refrigerators, ovens, and televisions. Electricity consumption of those appliances is classified as non-DHW power demand. Hot water is produced by means of an electric heater, which is sized according to the occupants’ demand. For the purpose of this study, all households have contracted with an electricity provider and benefit from advantageous electricity prices thanks to on-peak/off-peak hours tariffs (as detailed in Section 2.4.2). In addition, agreements are in place for selling the PV power generation surplus.

2.4.2. Power Purchase and CO₂ Emissions

Grid balancing, which is the mechanism of balancing energy supply and demand, is essential. This is the reason behind the introduction of on-peak/off-peak hours. Users are encouraged to shift their electricity usage to off-peak hours, which are decided based on operating conditions and the local capacity of the power distribution grid [53], by the French distribution network operator ENEDIS. In this research study, the choice of on-peak/off-peak hours is made to encompass all possible distributions during both the day and the night, as shown in

Table 6. Electricity purchase tariffs (${\theta }_b^{\mathrm{ON}/\mathrm{OFF}}$) according to EDF 2022 for contracts with on/off-peak hours over 24 $\mathrm{h}$ [53], in $\mathrm{EUR}/\mathrm{k}\mathrm{W}\mathrm{h}$.

Period of Time	Hour Type	${\mathit{\theta }}_{\mathit{b}}^{\mathbf{ON}/\mathbf{OFF}}$
00 h 00–05 h 50	Off-peak hours	14.70
06 h 00–07 h 50	On-peak hours	18.41
08 h 00–11 h 50	Off-peak hours	14.70
12 h 00–13 h 50	On-peak hours	18.41
14 h 00–15 h 50	Off-peak hours	14.70
16 h 00–21 h 50	On-peak hours	18.41
22 h 00–23 h 50	Off-peak hours	14.70

. The electricity purchase tariffs (${\theta }_b^{\mathrm{ON}/\mathrm{OFF}}$)—electricity tariffs evolve and generally increase over time—considered in this study are the ones set by EDF (Electricité de France) in 2022. The database quantifying how carbon-intensive (in $\mathrm{g}{\mathrm{CO}}_{2.\mathrm{eq}}/\mathrm{k}\mathrm{W}\mathrm{h})$ electricity in France is comes from the website “electricitymap.org” [54]. The ${\mathrm{CO}}_2$ emissions per unit of power extracted from the distribution grid are calculated using Equation (6), where $E_G$ is the energy consumed from the distribution grid and ${\mathrm{CI}}_{eq}$ is the carbon intensity in kilograms of ${\mathrm{CO}}_2$ equivalent per kilowatt-hour:

$${\mathrm{CO}}_2\left[\mathrm{k}\mathrm{g}{\mathrm{CO}}_{2eq}\right]=E_G\left[\mathrm{k}\mathrm{W}\mathrm{h}\right]{\mathrm{CI}}_{eq}[\mathrm{k}\mathrm{g}{\mathrm{CO}}_{2eq}/\mathrm{k}\mathrm{W}\mathrm{h}]$$

(6)

2.4.3. Typical Scenarios

The typical scenarios are based on the French household structure reported by INSEE in 2021 [55], where 37% of the French households consist of a single person, 33% consist of 2 persons, 13% consist of 3 persons, 11% consist of 4 persons, and 4% consist of 5 persons, as shown in Figure 1. After identifying the most representative household structures, the scenario parameters were identified by applying the median across all families of the same structure in the HSS database—as previously mentioned (see Section 2.2), the database is composed of 236 responses, which were collected from a survey between mid-April and mid-June 2021, i.e., during the COVID-19 pandemic—with the aim of developing representative scenarios of the treated population (see Table 1 and Table 2).

Table 7. Characteristics of the typical scenarios, where AgeHPR is the age of the reference person, NWH is the number of women over the age of 18 in the household, NCH is the number of children in the household, TSFD is the total square footage of the dwelling, and DOWD is the average daily occupancy of the dwelling during weekdays, from 8 a.m. to 8 p.m.

Typical Scenario	1	2	3	4
AgeHRP	38	48	47	42
NWH	1	1	1	1
NCH	0	0	1	2
TSFD	46	90	110	120
DOWD	2	3	5	6

summarizes the characteristics (AgeHRP, NWH, NCH, TSFD, and DOWD) of each typical scenario.

2.4.4. DHW Demand

The development of DHW demand profiles has two main parts. The first part involves the parametric modeling of the daily DHW demand (in liters at 40 °C) using the QR approach (Equations (3) and (4)). The second part deals with the development of a parametric time series for DHW demand (in liters at 60 °C).

Validation of the Parametric DHW Demand Models

Validation of the models, which is based on a bottom-up approach presented in Section 2.3.2 and Section 2.3.3, is performed using a guide to DHW demand in metropolitan France (from 2002 to 2015) from ADEME and a dataset consisting of more than 15,500 annual water meter readings over one year [56]. The guide provides daily ranges of DHW demand values for French consumers, which vary depending on the household size. These ranges of DHW demand values are used to validate the results obtained from the parametric QR models of the daily DHW demand during both the winter season and the summer season. The validation results are presented in

Table 8. Daily DHW demand (in $\mathrm{L}/\mathrm{d}$) scenarios (ADEME study and PROMES/ART-Dev study). $V^{\mathrm{ADEME}}$: ADEME daily DHW demand. $V_W^{\mathrm{QR}}$: QR daily DHW demand for the winter season. $V_S^{\mathrm{QR}}$: QR daily DHW demand for the summer season. ${\mathrm{NRMSE}}_W$: normalized root mean square error relative to ground-truth values for the winter season. ${\mathrm{NRMSE}}_S$: normalized root mean square error relative to ground-truth values for the summer season. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	1	2	3	4
$V^{\mathrm{ADEME}}$	$80\pm 35$	$120\pm 45$	$150\pm 50$	$170\pm 70$
$V_W^{\mathrm{QR}}$	107	172	190	290
${\mathrm{NRMSE}}_W$	0.00	0.62	0.00	0.36
$V_S^{\mathrm{QR}}$	94	163	171	265
${\mathrm{NRMSE}}_S$	0.00	0.00	0.00	0.35

, where the daily DHW demand (in liters per day at a temperature of 40 °C) given by the ADEME model ($V^{\mathrm{ADEME}}$), the daily DHW demand given by the parametric QR model for the winter season ($V_W^{\mathrm{ADEME}}$), and the daily DHW demand given by the parametric QR model for the summer season ($V_S^{\mathrm{ADEME}}$) are summarized. The normalized root mean square error (NRMSE) (Equation (7)), which allows the normalized average magnitude of the errors between the estimated values provided by the parametric QR models and the ground-truth values—DHW demand values provided by ADEME—to be evaluated, is provided as well:

$$\begin{array}{cc}\hfill & \mathrm{NRMSE}={\displaystyle \frac{\sqrt{\frac{1}{n}{\sum }_{i=1}^n{(x_i-{\hat{x}}_i)}^2}}{x^{max}-x^{min}}}\hfill \end{array}$$

(7)

Parametric DHW Demand Profiles

In this work, a model developed by ADEME [56] has been modified and used to define time-series profiles. The ADEME’s model takes into consideration the daily DHW demand at a temperature of 40 °C (Table 8), the hourly, weekly, and monthly coefficients of DHW demand at a temperature of 40 °C, and the cold water temperature ($T_{cw}$). The main limitation of the model is the daily DHW demand at 40 °C, which is defined based on a limited number of occupants. Therefore, in this research study, it has been replaced by the daily DHW demand estimated for both the summer season and the winter season using the QR models, which are parametrically defined, based on variables identified as major influencers of the daily DHW demand in southwest France (Section 2.3.2 and Section 2.3.3), and considered more realistic. The model is described by Equation (8):

$$Q_{60}=V_x^{\mathrm{QR}}{C}_H{C}_W{C}_M\left({\displaystyle \frac{40-T_{cw}}{60-T_{cw}}}\right)$$

(8)

where $V_x^{\mathrm{QR}}$ is the daily DHW demand (in liters per day) during the season $x\in \{\mathrm{S},\mathrm{W}\}$, where S indicates the summer season and W indicates the winter season, $C_H$, $C_W$, and $C_M$ are the hourly, weekly, and monthly coefficient of DHW demand at 40 °C, respectively, and $T_{cw}$ is the cold water temperature.

2.4.5. Non-DHW Power Demand

The development of non-DHW power demand profiles has two main parts. The first part involves the parametric modeling of the daily power demand using a bottom-up approach. The second part deals with the development of a parametric time-series profile for non-DHW power demand. This is achieved through a correlation study between the daily power demand calculated in the first part of the work and an averaged profile published by the French distribution network operator ENEDIS, which deals with a typical occupant who has a power purchasing contract with on-peak/off-peak hours, living in southwest France.

Validation of the Non-DHW Power Demand Model

Validation of the model (Equation (5)) is carried out thanks to the real power demand data collected from the HSS study and by applying the median of the samples with the same household structure. The results are presented in

Table 9. Power demand in $\mathrm{k}\mathrm{W}\mathrm{h}/\mathrm{year},$ where $E^R$ is the value from the HSS database used for validation, $E^{QR}$ is the value predicted by the HSS model, and ${\mathrm{NRMSE}}_E$ is the normalized root mean square error. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	1	2	3	4
$E^R$	3881	4163	5415	6762
$E^{QR}$	3325	4404	5298	6224
${\mathrm{NRMSE}}_E$	0.1055	0.0517	0.0251	0.1155

. Accuracy is assessed based on NRMSE (Equation (7)). $E^{QR}$ stands for estimates of the daily power demand provided by the QR model and $E^R$ refers to estimates of the daily power demand obtained by applying the median to the samples with the same household architecture in the HSS database. Overall, the results are acceptable. However, a tiny variation between the ground-truth values and the predicted values is observed, as shown in Table 9.

Parametric Time-Series Profile for Non-DHW Power Demand

Finally, a parametric time-series profile of the non-DHW power demand has been developed. An averaged profile published by ENEDIS [57] has been used as a basis to model the daily trends of non-DHW power demand in dwellings in southwest France, ensuring seasonal variability, such as an increase in power demand during the winter season due to a high energy usage. The profile deals with a typical occupant who has a power purchasing contract with on-peak/off-peak hours [53]. Such a contract allows taking advantage of off-peak hours for turning on electrical appliances, such as EWHs, and contributes to mitigating grid load during on-peak hours. A correlation study was conducted using the above-mentioned ENEDIS profile and the daily power demand estimated by the parametric QR model. This correlation study led to subtracting the annual power needed for DHW production, and correlating the parametric annual non-DHW power demand with the ENEDIS typical profile to obtain a representative time-series profile of the non-DHW power demand.

2.4.6. PV Power Generation

The PV power generation profile is obtained from measurements of global horizontal irradiance (GHI)—GHI refers to the total amount of shortwave radiation received from above by a surface horizontal to the ground—conducted by PROMES-CNRS in Perpignan (France) and is based on the models developed by Takilalte et al. [58] and Bressan et al. [59]. The database is made up of one year of GHI measurements.

3. EWH Modeling and Control

3.1. EWH Modeling

In this paper, the EWH model is chosen based on two criteria: an acceptable level of accuracy according to the application requirements and a reasonable computation time. Therefore, a model describing the heat transfer occurring in an EWH is considered [60], based on two submodels, as explained in the subsections below (Section 3.1.1 and Section 3.1.2). The transition between the two EWH submodels occurs by satisfying the different conditions mentioned in Algorithm 1, depending on the situation of the tank. Here, $m_{min}$ is the minimum hot water demand that can impact the hot water height in the tank.

Algorithm 1 Transition rules between the two EWH submodels.

1: Transition rules from the 2nd EWH submodel to the 1st EWH submodel   2:   if $h=H$ and ${\displaystyle \frac{dh}{dt}}\ge 0$:   3:     then the first EWH submodel (Equations (9) and (10)) = active   4:     and $h=H$ (constant)   5: $\to$ The transition occurs when the maximum tank height is reached and the hot water height h continues to increase, causing this energy to produce a temperature rise.   6: Transition rules from the 1st EWH submodel to the 2nd EWH submodel   7:   if $\dot{m}\ge m_{min}$   8:     then the second EWH submodel (Equations (11), (12) and (13)) = active   9:     then $T_w(k+1)=T_w\left(k\right)$ (constant) 10: $\to$ The transition occurs when the DHW demand flow rate impacts the hot water height in the tank significantly, so $h\ne H$ and ${\displaystyle \frac{dh}{dt}}<0$

3.1.1. First EWH Submodel

The first EWH submodel assumes that the water in the tank has a uniform homogeneous temperature, and the heat transfer process is modeled as a first-order differential Equation (Equations (9) and (10)):

$$C_w{\displaystyle \frac{d{T}_w}{dt}}=P_{\mathrm{EWH}}-\dot{m}{C}_p(T_w-T_{inlet})+U{A}_w(T_{amb}-T_w)$$

(9)

$$C_w=m{C}_p$$

(10)

where $C_w$ is the specific heat capacity of the tank, $T_w$ is the hot water temperature, $P_{\mathrm{EWH}}$ is the power of the EWH heating element, $\dot{m}$ is the hot water flow rate, $C_p$ is the thermal capacitance of water, $T_{inlet}$ is the cold water temperature at the tank inlet, $T_{amb}$ is the ambient temperature, $U{A}_w$ is the thermal conductance of the tank, and m is the mass of the tank.

3.1.2. Second EWH Submodel

The second EWH submodel assumes two distinct compartments of water in the tank: (i) the upper compartment has a uniform water temperature equal to the temperature provided by the first EWH submodel and (ii) the lower compartment has a uniform water temperature which is equal to the cold water temperature $T_{inlet}$ (Equation (11)):

$${\displaystyle \frac{dh}{dt}}=a-bh$$

(11)

where a is defined as follows (Equation (12)):

$$a=\left({\displaystyle \frac{P_{\mathrm{EWH}}+U{A}_w(T_{amb}-T_{lower})}{C_w(T_{upper}-T_{lower})}}-{\displaystyle \frac{\dot{m}{C}_p}{C_w}}\right)H$$

(12)

and b is defined as follows (Equation (13)):

$$b={\displaystyle \frac{U{A}_w}{C_w}}$$

(13)

where h is the hot water height, H is the height of the tank, $T_{lower}$ is the temperature of the water in the lower part of the tank, and $T_{upper}$ is the temperature of the water in the upper part of the tank.

3.2. Rule-Based Control Strategies

3.2.1. Standard Rule-Based Strategy

The reference strategy is rule-based. This strategy—denoted as the standard rule-based strategy (SRBS)—describes the classical behavior of EWHs in individual dwellings in France (Figure 2). Algorithm 2 summarizes the different rules. Water heating is performed during off-peak hours only. k is the actual time step, $T_w^{min}$ and $T_w^{max}$ are the minimum and maximum temperature control thresholds, respectively, and $h^{min}$ and $h^{max}$ are the minimum and maximum hot water height control thresholds, respectively. $P_c$ is the power consumed to meet the non-DHW power needs and $P_G$ is the power extracted from the distribution grid. To summarize, the main idea behind the SRBS strategy is to turn the EWH on when the temperature or the height of hot water in the tank is below a minimum control threshold, only during off-peak hours.

Algorithm 2 Standard rule-based strategy (SRBS).

1: EWH operation mode during off-peak hours 2:   if $T_w\left(k\right)\le T_w^{min}$ or $h\left(k\right)\le h^{min}$ 4:     then $P_{\mathrm{EWH}}(k+1)=P_G$ 4:   else if $T_w\left(k\right)\ge T_w^{max}$ or $h\left(k\right)\ge h^{max}$ 5:     then $P_{\mathrm{EWH}}(k+1)=\mathrm{OFF}$ 6: EWH operation mode during on-peak hours 7:   $\forall T_w\left(k\right)\mathrm{and}\forall h\left(k\right)$ 8:     $P_{\mathrm{EWH}}=\mathrm{OFF}$

3.2.2. Enhanced Rule-Based Strategy

An enhanced rule-based strategy (ERBS) is proposed as well with the aim of optimizing the utilization of the PV power generation surplus for DHW production without inflating the grid purchase cost (Figure 3). Algorithm 3 details the different rules the proposed ERBS strategy is based on.

Algorithm 3 Enhanced rule-based strategy (ERBS).

1: EWH operation mode during off-peak hours   2:  if $(P_{\mathrm{sPV}}(k+1)\ge P_{max})$   3:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$   4:    then $P_{\mathrm{EWH}}(k+1)=P_{\mathrm{sPV}}(k+1)$   5:  else if $0\le P_{\mathrm{sPV}}(k+1)<P_{max}$   6:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$   7:    then $P_{\mathrm{EWH}}(k+1)=P_{\mathrm{sPV}}(k+1)+P_G$   8:  else if $P_{\mathrm{sPV}}(k+1)<0$   9:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$ 10:    then $P_{\mathrm{EWH}}(k+1)=P_G$ 11: EWH operation mode during on-peak hours 12:  if $P_{\mathrm{sPV}}(k+1)\ge P_{max}$ 13:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$ 14:    then $P_{\mathrm{EWH}}(k+1)=P_{\mathrm{sPV}}(k+1)$ 15:  else if $S(k+1)\le P_{\mathrm{sPV}}(k+1)<P_{max}$ 16:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$ 17:    then $P_{\mathrm{EWH}}(k+1)=P_{\mathrm{sPV}}(k+1)+P_G$ 18:  else if $P_{\mathrm{sPV}}(k+1)<S(k+1)$ 19:   if $h\left(k\right)\le h^{min}\mathrm{or}{T}_w\left(k\right)\le T_w^{min}$ 20:    then $P_{\mathrm{EWH}}(k+1)=\mathrm{OFF}$

The core concept revolves around activating the EWH during both on-peak hours and off-peak hours, employing a specific threshold (Equation (14)) for ensuring that the cost of extracting power from the distribution grid during on-peak hours does not exceed the cost during off-peak hours, fostering a balanced equilibrium between power extraction and sales. The threshold is calculated after satisfying the non-DHW power demand thanks to the PV power generation, where $P_{max}$ is the maximum EWH power, $P_{\mathrm{sPV}}$ is the remaining PV power generation surplus after satisfying the non-DHW power demand calculated using Equation (15), ${\theta }^{\mathrm{ON}}$ and ${\theta }^{\mathrm{OFF}}$ are the electricity purchase tariffs during on-peak hours and off-peak hours, respectively, and ${\theta }_s$ is the selling price of the unused PV power generation:

$$S={\displaystyle \frac{P_{max}({\theta }^{\mathrm{ON}}-{\theta }^{\mathrm{OFF}})}{{\theta }^{\mathrm{ON}}-{\theta }_s}}$$

(14)

$$P_{\mathrm{sPV}}=P_{\mathrm{PV}}-P_c\mathrm{and}{P}_{\mathrm{sPV}}=P_{\mathrm{sPV}}(P_{\mathrm{sPV}}>0)$$

(15)

3.3. Economic-Model-Based Predictive Control Strategies

As mentioned above, economic-model-based predictive control (economic MPC) is the main control method chosen in this research work. Economic MPC is an advanced control method that combines predictive control with economic optimization. In other words, the method aims at optimizing a performance criterion related to economic objectives. In this research work, economic MPC allows for optimizing the EWH behavior with a one-day-ahead prediction horizon (H_p = 24 h), taking into consideration three perfect prediction vectors which influence the system’s behavior: DHW demand, non-DHW power demand, and PV power generation. An EWH model, a set of constraints (Section 3.3.3), and an optimizer tailored to the optimization problem (Section 3.3.2) are used, as shown in Figure 4. This essentially acts as a sliding window allowing the EWH’s behavior to be optimized in advance.

The choice of the optimization algorithm depends on the optimization variable. In this research work, this variable is the power of the EWH heating element ($P_{\mathrm{EWH}}$). The optimizer decides for the best times to heat the water in the tank to a certain temperature, based on an objective function to be minimized. Times with an available PV power generation surplus are prioritized and in case this surplus is not enough, power is extracted from the distribution grid during off-peak hours as much as possible. Additionally, the optimizer decides for instances that maximize the economic gain by increasing the PV power generation self-consumption rate while satisfying users’ comfort (i.e., ensuring the availability of hot water for daily activities).

3.3.1. Optimization Problem Category

The optimization problem falls under the category of MINLP. The nonlinearity is due to the EWH model, the objective function, and the optimization constraints. In addition, the variation of the optimization variable is binary (i.e., ON/OFF): $P_{\mathrm{EWH}}(k+i)\in [0;P_{max}]$, where $P_{max}$ is the maximum EWH heating power, k is the actual time step, and i is the time step in the prediction horizon.

3.3.2. Optimization Problem Formulation

The optimization problem aims at reducing the economic cost ($E_G$), increasing the PV power generation self-consumption rate while benefiting from sales of the PV power generation surplus, and upholding users’ comfort, which is described by nonlinear constraints. The objective function $F_{obj}$ is minimized using an optimizer—a genetic algorithm here—providing $P_{\mathrm{EWH}}^{*}$ (Equation (16)):

$$P_{\mathrm{EWH}}^{*}=\arg \min F_{obj}\left(P_{\mathrm{EWH}}\right)$$

(16)

$F_{obj}$ is defined as follows (Equation (17)), where $E_G$ is the energy consumed from the distribution grid, ${\theta }_b^{\mathrm{ON}/\mathrm{OFF}}$ is the cost associated with the power extracted from the distribution grid during on-peak/off-peak hours, and ${\theta }_s$ is the selling price of the PV power generation surplus:

$$\begin{array}{cc}\hfill F_{obj}& =\sum _{i=1}^{H_p}\left(E_G(k+i)(E_G(k+i)>0){\theta }_b^{\mathrm{ON}/\mathrm{OFF}}(k+i)\right)\hfill \\ & +\sum _{i=1}^{H_p}\left(E_G(k+i)(E_G(k+i)\le 0){\theta }_s(k+i)\right)\hfill \end{array}$$

(17)

The corresponding energy $E_{\mathrm{EWH}}^{*}$ is given by Equation (18), where $T_e=10$ min is the time step:

$$E_{\mathrm{EWH}}^{*}(k+i)=P_{\mathrm{EWH}}^{*}(k+i){\displaystyle \frac{1}{T_e}}$$

(18)

The power extracted from the distribution grid ($P_G$) is a function of the PV power generation ($P_{\mathrm{PV}}$), the non-DHW power demand ($P_c$), and the EWH power demand ($P_{\mathrm{EWH}}$) (Equation (19)):

$$P_G(k+i)=P_{\mathrm{EWH}}(k+i)-P_{\mathrm{PV}}(k+i)+P_c(k+i)$$

(19)

3.3.3. Optimization Constraints

As mentioned above, the optimization problem is nonlinear, primarily due to the nonlinear nature of the EWH model used and the objective function. The optimization constraints are nonlinear as well. These constraints deal with users’ comfort, i.e., the hot water temperature and the height of hot water in the tank. Both of these variables must remain above a minimum comfort level and below a maximum comfort level. The selection of the thresholds is paramount to ensure that the hot water temperature is above a minimum value ($T_w^{min}$), thus mitigating the risk of bacteria growth in the tank, such as Legionella pneumophila (L. pneumophila), and guarantee the availability of hot water. In addition, a maximum value is set to avoid the risk of scalding ($T_w^{max}$) (Equation (20)):

$$T_w^{min}\le T_w(k+i)\le T_w^{max}$$

(20)

The minimum and maximum heights of hot water in the tank allowing hot water requirements to be satisfied were determined after multiple trials and considering different demand profiles. The minimum comfort height is $h^{min}=60\%$ and the maximum comfort height is equal to the height of the tank ($h^{max}=H$) (Equation (21)):

$$h^{min}\le h(k+i)\le h^{max}$$

(21)

3.3.4. Optimization Problem Resolution

The heating power is either 0 or $P_{max}$. As a result, this makes the optimization problem discrete and requires a heuristic optimizer capable of solving such a complex optimization problem. In addition, two initialization methods (${\mathrm{M}}_1$ and ${\mathrm{M}}_2$) are proposed with the aim of improving performance and controlling the computation time needed to solve the optimization problem:

• Using the initialization method ${\mathrm{M}}_1$, the initialization vector is a vector of zeros, with no parallel computing used;
• Using the initialization method ${\mathrm{M}}_2$, the initialization vector is the EWH power vector provided by the ERBS strategy for the first time step ($P_{\mathrm{Init}}$) and then the optimal solution is used for the remaining simulation, with parallel computing used.

The discrete nature of the problem aims at finding an approximation of the global minimum of the objective function. As a result, the choice of the optimization algorithm is paramount. In the past decade, a plethora of evolutionary and heuristic-based search algorithms have demonstrated to be effective in addressing MINLP problems. This effectiveness stems from their simple implementation, independence from auxiliary information, adaptability across diverse problem domains, and compatibility with various problem formats. Furthermore, these algorithms evade local optima traps owing to their probabilistic search mechanisms [61]. Due to its ability to address complex problems with constraints that entail a mix of nonlinear constraints and continuous and discrete variables, the heuristic-based search algorithm chosen to solve the optimization problem is a genetic algorithm (GA) [62]. This preference stems from the efficiency of GAs in exploring the search space and attaining quality solutions. Additionally, they are resilient against local optima and have enhanced capability to handle problems with multiple conditioned solutions and constrains. Moreover, the way GAs work predisposes them to parallel computing.

Alternative optimization methods, such as binary particle swarm optimization (BPSO), were used in existing studies [63] to address MINLP problems in the context of energy optimization in residential dwellings. Specifically, the application pertains to the control of EWHs within individual dwellings and residential parks in France. However, these investigations have typically overlooked the integration of RES such as solar or wind energy. Findings of this research study indicate that BPSO optimization can effectively reduce power demand in the context of DHW production; however, users’ comfort may not always be ensured under certain circumstances. Therefore, the objective in this research study is to evaluate a similar GA-based optimization approach within the context of French residential dwellings, focusing on EWH control, to assess its impact on both energy optimization and users’ comfort. GAs are based on natural selection and genetics (the law of the survival of the fittest) [64]. GAs mainly possess chromosomal representation of variables such as binary representation. Besides this, GAs also incorporate decoder, random population generator, fitness evaluator from objective function value, and the three main operators: selection, cross-over, and mutation. These features modify the parent population to generate children having better fitness values while completing one generation.

Regarding the GA parameters, the goal is to decide for both the number of individuals in the population and the number of generations which give the best performance, i.e., the lower EWH cost associated with the power extracted from the distribution grid and the lower computational cost. The number of individuals is the population size in each generation, while the generation size specifically refers to the number of offspring produced in each generation. Adequately choosing these parameters is crucial to get good solutions and control the computational cost. In this research study, the computational cost is calculated using Equation (22), where $N_w=8$ is the number of workers chosen after several test cases and $C_t$ is the mean computation time which is defined as the mean optimization time:

$$C_c=N_w{C}_t$$

(22)

The optimization problem being MINLP, the crossover function used is the “Laplace crossover” function [65]. The mutation function used is the “Power mutation” function, which is based on a power distribution [65]. The simulation test was performed over two days and five configurations were tested in order to select the best one, as shown in

Table 10. Computational cost and economic cost associated with the power extracted from the distribution grid by varying both the number of individuals and the number of generations. Configuration 1: 2000 individuals and 1000 ${\mathrm{OV}}_{\mathrm{L}}$ generations. Configuration 2: 1000 individuals and 500 ${\mathrm{OV}}_{\mathrm{L}}$ generations. Configuration 3: 500 individuals and 250 ${\mathrm{OV}}_{\mathrm{L}}$ generations. Configuration 4: 900 individuals and 400 ${\mathrm{OV}}_{\mathrm{L}}$ generations. Configurations 5: 2500 individuals and 1500 ${\mathrm{OV}}_{\mathrm{L}}$ generations.

GA Configuration	1	2	3	4	5
Computational cost	101.60	60.40	23.90	38.40	134.20
Economic cost associated with the power extracted from the distribution grid [EUR]	3.97	3.96	3.98	4.01	3.97

. Configuration 1 is 2000 individuals and 1000${\mathrm{OV}}_{\mathrm{L}}$ generations, with ${\mathrm{OV}}_{\mathrm{L}}$ the optimization vector length. Configuration 2 is 1000 individuals and 500${\mathrm{OV}}_{\mathrm{L}}$ generations. Configuration 3 is 500 individuals and 250${\mathrm{OV}}_{\mathrm{L}}$ generations. Configuration 4 is 900 individuals and 400${\mathrm{OV}}_{\mathrm{L}}$ generations. Configurations 5 is 2500 individuals and 1500${\mathrm{OV}}_{\mathrm{L}}$ generations. The chosen configuration is 1000 individuals and 500${\mathrm{OV}}_{\mathrm{L}}$ generations, providing the best trade-off between computational cost and performance (

Table 11. GA parameters. The crossover fraction is the fraction of the population at the next generation, not including elite children, that the crossover function creates. The mutation function is the function that produces mutation children. ${\mathrm{OV}}_{\mathrm{L}}$ is the optimization vector length.

GA Parameter [66]	Value
Number of individuals	1000
Number of generations	500${\mathrm{OV}}_{\mathrm{L}}$
Crossover function	“Laplace crossover”
Crossover fraction	0.8
Mutation function	“Mutation power”

).

3.3.5. Linearization Approach

As mentioned above, the optimization problem is nonlinear, which is related to the EWH model used, the optimization variable, and the objective function. This subsection deals with presenting another economic MPC strategy based on a linear EWH model and linear constraints aiming at reducing the computational complexity associated with implementing the control strategy while trying to maintain optimization performance. The predictive strategy—denoted economic LMPC—has the same objectives as the economic MPC strategy (see Section 3.3.2–Section 3.3.4); the objective function is still nonlinear and has the same formulation as for the nonlinear control approach. The main idea behind making the model linear is to reduce complexity, thus reducing the computation time. Due to the fact that nonlinear constraints invoke the EWH model at each time step of the prediction horizon, which is computationally expensive, we have decided to make the constraints linear. The idea is to create a recurrence relation that allows calculating the remaining 143 values of the hot water height (h) from the first value, which is the last value calculated by the system, as shown in Figure 5.

Linear EWH Model

The submodel allowing the hot water height in the tank to be calculated is linear. The submodel allowing the hot water temperature to be calculated is nonlinear (Section 3). The idea is then to make this model linear according to $P_{\mathrm{EWH}}$, the optimization variable, by fixing the hot water temperature value ($T_{upper}$) at 60 °C, which is a temperature allowing users’ comfort constraints to be satisfied for all purposes and under various cold water temperatures. Only the hot water height variation is calculated. The new EWH model is governed by Equation (23):

$${\displaystyle \frac{dh}{dt}}=a-bh$$

(23)

where a is defined as follows (Equation (24)):

$$a=\left({\displaystyle \frac{P_{\mathrm{EWH}}+U{A}_w(T_{amb}-T_{lower})}{C_w(T_{upper}-T_{lower})}}-{\displaystyle \frac{\dot{m}{C}_p}{C_w}}\right)H$$

(24)

and b is defined as follows (Equation (25)):

$$b={\displaystyle \frac{U{A}_w}{C_w}}$$

(25)

where h is the hot water height, $P_{\mathrm{EWH}}$ is the power of the EWH heating element, $U{A}_w$ is the thermal conductance of the tank, $T_{amb}$ is the ambient temperature, $T_{lower}$ is the water temperature in the lower part of the tank, $C_w$ is the specific heat capacity of the tank, $\dot{m}$ is the hot water flow rate, $C_p$ is the thermal capacitance of water, and H is the height of the tank.

Linear Constraints

As explained above, the process of constraint linearization aims at canceling the nonlinearity by defining equations describing how the hot water height in the tank varies, without relying on the EWH model. The standard form of linear constraints—the MINLP optimization problem is solved using a GA—is as follows, where $P_{\mathrm{EWH}}$ is the optimization vector (Equation (26)):

$$A\times P_{\mathrm{EWH}}\le c$$

(26)

The constraints are depicted by Equations (27) and (28), where h is the hot water height and $h^{min}$ and $h^{max}$ are the minimum and maximum threshold of hot water, respectively:

$$h(k+i+1)\le h^{max}$$

(27)

$$-h(k+i+1)\le -h^{min}$$

(28)

Therefore, these linear constraints are formulated as follows (Equations (29) and (30)):

$$\begin{array}{cc}\hfill f_a^{k+i}{h}_0+\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_b{P}_{\mathrm{EWH}}\left(j\right)\right)& +\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_c\dot{m}\left(j\right)\right)+\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_d\right)\le h_{\max }\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill -\left(f_a^{k+i}{h}_0+\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_b{P}_{\mathrm{EWH}}\left(j\right)\right)+\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_c\dot{m}\left(j\right)\right)+\sum _{j=1}^{k+i}\left(f_a^{k+i-j}{f}_d\right)\right)\le -h_{\min }\end{array}$$

(30)

where $f_a$, $f_b$, $f_c$, and $f_d$ are defined by Equations (31), (32), (33) and (34), respectively:

$$f_a=1-{\displaystyle \frac{T_{e}U{A}_w}{C_w}}$$

(31)

$$f_b={\displaystyle \frac{H{T}_e}{C_w(T_{upper}-T_{lower})}}$$

(32)

$$f_c=-{\displaystyle \frac{H{T}_e{C}_p}{C_w}}$$

(33)

$$f_d={\displaystyle \frac{H{T}_{e}U{A}_w(T_{amb}-T_{lower})}{C_w(T_{upper}-T_{lower})}}$$

(34)

where $h_0$ is the initial value of the hot water height in the tank, $P_{\mathrm{EWH}}$ is the power of the EWH heating element, $\dot{m}$ is the hot water flow rate, $T_e$ is the time step, $U{A}_w$ is the thermal conductance of the tank, $C_w$ is the specific heat capacity of the tank, H is the height of the tank, $T_{lower}$ is the water temperature in the lower part of the tank, $C_p$ is the thermal capacitance of water, and $T_{amb}$ is the ambient temperature.

4. Results and Discussion

All the results presented below are based on France’s electricity purchase tariffs (Section 2.4.2 [53]). The database quantifying how carbon-intensive electricity is (in $\mathrm{g}{\mathrm{CO}}_{2.\mathrm{eq}}/\mathrm{k}\mathrm{W}\mathrm{h})$ comes from the website “electricitymap.org” [54]. Matlab R2020b was used to perform the simulations. Those simulations were run on a calculation server composed of two Intel Xeon Gold 6230 @ 2.10 $\mathrm{G}$$\mathrm{Hz}$ processors, with 20 cores and 40 threads, 512 GB of RAM, and an average CPU mark of 26,657. Parallel computing is launched using eight workers. For all the results presented below, six-day simulations were conducted for each of the four seasons of the year (winter, spring, summer and autumn), for each typical scenario (see Table 7 for the characteristics of the typical scenarios), in order to assess the seasonal variability in the decision-making process. The time step is 10 min and the prediction horizon is 24 h. DHW demand, non-DHW power demand, and PV power generation predictions are “perfect”, in the sense that data coming from the scenarios are used. No prediction module was developed to generate stochastic DHW demand, non-DHW power demand, and PV power generation profiles.

4.1. Mean Computation Time

The two initialization methods ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ were evaluated taking as a criterion the mean computation time—6-day simulations were conducted. The initialization method ${\mathrm{M}}_2$ was used with both the economic MPC and LMPC strategies to show the impact of the linear EWH model and the linear constraints on computational complexity. Regarding the mean computation time,

Table 12. Mean computation time [s] for 6-day simulations, with two initialization methods. ${\mathrm{M}}_1$: initialization method 1. ${\mathrm{M}}_2$: initialization method 2. MPC: model-based predictive control. LMPC: linear EWH model predictive control. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	Season	MPC (${\mathbf{M}}_1$)	MPC (${\mathbf{M}}_2$)	LMPC (${\mathbf{M}}_2$)
1	Winter	49	37	20
	Spring	42	17	21
	Summer	38	28	19
	Autumn	60	32	23
2	Winter	63	41	21
	Spring	53	33	22
	Summer	42	25	24
	Autumn	68	39	23
3	Winter	52	34	19
	Spring	46	27	19
	Summer	34	20	16
	Autumn	53	52	21
4	Winter	62	36	18
	Spring	49	32	20
	Summer	38	19	16
	Autumn	70	42	31

highlights the impact of parallel computing.

By taking a look at the results, one can observe that the two initialization methods manage to meet the time step T_e = 10 min, for a prediction horizon of 24 h. The economic LMPC strategy with initialization method M₂ provides reasonable computation time for all typical scenarios, with a reduction of 56% in the average time required for each forecast in case of the simulation of a dwelling occupied by one person, 60% for a dwelling occupied by two persons, 59% for a dwelling occupied by three persons, and 61% for a dwelling occupied by four persons. Also, the economic MPC strategy with initialization method M₂ allows for a significant reduction in computation time with a reduction of 40% in the average time required for each forecast in case of the simulation of a dwelling occupied by one person, 39% for a dwelling occupied by two persons, 28% for a dwelling occupied by three persons, and 41% for a dwelling occupied by four persons. A comparison of the control strategies is carried out in the next subsections using the economic MPC strategy and the economic LMPC strategy, initialized with the method M₂ (see Section 4.2, Section 4.3 and Section 4.4).

4.2. Economic Gain

Table 13. Economic gain [EUR] for 6-day simulations (the SRBS strategy provides reference results). M₂: initialization method 2. MPC: model-based predictive control. LMPC: linear model-based predictive control. ERBS: enhanced rule-based strategy. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	Season	ERBS	MPC (${\mathbf{M}}_2$)	LMPC (${\mathbf{M}}_2$)
1	Winter	0.14	1.00	0.48
	Spring	0.22	1.00	0.63
	Summer	0.56	0.85	0.79
	Autumn	0.38	0.78	0.48
2	Winter	0.51	1.00	1.10
	Spring	0.86	1.31	0.66
	Summer	1.18	1.45	1.55
	Autumn	0.66	1.34	0.74
3	Winter	0.37	0.96	0.50
	Spring	0.80	1.56	0.90
	Summer	1.08	1.60	1.50
	Autumn	0.60	1.58	1.78
4	Winter	1.03	2.02	2.70
	Spring	1.00	2.20	1.20
	Summer	1.65	2.30	2.50
	Autumn	1.74	2.20	1.80

summarizes the results dealing with the economic gain. The economic gain is defined as the difference in the overall electricity bill between economic MPC, economic LMPC or ERBS—these strategies allow reducing the cost associated with the power extracted from the distribution grid for DHW production by taking advantage of the PV power generation surplus—and SRBS (the reference strategy). Taking a look at the results, one can note that the ERBS strategy yields an average economic gain over 6 days that reaches 0.32 EUR for Scenario 1, 0.80 EUR for Scenario 2, 0.71 EUR for Scenario 3, and 1.35 EUR for Scenario 4. However, the economic LMPC achieves better results over 6 days: the average economic gain reaches 0.59 EUR for Scenario 1, 1.01 EUR for Scenario 2, 1.17 EUR for Scenario 3, and 2.05 EUR for Scenario 4. In addition, the economic MPC strategy has the higher economic gain: the average economic gain over 6 days reaches 0.90 EUR for Scenario 1, 1.27 EUR for Scenario 2, 1.42 EUR for Scenario 3, and 2.18 EUR for Scenario 4.

By examining the economic gain derived from the three strategies, we can observe three significant points:

• Firstly, the economic gain achieved by the ERBS strategy highlights the impact of enhanced rule-based control that leverage the unused PV power generation to produce DHW without compromising users’ comfort;
• Secondly, the economic gain achieved by both the economic MPC and LMPC underscores the merits of model-based predictive control in achieving superior results compared to rule-based control;
• Lastly, the economic LMPC demonstrates that linearization of the optimization model and the constraints allows optimization performance to be preserved.

By examining the results obtained for each month, we can observe the influence of seasonality on the decision-making processes and, consequently, on the control actions. Let us note that in the winter season, the PV power generation decreases, whereas both the non-DHW power demand and the DHW needs increase; thus, the optimization becomes more complex, which explain the degradation in performance observed with the economic LMPC compared to the economic MPC, as shown in Table 13. However, during the summer season, the PV power generation increases while the non-DHW power demand decreases, which make the optimization more efficient and the gain higher.

4.3. PV Power Generation Self-Consumption Rate

The PV power generation self-consumption (SC) rate is defined as the part of the PV power generation consumed locally in order to satisfy the power needs of the individual dwelling (Equation (35)):

$$\mathrm{SC}[\%]=100{\displaystyle \frac{P_{\mathrm{PV}}^{\mathrm{SC}}}{P_{\mathrm{PV}}}}$$

(35)

where $P_{\mathrm{PV}}^{\mathrm{SC}}$ is the PV power generation consumed locally and $P_{\mathrm{PV}}$ is the PV power generation.

Table 14. Increase in the PV power generation self-consumption (SC) rate [%ps] for 6-day simulations (the SRBS strategy provides reference results). M₂: initialization method 2. MPC: model-based predictive control. LMPC: linear model-based predictive control. ERBS: enhanced rule-based strategy. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	Season	ERBS	MPC (${\mathbf{M}}_2$)	LMPC (${\mathbf{M}}_2$)
1	Winter	3.00	13.00	4.80
	Spring	5.00	11.00	7.80
	Summer	8.00	10.00	9.60
	Autumn	8.00	13.00	10.20
2	Winter	5.00	11.00	11.50
	Spring	7.00	10.00	6.10
	Summer	7.00	8.30	9.00
	Autumn	9.00	13.00	9.53
3	Winter	3.00	12.00	5.00
	Spring	6.00	9.00	6.55
	Summer	7.00	9.00	7.57
	Autumn	7.00	13.00	14.30
4	Winter	6.00	12.00	13.60
	Spring	5.00	10.00	6.00
	Summer	4.00	7.00	7.30
	Autumn	10.00	13.00	11.00

indicates an increase of the PV power generation self-consumption rate over 6 days with the ERBS strategy, the economic MPC strategy, and the economic LMPC strategy. SRBS is the reference. It should be noted that for the SRBS strategy, the PV power generation self-consumption rate only concerns the PV power consumed to meet non-DHW power needs due to the fact that the need for water heating generally occurs when there is no PV power generation, and the EWH is usually turned on during off-peak hours in the night, as reported by ADEME [2]. For the ERBS, the economic MPC, and the economic LMPC strategies, the PV power generation self-consumption rate deals with the power consumed locally to meet both the non-DHW power needs and the DHW production needs. Taking a look at Table 14, one can remark that the economic MPC strategy increases the PV power generation self-consumption rate for all scenarios, up to 13%ps (in winter for Scenario 1 and in autumn for Scenario 1, Scenario 2, Scenario 3, and Scenario 4), with an average of 11.75%ps for Scenario 1, 10.57%ps for Scenario 2, 10.75%ps for Scenario 3, and 10.5%ps for Scenario 4. As a comparison, the ERBS strategy increases the PV power generation self-consumption rate up to 10%ps (in autumn, for Scenario 4), with an average of 6.00%ps for Scenario 1, 7.00%ps for Scenario 2, 5.75%ps for Scenario 3, and 6.25%ps for Scenario 4. In addition, the economic LMPC achieves significant results, close to those of the economic MPC, with an increase in the PV power generation self-consumption rate for all scenarios, up to 14.30%ps (in autumn, for Scenario 3), with an average of 8.10%ps for Scenario 1, 9.03%ps for Scenario 2, 8.35%ps for Scenario 3, and 9.47%ps for Scenario 4.

The results highlight that both the economic MPC and economic LMPC strategies increase the PV power generation self-consumption rate significantly and demonstrate that EWHs are a promising candidate for storing the unused PV power generation during the day in order to increase energy self-consumption. Indeed, during the winter season, the PV power generation self-consumption rate increases compared to the summer season, due to the fact that the quantity of PV power generated is not substantial, in comparison to the demand. Therefore, a significant part of the PV power generation can be used locally which proves the effectiveness of the proposed strategy, even for the most complex cases. However, during the summer season, the PV power generation increases while the power demand decreases, resulting in a decrease in the PV power generation self-consumption rate. However, this does not hinder the effectiveness of the strategy.

4.4. CO₂ Emissions

The CO₂ Emissions per unit of power extracted from the distribution grid are calculated using Equation (6). The reduced cost associated with the power extracted from the distribution grid (Section 4.2) and the increased PV power generation self-consumption rate (Section 4.3) both contribute to reducing the CO₂ Emissions with the ERBS, economic MPC and economic LMPC strategies, compared to the reference strategy (SRBS). By taking a look at the results summarized in

Table 15. Reduction in ${\mathrm{CO}}_2$ emissions [$\mathrm{k}\mathrm{g}{\mathrm{CO}}_{2.\mathrm{eq}}]$ for 6-day simulations (the SRBS strategy provides reference results). ${\mathrm{M}}_2$: initialization method 2. MPC: model-based predictive control. LMPC: linear model-based predictive control. ERBS: enhanced rule-based strategy. See Table 7 for the characteristics of the typical scenarios.

Typical Scenario	Season	ERBS	MPC (${\mathbf{M}}_2$)	LMPC (${\mathbf{M}}_2$)
1	Winter	0.07	0.27	0.08
	Spring	0.10	0.45	0.31
	Summer	0.22	0.34	0.33
	Autumn	0.02	0.32	0.16
2	Winter	0.14	0.19	0.21
	Spring	0.41	0.67	0.35
	Summer	0.61	0.74	0.77
	Autumn	0.35	0.62	0.42
3	Winter	0.13	1.74	0.34
	Spring	0.43	2.61	0.76
	Summer	0.57	1.56	0.75
	Autumn	0.37	0.81	1.23
4	Winter	0.16	0.45	2.68
	Spring	0.45	3.92	1.01
	Summer	0.69	1.01	2.25
	Autumn	0.60	3.20	0.90

, we can observe that the economic MPC strategy consistently remains the best strategy for reducing CO₂ Emissions, with a reduction ranging from 14.00% for Scenario 1 to 38.00% for Scenario 4. Moreover, the economic LMPC strategy reaches a significant reduction as well, ranging from 8.00% for Scenario 1 to 30.00% for Scenario 4. In addition, the ERBS strategy reduces the CO₂ Emissions from 9.37% for Scenario 1 to 14.00% for Scenario 4.

5. Conclusions and Perspectives

As the residential sector is energy-consuming and a major contributor to climate change, efficient solutions are needed to increasing energy efficiency and reducing greenhouse gas (GHG) emissions. In this context, active demand-side management strategies are proposed to the control of electric water heaters in individual dwellings equipped with photovoltaic panels. Nonlinear/linear model-based predictive control is used. An initialization method is proposed with the aim of improving performance and controlling the computation time needed to solve the optimization problem—the problem is solved using a genetic algorithm—by taking advantage of parallel computing. The simulation results (over 6 days) highlight that the MPC/LMPC strategies outperform the rule-based strategies, while guaranteeing users’ comfort. The MPC and LMPC strategies allows for a significant increase of both the economic gain (up to 2.70 EUR with LMPC and initialization method ${\mathrm{M}}_2$) and the PV power generation self-consumption rate (up to 14.30%ps with LMPC and initialization method ${\mathrm{M}}_2$), which in turn allows the ${\mathrm{CO}}_2$ emissions to be reduced (up to 3.92 $\mathrm{k}\mathrm{g}{\mathrm{CO}}_{2.\mathrm{eq}}$ with MPC and initialization method ${\mathrm{M}}_2$). Additionally, the LMPC strategy shows a significant reduction in the mean computation time (up to 60%) while performance remains close to the one of the MPC strategy. In addition, these results clearly demonstrate the benefits of using EWHs to store the PV power generation surplus, in the context of producing DHW in residential buildings.

Future work will focus on assessing the predictive strategies with different demand habits and different PV power generation profiles. In addition, the control algorithms will be adapted to take advantage of the information users are willing to provide. Another perspective is to develop algorithms to the intra-day prediction of DHW demand, non-DHW power demand, and PV power generation, using machine learning (ML) techniques and evaluate the robustness of the strategies to prediction errors. Finally, experimental validation of the developed algorithms will be carried out.