1. Introduction
The prevailing view of electricity generation is that the system will be most efficient when fluctuations are mitigated as much as possible. This is in contrast to the earlier view that the generation capacity must be able to increase in order to satisfy any demand [
1]. Techniques have been devised to smooth the demand profile of the grid and limit the power demand [
1,
2,
3,
4]. This is particularly important in the case of electric water heaters (EWHs), as they account for 40% of the residential sector’s energy consumption, and this sector accounts for 20% of greenhouse gas emissions [
5,
6,
7,
8,
9,
10].
South Africa currently lacks the generation capacity to meet the demand on the grid. This has led to frequent power outages, necessitating load-shedding [
11]. The energy required to operate the five million water heaters in the country is responsible for 30% to 50% of the peak loads on the grid [
5,
7]. The country’s carbon footprint is also significant since approximately 88% of the electricity is generated from coal [
12].
Smart grids make it possible to deliver electricity in controlled, intelligent ways and encourage improved energy efficiency between the user and the utility [
13,
14]. This can be achieved by load management and demand response strategies. Demand-side management aims to change the end-user’s electricity usage patterns to reduce their overall consumption or shift it to different times [
15]. Achieving the goals of demand-side management can benefit the environment by reducing the release of greenhouse gases [
16].
The EWH is a thermal storage device with the flexibility to achieve significant energy savings [
17,
18]. For smart grid applications, the energy efficiency of EWHs can be improved by using thermal models and advanced control algorithms that typically reduce heat loss, maintain user comfort and prevent the growth of
Legionella bacteria, which can cause the disease
Legionellosis [
19,
20,
21]. Such models and strategies have been extensively researched but few of them have been designed to reduce demand and overall energy [
22]. The typical, and most energy inefficient, control of EWHs, referred to as thermostat control (TC), is to leave the thermostat always on to maintain the water temperature at a set point. A better solution is schedule control (SC), in which the user decides when the water should be heated during the day to ensure user comfort while reducing thermal losses over the whole day [
23]. The demand profile for EWHs typically peaks in the morning and the evening [
24]. The profile reflects the water usage behaviour of each household and is influenced by weather and time of use factors [
25,
26,
27,
28].
The grid demand for EWH control can be regulated through various demand-side management programs, such as load-response, incentive-based and direct load control [
15]. In direct load control programs, the utility controls the operation of the user’s EWH remotely, via power line communication, to reduce the grid load during peak times.
From the perspective of the user, the desired strategy reduces the overall energy usage but avoids causing discomfort. From the perspective of the utility, the desired strategy reduces the overall energy usage while smoothing out peak loads on the grid. The ideal demand-side management strategy achieves both these aims.
Figure 1 shows how the energy usage, user discomfort levels and peak loads are distributed by existing strategies and by our new strategy, as explained in this paper. In the next section, a complete literature synthesis is performed on related work and is summarised in
Table 1. The table includes work that focuses on individual EWH control to minimise energy usage and user discomfort, as well as centralised EWH control that focuses on minimising peaks and user discomfort.
Optimal control strategies have been proven to greatly reduce EWH energy usage while ensuring that user comfort levels are preserved. However, they fail to consider what the grid can manage without compromising the user’s comfort or reducing the energy savings. We propose a centrally adapted control (CAC) model that minimises all three attributes: peak load, energy usage and user discomfort.
Figure 1.
Distribution of user discomfort levels, overall energy usage and peak load management for five control strategies.
Figure 1.
Distribution of user discomfort levels, overall energy usage and peak load management for five control strategies.
Table 1.
Table of related work for individual and centralised control of EWHs.
Table 1.
Table of related work for individual and centralised control of EWHs.
| [29] | [30] | [31] | [32] | [33] | [34] | [35] | [3] | [36] | This Paper |
---|
Control Type | | | | | | | | | | |
Individual Control | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ |
Centralised Control | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓ |
Control Objective | | | | | | | | | | |
Time of Use Optimisation | ✓ | ✗ | ✓ | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | ✗ |
Energy Optimisation | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ |
General | | | | | | | | | | |
Grid Peak Load Level Limits | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓ |
Field-Measured Hot Water | ✗ | ✓ | ✗ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ |
No User Interaction | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✓ | ✓ |
Temperature Matched Output | ✗ | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ |
Energy Matched Output | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ |
Legionella Prevention | ✓ | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ |
User Comfort | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓ | ✓ | ✓ |
Optimal Control | ✗ | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗ | ✓ |
In this paper, we present a two-stage optimisation system which first determines the optimal control for each EWH without consideration for the grid and then determines the optimal centrally adapted control for multiple EWHs to ensure that the grid demand does not exceed the grid’s power supply limit.
1.1. Literature Review
This section reviews work on control strategies for individual EWHs and centralised control to regulate grid demand.
Table 1 summarises the main findings and highlights the challenges that remain and that this paper addresses.
1.2. Individual EWH Control
Gholizadeh and Aravinthan [
29] assessed the benefits of adjusting the set-point temperature and using the day-ahead pricing to decrease the total cost of electricity for residential EWHs. The main factors they examined were consumer comfort, the reduction of temperature variations, and health concerns such as
Legionella. They determined cost savings of 5.9% to 6.4%. However, their study used synthetic water profiles associated with the ASHRAE standard rather than field-measured data.
Booysen and Cloete [
30] demonstrated how modern smart grid technologies can determine the impact of schedule control for large-scale experiments. Their experiment measured and analysed the energy usage, during two months’ schedule control, of five water heaters equipped with ETSI smart grid technology. The results were validated by a lab experiment with a two-node thermal model. Their work determined energy savings without accounting for temperature-matched or energy-matched outputs.
Kepplinger et al. [
31] proposed an incentive-driven demand-side management strategy that optimised the control of EWHs by predicting user behaviour. They solved the optimal control problem with a binary integer algorithm and estimated the future water consumption with a nearest-neighbour algorithm performed on historical time-series data. Using a night-tariff-switched EWH as a reference, they found the cost and energy savings with energy-matched outputs were both 12%.
Booysen et al. [
32] determined the theoretical energy savings that can be achieved by the optimal control of EWHs. They used a novel dynamic programming optimisation algorithm on a one-node EWH thermal model to determine the optimal scheduling of an EWH with knowledge of future water demand. Their simulations for 77 residential households for one month achieved energy savings of 8% for temperature-matching and 18% for energy-matching strategies. When they modified the energy matching strategy to prevent
Legionella growth, the energy savings were reduced to 13%.
Ritchie et al. [
33] built on this earlier work [
32] by incorporating a two-node EWH thermal model and determining the energy savings. They developed a novel A* optimisation algorithm to determine the optimal control sequence for a two-node EWH when water demand is predicted. They used a hot water usage model developed in [
37] to predict water usage based on historical water usage data and account for household uniqueness and factors that influence temporal variation. They achieved energy savings of 2.2% for temperature-matching and 9.6% for energy-matching strategies.
1.3. Centralised Control
Kondoh et al. [
34] investigated the potential of a direct-load control strategy that aggregates a load of water to provide a regulation service for the grid. They used a vertically oriented EWH thermal model that has two heating elements. The proposed algorithm controls the power usage of each EWH such that the total usage does not exceed the power limit of the grid. This was achieved by controlling the EWH target temperature and the grid load. A shortcoming of this study was that the algorithm failed to optimise the EWH energy storage to reduce thermal losses and instead tried to store as much thermal energy as possible.
Cui et al. [
35] presented a direct load control model that uses a cooperative gaming strategy in which the retailer lets the users compete indirectly in the market. This is achieved by dynamically adjusting the power scheduling and bidding strategy to correspond with market information. The model benefits the retailer as risks are reduced and profits increase. However, this model does not account for user comfort or the overall energy usage, and user interaction is required.
Roux et al. [
3] devised a centralised control of EWHs that takes into account user comfort, peak loads and overall energy usage. They simulated 34 EWHs with schedule control for 28 days. To ensure that the grid’s power limit was not exceeded, the model prioritised the delivery of electricity from the grid to the EWHs according to each EWH’s required time to reach the target temperature from the current temperature and expected time until the next hot water usage. By using a two-node EWH model, they successfully reduced a peak load of 62 kW to 50, 40, 30 and 20 kW with only a slight decline in user comfort.
Xiang et al. [
36] proposed a direct load control method for EWHs that produced a customer satisfaction prediction index based on a weight matrix. This matrix was calculated from hot water usage patterns and determined the user’s comfort levels. The proposed strategy produced a peak shifting service that ensured user comfort.
1.4. Contributions
This paper presents a novel scheduling model that centrally adapts the control of EWHs to reduce peaks and ensure that the demand on the grid does not exceed the grid’s generation capacity. This provides a balance between the requirements of the user and those of the utility—a need that is disregarded by existing approaches. Our proposed method uses measured hot water usage data from 77 EWHs over four seasons, one week per season, and produces the optimal plan when the model has perfect foreknowledge of water usage and then also when the water usages are predicted.
The rest of the paper is structured as follows:
Section 2 describes the system overview.
Section 3 and
Section 4 define the EWH model and the individual EWH optimisation algorithm.
Section 5 presents the scheduling model.
Section 6 describes the experimental setup.
Section 7 and
Section 8 present the results and conclusion.
3. EWH Model
The EWH is presented as a closed loop and the feedback is provided by a thermostat.
Figure 3 shows the modelling of a one-node EWH where the temperature inside the tank is assumed to be distributed uniformly. The hot water inside the tank can be represented as stored thermal energy, or
. When water is drawn from the tank at a volumetric flow rate,
, and at a higher temperature,
, than the cold water temperature,
, in the inlet pipe situated at the bottom of the tank, thermal energy is removed from the tank and replaced by cold water. A heating element supplies electrical energy to increase the tank’s water temperature,
, and the thermal energy. The thermostat measures the temperature of the water and determines the control state of the heating element to maintain the tank at the set-point temperature. Thermal energy loss,
, is energy lost to the surrounding environment due to standing losses and is caused by a temperature difference between the tank and the ambient temperature,
and the rate is determined by the thermal resistance of the tank wall,
.
The EWH is presented in this study as a lumped-parameter model and the equations that describe the thermal dynamics were originally formulated by Nel [
39]. The rate of change of thermal energy flowing from the tank is defined as follows:
where
is the electrical power supplied by the heating element,
is the power in the hot water that flows from the outlet pipe, and
is the power lost from the tank due to the standing losses. The power supplied by the heating element is defined as follows:
where the “off” or “on” state of the heating element corresponds to delivering either zero power or power equal to the heating element’s rated power,
. The power of the hot water drawn from the tank is defined as follows:
where
is the constant pressure-specific heat capacity of water and
is the density of water. This equation shows that the output power drawn is dependent on the temperature difference of the outlet and inlet water pipes. The power lost from the tank in the form of thermal losses is defined as follows:
This equation shows that the output power lost is dependent on the thermal resistance of the tank and the difference between the temperature of the water and the ambient temperature. Given the temperature of the water, the total thermal energy stored in the tank is calculated as follows:
where
is the volume of the tank.
4. EWH Optimisation
Optimal control theory is concerned with formulating an optimal control problem and then using an appropriate constrained optimisation solver to determine the state trajectory and the associated control signal to achieve a given optimality criterion subject to the system constraints. The optimal control problem is formulated by defining a cost function to be minimised or an objective function to be maximised, and by defining a set of equality and inequality constraints, which typically include differential constraints, state constraints, and input constraints. More information on optimal control theory can be found in [
40]. Various approaches to solving the optimal control problem are available, specifically, search-based (Dijkstra [
41], A* search [
38], dynamic programming [
32]) and optimisation-based (sequential quadratic programming [
42]) methods.
This section presents the optimisation algorithm that produces the optimal heating schedule for an individual EWH. We first define the optimal control problem then describe the A* algorithm. The output schedules for determining the grid optimisation of multiple EWHs are described in
Section 5.
The optimal temperature and heating plan for the one-node EWH are produced by an A* search algorithm. The algorithm was previously formulated for a stratified two-node EWH model by [
38] and is modified in this study to produce the optimal plan for a one-node EWH.
4.1. Optimal Control Problem
The optimal control problem is to determine the optimal switching sequence for the heating element to minimise the total electrical energy supplied to the EWH while satisfying a given hot water usage profile. The profile is satisfied if the user never experiences a temperature below the minimum usage temperature, , when hot water is drawn from the tank.
The system dynamics are defined by the differential equations that describe the thermodynamics of the one-node EWH, specified by Equations (
1)–(
5). The state variable of the system is represented by
, defined by the thermal energy of the tank, and is defined as follows:
The state constraints are defined by the physical limitations of the thermal energy of the tank. The lower and upper bounds are represented by and . These correspond to the minimum and maximum admissible temperatures, and , with a constant volume of .
The control input is represented by and is defined as , the electrical power supplied by the heating element. The control input is constrained to a power supply equal to either zero or .
The cost function has the objective of minimising the electrical energy and is defined as follows:
where
and
are the initial and final time instants. Temperature profile constraints are defined to ensure that the hot water usage profile is satisfied. The following inequality represents the objective of the temperature to satisfy the hot water usage profile:
where
are the constraints imposed and are expressed as follows:
where
is the temperature required to prevent the growth of
Legionella. These constraints show that the outlet water temperature cannot fall below
during water usage, that the entire tank must be sufficiently heated to ensure
Legionella prevention, and that the temperature of the water in the tank cannot exceed the minimum temperature,
.
4.2. A* Search Algorithm
The A* search algorithm is a popular and widely used shortest path search algorithm. It can solve non-linear optimal control problems by creating multiple node-based paths that originate at a starting position and navigate towards a desired final position. The algorithm performs efficiently as heuristics are introduced to help optimise the decision making.
Before the A* search algorithm can be applied to the optimal control problem, the problem must be broken down into discrete time instants and states to represent the decision stages and choices, respectively. Since the hot water usage data is presented with one-minute resolution, Equations (
1)–(
5) are discretised with a per-minute sampling period
.
The purpose of the algorithm is to find the shortest path from an initial state at time instant to a goal state at time instant . At the initial state, the temperature of the entire tank is assumed to be at the starting temperature . At the goal state, the temperature of the entire tank must be equal to or greater than the final temperature . The boundary conditions for the optimal path, and , are specified for the algorithm.
A binary search tree data structure is defined to aid the navigation of the search process (more information on the binary search tree data structure can be found in [
43]). The binary search tree comprises multiple search paths that navigate from the initial state to the final state. The paths are made up of nodes connecting states in the previous time instants to the calculated state in the next time instant as a result of the scenarios where the control input
and
.
Each search path ending calculates a cost that is the sum of two components: the amount of electrical energy that was required to reach its current position and the estimated amount of electrical energy that is still required to reach the final state at the final time instant. A priority queue keeps track of the cost of all the path endings, prioritises them from lowest to highest cost, determines which path end is currently the closest to the desired final state, and instructs that path (the optimal path) to extend further.
The first path that reaches the desired state at the final time instant is also the optimal path and the algorithm stops executing. With the optimal path having been reached, the optimal temperature trajectory and heating schedule are produced for the given hot water usage profile.
4.3. Heating Control Strategies
We present three heating control strategies in this study, labelled TC, TM and EML.
Thermostat control (TC): This is the baseline strategy against which we evaluate the energy savings achieved by our other two optimal control methods. This kind of control, the way an EWH typically operates, is extremely inefficient. The water temperature is maintained at the temperature at which the thermostat is set. We, therefore, expect high standing losses as the water in the tank is always hot, even when no water is drawn for long periods.
Temperature–matched optimal control (TM): This strategy is produced by the A* search algorithm and determines the optimal control of the EWH for a given hot water usage profile. We modify the temperature profile constraints so that the temperature of the water in the tank at the start of each water usage is exactly matched to the temperature that is expected for the same water usage in TC. This strategy ensures that the temperature and energy of the water drawn from the EWH are not compromised by the reduction in the overall electrical energy that is supplied.
Energy–matched optimal control with Legionella prevention (EML): This strategy is similar to the previous strategy, except that the temperature is only constrained to during water usages. However, the hot water flow rate is increased during water usages to ensure that an equivalent amount of energy is delivered, despite the lower temperature experienced by the user. However, as a lower temperature profile can enable the growth of Legionella inside the tank, we mitigate this health risk by ensuring that the EWH is heated to at least once a day.
7. Results
This section describes all the results obtained from the simulations.
Table 3 and
Table 4 show the results obtained with
perfect foreknowledge of hot water usages and with
predicted hot water usages.
Figure 5 compares the simulation results for an identical EWH for the TC, TM and EML heating control strategies for a typical day’s usage profile.
Thermostat control (TC): The EWH temperature is always maintained at the set-point temperature (with hysteresis). If the temperature drops below this point, the heating element is switched on to raise it again. When the first water usage occurs at , the temperature drops significantly. The heating element is switched on and provides electrical power until the set-point temperature is reached. This water usage also resulted in a cold event since the temperature dropped below the cold event threshold. The electrical power consumption over the 48 h for this strategy was .
Temperature-matched optimal control (TM): The EWH temperature matches the corresponding temperature for TC at the start of each water usage, but remains lower than the TC temperature between water usages. A cold event is observed at . The electrical power consumption over the 48 h for this strategy was .
Energy-matched optimal control (EML): The EWH temperature remains above the cold event threshold temperature during water usage. The outlet flow rate is increased so that more hot water is drawn from the EWH and the user mixes it with less cold water. The temperature remains lower than that of the TC and TM strategies between water usages. Since the temperature remains low for long periods, the EWH temperature is increased just before the largest water usage for the day to ensure the prevention of Legionella. It remains above 60 °C for 11 at . The electrical power consumption over the 48 h for this strategy was .
Figure 5.
Simulation results for identical EWHs for the TC, TM and EML heating control strategies over a 24-h period. Each plot shows the EWH temperature (black), the outlet flow rate (blue) and the heating element state (red). The temperature for TC is repeated in the other two plots for comparison. The cold event temperature (40 C) and the Legionella prevention temperature (60 C) are indicated with blue and red dashed lines, respectively. (a) TC. (b) TM. (c) EML.
Figure 5.
Simulation results for identical EWHs for the TC, TM and EML heating control strategies over a 24-h period. Each plot shows the EWH temperature (black), the outlet flow rate (blue) and the heating element state (red). The temperature for TC is repeated in the other two plots for comparison. The cold event temperature (40 C) and the Legionella prevention temperature (60 C) are indicated with blue and red dashed lines, respectively. (a) TC. (b) TM. (c) EML.
Figure 6 shows the temperature trajectory for an individual EWH for the optimal plan (red) and the time-shifted optimal plan (blue) when the grid power limit is 0.4 kW/EWH. The time-shifted optimal plan delays the heating schedule at 7 a.m., during the morning peak hours, because it increases the temperature earlier (at 3 a.m.) than the optimal plan, which increases it at 7 a.m. This results in the temperature of the tank for the time-shifted optimal plan remaining higher than the temperature for the non-shifted optimal plan and the energy savings are reduced because of higher standing losses.
Figure 7 shows the grid power demand over 24 h for various grid power limits for the TM and EML heating control strategies. For TM, only the morning peak is flattened when the grid limit is 0.6 kW/EWH or above. The evening peak only begins to flatten for a grid limit of 0.4 kW/EWH or below. For a grid limit of 0.2 kW/EWH, the power demand remains at a constant level equivalent to the grid limit. This shows that the full power supply is used throughout the day. Comparing all the grid power limits, we see that flattening a peak pushes the excess power demand further back in the day. The results for EML are similar to those for TM but because the evening peak is higher, it flattens when the grid limit is set as high as 0.6 kW/EWH.
Table 3 summarises the simulation results for all 77 EWHs when the three heating control strategies have perfect foreknowledge of water usage. A dash in the power limit column indicates simulations with no set power limit.
Simulation results for TC: The median electricity usage for TC was the highest of all the control strategies at 5.94 kWh/day. This is because it had the highest median energy losses of 2.22 kWh/day. The median usage temperature was 68.0 C, which shows that this strategy maintained the temperature at the 68.5 C set-point. The median energy used was 3.36 kWh/day. The PAPR for the overall grid power demand was 17.62 dB, which indicates that the profile must have had many high peaks during the day, given the average power it required throughout the day. With no set power limit, a total of seven cold events occurred and the maximum grid power peak was 1.1 kW/EWH.
Simulation results for TM: When there was no set power limit, the median energy used was 3.37 kWh/day. The median electricity used was 5.46 kWh/day. This resulted in a distribution of energy savings, given as [25th percentile, median, 75th percentile], of [0.28, 0.44, 0.64] kWh/day ([3.92, 6.22, 9.66] %). These savings were achieved because the median energy losses were reduced to 2.22 kWh/day. The distribution of the increased number of cold events, given as [min, 25th percentile, median, 75 percentile, max], was [0, 0, 0, 0, 3]. The PAPR decreased to 11.53 dB.
When the power limit was set to 1, 0.8, and 0.6 kW/EWH, the PAPR was reduced to 8.66, 6.82, and 4.63 dB. Despite the significant impact these power limits had on flattening the peaks, the rest of the results showed negligible changes, as can be seen in
Figure 7a, where only the tip of the morning peak is flattened, representing only a small portion of the overall demand. For a power limit of 0.4 kW/EWH, the energy savings are reduced to [
0.19,
0.38,
0.54] kWh/day ([
2.76,
5.03,
8.37] %) because the median usage temperature increased to 68.6
C. This is a consequence of time-shifting larger portions of the demand and rising usage temperatures. The PAPR decreased to 1.98 dB. We see the most significant changes when the power limit was set to 0.2 kW/EWH. The median usage temperature dropped to 44.5
C, increasing the energy savings to [
1.51,
2.19,
3.0] kWh/day ([
27.37,
34.08,
47.56] %). However, the significant drop in usage temperature increased the number of cold events to [
0,
3,
9,
24,
137]. The PAPR decreased to 0 dB, meaning that the demand profile was flat throughout the day.
Simulation results for EML: As with TM, the median energy used remained the same for all simulations. When there was no set power limit, the median electricity used was 4.95 kWh/day, which was less than that used by the simulations of TC and TM. The energy savings were [0.76, 0.86, 0.94] kWh/day ([7.88, 16.81, 21.27] %). Since the usage temperature did not have to match that of TC at the start of water usage, the median usage temperature was 54.2 C. This resulted in the median energy losses decreasing to 1.65 kWh/day. Again as with TM, the number of cold events increased to [0, 0, 0, 0, 3]. The PAPR was 11.98 dB.
When the power limit was set to 1, 0.8, 0.6, 0.4, and 0.2 kW/EWH, the PAPR reduced to 9.16, 7.28, 4.93, 2.22 and 0 dB. However, the simulation results begin to show changes when the power limit is 0.6 kW/EWH. This can be explained by observing
Figure 7b, where the evening peak is larger for EML than that of TM. Therefore, a larger portion of the grid is shifted for a power limit of 0.6 kW/EWH and below. When the power limit is set to 0.6 and 0.4 kW/EWH, the median usage temperature increased to 54.8
C and 62.6
C. The energy savings respectively decreased to [
0.72,
0.78,
0.87] kWh/day ([
7.56,
14.31,
20.12] %) and [
0.45,
0.64,
0.74] kWh/day ([
5.38,
11.72,
17.88] %). For a power limit of 0.2 kW/EWH, the median usage temperature dropped to 37.9
C and the energy savings increased to [
2.11,
2.62,
3.27] kWh/day ([
34.37,
45.83,
55.29] %). Similarly to TM, this is because there is insufficient power for the grid to supply to all the EWHs. The energy savings were [
1.51,
2.19,
3.0] kWh/day ([
27.37,
34.08,
47.56] %) and the number of cold events increased to [
0,
9,
21,
42,
130].
Table 3.
Simulation results when the planner has perfect foreknowledge of hot water usages for 77 EWHs.
Table 3.
Simulation results when the planner has perfect foreknowledge of hot water usages for 77 EWHs.
Power Limit (kW/E) | (°C) | Electricity Used (kWh/day) | Energy Used (kWh/day) | Energy Loss (kWh/day) | Energy Savings (kWh/day) | Energy Savings (%) | ΔCold Events * | PAPR (dB) |
---|
Thermostat control |
- | 66.0, 68.0, 68.6 | 4.09, 5.94, 10.02 | 1.63, 3.36, 7.37 | 2.41, 2.42, 2.42 | - | - | - | 17.62 |
Temperature matching |
- | 66.0, 68.0, 68.7 | 3.66, 5.46, 9.39 | 1.64, 3.37, 7.39 | 2.05, 2.22, 2.32 | 0.28, 0.44, 0.64 | 3.92, 6.22, 9.66 | 0, 0, 0, 0, 3 | 11.53 |
1 | 66.0, 68.1, 68.8 | 3.66, 5.46, 9.39 | 1.64, 3.37, 7.39 | 2.05, 2.22, 2.32 | 0.27, 0.44, 0.64 | 3.80, 6.25, 9.66 | 0, 0, 0, 0, 3 | 8.66 |
0.8 | 66.0, 68.2, 68.7 | 3.67, 5.46, 9.39 | 1.64, 3.37, 7.39 | 2.05, 2.21, 2.32 | 0.28, 0.44, 0.64 | 3.56, 6.22, 9.27 | 0, 0, 0, 0, 3 | 6.82 |
0.6 | 66.3, 68.1, 68.7 | 3.68, 5.46, 9.39 | 1.64, 3.39, 7.44 | 2.06, 2.20, 2.32 | 0.27, 0.45, 0.63 | 3.91, 6.22, 9.35 | 0, 0, 0, 0, 3 | 4.63 |
0.4 | 66.8, 68.6, 70.0 | 3.83, 5.59, 9.56 | 1.64, 3.42, 7.64 | 2.15, 2.31, 2.42 | 0.19, 0.38, 0.54 | 2.76, 5.03, 8.37 | 0, 0, 0, 0, 4 | 1.98 |
0.2 | 34.8, 44.5, 53.7 | 2.16, 3.91, 7.28 | 1.6, 3.29, 7.29 | 1.05, 1.25, 1.62 | 1.51, 2.19, 3.00 | 27.4, 34.1, 47.6 | 0, 3, 9, 24, 137 | 0.0 |
Energy matching with Legionella prevention |
- | 52.0, 54.2, 57.5 | 3.20, 4.95, 9.26 | 1.64, 3.36, 7.38 | 1.57, 1.65, 1.77 | 0.76, 0.86, 0.94 | 7.88, 16.81, 21.27 | 0, 0, 0, 0, 3 | 11.98 |
1 | 52.0, 54.2, 57.5 | 3.21, 4.96, 9.26 | 1.64, 3.36, 7.38 | 1.57, 1.65, 1.77 | 0.76, 0.86, 0.94 | 7.88, 16.69, 21.19 | 0, 0, 0, 0, 3 | 9.16 |
0.8 | 52.0, 54.2, 57.5 | 3.20, 4.96, 9.26 | 1.64, 3.36, 7.39 | 1.57, 1.65, 1.78 | 0.76, 0.84, 0.93 | 7.78, 16.69, 21.19 | 0, 0, 0, 0, 3 | 7.28 |
0.6 | 52.4, 54.8, 57.7 | 3.30, 5.0, 9.30 | 1.64, 3.39, 7.43 | 1.62, 1.69, 1.80 | 0.72, 0.78, 0.87 | 7.56, 14.31, 20.12 | 0, 0, 0, 0, 3 | 4.93 |
0.4 | 60.5, 62.6, 64.2 | 3.38, 5.15, 9.36 | 1.65, 3.43, 7.72 | 2.04, 2.10, 2.22 | 0.45, 0.64, 0.74 | 5.38, 11.72, 17.88 | 0, 0, 0, 0, 3 | 2.22 |
0.2 | 35.1, 37.9, 43.7 | 1.86, 3.05, 6.78 | 1.62, 3.35, 7.39 | 0.92, 1.08, 1.22 | 2.11, 2.62, 3.27 | 34.4, 45.83, 55.3 | 0, 9, 21, 42, 130 | 0.0 |
Table 4 shows similar results to the previous table with the exception that the three control strategies have
predicted water usages.
Simulation results for TM: When there was no set power limit, the median outlet temperature was 62.6 C. This is a decrease from the 68.1 C obtained for TC because of water usage predictions. If water usage occurs for the actual profile and not for the predicted profile, the EWH temperature may not be at the desired temperature at the start of a water usage. The energy savings were [0.34, 0.53, 0.89] kWh/day ([3.22, 4.68, 7.44] %). The number of cold events increased to [0, 0, 0, 0, 40]. When the power limit was set and lowered, the usage temperatures continued to drop and the number of cold events significantly increased. When the power limit was 0.2 kW/EWH, the number of cold events increased to [0, 14, 38, 75, 194].
Simulation results for EML: When there was no set power limit, the median outlet temperature was 48.7 C. The energy savings were [0.91, 1.15, 1.47] kWh/day ([8.86, 11.82, 15.51] %). Since the outlet temperature did not need to match that of TC at any time of the day, the number of cold events increased to [0, 0, 6, 29, 64]. We see a similar trend to that of TM as the power limit is set and lowered. When the power limit was 0.2 kW/EWH, the number of cold events increased to [0, 23, 39, 82, 195].
Table 4.
Simulation results when planning has predicted water usages for 77 EWHs.
Table 4.
Simulation results when planning has predicted water usages for 77 EWHs.
Power Limit (kW/E) | (°C) | Electricity Used (kWh/day) | Energy Used (kWh/day) | Energy Loss (kWh/day) | Energy Savings (kWh/day) | Energy Savings (%) | ΔCold Events * | PAPR (dB) |
---|
Thermostat Control |
- | 66.0, 68.1, 68.8 | 6.66, 11.01, 20.09 | 4.08, 8.4, 17.11 | 2.36, 2.41, 2.42 | - | - | - | 17.62 |
Temperature matching |
- | 59.1, 62.6, 64.7 | 6.22, 10.41, 18.47 | 4.11, 8.41, 17.22 | 2.01, 2.16, 2.28 | 0.34, 0.53, 0.89 | 3.22, 4.68, 7.44 | 0, 0, 0, 0, 40 | 6.94 |
1 | 57.5, 62.0, 64.2 | 6.21, 10.39, 18.46 | 4.10, 8.41, 17.18 | 1.98, 2.14, 2.26 | 0.37, 0.58, 0.91 | 3.32, 4.78, 7.38 | 0, 0, 0, 1, 41 | 4.19 |
0.8 | 56.0, 60.2, 62.7 | 6.19, 10.34, 18.41 | 4.08, 8.41, 17.15 | 1.93, 2.11, 2.22 | 0.44, 0.62, 0.97 | 3.67, 5.36, 8.09 | 0, 0, 0, 3, 43 | 2.52 |
0.6 | 50.1, 55.2, 58.8 | 5.45, 9.59, 17.93 | 4.08, 8.40, 17.11 | 1.73, 1.91, 2.06 | 0.68, 1.14, 1.71 | 5.75, 8.20, 16.8 | 0, 0, 3, 12, 77 | 0.94 |
0.4 | 28.1, 40.8, 52.6 | 4.41, 8.89, 14.78 | 4.08, 8.40, 17.08 | 0.84, 1.19, 1.6 | 1.51, 2.46, 5.14 | 19.6, 24.90, 36.8 | 0, 6, 15, 58, 194 | 0.08 |
0.2 | 11.1, 16.9, 38.7 | 3.67, 5.34, 6.34 | 4.10, 8.39, 17.21 | 0.82, 0.90, 1.03 | 2.54, 6.43, 13.66 | 40.9, 65.73, 72.7 | 0, 14, 38, 75, 194 | 0.0 |
Energy matching with Legionella prevention |
- | 44.5, 48.7, 52.3 | 5.61, 9.76, 18.28 | 4.08, 8.40, 17.09 | 1.32, 1.49, 1.67 | 0.91, 1.15, 1.47 | 8.86, 11.82, 15.51 | 0, 0, 6, 29, 64 | 7.49 |
1 | 44.4, 48.7, 52.3 | 5.61, 9.69, 18.26 | 4.08, 8.40, 17.09 | 1.31, 1.48, 1.66 | 0.98, 1.22, 1.68 | 9.40, 13.23, 16.41 | 0, 0, 6, 29, 64 | 4.71 |
0.8 | 44.0, 47.9, 51.5 | 5.52, 9.73, 18.10 | 4.08, 8.4, 17.09 | 1.28, 1.46, 1.65 | 1.14, 1.35, 1.79 | 10.1, 13.71, 18.7 | 0, 1, 7, 31, 68 | 3.03 |
0.6 | 40.4, 44.8, 49.4 | 5.04, 8.96, 17.31 | 4.08, 8.40, 17.08 | 1.14, 1.35, 1.59 | 1.26, 1.75, 2.29 | 12.7, 17.02, 25.0 | 0, 5, 11, 40, 93 | 1.31 |
0.4 | 26.8, 34.9, 40.5 | 3.97, 7.78, 14.22 | 4.08, 8.40, 17.08 | 0.80, 0.99, 1.18 | 2.24, 3.0, 5.59 | 26.0, 31.89, 37.7 | 0, 15, 31, 64, 135 | 0.18 |
0.2 | 11.7, 17.6, 33.2 | 2.89, 5.29, 6.64 | 4.08, 8.46, 17.20 | 0.78, 0.86, 0.95 | 3.17, 6.47, 12.91 | 51.1, 62.80, 69.9 | 0, 23, 39, 82, 195 | 0.0 |
Table 5 shows the percentage point reduction in energy savings for the simulation results of the CAC model when compared to simulations where there was no set power limit. The first column shows the grid power limits for the TM and EML heating control strategies, and the next three columns show the results for the worst, average and best cases for an EWH. For both of these heating control strategies, the average and best cases show no reductions in energy savings for grid limits of 0.8 kW/EWH and above. Otherwise, there is an increased reduction as the limit becomes lower. However, at grid limits of 0.4 kW/EWH and 0.2 kW/EWH, the reduction in energy savings becomes negative, indicating an increase in the energy savings. This is caused by an insufficient supply of power to satisfy all the EWH optimal temperature trajectories that have been time-shifted according to priority. Although we see the increased energy savings as a positive outcome, we acknowledge that they would also result in a higher risk of cold events, as can be seen in
Table 3 and
Table 4. These results represent a scenario where there is an insufficient power supply to satisfy the users comfort for all the EWHs, regardless of how the heating control plan is scheduled.
Table 5.
Percentage point reductions in energy savings for the simulation results of the time-shifted optimal plan compared to simulations where there was no set power limit.
Table 5.
Percentage point reductions in energy savings for the simulation results of the time-shifted optimal plan compared to simulations where there was no set power limit.
Power Limit (kW) | Worst Case | Average Case | Best Case |
---|
Temperature matching |
1 | 0.26 | 0.0 | 0.0 |
0.8 | 0.46 | 0.0 | 0.0 |
0.6 | 1.01 | 0.04 | 0.0 |
0.4 | 5.75 | 1.47 | −1.54 |
0.2 | −9.16 | −27.73 | −72.14 |
Energy matching with Legionella prevention |
1 | 0.41 | 0.0 | 0.0 |
0.8 | 1.09 | 0.10 | 0.0 |
0.6 | 4.54 | 0.59 | 0.0 |
0.4 | 11.58 | 3.81 | −3.08 |
0.2 | −6.63 | −27.64 | −52.92 |
Discussion of Results
Determining the optimal control for an individual EWH for a given water profile with
perfect foreknowledge of usages achieves the absolute best energy savings while not compromising the comfort of the user, as proven in [
32]. However, there are significant demand peaks in the morning and evening. These are evident in the PAPR of the grid demand for 77 EWHs when there is no set power limit.
Table 3 shows that the PAPR for TM and EML was 11.53 and 11.98 dB.
With the CAC model developed in this paper, the optimal control plan for each EWH can be adjusted to satisfy the power limits of the grid. There is no increase in the number of cold events when the power limit is set to as low as 0.4 kW/EWH. At this power limit, for TM the energy savings were [
0.19,
0.38,
0.54] kWh/day ([
2.76,
5.03,
8.37] %) and for EML [
0.45,
0.64,
0.74] kWh/day ([
5.38,
11.72,
17.88] %).
Table 5 shows that when a power limit is set, for TM there is a percentage point decrease in energy savings of 5.75 and 1.47 for the worst and average cases and an increase of 1.54 for the best case, and for EML there is a percentage point decrease in energy savings of 11.58 and 3.81 for the worst and average cases and an increase of 3.08 for the best case. If the power limit is set lower than 0.4 kW/EWH, both of these heating control strategies fail to satisfy the optimal temperature trajectory and the number of cold events significantly increases. The increase in the number of cold events was [
0,
3,
9,
24,
137] for TM and [
0,
9,
21,
42,
130] for EML.
When the optimal plan predicts water usages, the simulations already fail to satisfy the optimal temperature trajectory when there is no set power limit.
Table 4 shows that the number of cold events increased to [
0,
0,
0,
0,
40] for TM and [
0.0,
0,
6,
29,
64] for EML. The corresponding energy savings were [
0.34,
0.53,
0.89] kWh/day ([
3.22,
4.68,
7.44] %) for TM and [
0.91,
1.15,
1.47] kWh/day ([
8.86,
11.82,
15.51] %) for EML. When the power limit is set and lowered, the simulations perform even worse and the number of cold events drastically increases still further for both heating control strategies.
The results show that there is a performance guarantee for the power limit since the energy demand of all of the EWHs never surpassed this limit. However, a lower set power limit can reduce the user’s comfort due to an insufficient amount of available energy for all of the EWHs. In summary, the CAC model has been proven to centrally adapt the optimal schedules of the individual EWHs to satisfy the power limits of the grid and minimise the deviations of the adapted schedules to achieve nearly optimal energy savings.
8. Conclusions
In this paper, we fill the gap left by other models for optimal EWH control. Our CAC model creates a balance between the needs of the user and those of the utility by modifying the optimal control schedule of individual EWHs to spread the load into off-peak periods so that the power supply limit of the grid is not exceeded and the user’s comfort is not reduced.
We performed simulations for 77 EWHs over the four seasons—one week for each season. We developed a simulator that simultaneously controls all the EWHs with their corresponding optimal control schedules to obtain the results when the power limit is set to 1, 0.8, 0.6, 0.4 and 0.2 kW/EWH.
We first performed simulations for the scenario where the CAC model had perfect foreknowledge of water usage. The results show that the grid power limit can be set as low as 0.4 kW/EWH to ensure that the number of cold events does not increase. For this power limit, the median energy savings were 0.53 kWh/day (4.68%) for TM and 1.15 kWh/day (11.82%) for EML. Furthermore, the percentage point reduction in energy savings from simulations with no set power limit for the average case was small, at 1.47 for TM and 3.81 for EML.
We performed further simulations for the scenario where the CAC model predicted water usages. Even when there was no set power limit, the number of cold events increased from that of TC with a distribution of [0, 0, 0, 0, 40] for TM and [0, 0, 6, 29, 64] for EML, and the median energy savings were 0.53 kWh/day for TM and 1.15 kWh/day for EML. Setting a power limit further increased the number of cold events.
In future work, the CAC model, which minimises peak load, energy usage and user discomfort, could be implemented in a real-world scenario if hot water usage can be better predicted and the model is modified to improve its robustness to mispredictions.