Session 2: Advances in Sustainable Consumption
A Heuristic Algorithm for Operation Scheduling of Electric Water Heaters under Dynamic Pricing G. Safouri1, a and V. Kapsalis2 1
Hellenic Open University, School of Science and Technology, Parodos Aristotelous 18, 263 35, Patras, Greece 2 Technological Educational Institute of Western Greece, Department of Electrical Engineering, M. Alexandrou 1, Patras, 26334, Greece * Corresponding author:
[email protected]
Abstract This paper presents a heuristic operation scheduling algorithm applied to electric water heaters (EWHs), which, taking into account user’s preferences regarding cost and comfort as well as the daily hot water consumption rate, minimizes energy cost, without significant compromise of the perceived comfort level. The performance of the algorithm is evaluated through simulation under a day-ahead real-time pricing (DA-RTP) tariff. EWHs are devices with high nominal power ratings and in addition to that they provide high levels of scheduling flexibility, as they can shift their operation by storing hot water during time slots of low energy prices. Simulation results are presented under various realistic scenarios which study the effect of the upper temperature set point and the rated power on the energy cost. Furthermore, the user’s preferences regarding energy and comfort cost are incorporated in an objective function by means of proper weighting factors. This utility function is strived to be minimized by the algorithm. Regarding the implementation issue, the algorithm features a low computational complexity, extending its applicability to low-cost embedded controllers, which could directly control the ON/OFF operation of the heating elements of conventional EWHs. Keywords: heuristic scheduling algorithm, electric water heater, dynamic pricing, smart grid
Nomenclature Surface area of the water heater [m2] A Specific heat capacity of water [4187 J/(kgK)] C C = c1,..., cN Electricity tariff profile [cents/kWh] Energy cost for the whole period (24 hours) cost Mean value of temperatures’ difference from Tpref, when Tw < Tpref discomfort M = m1 ,..., mn Bath water usage profile [kg/s] Mass of tank water [kg] M Mass of bath water [kg] Mbath Mass of water that remains in the tank [kg] Mcurr Mass of water that is inserted in the tank [kg] Minlet Rated power of the water heater [W] Q Operation factor (takes values 0 or 1), (S(i) = operation factor for slot i) S S* = s(1)*,..., s(n) * Set of optimal operation factors for n time slots 288
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
Session 2: Advances in Sustainable Consumption
t T Tacc Tamb Tcurr Tpref Tw T0 TH TF U w wcost wdiscomfort Δt λcost λdiscomfort
1.
Time [sec] Water tank temperature [˚C] Minimum accepted water temperature [˚C] Ambient temperature [˚C] Temperature of water that remains in the tank [˚C] Preferred water temperature [˚C] Minimum achievable tank water temperature [˚C] Tank water temperature at the beginning of a slot [˚C] Tank water temperature at the end of a slot, before any water draw [˚C] Tank water temperature at the end of a slot, after a water draw [˚C] Stand by heat loss coefficient [W/(m2K)] Normalized objective function Normalized energy cost Normalized discomfort Time slot duration [sec] Weighting factor of energy cost Weighting factor of discomfort
INTRODUCTION
The building sector is responsible for up to 30% of global annual greenhouse gas emissions, as well as for up to 40% of total energy consumption (UNEP, 2009). While energy efficiency is the most prominent component of growing efforts to supply affordable, reliable, secure and clean electric power, demand response (DR) is a key pillar of utilities’ and independent system operators’ (ISOs’) resource plans towards the reduction of total energy consumption and peak demand. By applying DR, customers’ electricity consumption is constrained at critical time periods or in periods of high whole market prices. A considerably large number of research work has been carried out aiming at reducing the energy cost of residential appliances under price-based DR programs, such as dynamic pricing schemes. EWHs constitute an important type of load, featuring several advantages for implementing operation scheduling control, such as high power consumption as well as significant thermal storage capability. A large amount of models and control strategies for EWHs have been developed, divided in two main categories. The first category (utilitycentered) includes control strategies which deal with aggregate EWH load applied by utilities, aiming at peak-load reduction and balancing regulation services. The second category (customer-centered) deals with individual EWH control which aims at energy cost savings at the customer side. Since our paper belongs to the category of customer-centered control strategies, a brief presentation of representative papers belonging to the latter category follows. Lu & Katipamula (2005) have presented certain strategies for set point control of water heater loads aiming at shifting power consumption from the high-price to the low-price period in order to reduce the peak-load and energy cost, while keeping the tank water above a minimum temperature. Goh & Apt (2004) studied, among other strategies, the control of the set point temperature between a minimum and a maximum value, according to electricity price. However, this strategy does not always guarantee an acceptable tank water temperature. Du & Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
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Lu (2011) proposed a heuristic algorithm that schedules thermostatically controlled appliances (TCAs), such as EWHs, in such a manner that electricity bill is minimized, taking into consideration constraints set by comfort requirements. Zimmerman et al. (2011) presented a model of household water heating together with real-time pricing profiles and demonstrated a linear programming (LP) approach that generated cost-optimal solutions implemented by a client-side agent. Kepplinger et al. (2015) studied the potential of applying demand side management to electric water heaters. The optimization problem was formulated as a binary integer program, whereas actual system behavior was simulated with a multi-layer model. Shah et al. (2016) presented a greedy algorithm, which aimed at the reduction of the electrical water heating costs under a TOU pricing structure while adapting to varying user hot water consumption over time via thermal storage. The algorithm was keeping the tank water temperature within predefined limits. However, the algorithm uses a rather complicated process to tune an optimization factor that is used for the calculation of the standby loss. Kapsalis & Hadellis (2017) modeled the scheduling of EWHs as a single-source shortest path problem of a weighted directed acyclic graph (WDAG). The total weight (cost) of each edge was expressed as a function of both energy and comfort cost, while the user’s preferences were incorporated in the objective function in the form of relative weights of energy and comfort cost. The algorithm iteratively calculated the objective value by applying the Dijkstra’s algorithm for different weights of comfort and energy cost until an optimum was reached. This paper presents a heuristic algorithm (customer-centered) which deals with the operation scheduling of EWHs. The proposed algorithm takes into account the user’s preferences regarding comfort as well as predicted hot water consumption and electricity rates and it calculates the time slots at which the EWH must be on, in order to minimize energy cost without significant compromise of the perceived comfort level. Its performance is evaluated through simulation under a day-ahead real-time pricing (DA-RTP) tariff. The algorithm has been applied in various realistic scenarios, which study the effect of the maximum water tank temperature and of the EWH’s rated power on the energy cost. Regarding implementation aspects, the algorithm has low computational complexity and can be implemented either as a cloud-based service or locally by a low-cost embedded controller.
2. FORMULATION OF THE EWH OPERATION SCHEDULING PROBLEM The optimal operation schedule of the EWH requires an algorithm that computes an optimal water heating plan according to some criteria of optimality, a thermal model of the EWH, a model of the user’s preferences, a hot-water consumption model and an energy pricing tariff. These topics are explained in detail at the following subsections. 2.1 Modeling of the problem The water heater is regarded as a single mass with uniform temperature distribution, assuming perfect mixing (Goh & Apt, 2004), which is a good approximation compared to a multi-layer model described by A.H.Fanney (1996). The temperature change of the water is given by eq. 1. When part of hot water inside the tank is used, cold water from the water supply is inserted into the water tank. Therefore the final water temperature is given by eq. 2: 290
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
Session 2: Advances in Sustainable Consumption
(1)
(2)
The proposed algorithm takes into account the user’s preferences regarding both the preferred and the maximum tank water temperature although the latter may have an effect on the energy cost as shown by the simulations. Hot water consumption varies from one household to another, depending on the number of users, water-consumption habits (frequency of hot water usage and duration of water consumption) and the facilities of a house (bath, shower, etc.). Usually, several typical patterns of hot water demand per each time slot can be used for simulation purposes. Our simulations were based on hot water usage data for a 24-hour period (draw profile for a winter day) obtained from Hammerstrom et al. (2007) as illustrated in Figure 1. Real-Time Pricing (RTP) is a dynamic pricing scheme, under which, retail electricity prices change hourly. Our simulations were based on a DA-RTP tariff obtained from Illinois Power Company (Ameren, 2013). Under this scheme, the retail provider announces all 24 hourly prices for a given day at one time on the prior day.
80 60 40 20 22:00
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Time (hours)
12:00
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Hot water usage (litre/hour)
100
Figure 1: Hot water usage for one day
2.2 The proposed algorithm In order to solve the scheduling problem of an EWH, a new heuristic algorithm is proposed, the performance of which has been evaluated through simulation. First, a discretization of the problem was performed, which transformed the continuous model into a discrete model over various time intervals (time slots). For every time slot, three discrete temperatures are calculated: the initial temperature T0, i.e. the water temperature at the beginning of the time slot, the heating temperature TH, i.e. the water temperature after the heating procedure and before any water draw, and the final temperature TF, i.e. the water temperature at the end of the slot, after both the heating operation and the water usage have taken place. These temperatures are given by eq. 3 and 4: (3)
(4)
where, (5)
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
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The algorithm calculates the time slots that the EWH must be on in order to minimize the energy cost while satisfying the comfort preferences set by the user. The main restrictions of the optimization problem are the following: Τhe temperature of the water inside the tank must not exceed a defined maximum value at any time slot. For every slot where there exists hot water consumption, the temperature of the water inside the tank must not drop below the lower comfort limit, as set by the user. Given the hot water consumption pattern and the electricity pricing, the derived solution must fulfill the above restrictions with the minimum energy cost. The proposed algorithm is described by the following pseudocode: Algorithm 1 The operation scheduling algorithm inputs: M, U, A, Q, C, n, t, M = m1 ,..., mn , C = c1,..., cn , Tpref , Tinlet , Tamb , Tmax , Tinitial
outputs: S* = s(1)*,..., s(n) * for i = 1 to n do S(i) = 0 calculate T0(i), TH(i), TF(i) end for do tltp = false {tltp: temperature less than preferred} find the first slot k (from 1 to n) where TF(k) < Tpref if exists slot k then tltp = true for i = 1 to k do find slot j with minimum energy price, without exceeding Tmax at any slot if S(j)=1, has not been used before and j is the closest slot to k end for S(j) = 1 update S* = s(1)*,..., s(n) * for m = j to n do calculate T0(m), TH(m), TF(m) end for end if loop until tltp = false
3. SIMULATION RESULTS Two different types of time slots have been tested: 15-min slots and 5-min slots (closer to the continuous approach). The analysis has been performed for a total time of 24 hours, resulting to a total of 96 time slots of 15-min or a total of 288 time slots of 5-min. The operation schedule starts from 12 a.m. and runs for 24 hours to 12 a.m. the next day. The parameter values of the EWH model used in the simulations are given in Table 1. Table 1 Parameter values in the simulations Tpref [oC] 40.0 292
Tmax [oC] 75.0
Tamb [oC] 20.0
Tinlet [oC] Q [kW] M [kg] 20.0 5 190
C [J/(kg K))] 4187
UA [W/K] 1.52
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
Session 2: Advances in Sustainable Consumption
DA-RTP
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Figure 2 (a) Tank water temperature
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2 18:00
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4
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6
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8
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Energy price (cents/kWh)
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Temperature (˚C)
20 21:00
2 18:00
40
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4
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60
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6
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80
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8
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Energy price (cents/kWh)
Figure 2(a) illustrates the resulting tank water temperature at each slot (for 15-min and 5min duration), which minimizes the energy cost, subject to the previously mentioned constraints. Figure 2(b) illustrates the cumulative operation time, according to which, there is a shift of the operation of the EWH to time slots of low electricity prices. Also, both figures illustrate the used DA-RTP tariff (Ameren, 2013). The energy cost of the EWH for the whole period (24 hours) is 59.4 and 55.0 cents for 15-min and 5-min time slots, respectively. For comparison purposes, the energy cost of an EWH that is controlled by a conventional thermostat, set at 75 oC (deadband = 2 oC), is 95.6 cents. Thus, compared to it, the energy cost achieved by the proposed algorithm is decreased by 42.5% and 37.8% for 15-min and 5min time slots, respectively. A price-sensitive thermostat proposed by Goh & Apt (2004), which controls the temperature set point between a maximum and a minimum value equal to 75 oC and 40 oC, respectively, achieves an energy cost of 87.6 cents. Thus, the proposed algorithm for 15-min time slots achieves an energy cost reduction by 37.2.1%, compared to the price-sensitive thermostat. The comparison of the heuristic algorithm with an optimal scheduling algorithm, proposed by Kapsalis & Hadellis (2017) showed similar performance regarding the achieved energy cost reduction. Specifically, the optimal scheduling algorithm achieved an energy cost equal to 54.1 cents. Thus the energy cost achieved by the heuristic algorithm for 15-min time slots is 9.8% higher than the energy cost of the optimal algorithm and only 1.7% higher for 5-min time slots. However, this difference can be practically eliminated if the time slot duration gets shorter. In order to study the effect of the maximum water tank temperature as well as the EWH’s rated power on the energy cost, a short parametric analysis is performed, in which, the effect of each parameter is separately examined. According to the results, shown in Table 2, there is a strong correlation between the maximum water tank temperature and the resulting energy cost. Specifically, the energy cost is a monotonically decreasing function of the maximum water tank temperature due to the fact that a higher temperature leads to the storage of higher thermal energy in the form of hotter tank water. However, this benefit is mitigated by the higher standby heat loss, which is an increasing function of the difference between the tank water temperature and the ambient air temperature.
15-min
(b) Cumulative operation time
Table 2. Parametric analysis over maximum water tank temperature for 15-min slots Tmax [oC] Cost [cents]
55 70.25
60 65.00
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
70 61.50
75 59.38
80 57.50
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On the other hand, there seems to be no significant correlation between the heater’s power and the energy cost, as shown in Table 3. Table 3 Parametric analysis over EWH’s rated power for 15-min slots (maximum temperature = 75 oC) Q [W] Cost [cents]
2000 58.65
3000 57.98
4000 58.20
5000 59.38
4. COMBINED COST AND COMFORT APPROACH The outcome of the proposed algorithm is the set of time slots that the heater must be on, in order to satisfy the hot water needs with the minimum energy cost. The algorithm is a rigid one, meaning that it cannot let the water temperature drop below the preferred temperature Tpref for any slot. However, enabling lower temperatures than Tpref for some of the slots could lead to a more cost-efficient solution, whereas the consequent comfort would not be significantly affected. In order to incorporate this capability, the algorithm is modified as follows. 4.1. General assumptions For the implementation, two new temperatures are defined: Tw is the minimum water temperature recorded in the tank and Tacc is the temperature which corresponds to the minimum accepted comfort level (maximum discomfort). In practice, the minimum tank water temperature (Tw) is allowed to vary between Tacc and Tpref. According to this variation, both the energy cost and the provided comfort will also vary. Energy cost is given by eq. 6. The energy cost performance of each case is estimated by the normalized energy cost. Taking into account that the minimum and the maximum energy cost are achieved for Tacc and Tpref, respectively, the normalized energy cost is given by eq. 7. (6)
(7)
For every case, 0≤wcost≤1. Regarding the energy cost performance, the optimum case corresponds to wcost = 0, meaning that the energy cost equals to the minimum cost. The discomfort constitutes a function of the deviation of the water temperature from the preferred one. Specifically, discomfort is the mean value of temperatures’ difference from Tpref when temperature drops below Tpref (eq. 8). (8) The comfort performance of each case is estimated by the normalized discomfort. Taking into account that the minimum and the maximum discomfort are achieved for Tpref and Tacc, respectively, the normalized discomfort is given by eq. 9: (9) 294
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
Session 2: Advances in Sustainable Consumption
For Tw = Tpref, discomfort is equal to 0 (minimum), because in this case Tw is always above Tpref. Similarly, discomfort takes the maximum value (equal to 1) when Tw = Tacc. For every case, 0≤wdiscomfort≤1. Regarding the comfort cost performance, the optimum case corresponds to wdiscomfort=0. As the tank water temperature drops below Tpref the discomfort increases. Temperature values above Tpref are not taken into account in the calculation of discomfort, because in this case, the hot water at the outlet of the tank may reach to the preferred temperature by mixing with cold water. According to the described approach, every intermediate case partially fulfills the comfort requirements. When wcost reaches the optimum value (wcost = 0), wdiscomfort equals 1 and vice versa. In order to take into consideration both energy and comfort cost, a weighted normalized objective function is introduced, given by eq. 10. (10) where λcost and λdiscomfort are weighting factors, given by the user, in order to determine the importance of each parameter (i.e. cost and discomfort) in the final result. These weighting factors have the following limitations: (a) 0≤λcost≤1, (b) 0≤λdiscomfort≤1, (c) λcost + λdiscomorft = 1. A high value of λcost (λdiscomfort) denotes a high importance of cost (discomfort). Setting both weighting factors to 0.5 means that both cost and discomfort are of equal importance. 4.2. Implementation of the combined approach An implementation of the above described approach has been performed. The parameter values of the EWH model used in the simulations are given in Table 1. Tpref is set to 40˚C and Tacc is set to 36˚C. This means that water temperature between 36 ˚C and 40 ˚C is accepted. The algorithm has been run for different water temperatures (Tw) between 36 ˚C and 40 ˚C with a step of 0.5 ˚C, resulting into a total of 9 discrete cases. For each case, the algorithm calculates the total energy cost (eq. 6), the discomfort, (eq. 8), the normalized cost (eq. 7) and the normalized discomfort (eq. 9). The weighted normalized objective function w is calculated (eq. 10) for setting both weighting factors, λcost and λdiscomfort, equal to 0.5. The results are shown in Table 4. Table 4 Results of combined cost and comfort approach for 5-min slots Tset [oC] 36.0 36.5 37.0 37.5 38.0 38.5 39.0 39.5 40.0
Cost [cents] 50.71 50.87 52.04 52.21 53.54 53.54 54.96 55.04 55.04
Discomfort 2.29 2.06 1.62 1.36 0.89 0.77 0.49 0.25 0.00
Proceedings of the 18th European Roundtable for Sustainable Consumption and Production, 2017 Skiathos Island, Greece ● October 1-5, 2017 ΙSBN: 978-618-5271-24-4
wcost 0.00 0.04 0.31 0.34 0.65 0.65 0.98 1.00 1.00
wdiscomfort 1.00 0.89 0.71 0.59 0.39 0.34 0.21 0.11 0.00
w 0.50 0.47 0.51 0.47 0.52 0.50 0.60 0.56 0.50
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As shown in Table 4, the minimum value of w is achieved for water temperature equal to 36.5 ˚C. This means that for the specific optimization problem, as defined by the EWH’s characteristics, the electricity prices and the user’s hot water needs, the optimum case is achieved when letting temperature drop below Tpref (40 ˚C) until the lower limit of 36.5 ˚C or 37.5 ˚C. On the contrary, letting temperature drop until 39 ˚C yields to the worst case.
5. CONCLUSIONS In this paper, a heuristic algorithm was introduced, in order to solve the EWH operation scheduling problem under a DA-RTP tariff. The algorithm achieves the minimization of the energy cost by controlling the operation of the heating element of conventional EWHs. Several simulations were carried out in order to study the effect of parameter variation on the performance of the scheduling algorithm. The results showed that the energy cost is a monotonically decreasing function of the upper water tank temperature, whereas there is not any significant correlation between the heater’s power and the energy cost. A conventional thermostat, a price-based thermostat (Goh & Apt, 2004) and an optimal scheduling algorithm (Kapsalis & Hadellis, 2017) were also simulated to serve as references, regarding the achieved performance. The proposed algorithm showed practically equivalent performance compared to the optimal scheduling algorithm and considerably higher performance compared to both the conventional and the price-based thermostat. Furthermore, an enhancement of the initial algorithm introduced a weighted normalized objective function as a combination of both cost and discomfort preferences and subsequently, it calculated the optimum water temperature that minimized the objective function. Finally, the low complexity of the algorithm ensures its applicability to low-cost embedded controllers.
REFERENCES Ameren Illinois Power Co., Real-time prices for residential customers, 2013. URL: http://www.ameren.com/Residential/ADC_RTP_Res.asp (accessed 25.04.13). Du Pengwei and Lu Ning, 2011. Appliance Commitment for Household Load Scheduling. IEEE Trans. Smart Grid, 2, (2) 411-419. Fanney, A. H. and B. P. Dougherty, 1996. The thermal performance of residential electric water eaters subjected to various off-peak schedules. ASME Journal of Solar Energy Engineering, 118 (2), pp. 73-80. Goh C.H.K., J. Apt, 2005. Consumer Strategies for Controlling Electric Water Heaters under Dynamic Pricing. Carnegie Mellon Electricity Industry Center Working Paper (CEIC-04-02). Hammerstrom D. J., Brous J., Carlon T. A., Chassin D. P., Eustis C., Horst G. R., Järvegren O. M., Kajfasz R., Marek W., Michie P., Munson R. L., Oliver T., and Pratt R. G., 2007. Pacific Northwest Grid-Wise Testbed Projects: Part 2. Grid Friendly Appliance Project, PNNL-17079, Pacific Northwest Natl. Lab. Richland, WA. Kapsalis Vassilis, Hadellis Loukas, 2017. Optimal Operation Scheduling of Electric Water Heaters under Dynamic Pricing. Sustainable Cities and Society, Vol. 31, pp. 109–121. Kepplinger Peter, Huber Gerhard, Petrasch Jörg, 2015. Autonomous optimal control for demand side management with resistive domestic hot water heaters using linear optimization. Energy and Buildings 100 (1) 50–55.
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Session 2: Advances in Sustainable Consumption
Lu Ning and Katipamula Srinivas, 2005. Control Strategies of Thermostatically Controlled Appliances in a Competitive Electricity Market. IEEE Power Engineering Society General Meeting, pp. 202207. Shah Jigar J., Nielsen Matthew C., Shaffer Timothy S., and Fittro Roger L., 2016. Cost-Optimal Consumption-Aware Electric Water Heating Via Thermal Storage Under Time-of-Use Pricing. IEEE Transactions on Smart Grid, vol. 7, no. 2, pp. 592-599. Zimmerman, T. L., Smith, S. F. and Unahalekhaka, A, 2011. CONSERVE: Client side intelligent power scheduling, Proceedings of the Tenth International Conference on Autonomous Agents and Multiagent Systems, Taipei, Taiwan, 2–6 May.
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