Abstract
Due to the capacity of thermal storage, electric water heater (EWH) is one of the best candidates for demand response programs. However, few attentions are given to the modeling and optimization of EWHs with thermostaticallycontrolled automatic water mixer (TCAWM). In this paper, differential thermodynamic model is established for EWHs with TCAWM and a piecewise linear approximation method is performed for the nonlinear thermodynamic model. The multiobjective optimization model is established by introducing an index reflecting the comfort degree of users, so that the optimal energy usage of the EWH can be obtained by mixed integer linear programming. Testing examples verify the effectiveness of the proposed method.
Introduction
With the increasing penetration of renewable energy, the safety and stability of the power system are facing more challenges. In recent years, the rapid development of smart grid technologies has provided the basis for the rapid response of demand side resources [1,2,3,4]. Demand response (DR) plays an important role in the optimal allocation of demand side resources, and is effective in balancing the active power of the power systems [5,6,7]. The proper scheduling of residential loads is vital for saving energy costs without affecting daily power usage service [8,9,10].
Electric water heater (EWH) is one of the main contributors to electricity consumption, which accounts for as much as 30% of the household electricity [11, 12]. EWHs have good thermal energy storage capacities, so electric energy can be stored as heat energy in the EWH tank. EWHs are often considered in DR projects because of the capacity of load profiling by saving energy during peak hours that was previously stored during offpeak hours [13, 14].
Many EWH models and control strategies have been proposed for optimizing and controlling the energy usage of the EWH [15,16,17,18]. The performance of several different control strategies of an EWH under dynamic prices is analyzed in [15]. Reference [16] proposes a system for the realtime control of EWHs with wireless communications and machinetomachine networking. A twonode EWH model, proposed in [17], can realize the control of an EWH in a computationally inexpensive and accurate way. Reference [18] introduces a systematic way to formulate an optimal load scheduling model of an EWH considering temperature constraints.
However, the details of the coldwater mixing process in many research works are usually neglected. The coldwater mixing process includes two processes. In the first process, the cold water entering the tank is mixed with hot water inside the EWH tank. And in the second process, when the heated water leaves the tank through the outlet pipe, it still needs to be fixed with cold water by the thermostaticallycontrolled automatic water mixer (TCAWM), so that the tap water temperature can keep constant. Many works [13, 17, 18] consider the coldwater mixing of the first process, but neglect the second one.
Reference [19] considers the equations of EWHs for two processes of coldwater mixing. The flow rate of the output water is controlled by TCAWM in order to keep the tap water temperature constant. However, when considering TCAWM, the dynamics of the EWHs becomes complex and nonlinear, and cannot be solved using standard optimization algorithms such as mixed integer linear programming (MILP). Heuristic algorithms, such as genetic algorithm (GA) and particle swarm optimization (PSO), are common optimization algorithms for nonlinear relations. However, the heuristic algorithms are direct, random, and easy to fall into local optimum, and have problems such as the long calculation time, weak global search ability [20].
To fill this gap, this paper detailedly formulates the piecewise linear approximation functions of the nonlinear thermodynamic model and then adopts a MILP method to optimize the energy usage of an EWH with TCAWM. By this way, the electricity cost can be saved and specific requests of users can be met.
The rest of the paper is organized as follows. Section 2 presents all mathematical formulas of the EWH with TCAWM multiobjective optimization model. Section 3 introduces the piecewise linear approximation method and further proposes the EWH with TCAWM multiobjective optimization model based on MILP. The testing results are discussed in Section 4. The conclusions are provided in Section 5.
An EWH with TCAWM multiobjective optimization model
The EWH with TCAWM consists of water tank heating system and automatic water mixing system. These two parts respectively reflect the heating process of water inside the tank and the mixing process of outflowing water, which has a great impact on electricity cost and user comfort level, as shown in Fig. 1.
Nonlinear thermodynamic model of EWH with TCAWM
The accurate simulations of temperature and energy flow are essential. The thermodynamic behavior of the EWH with TCAWM can be detailed introduced by formulas below.
We treat the water inside the EWH as a single body with uniform temperature, and assume that the process of heat transferring is instantaneous. Based on the above assumptions, the energy balance equation can be written as [17]:
where ΔE_{in} is the total energy change inside the EWH tank during time interval [t, t+1]; ΔE_{input} is the energy change used to heat water during time interval [t, t+1]; ΔE_{loss} is the standing loss caused by temperature difference at time interval [t, t+1]; and ΔE_{usage} is the heat lost because of water usage during time interval [t, t+1].
The temperature relationship obtained from the energy balance relationship is:
where T_{in}(t+1) is the water temperature in the EWH tank at sample t+1, after the hot water activity (WA) that occurs from sample t to sample t+1; V_{usage}(t+1) is the volume of water used during time interval [t, t+1]; V_{tank} is the volume of water in the full EWH tank; and T_{inlet} is the temperature of cold water flowing into the tank. And T_{inlet} is set to a constant.
Water temperature increases due to energy input, which can be expressed as:
where η is the heating efficiency; P_{0} is the rated heating power; u(t) is the equal to 1 if the EWH is on at sample t and to 0 if it is not; and Δt is the time step size.
The energy lost during the WA period is calculated as:
where c is the specific heat capacity of water; and ρ is the density of water.
In addition to the energy loss caused by the water usage, (5) represents the heat dissipation from the tank to the environment in one sampling interval [14].
where T_{amb} is the air temperature outside the EWH tank; and R is the water heater thermal resistance.
(1)–(5) can be substituted in ΔE=cmΔT, where ΔE is the difference in thermal energy, which is equivalent to ΔE_{in}; m is the mass of the water in the tank; and ΔT is the water temperature difference generated when ΔE is supplied. The water temperature inside EWH tank equation is obtained as:
where B(t) is the flow rate of hot water leaving the EWH tank at sample t.
It can be seen from (6) that variables determining the water temperature inside the EWH tank include the previous temperature, ambient temperature, thermodynamic parameters, hot water consumption and so on.
As shown in Fig. 2, the mixing valve can automatically control the water valve to ensure the fixed tap water temperature, the following formula is available [21]:
where ΔE_{tap} is the energy change of the tap water during time interval [t, t+1]; ΔE_{cw} is the energy supplied by cold water during time interval [t, t+1]; B_{tap} is the fixed flow rate of the tap water; and B_{cw}(t) is the flow rate of cold water in automatic water mixing system at sample t.
Substituting ΔE=cmΔT into (8), the relationship between the water flow rates can be obtained:
where T_{cw} is the temperature of cold water used to mix hot water flowing from the tank; and T_{tap}(t) is the current tap water temperature at sample t. According to the above analysis, T_{cw} is equal to T_{inlet}, and both of them will be treated as a constant.
Different water events are considered as having different requirements for the tap temperature. To a certain extent, the temperature expected by the user T_{exp} reflects comfort demands. With the outflowing hot water and the inflowing cold water, the temperature in the EWH tank T_{in}(t) gradually decreases.
Initially T_{in}(t)≥T_{exp}, the control valve is used to adjust the flow rate of hot water and cold water for maintaining the fixed tap temperature. Under this condition, T_{tap}(t)=T_{exp,} indicating that there is no user thermal discomfort. With the outflow of hot water in the tank and the injection of cold water, the water temperature inside the tank T_{in}(t) gradually decreases until T_{in}(t)≤T_{exp}, then the cold water valve is switched off so that the hot water is no longer mixed with cold water. The current tap water is all supplied by the hot water in the tank. Under this condition, T_{tap}(t)=T_{in}(t), which leads to user thermal discomfort.
The relationship between the tap water flow rate and hot water flow rate can be obtained by transforming (9) as following:
Equations (6) and (10) are the formulas of the nonlinear thermodynamic model for the EWH with TCAWM.
Objective function
Cost and user comfort level are both important during the implementation of DR. Optimizations for the operation of an EWH with TCAWM have to account for user comfort level. The difference between the current tap water temperature T_{tap}(t) and the most desirable temperature T_{exp} offered by users during the WA period determines the comfort level. To simplify our presentation, we use the comfort index D to quantify user thermal discomfort during the WA period. The smaller the deviation, the higher the user comfort level.
Based on the analysis above, we take the following two objectives into account when we establish the multiobjective optimization model.
The objective function for EWH electricity cost optimization can be expressed as following:
where C is the electricity cost of a single EWH over 24 hours; p(t) is the electricity price at sample t; and P(t) is the power consumption of the EWH at sample t.
The objective function for user comfort level optimization can be expressed as follows:
where t_{n} represents the time samples during the WA period.
Objective function transformation
There are many methods for dealing with multiobjective problems. The most common methods are the weighted summation method and the εconstraint method [22]. The weighted summation method can obtain the solution in different situations by changing weighting factors, which is also the most simple and effective method for solving multiobjective optimization problems [23, 24]. Then we adopt the weighted summation method to combine the objective functions of multiobjective optimization problem to construct a single objective function. The objective function can be written as:
where α and β are the weighting factors, α+β=1; and μ is the scale factor, μ>0. α is also called user preference factor, 0≤α≤1. The smaller the value of α, the higher requirement for comfort, the larger the value of α, the more attention the user pays to the electricity saving. α is only set by the user. If the user is not satisfied with the control result for the given α, it can be reset.
Constraints

1)
Electric power constraint
$$P(t)= P_{0} u(t)$$(14)

2)
Temperature limit constraints
The water temperature at the beginning of the WA is not lower than the most comfortable temperature:
where T_{max} is the maximum temperature that the EWH tank can withstand.

3)
Switch on/off signal constraints
The switch on/off signal is determined by the on/off status of the EWH. At the same time, only one of signals status is equal to 1 and the other is equal to 0. Taking Fig. 3 as an example to explain the relationship between the switch on/off signal and the on/off status.
As shown in the Fig. 3, in the third period, u(t) changes from 1 to 0, at this moment the switchingoff signal y(3)=1 is executed. In the seventh period, u(t) changes from 0 to 1, and the switchingon signal x(3)=1 is executed. Under other situations, x(t)=y(t)=0. Equation (16) explicitly shows the linear relation of switch on/off signals x(t) and y(t) using the on/off status u(t).
where x(t) is equal to 1 if the EWH is switched on at sample t and to 0 if it does not; and y(t) is equal to 1 if the EWH is switched off at sample t and to 0 if it does not.

4)
Minimum time for continuous heating and closure constraints
Frequency of switch on/off signal will affect the EWH lifetime and operating costs. Limiting the minimum time for continuous heating and closure is necessary [25]:
where T_{on} is the minimum time of continuous heating; and T_{off} is the minimum time of continuous closure.
Since the switch on/off signal occurs in pair, the longer the minimum time for continuous heating or closure, the less the number of actioning, decreasing the impact on the EWH lifetime. Since j=max(t−T_{on}+1,1) in (17), j=1 indicates that the continuous heating time of EWH is less than T_{on}, and the minimum heating time constraint is required in this case, whereas j=t−T_{on}+1 indicates that the continuous heating time has exceeded T_{on} with the start of next cycle. The constraint is satisfied within the next T_{on} period. The principle of (18) is the same as described above.
Multiobjective optimization model based on MILP
Piecewise linear approximation of thermodynamic model
The EWH with TCAWM multiobjective optimization problem described above is a complex nonlinear optimization problem. In order to solve the optimization problem, the nonlinear relationship of the water temperature in the tank is converted into linear relationship by piecewise linearization.
Equation (6) can be transformed into the following equation:
We perform Taylor series expansion on the exponential function in (19), that is e^{x}≈x+1. Equation (19) can be modified into the following equation:
Combining (10) with (20), the temperature in the EWH tank can be expressed as the following piecewise function:
Now, to transform (21) into a linear form, we first define a new set of binary variables δ(t) as follows:
The definition of δ(t) in (22) has the following explicit equivalent linear form:
where M is a coefficient that has sufficient large positive value.
The equivalence between the implicit relation (22) and its explicit form (23) can be verified as follows: if T_{in}(t)<T_{exp}, then from (23), we see that −ε≤δ(t)≤1−ε, where ε is a positive number less than 1. Since δ(t) is a binary variable, the only choice here is δ(t)=0. Similarly, when T_{in}(t)>T_{exp}, we can come to conclusion that ε≤δ(t)≤1+ε, implying that δ(t)=1.
With the explicit characterization of δ(t) in (23), it is now possible to express T_{in}(t) through the equivalent explicit inequalities. The linear form of tank water temperature T_{in}(t) can be written as:
Equation (21) is equivalent to (24), (25). If δ(t)=0 (or T_{in}(t)<T_{exp}), (24) is zero on both sides, which implies that T_{in}(t+1)=T_{in}(t)[1−Δt/(cρRV_{tank})]+Δt(ηP_{0}u(t)+T_{amb}/R)/(cρV_{tank})+Δt(T_{inlet}−T_{in}(t))B_{tap}/V_{tank}, since M is an arbitrary large positive value, (25) is always satisfied. On the contrary, If δ(t)=1 (or T_{in}(t)>T_{exp}), (25) is zero on both sides, which implies that T_{in}(t+1)=T_{in}(t)[1−Δt/(cρRV_{tank})]+Δt(ηP_{0}u(t)+T_{amb}/R)/(cρV_{tank})+Δt(T_{inlet}−T_{exp})B_{tap}/V_{tank}, since M is large enough, (24) is always satisfied.
MILP
Generally, linear programming with integer requirements is called integer linear programming (ILP). MILP is that the required solution has both real and integer requirements. Efficient MILP software such as the branchandcut algorithm has been developed, and optimized commercial solvers with largescale capabilities are currently available. The mainstream commercial solvers are CPLEX and GUROB.
The difficulty of solving MILP problems, however, is caused by the presence of integer decision variables. Integer variables lead to combinatorial complexity, which becomes very high when the problem size is very large. As a result, the difficulty of working on finding the best solution for the MILP problem has grown exponentially.
MILP is used to solve planning problems with both linear real variables and linear integer variables, as is shown in (26).
where \(\varvec{f}^{\text{T}} (\varvec{w},\varvec{v})\) is the objective function of a MILP problem; f is a linear function vector whose elements are constant; w, v are the solution vectors, the elements of w are integers; A_{ineq1} and A_{ineq2} are the inequality matrices; A_{eq1} and A_{eq2} are the equality matrices; b_{ineq} is inequality vector; b_{eq} is equality vector; l_{b1} and l_{b2} are the lowerbounds; and u_{b1} and u_{b2} are the upperbounds. In particular, there are restrictions on the variables w to be the integer, both the equality constraints and the inequality constraints to be linear.
Formulation of optimization model based on MILP
Mathematical formulas of the nonlinear thermodynamic model for the EWH with TCAWM are listed in (1) to (10). Formulas (20) to (25) describe the piecewise linear approximation of nonlinear thermodynamic equations. The linear formulation of the multiobjective optimization model based on MILP is designed to optimize the energy usage of an EWH with TCAWM, which can be summarized by the following: ① a linearized version of the objective function (13); ② subject to (14)–(18), (23)–(25).
Decision variables in the proposed multiobjective model include not only real variables (temperature inside the tank T_{in}(t)), but also integer variables (on/off status variable u(t), switchon signal x(t), switchoff signal y(t), and the auxiliary variable δ(t)). Then the MILP method is adopted. By introducing the auxiliary variable, the nonlinear thermodynamic characteristics of water temperature can be expressed in piecewise linearized form. Using electric power constraint, temperature limit constraints, switch on/off signal constraints, and minimum time for continuous heating and closure constraints, a multiobjective optimization for operating an EWH with TCAWM is established. A MILP formulation is proposed in order to solve the optimization problem more efficiently. Moreover, the multiobjective optimization model based on MILP is solved by CPLEX optimization toolbox.
Testing results
In the following testing examples, the multiobjective optimization model based on MILP optimizes the operation of an EWH over 24 hours. The value of the parameter R is the same as that in [17]. Other parameters are as follows: V_{tank} is 150 L, P_{0} is 3.5 kW, c is 4200 J/(kg·°C), R is 17.922 K·day/kWh, Η is 100%.
Electricity price signal in the testing examples uses timeofuse (TOU) pricing implemented in Jiangsu province. The price is 0.5583 RMB during the peak hours from 08:00 a.m. to 21:00 p.m. and the price is 0.3583 RMB during offpeak hours from 21:00 p.m. to 08:00 a.m. of the next day.
Assume that the WA1 occurs during the period from 12:00 p.m. to 12:30 p.m. for kitchen water, the WA2 occurs from 18:00 p.m. and 19:00 p.m. in the evening, and the most desirable temperature T_{exp} is 40 °C.
The research objective of this paper is the EWH considering automatic water mixing system placed indoors. T_{amb} can be obtained by collecting the value of air temperature outside the EWH tank in the past one day through the temperature sensor installed in the room, and calculating the average value. Because water temperature does not change obviously during the day, the water temperature outside the tank T_{inlet} takes a constant value of 15 °C.
Linear form and nonlinear form of tank water temperature
In order to examine the accuracy of linear form (24), (25), Figure 4 illustrates the curves of water temperature obtained, respectively, by applying the nonlinear form (6) and the linear forms (24), (25) with the same EWH power consumption and WAs. The consumer hot water demand period is WA2 and Δt takes 2 min.
A magnified view of the curves shows that the simulation results under different tank temperature formulas have small differences and the temperature curves almost coincide, which shows effectiveness of the piecewise linear approximation method for the EWH with TCAWM thermodynamic function.
Optimization results of EWH based on MILP
The performance of the proposed multiobjective optimization model is examined in this section. The hot water demand period includes WA1 and WA2. Δt takes 6 min. In order to test the optimization model considering both electricity savings and user comfort demands, Table 1 and Fig. 5 respectively show the optimization results while the user preference factor α takes different values.
In Table 1, the electricity cost is as high as 16.4 RMB when the optimal comfort scheme is applied without considering cost savings. The electricity cost is obviously saved while adopting the optimal electricity cost scheme without considering user comfort demands, and the electricity cost is only 4.9 RMB.
With the multiobjective optimization scheme, different requirements for user comfort level are achieved by changing the user preference factor α. It can be seen from Table 1 that the electricity cost gradually decreases and the comfort index value D gradually increases with the increase of α.
It can be seen from Fig. 5 that the optimal comfort scheme results in the most power consumption. Consequently, the water tank temperature is maintained within a higher temperature limit, and the tap water temperature can be completely maintained at the most desirable temperature T_{exp} during the hot water demand period.
Under multiobjective optimization scheme, we assume that α is equal to 0.5, indicating that both the cost saving requirements and user comfort needs should be satisfied. Figure 5 shows that the EWH heating process is mainly concentrated on the morning and afternoon when the electricity price is low. Considering the user comfort requirements, the EWH is heated again before water usage happened, therefore the tap water temperature can be maintained at 40 °C for a comparatively long time.
Curve of the optimal cost scheme is special. Compared with the other two schemes, water tank temperature curve is generally lower and tap temperature is also decreased faster, which caused by the lowest requirement for user comfort.
Comparison with heuristic algorithms
GA is a typical heuristic algorithm with a wide range of applications and the strong search ability, which is suitable for solving complex nonlinear optimization problems. Therefore, taking GA as an example to solve the EWH with TCAWM multiobjective optimization problem with nonlinear equations. The objective function is also used as the fitness function, and the coding method adopts real number coding to control EWH start from 12:00 a.m. and run for 24 hours to 12:00 a.m. of the next day, with water events and user preference factors the same as that of the model using MILP.
In order to ensure the convergence of the multiobjective optimization model based on GA testing results below, we compare the value of termination condition V_{exitflag} (if the optimization result converges, V_{exitflag} is equal to 1; otherwise, V_{exitflag} is equal to 0) to select the proper result. Considering the inconsistent convergence of GA, the results listed below are the average of the testing results that the program runs 5 times independently.
Figure 6 shows the power consumption curves and tap water temperature curves of the two optimization models with MILP and GA, respectively. Δt takes 6 min.
As seen from Fig. 6, no matter which algorithm is selected for modeling, EWH usually heats for a long time before 08:00 a.m..
As observed from Fig. 6b, d, f, the model using MILP can better meet the requirement for user comfort, that is, the tap water temperature remains within a higher temperature level.
For example, the model using MILP considering both electricity cost and user comfort level can ensure that the change of tap water temperature during the WA2 period is not obvious, although the tap water temperature is much lower than that in the WA1. The cost of the MILP model is 4.9 RMB, and the cost of the GA model is 4.7 RMB, respectively.
Additionally, a comparative study on the MILP and GA to find out an optimal solution is presented by evaluating the performance of the solution and the execution time of computation. Δt is 2 min.

1)
Comparison in optimization performance
The optimization is to solve the extreme value problem including constraints. The objective function is minimized with the premise of all constraints. The smaller the objective function obtained by the optimization, the better the effect of optimizations. According to the modeling process, the performance of the optimization can be judged by comparing the value of the objective function. Figure 7 shows the objective function values optimized by different modeling methods for different economic needs and comfort requirements.
It can be seen from Fig. 7 that: ① regardless of whether users prefer savings in electricity bills or comfort demands, the multiobjective model based on MILP performs better than the multiobjective model based on GA; ② the higher requirement for user comfort, i.e. the smaller the user preference factor α, the better the effect of multiobjective model based on MILP; ③ when users pay more attention to electricity cost, i.e. the bigger the user preference factor α, the performance of solution using GA and MILP tends to be more similar.

2)
Comparison in execution time
The execution time is also an important element to test the optimization effect apart from the objective function. Figure 8 demonstrates the execution time of the optimization solutions using GA and MILP with different user comfort requirements.
We can clearly see that MILP has shorter execution time than GA in Fig. 8. Especially, when the user preference factor α is 1/5, 1/2, 4/5, respectively, the effect of multiobjective optimization model based on MILP is better because the execution time is reduced by more than half. While α is equal to 1/2, the execution time of the solution using GA is 54 s with the execution time in MILP modeling only about 10 s.
Aggregated power of EWH load group
In order to evaluate the effectiveness of the proposed multiobjective optimization scheme, this section takes the load group composed of 20 EWHs as an example. Both the most desirable temperature and time of the hot water usage, are uniformly distributed. In addition, the consumer hot water demand period includes WA1 and WA2, and Δt takes 6 min.
Figure 9 shows curves of the aggregated power obtained by applying two control methods. The following two control schemes are considered in this simulation.

1)
Multiobjective optimization scheme: α takes 1/2 when the operation of EWHs is optimized.

2)
Unoptimized scheme: EWHs work without adopting the proposed multiobjective optimization scheme.
As shown in Fig. 9, it is clear that, compared to unoptimized scheme, the aggregated power has been shifted from the peak hours to the offpeak hours when the multiobjective optimization scheme is applied. Electricity price (i.e. TOU pricing) is higher prices during peak hours, and lower during offpeak hours. Due to the users’ desire for economic requirement, the proposed scheme optimizes the on/off status of an EWH by reasonably transferring power demand to offpeak hours, which realizes load shedding and peak load shaving. In this way, all users participate in the DR program by responding to electricity price signal. Therefore, we can believe that the proposed multiobjective optimization scheme is beneficial to both the electric power operators and users.
Conclusion
In this paper, multiobjective optimization model of an EWH with TCAWM based on MILP is proposed, satisfying users’ desire for both economic need and comfort requirement. The main contributions of this paper can be summarized as follows:

1)
The thermodynamic model of an EWH with TCAWM not only can reflect change of the current tap water temperature but also is useful to quantify the user thermal discomfort.

2)
A piecewise linear approximation method is proposed in order to establish the multiobjective optimization model using MILP.

3)
Compared with the multiobjective optimization model using GA, the proposed model using MILP can remarkedly decrease the execution time with better performance.
The simulation results have verified that this proposed method can greatly improve the performance of optimizing the operation of an EWH with TCAWM, which can make up for the shortage of the existing optimization model. By considering the TCAWM and adopting an efficient optimization method, the multiobjective optimization model is able to balance the costsaving and user comfort.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (No. 51707099), Natural Science Fund for Colleges and Universities of Jiangsu Province (No. 16KJB470009), and China Postdoctoral Science Foundation (No. 2017M611859).
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CrossCheck date: 4 April 2019
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WU, M., BAO, YQ., ZHANG, J. et al. Multiobjective optimization for electric water heater using mixed integer linear programming. J. Mod. Power Syst. Clean Energy 7, 1256–1266 (2019). https://doi.org/10.1007/s4056501905425
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Keywords
 Demand response
 Electric water heater
 Thermostaticallycontrolled automatic water mixer
 Mixed integer linear programming