*2.1. SETS Structure*

Figure 1 shows a schematic diagram of SETS. It contains electric heating wires, magnesia bricks, the temperature sensor of bricks, two layers of thermal isolation (including perlite and ceramic fiber) wrapped in steel plates, heat exchanger, and fan. When the SETS operates, the resistive heating wires generate thermal energy, which will be stored in the bricks for discharging. The cold air is blown into the interior by the fan, and transferred into hot air by the heating wires for changing the entrance temperature of the heat exchanger. The cold water is circulated into the heat exchanger through the return water pipe and heated by the hot air to improve the customers' room interior temperature. SETS becomes a convenient DSM tool for utility companies.

**Figure 1.** Schematic diagram of solid electric thermal storage (SETS). (**a**) Cold air. (**b**) Perlite. (**c**) Ceramic fiber. (**d**) Steel plates. (**e**) Thermocouple. (**f**) Flow channel. (**g**) Bricks. (**h**) Heating wires. (**i**) Partition wall. (**j**) Hot air. (**k**) Exchanger. (**l**) Supply water. (**m**) Return water. (**n**) Fan.

#### *2.2. Principle*

The basic assumption of the proposed PM is that the temperature of the bricks is spatially uniform in any transient process. This assumption implies that the temperature gradients within the bricks are negligible. The proposed model is based on the following assumptions: all the electric energy consumed by the electric heating wires is stored in the bricks in the form of thermal energy. The thermal energy stored in the SETS is mainly released by thermal radiation and convection. The isolation layer can prevent the thermal energy of hot air in the SETS from flowing outside. In heat transfer of the heat

exchanger, the heat loss between the heat exchanger and the ambient environment, the pipe thermal resistance and fouling effect are ignored. In Figure 2, the PM of SETS in this paper is based on the basic principle of heat transfer instead of entransy dissipation-based thermal resistance theory presented in [22].

**Figure 2.** The structure of the temperature-based energy flow of the SETS. (**a**) Heat storage. Electric energy is consumed by the heating wires and stored in the thermal brick in the form of thermal energy. (**b**) Heat transfer. The stored thermal energy is transferred to the circulating water through the heat exchanger. (**c**) Customers heating. Uses the thermal energy to keep the room warm.

The initial and final temperatures of the bricks are assumed to be *T*ini and *T*fin, respectively. The internal and external temperatures of the isolation layer are *T*int and *T*ext, respectively. *T*ini is equal to *T*int before the electric heating wires are energized. The *T*ini is increased as the heating wires are energized. In the heating process, thermal energy always transfers heat to the surroundings through the isolation layer, so *T*int can be calculated by the average temperature of the bricks. According to a large number of survey and analysis of SETS operation, the *T*ext approximation is 50 ◦C. The boundary and initial conditions of the PM are described in Equation (1).

$$\begin{cases} \begin{aligned} T\_{\text{ini}} &= T\_{\text{int}} & (t=0) \\ T\_{\text{fin}} &\ge T\_{\text{int}} > T\_{\text{ini}} & (t>0) \\ T\_{\text{int}} &= (T\_{\text{fin}} + T\_{\text{ini}})/2 \\ T\_{\text{ext}} &= 50 \end{aligned} \end{cases} \tag{1}$$

The inlet and outlet temperatures of the hot air through the heat exchanger are *T*a,in and *T*a,o, respectively. The inlet and outlet temperatures of the circulating water through the heat exchanger are *T*w,in and *T*w,o, respectively. *T*cri a,room is the criterion minimum temperature of customers' room. The room temperature *T*a,room is influenced by the environmental conditions (e.g., ambient temperature *T*a,amb. In Section 2.7. (Influencing Factors) explains why ambient temperature is selected). In order to keep the room temperature close to *T*cri a,room, the supply and return water temperatures *T*w,in and *T*w,o can be increased correspondingly. In the inlet of the heat exchanger, a high-temperature air is required to be input, so the stored thermal energy final temperature *T*fin needs to be higher (about 700 ◦C). *T*fin can be deduced by the change of ambient temperature *T*a,amb, and the average power consumption of the SETS can be obtained by the *T*fin. According to engineering experience, the variation ranges of *T*ini and *T*a,o are very little that they are assumed to be constant. *T*w,o is affected by the ambient temperature *T*a,amb, so the input and control variables of the heat exchanger *T*w,o and *T*w,in are deduced based on Equation (12).

Without considering the temperature gradients within the bricks, the initial temperature *T*ini and final temperature *T*fin changes in the SETS are used to predict the thermal energy consumption. The proposed SETS PM prediction of the consumed thermal energy is equal to the electricity by the ambient temperature change.

#### *2.3. Thermal Energy Storage*

According to the above principle, the electric energy consumption *E*pro,*<sup>t</sup>* is equal to the sum of thermal energy *E*sto,*<sup>t</sup>* stored in the bricks and the thermal energy loss of the SETS *E*los,*<sup>t</sup>* as shown in Equation (2).

$$E\_{\rm pro,t} = E\_{\rm sto,t} + E\_{\rm los,t}.\tag{2}$$

The *E*los,*<sup>t</sup>* is not considered in the modeling. So the Equation (2) can be a simplify as *E*pro,*<sup>t</sup>* = *E*sto,*t*. The thermal energy *E*sto,*<sup>t</sup>* is transferred to the hot air by thermal convection *φ*conv, thermal radiation *φ*rad, and thermal conduction *φ*cond. The three heat transfer formulas are introduced as follows:
