Optimal scheduling method of regenerative electric heating for emergency residential building heating: An affine arithmetic‐based model predictive control approach

Abstract

Residential heating faces the challenge of heating interruption when an electric power outage occurs. As a promising heating electrification form, regenerative electric heating (REH) equipped with thermal energy storage (TES) has the flexibility of maintaining the building indoor temperature within the desired range during power outages and reducing the operation cost during normal operation states. However, the allocation and scheduling of the limited thermal energy in TES for the above two purposes is impacted by many uncertainties, for example, outdoor temperature, irradiation, and duration of power outages. Overestimation of the thermal energy required for power outages in the TES can improve the heating supply reliability, but it will also increase the REH operation cost to some extent, and vice versa. To address this problem, an affine arithmetic‐based model predictive control approach (AA‐MPC) for an optimal REH scheduling method is proposed to balance the heating supply reliability during power outages and operation economy of REH at the same time. An REH‐based residential building energy system model is developed to describe the building thermal load associated with the outdoor temperature and irradiation. Then, the required thermal energy for emergency building heating provided by the hot water tank (HWT) is determined using the minimum thermal demand of residents during a power outage, which is constrained by the minimum comfort temperature threshold. Based on this, an AA‐MPC approach that takes the thermal energy for emergency building heating as a time‐varying constraint of the HWT is developed to determine the optimal REH scheduling that considers emergency residential building heating under the above uncertainties. Numerical studies show that the proposed method can maintain minimum thermal demand for at least 2 h when a power outage occurs under uncertainties. At the same time, it can reduce the impact of uncertainties on the operation cost and reduce economic problems caused by emergency heating to a certain extent. Compared to the interval arithmetic‐based model predictive control approach, the operation cost intervals of the proposed method are reduced by 57.3%, 0.3%, and 32.5% under low, middle, and high prediction error levels respectively.

INTRODUCTION

Conventional energy use from fossil fuels has impacted the physical and social sectors, motivating the deployment of low-carbon and clean energy [1]. Many countries have established energy-saving and emission reduction targets for future decades [2, 3]. The conversion to decarbonisation and electrification energy in building energy systems (BES) has accelerated rapidly to reach these targets. The residential space heating energy consumption reached 40% of China's total building energy consumption in 2020 [4]. With the development of electric heating comes new threats, such as excessive operation costs and heating interruptions caused by electric power outages [5]. A policy was issued by the State Grid Corporation of China in 2018 to avoid jeopardising the indoor thermal comfort of residents, and 12°C was set as the minimum comfort temperature threshold during power outages [6]. Consequently, emergency building heating should be considered in the economic scheduling of electric heating to improve heating supply reliability when a power outage happens.

Regenerative electric heating (REH) equipped with hot water tanks (HWTs) as thermal energy storage (TES) is a clean and low-carbon form of electric heating for residential buildings [7]. With REH participating in the thermal supply, two main advantages can be achieved. ① REH is suited to load shifting with TES [8, 9]. Relevant work focussed on reducing the operation cost of REH with the integration of TES and time-of-use (ToU) tariffs [10, 11]. ② Emergency building heating, described as meeting the minimum thermal demand of residents during a power outage, can be accomplished using REH with the thermal energy stored in TES. Currently, most of the existing research focuses on the use of electrical energy storage in dealing with emergent energy outages. Similar to electrical energy storage, the use of TES also has the ability to solve the thermal supply-demand mismatch caused by external energy supply interruption [12]. On the other hand, rather than electric energy storage, TES in REH is more economical [13]. Similar to the studies in ref. [14], the TES of REH was used to keep the indoor thermal comfort of users within the desired temperature range in the event of thermal power shortage during power outages. Day-ahead optimal scheduling of REH considering emergency residential building heating was proposed in ref. [15], where the thermal energy for emergency heating was taken as a constraint of TES. A study in ref. [16] proposed a multistage scheduling strategy for thermal supply enhancement in which thermal storage can reduce user losses in the event of a power outage as an emergency response resource. In ref. [12], a two-level rolling optimisation method for TES was proposed to ensure load supply in the case of external energy interruption or equipment failure.

However, the thermal energy for emergency building heating provided by an HWT is determined by the minimum thermal demand of residents during a power outage. The unpredicted power outage duration and weather, mainly including the outdoor temperature and irradiation, are the main factors that impact the optimal scheduling of REH, especially emergency heating [12, 17, 18]. Under this circumstance, overestimating the thermal energy for emergency heating will cause excessive energy consumption. In contrast, the indoor comfort of residents will be influenced if the thermal energy for emergency heating is underestimated. Thus, the above uncertainties should be considered in the economic scheduling of REH for use as emergency heating.

Many methods have been proposed to address the uncertainties in BES. An adaptive robust optimal scheduling method was introduced in ref. [19], which ensured the security and economic operation of a BES. Nevertheless, the robust optimisation methods are based on the worst-case analysis, and the obtained solutions are relatively conservative. Considering the uncertainty of the weather, a stochastic optimal scheduling method was proposed in ref. [20] to reduce the energy costs of buildings. However, sufficient historical data are required to establish precise probability density functions in stochastic optimisation. Fuzzy optimisation was suggested to address the power supply security in an isolated BES under power generation and system load uncertainties in ref. [21]. Power outages may be caused by many unexpected events, such as meteorological disasters and man-made damage. [22, 23]. Thus, it is difficult to obtain precise distribution functions of a power outage duration or weather information. In ref. [24], an interval optimisation method was used to cope with the uncertainties of electricity prices in a BES, where the upper and lower bounds of the uncertainties were considered to derive optimistic and pessimistic solutions. However, an error explosion phenomenon in the interval optimisation method will make the bounds of the results wider than the true solution bounds. A model predictive control (MPC) method was used in ref. [25] to reduce the impact of errors between predicting data and actual data on scheduling results. However, if the uncertainties are not considered in MPC models, they are treated as deterministic and may result in disadvantageous dynamic performance [26].

Affine arithmetic (AA) optimisation, as an improved interval optimisation method, only requires the ranges of uncertainties rather than precise distribution functions or membership functions of unpredicted power outage duration or weather information required in stochastic optimisation and fuzzy optimisation methods [27, 28]. In addition, an MPC method is further incorporated within AA optimisation to address uncertainties [29]. The affine arithmetic-based MPC (AA-MPC) method can not only maintain the advantage of AA optimisation but also consider the future behaviour of uncertainties in prediction horizons, making the model suitable for the optimal scheduling of REH for emergency heating.

To address the aforementioned challenges, an AA-MPC approach for the economic scheduling of REH considering emergency residential building heating is proposed to reduce the operation cost and ensure resident indoor thermal comfort both in the normal operation state and during power outages under uncertainties. The main contributions of this paper are summarised as follows:

• (1)

  Thermal energy for emergency residential building heating is determined by the minimum thermal demand of residents. To quantify the minimum thermal demand during a power outage, the power outage duration and the minimum indoor comfort temperature threshold obtained by the predicted mean vote (PMV) are considered. In the scheduling of REH, the thermal energy for emergency heating is regarded as a time-varying constraint of the HWT.

• (2)

  Considering the uncertainties of outdoor temperature, irradiation, and power outage duration, an affine arithmetic-based model predictive control (AA-MPC) approach is developed to determine the optimal scheduling of REH for emergency residential building heating, which indicates the HP output interval and operation cost interval of REH under uncertain conditions.

MODELLING OF AN REH-BASED BUILDING ENERGY SYSTEM FOR EMERGENCY HEATING

As shown in Figure 1, the main components of REH consist of an HP, an HWT, an HP cycling pump, a heating network cycling pump, a battery, and radiators. For emergency heating during a power outage, the heating network cycling pump is driven by the battery to maintain thermal supply during the power outage. To guarantee thermal energy supply under normal and emergency conditions, the thermal energy stored in the HWT is divided into two parts: (1) thermal energy for daily thermal supply, which is used to participate in the daily REH scheduling, and (2) thermal energy for emergency heating.

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Mathematical models of the REH system

In this study, the main mathematical model of the REH system includes the HP and HWT. To avoid diluting the theme of this research by determining the battery storage, pump models are not considered in this paper.

Heat pump

The electrical energy from the grid is transformed into thermal energy by the HP. The performance of the HP can be quantified by the coefficient of performance (COP), which is defined as the ratio between the thermal power and the electric power of the HPas in Equation (1). The COP of the HP depends on the outdoor temperature, as shown in Equation (2) [27]: 1 $${P}_{\text{HP}}={P}_{\text{COP}}/{\lambda }_{\text{COP}}$$ 2 $${\lambda }_{\text{COP}}={\alpha }_{\text{COP}}{T}_{\text{out}}^{2}+{\beta }_{\text{COP}}{T}_{\text{out}}+{\chi }_{\text{COP}}$$

Hot water tank

The thermal energy from the HP can be stored in the HWT. The thermal storage of the HWT is characterised by the charging/discharging heat power, which is defined in Equation (3) [31]: 3 $${Q}_{\text{TS},t+1}=\left(1-{\sigma }_{\text{hl}}\right){Q}_{\text{TS},t}+\left({Q}_{\text{ch},t}{\eta }_{\text{ch}}-{Q}_{\text{dch},t}/{\eta }_{\text{dch}}\right){\Delta }t$$

Thermal model of the building

The thermal dynamics of the residential building are modelled as a resistor-capacitor (RC) thermal network model [31]. As shown in Figure 2, two kinds of nodes, temperatures on building surfaces and indoor/outdoor temperatures, are considered in the RC thermal model. All temperature nodes are linked with each other using thermal resistors and grounded by thermal capacitors. The thermal model of the building is described in state-space form, as shown in Equation (4). 4 $$\left\{\begin{array}{l}\dot{x}=\boldsymbol{A}x+u+\boldsymbol{B}{T}_{\text{out}}+\boldsymbol{C}{I}_{\text{solar}}+\boldsymbol{D}\\ y=\boldsymbol{F}x\end{array}\right.$$ where x is the state variable, which represents the temperatures of each node in the building; u is the input variable, which represents the thermal supply from the REH; and y is the output variable, which represents the indoor temperature. The state variables and input variables are shown in Equations (5) and (6) respectively. A, B, C, D, and F are relevant state-space matrices, as shown in Equations (7)–(11). 5 $$x={\left[\begin{array}{cccccc}{T}_{\text{in}}& {T}_{r,\text{out}}& {T}_{r,\text{in}}& {T}_{w,\text{out}}& {T}_{w,\text{in}}& {T}_{f,\text{in}}\end{array}\right]}^{\mathrm{T}}$$ 6 $$u={\left[\begin{array}{cccccc}\frac{{Q}_{\text{heat}}}{{C}_{a}}& 0& 0& 0& 0& 0\end{array}\right]}^{\mathrm{T}}$$ 7 $$\boldsymbol{A}=\left[\begin{array}{cccccc}-\frac{S}{{C}_{a}}& 0& \frac{1}{{C}_{a}{R}_{r,\text{in}}}& 0& \frac{1}{{C}_{a}{R}_{w,\text{in}}}& \frac{1}{{C}_{a}{R}_{f,\text{in}}}\\ 0& -\left(\frac{1}{{C}_{r,\text{out}}{R}_{r,\text{out}}}+\frac{1}{{C}_{r,\text{out}}{R}_{r,\text{en}}}\right)& \frac{1}{{C}_{r,\text{out}}{R}_{r,\text{en}}}& 0& 0& 0\\ \frac{1}{{C}_{r,\text{in}}{R}_{r,\text{in}}}& \frac{1}{{C}_{r,\text{in}}{R}_{r,\text{en}}}& -\left(\frac{1}{{C}_{r,\text{in}}{R}_{r,\text{in}}}+\frac{1}{{C}_{r,\text{in}}{R}_{r,\text{en}}}\right)& 0& 0& 0\\ 0& 0& 0& -\left(\frac{1}{{C}_{w,\text{out}}{R}_{w,\text{out}}}+\frac{1}{{C}_{w,\text{out}}{R}_{w,\text{en}}}\right)& \frac{1}{{C}_{w,\text{out}}{R}_{w,\text{en}}}& 0\\ \frac{1}{{C}_{w,\text{in}}{R}_{w,\text{in}}}& 0& 0& \frac{1}{{C}_{w,\text{in}}{R}_{w,\text{en}}}& -\left(\frac{1}{{C}_{w,\text{in}}{R}_{w,\text{in}}}+\frac{1}{{C}_{w,\text{in}}{R}_{w,\text{en}}}\right)& 0\\ \frac{1}{{C}_{f,\text{in}}{R}_{f,\text{in}}}& 0& 0& 0& 0& -\left(\frac{1}{{C}_{f,\text{in}}{R}_{f,\text{in}}}+\frac{1}{{C}_{f,\text{in}}{R}_{f,\text{en}}}\right)\end{array}\right]$$ where $$S=\frac{1}{{R}_{r,\text{in}}}+\frac{1}{{R}_{\text{window}}}+\frac{1}{{R}_{w,\text{in}}}+\frac{1}{{R}_{f,\text{in}}}$$ 8 $$\boldsymbol{B}={\left[\begin{array}{cccccc}\frac{1}{{C}_{a}{R}_{\text{win}}}& \frac{1}{{C}_{r,\text{out}}{R}_{r,\text{out}}}& 0& \frac{1}{{C}_{w,\text{out}}{R}_{w,\text{out}}}& 0& \frac{1}{{C}_{f,\text{in}}{R}_{f,\text{en}}}\end{array}\right]}^{T}$$ 9 $$\boldsymbol{C}={\left[\begin{array}{cccccc}\frac{{\lambda }_{\text{win}}{A}_{\text{win}}}{{C}_{\mathrm{a}}}& \frac{{\alpha }_{\mathrm{r}}\cdot \frac{{R}_{\mathrm{r},\text{out}}}{{R}_{\mathrm{r}}}{A}_{\mathrm{r}}\cdot \mathrm{sin}\theta }{{C}_{\mathrm{r},\text{out}}}& 0& \frac{{\alpha }_{\mathrm{w}}\cdot \frac{{R}_{\mathrm{w},\text{out}}}{{R}_{\mathrm{w}}}\cdot {A}_{\mathrm{w}}\cdot \mathrm{sin}\theta }{{C}_{\mathrm{w},\text{out}}}& 0& 0\end{array}\right]}^{T}$$ 10 $$\boldsymbol{D}={\left[\begin{array}{cccccc}\frac{{Q}_{\text{vent}}+{Q}_{\text{man}}}{{C}_{a}}& 0& 0& 0& 0& 0\end{array}\right]}^{T}$$ 11 $$\boldsymbol{F}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\end{array}\right]$$

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Model of emergency heating during a power outage

Thermal energy for emergency heating

The thermal energy for emergency heating at each moment is isolated from the capacity of the HWT. Thus, the thermal energy of the HWT Q_TS,t at time t is divided into the thermal energy for daily operation Q_use,t and the thermal energy for emergency heating Q_em,t as shown in Equation (12). 12 $${Q}_{\text{TS},t}={Q}_{\text{use},t}+{Q}_{\text{em},t}$$

Different indoor set temperatures, which are obtained in Section 2.3.2, are utilised to obtain the thermal demands during normal operation $${Q}_{\text{heat}}$$ and emergency heating $${Q}_{\text{heat}}^{\text{em}}$$ in Section 2.2. The minimum thermal demand during a power outage for at least t_E hours is fulfilled by the thermal energy for emergency heating, as shown in Equation (13). 13 $${Q}_{\text{em},t}=\sum\limits _{i=t}^{t+{t}_{\mathrm{E}}}{Q}_{\text{heat},i}^{\text{em}}{\Delta }t$$

Indoor set temperature under normal conditions and during a power outage

The indoor temperature setting range is closely related to the thermal comfort of residents. The PMV is used to determine the indoor comfort temperature [29]. According to ISO 7730, the PMV is 0 in daily thermal supply operations. Considering the minimum thermal demand during power outages, the PMV is taken as −0.5, which is the minimum indoor comfort threshold [30]. As shown in Equations (14) and (15), the indoor set temperature $${T}_{\text{in}}^{set}$$ under normal conditions and during power outages is calculated according to the PMV setting value. 14 $$PMV=\left(0.028+0.3033{e}^{-0.036M}\right)\times L$$ 15 $$L=M-W-3.05\times {10}^{-3}\times \left[5733-6.99(M-W)-{P}_{a}\right]-0.42\times [(M-W)-58.15]-1.7\times {10}^{-5}M\left(5867-{P}_{a}\right)-0.0014M\left(34-{T}_{\text{in}}^{set}\right)-3.96\times {10}^{-8}{f}_{\text{cl}}\left[{\left({T}_{\text{cl}}+273\right)}^{4}-{\left({T}_{\text{ra}}+273\right)}^{4}\right]-{f}_{\text{cl}}{h}_{\mathrm{c}}\left({T}_{\text{cl}}-{T}_{\text{in}}^{set}\right)$$

THE AA-MPC-BASED OPTIMAL REH SCHEDULING METHOD FOR EMERGENCY RESIDENTIAL BUILDING HEATING

Affine representation of variables in the REH model

The irradiation, outdoor temperature, and duration of the power outage uncertainties are shown in affine form in Equations (16)–(18). 16 $${\widehat{I}}_{\text{solar},t}={I}_{\text{solar},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{I}_{\text{solar},i,t}{\varepsilon }_{i,t}$$ 17 $${\widehat{T}}_{\text{out},t}={T}_{\text{out},0,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{T}_{\text{out},j,t}{\varepsilon }_{j,t}$$ 18 $${\widehat{t}}_{\mathrm{E},t}={t}_{\mathrm{E},0,t}+\sum\limits _{k=1}^{{n}_{\mathrm{E}}}{t}_{\mathrm{E},k,t}{\varepsilon }_{k,t}$$ where the first term of each affine form is the central value of the variable. The noise symbol $$\varepsilon $$ is a symbolic real variable whose value is restricted to $$\varepsilon \in [-1,1]$$.

As shown in Section 2.2, I_solar,t and T_out,t are the main influencing factors of $${Q}_{\text{heat},t}$$ and $${Q}_{\text{heat},t}^{\text{em}}$$. The affine forms of $${Q}_{\text{heat},t}$$ and $${Q}_{\text{heat},t}^{\text{em}}$$ are shown in Equations (19) and (20) respectively. 19 $${\widehat{Q}}_{\text{heat},t}={Q}_{\text{heat},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{Q}_{\text{heat},i,t}{\varepsilon }_{i,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{Q}_{\text{heat},j,t}{\varepsilon }_{j,t}$$ 20 $${\widehat{Q}}_{\text{heat},t}^{\text{em}}={Q}_{\text{heat},0,t}^{\text{em}}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{Q}_{\text{heat},i,t}^{\text{em}}{\varepsilon }_{i,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{Q}_{\text{heat},j,t}^{\text{em}}{\varepsilon }_{j,t}$$ where the second and third terms are the forecast deviations of irradiation and outdoor temperature.

Based on Equation (13), the thermal energy for emergency heating in affine form is shown in Equation (21): 21 $${\widehat{Q}}_{\text{em},t}=\sum\limits _{l=t}^{t+{t}_{\mathrm{E},\varepsilon ,t}}\left({Q}_{\text{heat},0,l}^{\text{em}}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{Q}_{\text{heat},i,l}^{\text{em}}{\varepsilon }_{i,l}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{Q}_{\text{heat},j,l}^{\text{em}}{\varepsilon }_{j,l}\right)$$ where the duration of the power outage in affine form is shown in Equation (22): 22 $${t}_{\mathrm{E},\varepsilon ,t}=\lceil {t}_{\mathrm{E},0,t}+\sum\limits _{k=1}^{{n}_{\mathrm{E}}}{t}_{\mathrm{E},k,t}{\varepsilon }_{k,t}\rceil $$

Furthermore, the affine forms of the HP electric power, HWT charging and discharging power and HWT thermal energy are represented by Equations (23)–(25): 23 $${\widehat{P}}_{\text{HP},t}={P}_{\text{HP},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{P}_{\text{HP},i,t}{\varepsilon }_{i,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{P}_{\text{HP},j,t}{\varepsilon }_{j,t}+\sum\limits _{k=1}^{{n}_{\mathrm{E}}}{P}_{\text{HP},k,t}{\varepsilon }_{k,t}$$ 24 $${\widehat{Q}}_{\text{ch}/\text{dch},t}={Q}_{\text{ch}/\text{dch},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{Q}_{\text{ch}/\text{dch},i,t}{\varepsilon }_{i,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{Q}_{\text{ch}/\text{dch},j,t}{\varepsilon }_{j,t}+\sum\limits _{k=1}^{{n}_{\mathrm{E}}}{Q}_{\text{ch}/\text{dch},k,t}{\varepsilon }_{k,t}$$ 25 $${\widehat{Q}}_{\text{TS},t}={Q}_{\text{TS},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}{Q}_{\text{TS},i,t}{\varepsilon }_{i,t}+\sum\limits _{j=1}^{{n}_{\mathrm{T}}}{Q}_{\text{TS},j,t}{\varepsilon }_{j,t}+\sum\limits _{k=1}^{{n}_{\mathrm{E}}}{Q}_{\text{TS},k,t}{\varepsilon }_{k,t}$$ where the first term of each affine form is the central value of the corresponding variable and the other terms are the deviations of the variable due to the forecast errors of irradiation, outdoor temperature, and duration of power outages.

Affine arithmetic-based optimal REH scheduling method

The proposed AA-MPC-based REH scheduling method seeks to find the parameters of the affine forms associated with the different continuous variables that minimise the REH system operation cost while meeting all equipment constraints and satisfying thermal demand. The objective function of the proposed REH scheduling method is stated in Equation (26). 26 $$\mathrm{min}\sum\limits _{t=1}^{T}{C}_{\text{ToU},t}{\widehat{P}}_{\text{HP},t}{\Delta }t$$

The proposed optimal REH scheduling method in this study includes an energy balance constraint, HP constraints, and HWT constraints, as follows:

Energy balance constraint

The energy balance in the process of REH energy scheduling must be ensured (see Figure 1), as shown in the constraint. 27 $$\sum\limits _{t=1}^{T}\left[{\eta }_{\text{ch}}{\widehat{P}}_{\text{COP},t}-\left(1-{\sigma }_{\text{hl}}\right){\widehat{Q}}_{\text{TS},t}/{\Delta }t\right]/{\eta }_{\text{dch}}={\widehat{Q}}_{\text{heat},t}$$

HP constraints

The HP electricity power is constrained by the upper and lower electricity power limits of the HP, as shown in affine forms in Equation (28). The ramp-rate constraint is formulated in affine form in Equation (29). 28 $${P}_{\text{HP},\mathrm{min}}\le {\widehat{P}}_{\text{HP},t}\le {P}_{\text{HP},\mathrm{max}}$$ 29 $${D}_{\text{HP}}\le \left({\widehat{P}}_{\text{HP},t+1}-{\widehat{P}}_{\text{HP},t}\right)/{\Delta }t\le {U}_{\text{HP}}$$

HWT constraints

The energy constraint of the HWT is described in affine form in Equation (30). Equation (31) is the thermal energy limit of the HWT, which ensures that the thermal energy for emergency heating fulfils the minimum thermal demand during a power outage. Equations (32) and (33) indicate the charging and discharging constraints of the HWT respectively. 30 $${\widehat{Q}}_{\text{TS},t+1}=\left(1-{\sigma }_{\text{hl}}\right){\widehat{Q}}_{\text{TS},t}+\left({\widehat{Q}}_{\text{ch},t}{\eta }_{\text{ch}}-{\widehat{Q}}_{\text{dch},t}/{\eta }_{\text{dch}}\right){\Delta }t$$ 31 $$\mathrm{max}\left(0,\sum\limits _{l=t}^{t+{\hat{t}}_{\mathrm{E},t}}{\widehat{Q}}_{\text{heat},l}^{\text{em}}\right)\le {\widehat{Q}}_{\text{TS},t}\le {Q}_{\text{TS},\mathrm{max}}$$ 32 $${\widehat{Q}}_{t}^{\text{ch}}=\mathrm{min}\left({\widehat{P}}_{\text{COP},t},{Q}_{\text{ch},\mathrm{max}}\right)$$ 33 $${\widehat{Q}}_{t}^{\text{dch}}=\mathrm{min}\left({\widehat{Q}}_{\text{heat},t},{Q}_{\text{dch},\mathrm{max}}\right)$$

Implementation of the proposed AA-MPC

Since the proposed affine arithmetic-based optimal REH scheduling method is a non-linear programme with uncertain information, it cannot be solved directly by solvers [31]. The optimisation needs to be transformed into a deterministic optimisation method. According to AA theory, the noise symbols in this research $${\varepsilon }_{i}$$, $${\varepsilon }_{j}$$, and $${\varepsilon }_{k}$$ are equal to 1 or −1 to obtain the maximum or the minimum value of uncertainties [24]. The maximum operation cost corresponds to the pessimistic solution, while the minimum operation cost corresponds to the optimistic solution.

The constraints of the proposed scheduling are divided into non-intertemporal and intertemporal constraints, which are intrinsically coupled with the uncertainties mentioned above and should be treated differently. Non-intertemporal constraints are formulated in affine forms on AA mathematical operators, such as Equations (27), (28) and (31)–(33). For intertemporal constraints, such as Equations (27) and (29)–(30), since the noise symbols $${\varepsilon }_{t}$$ and $${\varepsilon }_{t+1}$$ are not necessarily equal, they cannot be formulated directly according to AA rules. Based on the correlation between AA and IA theory, several definitions have been used in this study [25]. 34 $$\sum\limits _{t=1}^{T}\left\{\left[{\eta }_{\text{ch}}{P}_{\text{COP},0,t}-\left(1-{\sigma }_{\text{hl}}\right){Q}_{\text{TS},0,t}/{\Delta }t\right]/{\eta }_{\text{dch}}-\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert \left[{\eta }_{\text{ch}}{P}_{\text{COP},i,t}-\left(1-{\sigma }_{\text{hl}}\right){Q}_{\text{TS},i,t}/{\Delta }t\right]/{\eta }_{\text{dch}}\vert \right\}\ge \sum\limits _{t=1}^{T}\left({Q}_{\text{heat},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {Q}_{\text{heat},i,t}\vert \right)$$ 35 $${P}_{\text{HP},\mathrm{min}}\le {P}_{\text{HP},0,t}-\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t}\vert $$ 36 $${P}_{\text{HP},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t}\vert \le {P}_{\text{HP},\mathrm{max}}$$ 37 $${Q}_{\text{TS},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {Q}_{\text{TS},i,t}\vert \le {Q}_{\text{TS},\mathrm{max}}$$ 38 $$\mathrm{max}\left(0,\sum\limits _{l=t}^{t+{t}_{\mathrm{E},\varepsilon ,t}}\left({Q}_{\text{heat},0,l}^{\text{em}}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}}\vert {Q}_{\text{heat},i,l}^{\text{em}}\vert +\sum\limits _{j=1}^{{n}_{\mathrm{T}}}\vert {Q}_{\text{heat},j,l}^{\text{em}}\vert \right)\right)\le \left({Q}_{\text{TS},0,t}-\sum\limits _{l=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {Q}_{\text{TS},l,t}\vert \right)$$ 39 $${\widehat{Q}}_{t}^{\text{ch}}=\left\{\begin{array}{cc}{\widehat{P}}_{\text{COP},,t}& {P}_{\text{COP},0,t}+\sum\limits _{l=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{COP},l,t}\vert \le {Q}_{\text{ch},\mathrm{max}}\\ {Q}_{\text{ch},\mathrm{max}}& {P}_{\text{COP},0,t}+\sum\limits _{l=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{COP},l,t}\vert > {Q}_{\text{ch},\mathrm{max}}\end{array}\right.$$ 40 $${\widehat{Q}}_{t}^{\text{dch}}=\left\{\begin{array}{cc}{\widehat{Q}}_{\text{heat},t}& {\widehat{Q}}_{\text{heat},0,t}+\sum\limits _{i=1}^{{n}_{s}+{n}_{T}}\vert {Q}_{\text{heat},i,t}\vert \le {Q}_{\text{dch},\mathrm{max}}\\ {Q}_{\text{dch},\mathrm{max}}& {\widehat{Q}}_{\text{heat},0,t}+\sum\limits _{i=1}^{{n}_{s}+{n}_{T}}\vert {Q}_{\text{heat},i,t}\vert > {Q}_{\text{dch},\mathrm{max}}\end{array}\right.$$ 41 $${D}_{\text{HP}}\le \left\{\left({P}_{\text{HP},0,t+1}-\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t+1}\vert \right)-\left({P}_{\text{HP},0,t}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t}\vert \right)\right\}$$ 42 $$\left\{\left({P}_{\text{HP},0,t+1}+\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t+1}\vert \right)-\left({P}_{\text{HP},0,t}-\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {P}_{\text{HP},i,t}\vert \right)\right\}\le {U}_{\text{HP}}$$ 43 $${Q}_{\text{TS},0,t+1}=\left(1-{\sigma }_{\text{hl}}\right){Q}_{\text{TS},0,t}+\left({Q}_{\text{ch},0,t}{\eta }_{\text{ch}}-{Q}_{\text{dch},0,t}/{\eta }_{\text{dch}}\right){\Delta }t$$ 44 $$\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert {Q}_{\text{TS},i,t+1}\vert =\sum\limits _{i=1}^{{n}_{\mathrm{S}}+{n}_{\mathrm{T}}+{n}_{\mathrm{E}}}\vert \left(1-{\sigma }_{\text{hl}}\right){Q}_{\text{TS},0,t}+\left({Q}_{\text{ch},0,t}{\eta }_{\text{ch}}-{Q}_{\text{dch},0,t}/{\eta }_{\text{dch}}\right){\Delta }t\vert $$

Non-intertemporal constraints

For the energy balance constraint proposed in Equation (27), the lower bound of the difference between the thermal power of the HP and the loss of the HWT should exceed the upper bound of the thermal demand of the building, as shown in Equation (34). According to the definition for non-intertemporal inequality constraints mentioned in ref. [32], the electricity power constraint of the HP (Equation (28)) is depicted in Equations (35) and (36). Similarly, the non-intertemporal inequality constraints of the HWT (Equations (31)–(33)) are shown in Equations (37)–(40).

Intertemporal constraints

For the intertemporal inequality constraint (Equation (29)), based on basic interval theory, the difference between the maximum value of P_HP at time t + 1 and the minimum value at time t should be smaller than the upper limit of the HP ramp rate, and vice versa. The ramp-rate constraints are formulated in affine form in Equations (41) and (42). For intertemporal equality constraints, the HWT energy constraint (Equation (30)) is approximated in the AA domain by equalising the central values and the radius of the affine forms [25], as shown in Equations (43) and (44).

Implementation process

An illustration of the implementation of the proposed AA-MPC-based optimal scheduling is shown in Figure 3. At the current scheduling time t, the data of T_out, I_solar, and t_E are obtained over k_p prediction steps. Q_em is calculated as a time-varying constraint of the HWT. The optimal REH scheduling is solved over k_s scheduling steps to obtain P_HP in each scheduling step. The optimised schedules for the first-time slot at t obtained are applied. At the next time t + 1, the data are updated for the next k_p prediction steps. The prediction and scheduling time horizons move forward by a one-time slot for the new forward-looking optimisation. The optimal REH scheduling is implemented until all scheduling horizons are determined during the entire timeline. The MPC method for REH scheduling is realised in Figure 4. The above deterministic optimisation model is a linear optimisation problem. The optimisation problem is solved using the Gurobi optimisation solver [33].

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NUMERICAL RESULTS

Input data and scenario setting

An REH-based residential building energy system is used to verify the proposed method. The parameters of the REH are shown in Table 1 [30]. The parameters of the building are shown in Appendix Table A1 and [34]. The PMV index parameters are shown in Appendix Table A2 [35]. The outdoor temperature and irradiation are shown in Appendix Figures A1 and A2 respectively [15]. The ToU electricity price is shown in Appendix Figure A3 [15]. The prediction and scheduling horizons are set to 24 h, and the scheduling interval is 30 min (k_p = k_s = 48). The power outage duration is 2 h [36].

TABLE 1 REH parameters [30]

Parameter	Data	Parameter	Data
$${P}_{\text{HP},\mathrm{max}}/{P}_{\text{HP},\mathrm{min}}(\text{kW})$$	4.9/0	UHP/DHP(kW/h)	1.6/1.6
$${\alpha }_{\text{COP}}$$	0.0005	$${\sigma }_{\text{hl}}$$	1%
$${\beta }_{\text{COP}}$$	0.0756	$${\eta }_{\text{ch}}$$	99%
$${\chi }_{\text{COP}}$$	2.2216	$${\eta }_{\text{dch}}$$	99%
QTS,max (kWh)	21

Scheduling results

In this section, the proposed AA-MPC-based REH scheduling method is compared with three other scheduling methods, namely, the day-ahead optimisation method [15], the MPC method [37], and the interval arithmetic-based model predictive control (IA-MPC) method [38]. Three types of prediction error levels are used to illustrate the effectiveness of the methods, as shown in Figure 5 [39].

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Scenario I (Day-ahead optimisation): Forecast errors of the outdoor temperature, irradiation, and the duration of the power outage are ignored in Scenario I.

Scenario II (MPC method): The MPC method is referred to [37].

Scenario III (IA-MPC method): The IA-MPC method is referred to [38].

Scenario IV Proposed AA-MPC method.

The comparison results of the HP electric power range at different prediction errors are shown in Figure 6. In both Scenario I and Scenario II, the HP runs at full power during the low electricity price period. Due to the low outdoor temperature from 4:00–6:00, according to Equation (2), the value of the COP is small during this period, so the output decreases. The HP turns off, and the thermal energy in the HWT is fully utilised to meet the user demand at 6:00–8:00. However, due to the high outdoor temperature from 15:00 to 17:00, the output of the equipment increases. The output of the heat pump decreases because of the decrease in the COP at 18:00–20:00.

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As seen from the figures, with the gradual increase in the prediction error level, both the electric power ranges of the HP electric power in Scenario III and Scenario IV increase. The scheduling result value of Scenario II is within the range of Scenario III and that of Scenario IV. However, the ranges of Scenario III grow larger than those of Scenario IV because of the problem known as the dependency problem in the IA theory [29].

Table 2 shows the operation cost ranges in different scenarios. The operation cost intervals in Scenario III and Scenario IV experience an increase with increasing prediction error levels. Comparing the central values of different scenarios, it can be concluded that the central values of Scenario III and Scenario IV are both smaller than that of Scenario I. The reason is that in optimistic scheduling results, the duration of outages and thermal energy for emergency heating in Scenario III and Scenario IV are smaller than those in Scenario I. In addition, the outdoor temperature is higher in the optimistic scheduling result compared with the outdoor temperature in pessimistic scheduling, so the operation cost decreases. Compared with Scenario III, Scenario IV can further reduce the impact of uncertainties on the operation costs. The cost of Scenario II is in the range of that of Scenario IV. The proposed AA-MPC method can calculate a more accurate assessment result interval of the resident operation cost after considering emergency heating. The operation cost interval width of the AA-MPC optimisation method is reduced by 57.3%, 0.3% and 32.5% under the three prediction error levels compared with that of the IA-MPC optimisation method.

TABLE 2 Operation cost in different scenarios

Scenario	Prediction error levels	Optimistic operation cost (¥)	Pessimistic operation cost (¥)	Interval width (¥)	Central value (¥)
I	—	—	—	—	22.47
II	Low	—	—	—	22.08
Middle	—	—	—	22.14
High	—	—	—	21.93
II	Low	21.18	23.55	2.37	22.37
Middle	21.02	23.90	2.88	22.46
High	18.90	24.71	5.81	21.81
III	Low	21.65	22.66	1.01	22.16
Middle	21.05	23.92	2.87	22.48
High	20.56	24.48	3.92	22.52

Error discussion

The proposed method provides users with consideration of the electric power of the REH and operation cost intervals after considering emergency heating under uncertainties. Therefore, in this section, a total of 300 scenarios are randomly generated by Monte Carlo sampling with three types of uncertainties. Each scenario represents one case of uncertainty realisation. Violation rates of electric power of the REH and operation cost comparisons of Scenario III and Scenario IV are shown in Table 3. The calculation of the violation rate is shown in Ref. [25]. In comparison, AA-MPC (Scenario III) has the highest power violation rate due to the linearisation of non-linear optimisation in Section 3.3 and [39]. Although the proposed method has a power violation rate, its operation cost has no violations, and the method can still provide residents with a reasonable operation cost interval for reference.

TABLE 3 Violation rate comparisons after Monte Carlo simulation

Scenario	Prediction error levels	Power violation rate	Cost violation rate
II	Low	0	0
Middle	0	0
High	0	0
III	Low	0	0
Middle	0.37%	0
High	1.66%	0

Influence on the effect of emergency heating

To verify the influence on the effect of emergency heating, Scenario I mentioned above is used. It is assumed that a power outage lasts from 18:00 to 20:00, according to the data in ref. [33]. If a power outage occurs during this period, the limited thermal energy in the HWT will cause heating interruption (see Figure 7b).

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The results without considering emergency heating both during the normal condition and during a power outage are shown in Figure 7. The HP is scheduled at its maximum electric power during most low electricity price periods (00:00–02:00, 04:00–06:00, 22:00–24:00). However, the HP turns off because of the lower outdoor temperature throughout the day from 02:00 to 04:00. Considering emergency heating during power outages, Q_use is lower than that without considering emergency heating (Equation (12)), as shown in Figure 8a. Due to the greater HP output during the high electricity price period, considering emergency heating will cause operating cost increases. The operation cost of Scenario I is ¥19.28. The operation cost is ¥22.47 after considering emergency heating, which is 16.54% higher.

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Considering emergency heating, the optimised scheduling results are shown in Figure 8. As can be observed from the circled areas with red lines in Figure 7b, once the power outage occurs at 18:00, the HP shuts down, and the thermal energy of the HWT is almost empty at the same time. Compared with Figure 7b, considering emergency heating during the power outage (see Figure 8b), the HWT utilises the stored thermal energy to cover the thermal shortage during the power outage.

The indoor temperatures from 16:00 to 21:30 considering emergency heating are shown in Figure 9. Without considering emergency heating, while the power outage occurs, the indoor temperature drops from 21.96°C (PMV = 0) at 18:00–12.47°C at 20:00 (PMV = −2, which means cold in ISO 7730 [30]). Although the HP turns on when the power is restored, the indoor temperature remains at a relatively low level. After considering emergency heating, the indoor temperature comfort is always maintained above 19.59°C (PMV = −0.5). As shown in Figure 10, after considering emergency heating, thermal storage in the HWT can maintain the thermal supply for at least 2 h when a power outage occurs. In conclusion, the proposed emergency heating model guarantees the realisation of thermal supply during the power outage.

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CONCLUSIONS

In this paper, an AA-MPC approach for the optimal scheduling of REH considering emergency residential building heating is proposed. In the face of heat interruption caused by power outages, thermal energy for emergency heating is proposed and regarded as a time-varying HWT constraint to participate in the economic scheduling of a residential REH in normal conditions. To minimise the REH operation costs and improve thermal supply reliability under uncertainties, an AA-MPC approach for economic scheduling of the REH considering emergency heating is proposed, taking the irradiation, outdoor temperature, and power outage duration uncertainties into consideration. The conclusions are as follows:

• (1)

  Thermal energy for emergency heating is developed associated with the duration of a power outage and the minimum thermal comfort temperature of residents. The proposed emergency heating method can supply thermal energy to cover thermal shortages and maintain indoor temperatures above the minimum comfortable temperature threshold during a power outage.

• (2)

  Considering the outdoor temperature, irradiation, and power outage duration uncertainties, the proposed AA-MPC method is conducive to providing an operation cost interval for residents after considering emergency heating and avoiding an excessive deviation between the predicted operation cost and the actual cost of the REH, compared with day-ahead optimisation, the model predictive control method, and the interval arithmetic-based model predictive control method.

Further consideration will be given to the impact of uncertainty on resident behaviours.

ACKNOWLEDGEMENT

This paper was funded by the Science and Technology Foundation of Global Energy Interconnection Group Co., Ltd. (No. SGGEIG00JYJS2100033).

CONFLICT OF INTEREST

The author declares that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported.

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Nomenclature

Acronyms

• AA
• affine arithmetic
• BES
• building energy system
• HP
• heat pump
• HWT
• hot water tank
• MPC
• model predictive control
• PMV
• predicted mean vote
• REH
• regenerative electric heating
• TES
• thermal energy storage
• ToU
• time of use

Indices

• 0
• central value
• i
• irradiation uncertainties
• j
• outdoor temperature uncertainties
• k
• duration of power outage uncertainties
• $$t$$
• the tth time horizon

Variables

• P_COP
• HP thermal power (kW)
• P_HP
• HP electrical power (kW)
• $${\lambda }_{\text{COP}}$$
• HP coefficient of performance
• $${Q}_{\text{TS}}$$
• HWT thermal energy (kW)
• $${Q}_{\text{heat}}$$, $${Q}_{\text{heat}}^{\text{em}}$$
• thermal demand for daily operation state/emergency heating (kW)
• $${Q}_{\text{em}}$$
• thermal energy for emergency heating (kW)
• $${Q}_{\text{use}}$$
• thermal energy for normal operation state (kW)
• Q_ch, Q_dch
• HWT charging/discharging power (kW)
• T_in
• indoor temperature (°C)
• T_r,in, T_r,out
• internal/external roof surface temperature (°C)
• T_w,in, T_w,out
• internal/external wall surface temperature (°C)
• T_f,in
• internal floor surface temperature (°C)
• Q_r,solar, Q_w,solar, Q_window,solar
• solar heat gains absorbed by roof/external wall surface/window (kW)
• $${T}_{\text{in}}^{\mathrm{s}\mathrm{e}\mathrm{t}}$$
• indoor set temperature (°C)
• $$\varepsilon $$
• noise symbol
• C_ToU
• ToU electricity price (CNY/kWh)

Parameters

• $${\Delta }t$$
• scheduling interval (h)
• n
• run time (h)
• I_solar
• irradiation (W/m²)
• T_out
• outdoor temperature (°C)
• t_E
• duration of power outage (h)
• $${\alpha }_{\text{COP}}$$, $${\beta }_{\text{COP}}$$, $${\chi }_{\text{COP}}$$
• COP coefficients
• P_HP,max, P_HP,min
• Upper and lower HP electricity power limits (kW)
• U_HP, D_HP
• HP ramp-up/ramp-down rate limit (kW/h)
• $${\eta }_{\text{ch}}$$, $${\eta }_{\text{dch}}$$
• HWT storing/releasing efficiency (%)
• $${\sigma }_{\text{hl}}$$
• HWT energy loss rate (%)
• Q_ch, Q_dch
• HWT charging/discharging heat power (kW)
• Q_ch,max, Q_dch,max
• HWT maximum charging/discharging heat power limit (kW)
• Q_TS,max
• HWT maximum thermal energy (kWh)
• A
• area of the building (m²)
• h
• height of the building (m)
• C_a
• thermal capacitance of air (kJ/kg⋅K)
• C_r, C_w, C_f
• thermal capacitance of roof/wall/floor (kJ/kg⋅K)
• R_r,in, R_r,en, R_r,out
• thermal resistance of roof for internal convective/conduction/external convective heat transfer (K/W)
• R_i,w,in, R_i,w,en, R_i,w,out
• thermal resistance of wall i for internal convective/conduction/external convective heat transfer (K/W)
• R_f,in, R_f,en
• Thermal resistance of floor for internal convective/conduction heat transfer (K/W)
• R_window
• thermal resistance of glass in window (K/W)
• Q_vent, Q_man
• heat gain from air penetration/user behaviour (kW)
• $${\lambda }_{\text{win}}$$
• window transmission coefficient (%)
• $$\theta $$
• solar altitude angle (°)
• $${\alpha }_{\mathrm{r}}$$, $${\alpha }_{\mathrm{w}}$$
• solar radiation absorptance of roof/external wall surface (%)
• M
• metabolic rate (W/m²)
• W
• external work (W/m²)
• $${f}_{\text{cl}}$$
• ratio of body surface area covered by clothes (%)
• $${T}_{\text{cl}}$$
• surface temperature of clothing (°C)
• $${T}_{\text{ra}}$$
• mean radiant temperature (°C)
• $${h}_{\mathrm{c}}$$
• convective heat transfer coefficient (W/m²⋅K)
• k_p, k_s
• prediction/scheduling step
• n_S, n_T, n_E
• number of noise symbols for irradiation, outdoor temperature, and duration of power outage
• $$\widehat{\cdot }$$
• affine representation of an uncertain variable

Appendix

APPENDIX

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TABLE A1 Building parameters [30]

Parameter	Data	Parameter	Data
$$\theta {(}^{\circ })$$	27	$${c}_{\mathrm{a}}(\text{kJ}/(\text{kg}/{}^{\circ}\mathrm{C}))$$	1.0
$${\alpha }_{\mathrm{r},\text{solar}}$$	70%	$$A\left({\mathrm{m}}^{2}\right)$$	100
$${\alpha }_{\mathrm{w},\text{solar}}$$	40%	$$h(\mathrm{m})$$	2.5
$${Q}_{\text{vent}}(\text{kW})$$	0.05	$${Q}_{\text{man}}(\text{kW})$$	0.05

TABLE A2 PMV index parameters [35]

Parameter	Data
M/(W/m2)	70.0
W/(W/m2)	0.0
Pa/(Pa)	2000.0
Tr/(°C)	29.7
Tcl/(°C)	32.0
$${f}_{\text{cl}}$$	110%
$${h}_{\mathrm{c}}$$ (W/m2⋅K)	5

References

Tarroja, B., et al.: Translating climate change and heating system electrification impacts on building energy use to future greenhouse gas emissions and electric grid capacity requirements in California. Appl. Energy. 225, 522–534 (2018). [DOI: https://dx.doi.org/10.1016/j.apenergy.2018.05.003]