Distributionally robust optimization of home energy management system based on receding horizon optimization

Abstract

This paper investigates the scheduling strategy of schedulable load in home energy management system (HEMS) under uncertain environment by proposing a distributionally robust optimization (DRO) method based on receding horizon optimization (RHO-DRO). First, the optimization model of HEMS, which contains uncertain variable outdoor temperature and hot water demand, is established and the scheduling problem is developed into a mixed integer linear programming (MILP) by using the DRO method based on the ambiguity sets of the probability distribution of uncertain variables. Combined with RHO, the MILP is solved in a rolling fashion using the latest update data related to uncertain variables. The simulation results demonstrate that the scheduling results are robust under uncertain environment while satisfying all operating constraints with little violation of user thermal comfort. Furthermore, compared with the robust optimization (RO) method, the RHO-DRO method proposed in this paper has a lower conservation and can save more electricity for users.

Keywords

distributionally robust optimization (DRO) / home energy management system (HEMS) / receding horizon optimization (RHO) / uncertainties

Introduction

Home energy management system (HEMS) has attracted extensive attention in recent years, with the development of advanced metering infrastructure, smart sensor technologies, smart home appliances, and home area network, etc [¹]. By using HEMS, users can have two-way communication with smart grids and actively participate in demand response based on incentives from the grid, e.g., real-time price (RTP). Since the static stability problem of the grid can result from the high penetration of renewable energy resources contained in the HEMS, a novel small signal modeling approach based on characteristic equation for converter-dominated as microgrids is proposed to assess the low-frequency stability of the system [²].

It is crucial to design an efficient scheduling strategy for HEMS [³] and research on the scheduling strategy has received great interest. Hu et al. presented a hardware design of HEMS [⁴] and used the machine learning algorithm to achieve a control strategy, which contains three modes, based on RTP. A combined pricing model of RTP and inclining block rate was proposed [⁵] to reduce both the electricity cost and peak-to-average ratio. Reference [⁶] optimized the operation of heating, ventilation, and air conditioning (HVAC) based on RTP and significantly reduced peak loads and electricity cost with a modest variation in thermal comfort. A mixed integer nonlinear programming model which considers electrical cost, user’s convenience rate and thermal comfort as objectives, was developed [⁷] to solve the optimal problem of HEMS and reach a meaningful balance between energy saving and a comfortable lifestyle. Reference [⁸] developed a mixed integer linear programming (MILP) with dynamic electricity constraints to reduce the electricity cost, while simultaneously ensuring the thermal comfort. A novel traversal-and-pruning algorithm named “inferior pruning” was proposed [⁹] to address the optimization problem of electric water heater (EWH) scheduling. Reference [¹⁰] formulated the integrated demand side management among smart energy hubs as a non-cooperative game and presented a cloud-based architecture to implement cloud computing. A generic MILP model considering renewable energy and energy storage was proposed to optimize the structures and energy management strategies of multi energy systems (MESs) simultaneously [¹¹].

Due to various factors, the forecast data are far from accurate and the uncertainty of forecast data may lead to a significant economic loss and violations in customer comfort [¹²]. To deal with such a problem, stochastic optimization (SO) was used [¹³,¹⁴]. Reference [³], considering the uncertainties of electricity prices and loads, formulated a chance constrained optimization model which was solved by particle swarm optimization. However, SO requires the exact probability distribution of the uncertain variable [¹⁵] which is often difficult and costly in practice. Robust optimization (RO), which denotes the uncertain variable by the interval number and only needs the boundary of the uncertain variable, was used [¹²,¹⁶–²⁰]. The robust-index method was proposed [¹²] to tackle the uncertainty of customer behavior. Reference [¹⁶] developed a MILP based on RO with different levels to optimize the operation of EWH while considering the uncertainties of outdoor temperature and water demand. However, RO has an over-conservativeness feature because it ignores probability distribution information. Recently, a distributionally robust optimization (DRO) method, which only needs the ambiguity set of the probability distribution, combines the advantages of both the SO and the RO. The DRO approach has been applied to power system optimization problems including unit commitment [²¹] and energy scheduling in microgrid [²²]. In Ref. [²³], the DRO approach was used to reform the scheduling problem of home energy into a second order conic programming problem considering the uncertainty of photovoltaic (PV) generation. However, the uncertainties of hot water demand and outdoor temperature, which have a direct and significant impact on user thermal comfort, are not considered.

While the methods used above incorporate the uncertainty within the mathematical model, model predictive control (MPC) used in Ref. [²⁴], which relies on updating algorithms and using the updated information, is another efficient tool to deal with uncertainties [²⁵]. In Ref. [²⁶], an event-triggered-based distributed algorithm was proposed to address cooperative energy management problems for multi-energy systems in both day-ahead and real-time scale. As the core of MPC, the receding horizon optimization (RHO) was used [²⁷,²⁸] to optimize the operation of HEMS in a rolling fashion according to the updated information. It was proved in Refs. [²⁷,²⁹] that the new optimization results with the updated information of uncertain variable was better than the previous optimization results.

In this paper, a DRO method based on RHO (RHO-DRO) method is used to develop the scheduling problem of HEMS into a MILP that is solved in a rolling fashion. The objective is to minimize the electricity cost while satisfying the thermal comfort constraints under uncertain environments. Besides, based on the established optimization model of HEMS, the DRO method, which only requires the historical data of uncertain variables to construct the ambiguity sets of probability distribution, is proposed to solve the uncertainties of outdoor temperature and hot water demand. In addition, the application of the DRO method transformed the scheduling problem of HEMS into a MILP problem which can be easily solved by mature tools. Moreover, by integrating the RHO with the MILP, the effect of uncertainties in the previous time slots can be effectively eliminated. Furthermore, since the data related to uncertain variables can be updated at the beginning of each time slot, a more robust and economical scheduling results can be obtained. The simulation results indicate that the scheduling results obtained by the RHO-DRO approach are robust to satisfy the thermal comfort constraints in different scenarios of uncertain variables. Compared with the integration of the RO with the RHO (RHO-RO), the RHO-DRO approach can bring more economic benefit to users because it is less conservative. In addition, the influence of different optimization time domain is discussed.

Optimization model of HEMS

A mathematical optimization model of the HEMS is established, as shown in Fig. 1. The household loads, which obtain power from the grid and PV, are divided into the non-controllable load (NCL), the non-interruptible load (NIL), the interruptible load (IL), and the thermostatically controlled load (TCL) according to their different working characteristics. The main appliances that are scheduled in this paper can also be seen in Fig. 1. Based on the electrical equipment information and data received, the control center (CC) optimizes the scheduling result of the electrical equipment through its embedded method and sends out the control signal. In addition, the HEMS can perform two-way power interaction with the grid and can obtain the required forecast data, e.g., RTP and outdoor temperature, via the internet.

Objective function

In this paper, the objective is to minimize the electricity cost of users based on RTP. The objective function is formulated as

(1)\min {\displaystyle \underset{t=1}{\overset{T}{\mathrm{\Sigma }}}{\mathrm{price}}_t}{P}_{\mathrm{grid},t}\mathrm{\Delta }t,

(2)

where P_{grid,tis the interactive power between HEMS and the grid at time slot t, Δt is the time period of a single time slot, pricebuy,t and pricesell,t denote the price of buying and selling electricity at time slot t, respectively, and T denotes the total time slots.}

Constraints

The constraints of the optimization model are consisted by different kinds of schedulable load constraints and power balance constraints.

Constraints for NIL

NIL, e.g., washing machine (WM) and dishwasher (DW), can be scheduled between the time slot b_NIL and e_NIL. Once the device starts working with the rated power P_NIL,rate, it cannot be stopped until the working time slots meet the requirement L_NIL. Given the above, the constraints of NIL are presented as

(3a)P_{\mathrm{NIL,}\hspace{0.17em}t}=X_{\mathrm{NIL,}\hspace{0.17em}t}\cdot P_{\mathrm{NIL,}\hspace{0.17em}\mathrm{rate}},

(3b)X_{\mathrm{NIL,}\hspace{0.17em}t}=\{\begin{array}{l}\mathrm{0}or\mathrm{1}\begin{array}{ll}& \forall t\in [b_{\mathrm{NIL}},e_{\mathrm{NIL}}],\hspace{0.17em}t\in N^{+}\end{array}\\ \mathrm{0}\begin{array}{ll}\begin{array}{ll}& \end{array}& \forall t\in [\mathrm{1},b_{\mathrm{NIL}})\cup (e_{\mathrm{NIL}},T],\hspace{0.17em}t\in N^{+}\end{array}\end{array},

(3c){\displaystyle \underset{t=b_{\mathrm{NIL}}}{\overset{e_{\mathrm{NIL}}}{\mathrm{\Sigma }}}{X}_{\mathrm{NIL,}\hspace{0.17em}t}}=L_{\mathrm{NIL}},

(3d){\displaystyle \underset{t=a}{\overset{a+L_{\mathrm{NIL}}-\mathrm{1}}{\mathrm{\Sigma }}}{X}_{\mathrm{NIL,}\hspace{0.17em}t}=(X_{\mathrm{NIL,}\hspace{0.17em}t}-X_{\mathrm{NIL,}\hspace{0.17em}t-\mathrm{1}})\cdot L_{\mathrm{NIL}}},\forall a\in (b_{\mathrm{NIL}},e_{\mathrm{NIL}}-L_{\mathrm{NIL}}+\mathrm{1}],

where X_NIL,t and P_NIL,t denote the working status (0–off, 1–on) and the power at time slot t, respectively, and a represents a positive integer. Equation (3b) indicates that NIL can be worked in [b_NIL,e_NIL], which is specified by the user, and can only be off during other time slots. It is noted that among the constraints, Eq. (3d) describes the uninterruptible feature of NIL.

Constraints for IL

The characteristics of IL are similar to NIL whereas IL is not constrained by working continuously. The constraints of IL can be formulated as

(4a)P_{\mathrm{IL,}\hspace{0.17em}t}=X_{\mathrm{IL,}\hspace{0.17em}t}\cdot P_{\mathrm{IL,}\hspace{0.17em}\mathrm{rate}},

(4b)X_{\mathrm{IL,}\hspace{0.17em}t}=\{\begin{array}{l}\mathrm{0}\mathrm{o}\mathrm{r}\mathrm{1}\begin{array}{ll}& \forall t\in [b_{\mathrm{IL}},e_{\mathrm{IL}}],\hspace{0.17em}t\in N^{+}\end{array}\\ \mathrm{0}\begin{array}{ll}\begin{array}{ll}& \end{array}& \forall t\in [\mathrm{1},b_{\mathrm{IL}})\cup (e_{\mathrm{IL}},N],\hspace{0.17em}t\in N^{+}\end{array}\end{array},

(4c){\displaystyle \underset{t=b_{\mathrm{IL}}}{\overset{e_{\mathrm{IL}}}{\mathrm{\Sigma }}}{X}_{\mathrm{IL,}\hspace{0.17em}t}}=L_{\mathrm{IL}},

where X_IL,t and P_IL,t denote the working status of IL (0–off, 1–on) and the rated power at time slot t, respectively; P_IL,rate is the rated power; and L_IL is the total working time slots between the time slot b_IL and e_IL.

Constraints for TCL

TCL is interruptible and schedulable. In addition, its working status directly influences the user thermal comfort. Air conditioner (AC) and EWH are considered as TCLs in this paper. It is assumed that the power of AC and EWH are less than the upper limit P_AC,max and P_WH,max, respectively [³⁰].

Constraints for AC

Since the building has been conducting thermal exchange with the outdoor environment all the time, the uncertain variable outdoor temperature θ_out is included in the constraints for AC. The constraints for AC are formulated as

(5a){\theta }_{\mathrm{room,}\hspace{0.17em}t}={\theta }_{\mathrm{room,}\hspace{0.17em}t-1}\cdot e^{-\mathrm{\Delta }t/(R\cdot C)}+R\cdot P_{\mathrm{AC},\hspace{0.17em}t}\cdot (\mathrm{1}-e^{-\mathrm{\Delta }t/(R\cdot C)})+{\theta }_{\mathrm{out,}\hspace{0.17em}t}\cdot (\mathrm{1}-e^{-\mathrm{\Delta }t/(R\cdot C)}),\forall i\in \{\mathrm{1},\mathrm{...},T\},

(5b){\theta }_{\mathrm{room}}^{\min }\le {\theta }_{\mathrm{room,}\hspace{0.17em}t}\le {\theta }_{\mathrm{room}}^{\max },

(5c)\mathrm{0}\le P_{\mathrm{AC,}\hspace{0.17em}t}\le P_{\mathrm{AC,}\hspace{0.17em}\max },

where the constants R and C are the equivalent heat resistance and capacity. Equation (5a) represents the thermal dynamic equation of indoor temperature and indicates that the indoor temperature at time slot t denoted by θ_room,t is decided by the indoor temperature in the previous time slot θ_room,t–1, power P_AC,t and outdoor temperature θ_out,t [¹⁹]; θ_room,0represents the initial indoor temperature; and {\theta }_{\mathrm{room}}^{\min } and {\theta }_{\mathrm{room}}^{\max } denote the lower and upper boundaries of the preset comfort range of indoor temperature.

Constraints for EWH

The temperature of hot water is tightly associated with the working state of EWH. In addition, the uncertain variable hot water demand d_WH has a significant effect on hot water temperature. The operational constraints of EWH are represented as

(6a){\theta }_{\mathrm{water,}\hspace{0.17em}t}={\theta }_{\mathrm{water,}\hspace{0.17em}t-1}-{\beta }_{\mathrm{WH}}{d}_{\mathrm{WH,}\hspace{0.17em}t}+{\displaystyle \frac{\mathrm{\Delta }t}{M\cdot C_{\mathrm{p}}}}{P}_{\mathrm{WH,}\hspace{0.17em}t}-{\gamma }_{\mathrm{WH}},\forall i\in \{\mathrm{1},\mathrm{...},T\},

(6b){\theta }_{\mathrm{water}}^{\min }\le {\theta }_{\mathrm{water,}\hspace{0.17em}t}\le {\theta }_{\mathrm{water}}^{\max },

(6c)\mathrm{0}\le P_{\mathrm{WH,}\hspace{0.17em}t}\le P_{\mathrm{WH,}\hspace{0.17em}\max },

where C_p represents the specific heat capacity of water, γ_WH denotes the heat loss in a time slot which is related to outdoor temperature, β_WH is the change in hot water temperature for each kilogram of hot water used by the user, and M is the mass of the water in full tank. Similar to the constraints for AC, Eq. (6a) represents the thermal dynamic equation of hot water temperature where θ_water,t is the hot water temperature at time slot t which is decided by the hot water temperature in the previous time slot θ_water,t–1, power rate P_{WH,tand hot water demand dWH,t [³¹];θwater,0 represents the initial hot water temperature; {\theta }_{\mathrm{water}}^{\min } and {\theta }_{\mathrm{water}}^{\max } denote the lower and upper boundaries of the preset comfort range of hot water temperature, respectively.}

Constraints for power balance

The difference between the power obtained from PV and the total power consumption of all kinds of loads should be equal to the interactive power with the grid at each time slot. The constraint for power balance is presented as

(7)P_{\mathrm{grid,}\hspace{0.17em}t}+P_{\mathrm{PV,}\hspace{0.17em}t}=P_{\mathrm{WH,}\hspace{0.17em}t}+P_{\mathrm{AC,}\hspace{0.17em}t}+P_{\mathrm{IL,}\hspace{0.17em}t}+P_{\mathrm{NIL,}\hspace{0.17em}t}+P_{\mathrm{NCL,}\hspace{0.17em}t},

where P_PV,t and P_NCL,t denote the power of PV and NCLs at time slot t, respectively.

RHO-DRO method for solving uncertainties

The RHO-DRO method is proposed to solve the uncertainties of outdoor temperature and hot water demand contained in Eqs. (5a)–(5b) and Eqs. (6a)–(6b). The optimization model of HEMS described above is transformed into a MILP problem which is solved in a rolling fashion.

Model of uncertain variables

First, the uncertain variables are modeled as interval numbers with several levels based on forecast values. {\theta }_{\mathrm{out,}\hspace{0.17em}t}^f and d_{\mathrm{WH,}\hspace{0.17em}t}^f denote the forecast values of outdoor temperature and hot water demand, respectively. Next, the outdoor temperature is taken as an example to illustrate the model of the uncertain variable. The model of hot water demand is similar. The interval number of outdoor temperature at level i is denoted as {\theta }_{\mathrm{out,}\hspace{0.17em}t}^i=[L_{\theta ,\hspace{0.17em}t}^i,U_{\theta ,\hspace{0.17em}t}^i],\forall i=\mathrm{1},\mathrm{...},m; L_{\theta ,\hspace{0.17em}t}^i and U_{\theta ,t}^i denote the upper bound and the lower bound at time slot t, respectively; and [L_{\theta ,\hspace{0.17em}t}^m,U_{\theta ,\hspace{0.17em}t}^m] denotes the maximum interval while the bound value of subinterval are represented as

(8a)L_{\theta ,\hspace{0.17em}t}^{m-i}=L_{\theta ,\hspace{0.17em}t}^m+i\cdot {\displaystyle \frac{U_{\theta ,\hspace{0.17em}t}^m-L_{\theta ,\hspace{0.17em}t}^m}{\mathrm{2}m-\mathrm{1}}}i=\mathrm{1},\mathrm{2},\mathrm{...},m-\mathrm{1}\hspace{0.17em}\hspace{0.17em},

(8b)U_{\theta ,\hspace{0.17em}t}^{m-i}=U_{\theta ,\hspace{0.17em}t}^m-i\cdot {\displaystyle \frac{U_{\theta ,\hspace{0.17em}t}^m-L_{\theta ,\hspace{0.17em}t}^m}{\mathrm{2}m-\mathrm{1}}}i=\mathrm{1},\mathrm{2},\mathrm{...},m-\mathrm{1.}

The m, which denotes the total level, is 5 in this paper. The interval number of outdoor temperature with 5 levels is demonstrated in Fig. 2.

Based on sufficient history data, the ambiguity set of the probability distribution information of the interval number at each level is constructed as [³⁰]

(9){{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}}=\{{\mathbb{P}}_{\theta ,t}\in {{\displaystyle \mathcal{P}}}_{\theta ,t}^0\left({\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}^m\right)|{\mathbb{P}}_{\theta ,t}\begin{array}{c}\begin{array}{c}{E}_{{\mathbb{P}}_{\theta ,t}}\left\{{\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}\right\}={\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}^{\mathrm{f}}\\ {\mathbb{P}}_{\theta ,t}\{{\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}\in {\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}^i\}=p_{\theta ,t}^i\end{array}\\ p_{\theta ,t}^m=1\end{array}\},

where {{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}} denotes the ambiguity set of the probability distribution, p_{\theta ,\hspace{0.17em}t}^i denotes the probability of {\theta }_{\mathrm{out,}\hspace{0.17em}t}\in {\theta }_{\mathrm{out,}\hspace{0.17em}t}^i, {\mathbb{P}}_{\theta ,\hspace{0.17em}t} denotes the probability distribution of outdoor temperature, and {{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}}\left({\theta }_{\mathrm{o}\mathrm{u}\mathrm{t},t}^m\right) denotes the set of all the probability distributions supported on {\theta }_{\mathrm{out,}\hspace{0.17em}t}^m.

Similarly, the upper bound and the lower bound of hot water demand at level i are represented by U_{d,\hspace{0.17em}t}^i, L_{\mathrm{d,}\hspace{0.17em}t}^i, respectively; the interval number of hot water demand is denoted as d_{\mathrm{WH},\hspace{0.17em}t}^i=[L_{\mathrm{d,}\hspace{0.17em}t}^i,U_{\mathrm{d,}\hspace{0.17em}t}^i],\forall i=\mathrm{1},\mathrm{...},m; and the ambiguity set is expressed as

(10){{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}}=\{{\mathbb{P}}_{\mathrm{d},t}\in {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^0\left(d_{\mathrm{W}\mathrm{H},t}^m\right)|{\mathbb{P}}_{\mathrm{d},t}\begin{array}{c}\begin{array}{c}{E}_{{\mathbb{P}}_{\mathrm{d},t}}\left\{{\theta }_{\mathrm{W}\mathrm{H},t}\right\}={\theta }_{\mathrm{W}\mathrm{H},t}^{\mathrm{f}}\\ {\mathbb{P}}_{\mathrm{d},t}\{{\theta }_{\mathrm{W}\mathrm{H},t}\in {\theta }_{\mathrm{W}\mathrm{H},t}^i\}=p_{\mathrm{d},t}^i\end{array}\\ p_{\mathrm{d},t}^m=1\end{array}\},

where {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}} denotes the ambiguity set of the probability distribution, p_{d,\hspace{0.17em}t}^i denotes the probability of

d_{\mathrm{WH,}\hspace{0.17em}t}\in d_{{}_{\mathrm{WH,}\hspace{0.17em}\hspace{0.17em}t}}^i

, {\mathbb{P}}_{d,\hspace{0.17em}t} denotes the probability distribution of hot water demand, and {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}}\left(d_{\mathrm{W}\mathrm{H},t}^m\right) denotes the set of all the probability distributions supported on d_{{}_{\mathrm{WH,}\hspace{0.17em}t}}^m.

DRO model

The DRO model of HEMS is established based on the uncertain variable model described above and the scheduling problem of HEMS is transformed into a MILP.

First, the thermal comfortable constrains described in Eqs. (5b) and (6b) are developed into distributionally robust chance constraints as

(11a){{\displaystyle \mathbb{P}}}_{\theta ,t}\{{{\displaystyle \theta }}_{\mathrm{room}}^{\min }\le {{\displaystyle \theta }}_{\mathrm{room},t}\le {{\displaystyle \theta }}_{\mathrm{room}}^{\max }\}\ge \mathrm{1}-{{\displaystyle \epsilon }}_{\mathrm{room}}\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\theta ,t}\in {{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}},

(11b){{\displaystyle \mathbb{P}}}_{\mathrm{d},t}\{{{\displaystyle \theta }}_{\mathrm{water}}^{\min }\le {{\displaystyle \theta }}_{\mathrm{water},t}\le {{\displaystyle \theta }}_{\mathrm{water}}^{\max }\}\ge \mathrm{1}-{{\displaystyle \epsilon }}_{\mathrm{water}}\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\mathrm{d},t}\in {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}}.

Equations (11a) and (11b) represent that the probabilities of indoor temperature and hot water temperature satisfying the preset comfort range must be no less than 1–ε_room and 1–ε_water for all the probability distributions satisfying the ambiguity set of the probabilities distribution described in Eqs. (9) and (10), respectively; and ε_room and ε_water are small positive numbers.

Then the distributionally robust chance constraints can be simplified to Eqs. (12a)–(12b) and Eqs. (13a)–(13b) using the conditional value at risk (CVaR) proposed in Ref. [³²]. Taking Eq. (12a) as an example, {\mathrm{CVaR}}_{{\epsilon }_{\mathrm{room}}}is introduced based on value at risk (VaR); VaR is the value satisfying the probability that {\theta }_{\mathrm{room}}^{\min }-{\theta }_{\mathrm{room,}\hspace{0.17em}t} is above VaR by at most ε_room; and CVaR is defined as the mean of {\theta }_{\mathrm{room}}^{\min }-{\theta }_{\mathrm{room,}\hspace{0.17em}t} on the tail distribution exceeding VaR.

(12a){{\displaystyle \mathbb{P}}}_{\theta ,t}-{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{{ϵ}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m}}}\{{{\displaystyle \theta }}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m}}^{\min }-{{\displaystyle \theta }}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m},t}\}\le 0\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\theta ,t}\in {{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}},

(12b){{\displaystyle \mathbb{P}}}_{\theta ,t}-{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{{ϵ}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m}}}\{{{\displaystyle \theta }}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m},t}-{{\displaystyle \theta }}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{m}}^{\max }\}\le 0\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\theta ,t}\in {{\displaystyle \mathcal{P}}}_{\theta ,t}^{\mathrm{1}},

(13a){{\displaystyle \mathbb{P}}}_{\mathrm{d},t}-{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{{ϵ}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}\{{{\displaystyle \theta }}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}^{\min }-{{\displaystyle \theta }}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r},t}\}\le 0\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\mathrm{d},t}\in {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}},

(13b){{\displaystyle \mathbb{P}}}_{\mathrm{d},t}-{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{{ϵ}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}\{{{\displaystyle \theta }}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r},t}-{{\displaystyle \theta }}_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}^{\max }\}\le 0\hspace{1em}{{\displaystyle \forall {\displaystyle \mathbb{P}}}}_{\mathrm{d},t}\in {{\displaystyle \mathcal{P}}}_{\mathrm{d},t}^{\mathrm{1}}\mathrm{.}

Reference [³⁰] proved that the constraint Eqs. (12a)–(12b) and Eqs. (13a)–(13b) are satisfied if and only if there exist constants z_θ, β_θ, λ_θ,j , and z_d, β_d, λ_d,j such that

(14a){\beta }_{\theta }+{\displaystyle \frac{\mathrm{1}}{{\epsilon }_{\mathrm{room}}}}\cdot \left({\theta }_{\mathrm{out},t}^f\cdot z_{\theta }+{\displaystyle \underset{i=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,i}\cdot p_{\theta ,t}^i}\right)\le \mathrm{0},

\forall i=\mathrm{1},\mathrm{2},\cdot \cdot \cdot ,m,

(14b)z_{\theta }\cdot L_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}\ge \mathrm{0},

(14c)z_{\theta }\cdot U_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}\ge \mathrm{0},

(14d)z_{\theta }\cdot L_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}-({\theta }_{\mathrm{room}}^{\min }-{\theta }_{\mathrm{room,}t}(L_{\theta ,t}^i)-{\beta }_{\theta })\ge \mathrm{0},

(14e)z_{\theta }\cdot U_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}-({\theta }_{\mathrm{room}}^{\min }-{\theta }_{\mathrm{room,}t}(U_{\theta ,t}^i)-{\beta }_{\theta })\ge \mathrm{0},

(14f)z_{\theta }\cdot L_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}-({\theta }_{\mathrm{room,}t}(L_{\theta ,t}^i)-{\theta }_{\mathrm{room}}^{\max }-{\beta }_{\theta })\ge \mathrm{0},

(14g)z_{\theta }\cdot U_{\theta ,t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\theta ,j}}-({\theta }_{\mathrm{room,}t}(U_{\theta ,t}^i)-{\theta }_{\mathrm{room}}^{\max }-{\beta }_{\theta })\ge \mathrm{0.}

(15a){\beta }_{\mathrm{d}}+{\displaystyle \frac{\mathrm{1}}{{\epsilon }_{\mathrm{water}}}}\cdot \left(d_{\mathrm{WH,}t}^f\cdot z_{\mathrm{d}}+{\displaystyle \underset{i=\mathrm{1}}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},i}\cdot p_{\mathrm{d},t}^i}\right)\le \mathrm{0},

\forall i=\mathrm{1},\mathrm{2},\cdot \cdot \cdot ,m.

(15b)z_{\mathrm{d}}\cdot L_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}\ge \mathrm{0},

(15c)z_{\mathrm{d}}\cdot U_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}\ge \mathrm{0},

(15d)z_{\mathrm{d}}\cdot L_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}-({\theta }_{\mathrm{water}}^{\min }-{\theta }_{\mathrm{water,}t}(L_{\mathrm{d},t}^i)-{\beta }_{\mathrm{d}})\ge \mathrm{0},

(15e)z_{\mathrm{d}}\cdot U_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}-({\theta }_{\mathrm{water}}^{\min }-{\theta }_{\mathrm{water,}t}(U_{\mathrm{d},t}^i)-{\beta }_{\mathrm{d}})\ge \mathrm{0},

(15f)z_{\mathrm{d}}\cdot L_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}-({\theta }_{\mathrm{water,}t}(L_{\mathrm{d},t}^i)-{\theta }_{\mathrm{water}}^{\max }-{\beta }_{\mathrm{d}})\ge \mathrm{0},

(15g)z_{\mathrm{d}}\cdot U_{\mathrm{d},t}^i+{\displaystyle \underset{j=i}{\overset{m}{\mathrm{\Sigma }}}{\lambda }_{\mathrm{d},j}}-({\theta }_{\mathrm{water,}t}(U_{\mathrm{d},t}^i)-{\theta }_{\mathrm{water}}^{\max }-{\beta }_{\mathrm{d}})\ge \mathrm{0},

where {\theta }_{\mathrm{room,}t}(L_{\theta ,t}^i) and {\theta }_{\mathrm{room,}t}(U_{\theta ,t}^i) are the indoor temperature calculated by Eq. (5a) with the outdoor temperature equaling L_{\theta ,t}^i and U_{\theta ,t}^i, respectively. Similarly, {\theta }_{\mathrm{water,}t}(L_{\mathrm{d},t}^i) and {\theta }_{\mathrm{water,}t}(U_{\mathrm{d},t}^i) are the hot water temperature calculated by Eq. (6a) with the hot water demand equaling L_{\mathrm{d},t}^i and U_{\mathrm{d},t}^i, respectively.

Till then, the constrain Eqs. (5b) and (6b), which contain uncertain variables, are transformed into linear constraint Eqs. (14a–14g) and Eqs. (15a–15g), respectively. By replacing Eqs. (5b) and (6b) with linear constraint Eqs. (14a–14g) and Eqs. (15a–15g) and using the forecast values of outdoor temperature and hot water demand to calculate the indoor temperature and hot water temperature according to Eqs. (5a) and (6a), respectively, the optimization model of HEMS described in Section 2 is transformed into a optimization model of the MILP problem which can be easily solved.

Overall RHO-DRO method

RHO is a typical local optimization which just involves a subset of the whole time horizon, called optimization time domain corresponding to the T in Eq. (1).

The time frame of RHO is depicted in Fig. 3 [³³,³⁴] and the RHO-DRO method works as follows: at the beginning of any time slot n, the CC receives predicted data of the next optimization time slot T, which include the forecast data of hot water demand and outdoor temperature. Then, according to the history data of hot water demand and outdoor temperature, the ambiguity sets of probability distribution of hot water demand and outdoor temperature are constructed. Based on the above data, the scheduling results of the next optimization time domain can be obtained by solving the MILP model described in Section 3.2. The electrical equipment in HEMS will operate in time slot n according to the optimization results. After the end of time slot n, the forecast data of hot water demand and outdoor temperature for the next optimization time domain and real-time indoor temperature and hot water temperature will be updated. Repeat the above operation based on the updated data until the total optimization time slots required by the user is reached. The detailed flowchart of optimization time process is presented in Fig. 4 and the total optimization time slots is denoted by N.

Simulation results

Simulation design

A HEMS of a household in autumn or winter is chosen as an example to simulate on a certain day (from 0:00 to 24:00). The time length of each time slot Δt is set as 15 min and the total number of time slots in a certain day is equal to 96. Due to the application of RHO, when optimizing the scheduling results in the later time slots of a certain day, it is inevitable that the data of the next day will be need and it is assumed that it is the same as the data of the previous day. The forecast RTP shown in Fig. 5(a) is obtained from Illinois Electric Power Company [³⁵] and the price of selling electricity is assumed as half of the RTP. The power of PV at each slot according to the data from the solar power plant located in Ashland, Oregon with 5 kW capacity [36] is shown in Fig. 5(b), so is the power of NCLs at each time slot.

WM and DW are considered as NILs and electric vehicle (EV), while pool pump (PP) is considered as ILs. The data for ILs and NILs are given in Table 1, while the parameters of EWH and AC are listed in Table 2 [³⁵] and Table 3 [¹⁹]. The initial indoor temperature and hot water temperature of the first time slot are assumed as 22°C and 65°C, respectively. ε_room and ε_water in chance constrains of DRO are both 0.05.

The boundary values of maximum interval number and the data of the normal distributions, which are used to generate the 1000 history data of the outdoor temperature and hot water demand, respectively, to construct the ambiguity set of the distribution information, are tabulated in Table 4 where E_θ and E_d denote the expected values of normal distribution of outdoor temperature and hot water demand, respectively; and σ_θ and σ_d denote the variance values. The data of outdoor temperature and hot water demand are exhibited in Fig. 6 where the solid and dotted dark lines represent predicted values and the boundary values of maximum interval number, respectively. In addition, a set of test data shown in red line is to verify the performance of the scheduling results.

All the simulations are implemented in MATLAB with YALMIP as the modeling tool and CPLEX which can stably and efficiently solve MILP as the solver running on an Intel Core-i5 personal computer with 4 GB RAM.

Scheduling results of HEMS using RHO-DRO

To facilitate the discussion of the performance of the RHO-DRO method, three different scenarios are proposed.

Scenario 1: Scenario 1 uses the RHO-DRO method in the worst scenario of realistic outdoor temperature and hot water demand, i.e., the outdoor temperature being the lower boundary value and the hot water demand being the upper boundary value.

Scenario 2: Scenario 2 uses the RHO-DRO method in the best scenario of realistic outdoor temperature and hot water demand, i.e., the outdoor temperature being the upper boundary value and the hot water demand being the lower boundary value.

Scenario 3: Scenario 3 obtains the scheduling results by only implementing RHO with the forecast value of outdoor temperature and hot water demand but calculates the indoor temperature and hot water temperature in the worst scenario of realistic outdoor temperature and hot water demand.

It is noted that when the uncertainties are not considered and the operation of HEMS is optimized in the day ahead mode, it will have a serious adverse effect on the comfort constraints of users, which has been demonstrated in Ref. [¹⁸]. In addition, as mentioned previously, the data of the next day will be considered when obtaining the scheduling results of a certain day by using RHO, so the optimization results of day ahead optimization and RHO is not comparable. Therefore, the scenario of day ahead optimization without considering uncertainty is not set to compare the RHO-DRO method. The optimized time domain is set to 24 time slots in the three scenarios. The scheduling results of 3 scenarios are illustrated in Fig. 7.

It can be observed from Fig. 7 that NILs and ILs all operate within the specified working hours with the rated power while the power of AC and EWH satisfies the power limit at every time slot in all the 3 scenarios. In scenarios 1 and 3, the power consumed by the AC and EWH is obviously higher than that consumed in scenario 2. The reason for this is that in the worst scenario of realistic data, AC and EWH need to consume more power to satisfy the user’s temperature comfort constraints. The results of indoor temperature and hot water temperature are shown in Fig. 8.

It can be seen from Fig. 8 that in all three scenarios, the TCLs utilizes its own heat storage characteristics to minimize the electricity cost. During the time slots of lower electricity price, AC and EWH are operated to make the indoor temperature and the hot water temperature reach a high level, e.g., the time slots near the 20th and 64th time slot. In the time slots when the electricity price is higher, the indoor temperature and the hot water temperature are maintained at the lower limit set by the user to save electricity costs. In addition, while using the RHO-DRO method in scenarios 1 and 2, the indoor temperature and hot water temperature are basically within the range set by the user, and there is no obvious violation of user comfort constraints both in the worst and best scenario of realistic data. However, when only the RHO method is used in scenario 3, it can be seen from Fig. 7(c) that the indoor temperature and the hot water temperature are under the lower limit set by the user in several time slots, in the worst case of the realistic data. The electricity cost and the violation value obtained by adding all the violation values of temperature in all time slots are given in Table 5. It can be seen from Table 5, the electricity cost in scenarios 1 and 2 are much higher than that in scenario 3, because AC and EWH need more power to satisfy the comfort constrains as mentioned earlier. Since the indoor temperature and hot water temperature are maintained near the lower temperature limit in most of the time slots in order to minimize the electricity cost, the violation in scenario 1 is higher than that in scenario 2. The electricity cost in scenario 1, which uses the RHO-DRO approach, is only 0.606 cents more than that in scenario 3, which only uses RHO, whereas the violation of comfort constrains reduced from 12.7643°C in scenario 3 to 0.6745°C. It indicates that the RHO-DRO approach significantly reduce the comfort violation with little electricity cost compared with RHO.

Comparison of RHO-RO in different optimization time domains

The RHO-DRO method is compared with the RHO-RO method in different optimization time domains. Similar to the RHO-DRO method, first, RHO-RO establishes the RO model for HEMS, and then combines the model with RHO to solve the scheduling problem in a rolling fashion. The detailed formulation of the RO model is described in Appendix. In addition, the realistic data of outdoor temperature and hot water demand use the test value described in Figs. 6(a) and 6(b), respectively. The optimization results of electricity cost in different optimization time domains according to the test value described in Fig. 6 are shown in Fig. 9.

It can be seen from Fig. 9 that the electricity cost of RHO-DRO and RHO-RO appears to be almost the same when the optimization time domain is changed. When the optimization time domain T is between 4 and 24, the electricity cost gradually decreases with the increase in optimization time domains, whereas the electricity cost gradually increases and level off ultimately when T is between 24 and 48. Hence, there is a minimum electricity cost at T = 24. The reason for this is that when T is small, the future information is not sufficient enough and the variation in future data cannot be adequately considered with a short optimization horizon. However, with the increase in T, the uncertainties of outdoor temperature and hot water demand in more time slots should be considered so that the electricity cost is increased. Therefore, a larger T can obtain more decision information and prediction horizons, but more uncertainties need to be considered and a best scheduling result can be obtained when the optimization time domain is equal to 24.

In addition, the electricity cost of RHO-DRO is smaller than that of RHO-RO in any optimization time domain. Compared with RO, DRO relaxes the constraints of the absolute bounds of the interval number and relies on enough historical data to obtain the probability distribution ambiguity set based on the forecast data. Besides, the electricity cost is smaller since the optimization results are not too conservative.

Conclusions

In this paper, the optimization problem of HEMS with uncertain parameters is studied. First, a DRO model of HEMS is formulated to enhance the robustness of scheduling results with the forecast data in the next optimization time domain and the scheduling problem is developed to a MILP. After that, RHO is combined with the MILP to solve the MILP by CLPEX in a rolling fashion. The simulation results show that no matter it is in the best scenario or the worst scenario of realistic data of uncertain parameters, the violation of user’s comfort constrains is quite small when using the RHO-DRO approach. At the same time, compared with RO-DRO, RHO-DRO has less conservation. In addition, the effect of optimization time domain on scheduling results is analyzed and the best scheduling result is obtained when the optimization time domain is equal to 24.

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