A comprehensive energy solution for households employing a micro combined cooling, heating and power generation system

Abstract

In recent years, micro combined cooling, heating and power generation (mCCHP) systems have attracted much attention in the energy demand side sector. The input energy of a mCCHP system is natural gas, while the outputs include heating, cooling and electricity energy. The mCCHP system is deemed as a possible solution for households with multiple energy demands. Given this background, a mCCHP based comprehensive energy solution for households is proposed in this paper. First, the mathematical model of a home energy hub (HEH) is presented to describe the inputs, outputs, conversion and consumption process of multiple energies in households. Then, electrical loads and thermal demands are classified and modeled in detail, and the coordination and complementation between electricity and natural gas are studied. Afterwards, the concept of thermal comfort is introduced and a robust optimization model for HEH is developed considering electricity price uncertainties. Finally, a household using a mCCHP as the energy conversion device is studied. The simulation results show that the comprehensive energy solution proposed in this work can realize multiple kinds of energy supplies for households with the minimized total energy cost.

Keywords

energy hub / micro combined cooling / heating and power generation (mCCHP) / thermal comfort / robust optimization

Introduction

The third industrial revolution, with the energy internet as the core technology, promotes the emergence of multiple kinds of primary energies and end-use energies in a complex multi-energy system, and changes the pattern of energy utilization [¹]. In the energy supply side, natural gas network and electric grid are the most important parts of an energy internet. In the energy demand side, energy utilization technology makes energy management more flexible. An energy management system not only controls flexible loads, but also chooses alternative energy among electricity, gas and other forms of energies. In recent years, micro combined cooling, heating and power generation (mCCHP) systems have attracted much attention in the energy demand side sector.

An energy hub represents a model that combines the demand side and the supply side, describes the conversion among different forms of energies, the exchanges of energy among multiple energy networks, and the coupling relationship among electrical loads, thermal loads and other kind of loads [²–³]. In an energy hub, energies can be consumed by loads, or transformed to other forms of energies. Residential energy consumption usually accounts for a great proportion in the energy demand side [⁴]. Households usually purchase electricity from the distribution system concerned, gas from the connected natural gas system, and transform these two kinds of energies though energy conversion devices so as to meet their own cooling/heating and electricity demands. Thus, the mCCHP can be a comprehensive energy solution for households with multiple kinds of energy demands. A comprehensive energy solution for heat, electricity and natural gas markets based on a network-connected CHP is established in Refs. [⁵–⁶]. In Ref. [⁵], different load conditions are considered, while in Ref. [6] the heat to electricity ratio of a CHP is assumed to vary with different load conditions. However, thermal comfort is not considered in Refs. [⁵–⁶].

A mCCHP based household can be regarded as an energy hub, that is, the home energy hub (HEH). Some publications on HEH are available. An appliance commitment algorithm for scheduling household loads is presented in Ref. [⁷], where electricity is the only input energy of each household. A direct load control scheme for residential demand response is established in Ref. [⁸] to alleviate the mismatch between the actual demand and the desired demand. An optimal scheduling of smart home power consumption is addressed in Ref. [⁹] to reduce the peak demand and minimize energy consumption cost. An electrical model considering household occupancy and customer preference is employed in developing the HEH model in Ref. [¹⁰]. However, heating and cooling demands are not considered in Refs. [⁷–¹⁰]. A micro combined heating and power generation (mCHP) system based HEH model is presented in Ref. [¹¹]. A mCHP-based residential building model is developed in Ref. [¹²] to coordinate thermal and electric loads, but cooling demands, temperature-controlling electrical loads and convertible loads between electricity and gas are not considered. In Ref. [¹³], a residential energy hub model is proposed to supply heating, cooling and electrical demands, without considering thermal comfort.

The mCCHP is a distributed generation system developed by the mCHP. By adding chillers to a mCHP, the mCCHP can meet the cooling/heating and electrical demands of single/multiple households or small office buildings [¹⁴], and provide a comprehensive energy solution for households. Compared to the way of supplying heating and electricity energy separately, a mCCHP can produce demanded forms of energies for households, and the comprehensive energy solution employing a mCCHP could avoid energy losses from long-distance transmission and attain high efficiency [¹⁵].The majority of electricity that the mCCHP generates is consumed by the HEH, while the residual part can be sold to the distribution system concerned so as to get some revenue. Besides, the comprehensive energy solution employing a mCCHP can provide a comfortable environment for households with less money spent.

Given the above background, a comprehensive energy solution for households is proposed in this paper. The mathematical model of a HEH is first presented, with the mCCHP as an energy conversion device. Then, the mathematical model of electrical loads and thermal loads in a HEH are developed. Specifically, electrical loads are classified into inflexible and flexible loads while thermal loads are classified into hot water loads, air heating/cooling loads and convertible loads between electricity and gas. Afterwards, the concept of thermal comfort is introduced and employed to control the temperature in the HEH, based on which, a robust optimization model for HEH is developed considering thermal comfort and uncertainties of electricity prices and formulated as a mixed integer linear programming problem with minimizing the total energy costs as the objective, and next solved by the CPLEX solver based on the AMPL platform. Finally, a typical winter day and a typical summer day of a household are employed to demonstrate the essential characteristics of the developed method.

Framework of a HEH

A HEH describes the inputs, outputs, conversion, and consumption process of multiple forms of energy in a household. In this work, it is assumed that a HEH is connected with a distribution system and a gas network. The inputs of the HEH are electricity and natural gas, while the energy conversion device is mCCHP, including some devices like the micro gas turbine, a heat recovery device, absorption chiller, electric chiller, storage water tank and heat exchanger. The energy management system receives energy prices and temperature information, and schedules the HEH operation, as shown in Fig. 1.

A mCCHP is supposed to work with the priority of supplying heating energy, and the remaining capability for providing electricity in this work. The mCCHP outputs heating power to meet the hot water demand and air cooling/heating demand, and the redundant amount is stored in a heat storage device (a hot water storage tank is regarded as the heat storage device in this work). The HEH purchases electricity from the associate distribution system when the electrical demand exceeds the maximum output of the mCCHP, and sells the redundant electricity to the distribution system when the electricity generated by the mCCHP exceeds the demand.

The model developed in this work is based on the discrete-event system. The operation cycle is divided into T stages, and the operation state in each period is assumed to remain unchanged. Therefore, the operation parameters at the instant t can be used to describe the state in the time interval (t–1, t).

Mathematical models of electrical demands in a HEH

Generally, the heating/cooling demands and the flexible part of electrical loads are time-adjustable in a HEH [5]. The electrical loads include a variety of electrical appliances, the various characteristics of which need to be analyzed before optimizing commitment strategies.

Flexible loads are transferable and usually sensitive to electricity prices. In a certain range, the working time of flexible loads can be reasonably scheduled, but each appliance concerned should be continuously operated in specified consecutive periods. The flexible loads in a household generally refer to the cleaner, dishwasher, washer, dryer, iron and others. The mathematical model of flexible loads can be described as

(1)[u_i(t-\mathrm{1})-u_i(t)]\times M_i\le {\displaystyle \sum _{k=\mathrm{1}}^{M_i}{u}_i(t-k)}, u_i\in (\mathrm{0},\mathrm{1}), t\in (E_i,L_i),

where E_i and L_i are respectively the starting and end time of the schedulable periods of inflexible load i, M_i is the minimum operating time of inflexible load i, and u_i(t) is the On/Off status of inflexible load i at time t; when u_i(t) =1/0, inflexible load i starts/stops working.

Mathematical models of thermal loads in a HEH

A mCCHP heats the water in the storage tank by burning natural gas and maintains the water temperature at a certain level to ensure hot water supply at any time. When a certain amount of hot water is consumed, the same amount of cold water is supposed to be injected to the tank accordingly. By applying the second law of thermodynamics, the water temperature can be calculated by [¹³]

(2)\begin{array}{l}{\theta }_{t+\mathrm{1}}^{\mathrm{w}\mathrm{s}}=[{\rho }_{\mathrm{w}}{v}_t^{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}}({\theta }_{\mathrm{c}\mathrm{w}}-{\theta }_t^{\mathrm{w}\mathrm{s}})+{\rho }_{\mathrm{w}}{V}_{\mathrm{w}}{\theta }_t^{\mathrm{w}\mathrm{s}}]/{\rho }_{\mathrm{w}}{V}_{\mathrm{w}}\\ +h_t^{\mathrm{w}\mathrm{s}}\cdot \mathrm{\Delta }T/{\rho }_{\mathrm{w}}{V}_{\mathrm{w}}{C}_{\mathrm{w}},\end{array}

where {\theta }_t^{\mathrm{w}\mathrm{s}} and v_t^{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}} are respectively the water temperature in the storage water tank and the cold water volume injected at time t, h_t^{\mathrm{w}\mathrm{s}} is the heating power that the mCCHP provides to the water storage tank at time t, r_w, and C_w are respectively the density and specific heat capacity of water, and V_w is the capacity of the water storage tank, q_cw is the temperature of the cold water injected to the tank. The first part on the right hand side of Eq. (2) represents the balancing temperature after cold water is injected to the water storage tank, while the second part is the changed value of the water temperature when the mCCHP provides the water storage tank with h_t^{\mathrm{w}\mathrm{s}}\cdot \mathrm{\Delta }Tamount of heat.

The water temperature in the water storage tank needs to be maintained within a range and can be described by

(3){\theta }_{\min }^{\mathrm{w}\mathrm{s}}\le {\theta }_t^{\mathrm{w}\mathrm{s}}\le {\theta }_{\max }^{\mathrm{w}\mathrm{s}}.

The indoor temperature should be controlled according to the comfort level of the users. When supplying heating load, in order to maintain the indoor temperature at a comfortable level, the mCCHP provides the indoor air with h_t^{\mathrm{w}\mathrm{a}}\cdot \mathrm{\Delta }Tamount of heat, and the indoor temperature at time t+1 in winter can be attained by [¹⁶]

(4){\theta }_{t+\mathrm{1}}^{\mathrm{i}\mathrm{n}}={\theta }_t^{\mathrm{i}\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{\Delta }T}{R{C}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}+(R\cdot h_t^{\mathrm{w}\mathrm{a}}\cdot \mathrm{\Delta }T+{\theta }_t^{\mathrm{o}\mathrm{u}\mathrm{t}})\cdot (\mathrm{1}-{\mathrm{e}}^{-\frac{\mathrm{\Delta }T}{R{C}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}),

where {\theta }_t^{\mathrm{i}\mathrm{n}} and {\theta }_t^{\mathrm{o}\mathrm{u}\mathrm{t}} are respectively the indoor and outdoor temperature at time t, R is the thermal resistance of building materials of the household, C_air is the specific heat capacity of the air, and h_t^{\mathrm{w}\mathrm{a}} is the heating power that the mCCHP provides to the indoor air.

When supplying the cooling load, the chillers absorb h_t^{\mathrm{c}\mathrm{a}}\cdot \mathrm{\Delta }Tamount of heat from the indoor air. The motive power of the absorption chiller comes from the mCCHP by burning gas (“motive power” is also called “heating power”), while the motive power of the electric chiller comes from electricity. The indoor temperature at time t+1 in summer can be attained by

(5){\theta }_{t+\mathrm{1}}^{\mathrm{i}\mathrm{n}}={\theta }_t^{\mathrm{i}\mathrm{n}}{\mathrm{e}}^{-\frac{\mathrm{\Delta }t}{R{C}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}+(-R\cdot h_t^{\mathrm{c}\mathrm{a}}\cdot \mathrm{\Delta }T+{\theta }_t^{\mathrm{o}\mathrm{u}\mathrm{t}})\cdot (\mathrm{1}-{\mathrm{e}}^{\frac{\mathrm{\Delta }t}{R{C}_{\mathrm{a}\mathrm{i}\mathrm{r}}}}),

where h_t^{\mathrm{c}\mathrm{a}} is the cooling power from the chillers in the mCCHP. The cooling and heating functions of the mCCHP cannot be acting at the same time. h_t^{\mathrm{c}\mathrm{a}} and h_t^{\mathrm{w}\mathrm{a}} should respect the following constraint.

(6)h_t^{\mathrm{w}\mathrm{a}}\cdot h_t^{\mathrm{c}\mathrm{a}}=\mathrm{0.}

The functions of electricity generation and air heating/cooling are integrated in the mCCHP. The electricity and heating power generated by the mCCHP should respect the following constraints:

(7)p_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}={\eta }_{\mathrm{e}}{\kappa }_{\mathrm{g}\mathrm{a}\mathrm{s}}{f}_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}},

(8)h_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}={\eta }_{\mathrm{h}}{\kappa }_{\mathrm{g}\mathrm{a}\mathrm{s}}{f}_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}},

where p_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}} and h_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}} are respectively the electricity and heating power from the mCCHP at time t,f_t^{CCHP} is the quantity (expressed by cubic meters) of natural gas consumed by the mCCHP at time t, h_e and h_h are respectively the efficiencies of natural gas converting to electricity and thermal energy, k_gas is the calorific value of natural gas, and DT is the time step.

For a HEH, a part of its load can be regarded as convertible between electricity and gas, because the energy management system in the HEH can choose to burn natural gas in the mCCHP to produce electricity or buy electricity from the distribution system concerned. The energy costs in the HEH at time t can be attained by

(9)\begin{array}{l}{C}_t={\lambda }_t^{\mathrm{g}}\cdot f_t^{\mathrm{CCHP}}\mathrm{\Delta }T+{\lambda }_t^{\mathrm{e}}\cdot p_t^{\mathrm{e}}\cdot \mathrm{\Delta }T\\ ={\lambda }_t^{\mathrm{g}}\cdot (h_t^{\mathrm{CCHP}}+p_t^{\mathrm{CCHP}})\cdot \mathrm{\Delta }T/{\kappa }_{\mathrm{gas}}+{\lambda }_t^{\mathrm{e}}\cdot p_t^{\mathrm{e}} \cdot \mathrm{\Delta }T \\ ={\lambda }_t^{\mathrm{g}}\cdot [({\eta }_{\mathrm{h}}/{\eta }_{\mathrm{e}})+\mathrm{1}]\cdot p_t^{\mathrm{CCHP}}\cdot \mathrm{\Delta }T/{\kappa }_{\mathrm{gas}}+{\lambda }_t^{\mathrm{e}}\cdot p_t^{\mathrm{e}}\cdot \mathrm{\Delta }T \\ ={\lambda }_t^{\mathrm{g}}\cdot [({\eta }_{\mathrm{h}}/{\eta }_{\mathrm{e}})+\mathrm{1}]\cdot p_t^{\mathrm{CCHP}}\cdot \mathrm{\Delta }T/{\kappa }_{\mathrm{gas}}\\ +{\lambda }_t^{\mathrm{e}}\cdot (e_t+p_t^{\mathrm{EC}}-p_t^{\mathrm{CCHP}})\cdot \mathrm{\Delta }T,\end{array}

where {\lambda }_t^{\mathrm{g}} and {\lambda }_t^{\mathrm{e}} are respectively the natural gas price and electricity price at time t, p_t^{\mathrm{e}} is the electricity amount purchased from the distribution system during period t, and e_t is the total electrical demand at time t.

The first part on the right hand side of Eq. (9) represents the natural gas cost and the second part is the electricity cost.

Taking the derivative of C_t with respect to p_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}} in Eq. (9) yields

(10)\mathrm{d}{C}_t/\mathrm{d}{p}_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}=({\lambda }_t^{\mathrm{g}}/{\eta }_{\mathrm{e}}{\kappa }_{\mathrm{g}\mathrm{a}\mathrm{s}}+{\lambda }_t^{\mathrm{g}}{\eta }_{\mathrm{h}}/{\eta }_{\mathrm{e}}{\kappa }_{\mathrm{g}\mathrm{a}\mathrm{s}})-{\lambda }_t^{\mathrm{e}}\cdot \mathrm{\Delta }T.

The first part on the right hand side of Eq. (10) is the per unit cost of generating electricity by the mCCHP, and the second part is the per unit cost of purchasing electricity from the distribution system.

When , generating electricity by burning natural gas will reduce the energy cost, because the unit cost of generating electricity by the mCCHP is lower than that of purchasing electricity from the distribution system. The energy management system dispatches the mCCHP to generate more electricity not only to meet the electrical demand, but also to sell the redundant amount to the distribution system for gaining revenue. In this case, the HEH can be regarded as a gas load.

When \mathrm{d}{C}_t/\mathrm{d}{p}_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}\ge \mathrm{0}, generating electricity by burning natural gas will increase the energy cost, because the unit cost of generating electricity by the mCCHP is higher than that of purchasing electricity from the distribution system. The energy management system dispatches the mCCHP to generate less electricity, and the shortage is purchased from the distribution system. In this situation, the HEH can be regarded as an electrical-gas load.

In addition, the chillers in the mCCHP include two parts, one being the electric chiller and the other, the absorption chiller. The electric chiller refrigerates by driving the electric compressor, whose performance coefficient is higher than that of the low-grade heat driven absorption chiller [17]. The unit cooling cost of the absorption chiller and the electric chiller can be formulated as

(11)c_{\mathrm{A}\mathrm{C}}=(\mathrm{\Delta }Q\cdot {\lambda }_t^{\mathrm{g}})/(Z_{\mathrm{A}\mathrm{C}}\cdot {\kappa }_{\mathrm{g}\mathrm{a}\mathrm{s}}),

(12)c_{\mathrm{E}\mathrm{C}}=(\mathrm{\Delta }Q\cdot {\lambda }_t^{\mathrm{e}})/Z_{\mathrm{E}\mathrm{C}},

where Z_AC and Z_EC are respectively the performance coefficients of the absorption chiller and electric chiller, and DQ is the unit cooling power.

When the electricity price is low, the energy management system dispatches the electric chiller prior to the absorption chiller, because the performance coefficient of the electric chiller is higher than that of the absorption chiller. In other words, the unit cooling cost of the electric chiller is lower than that of the absorption chiller. The absorption chiller will be dispatched when the cooling power from the electric chiller is insufficient. When the electricity price is high, the energy management system dispatches the mCCHP to generate more electricity and more heating power accordingly, thus the energy management system dispatches the absorption chiller prior to the electric chiller. Therefore, the cooling demand in the HEH can be regarded as a convertible load between electricity and gas.

Modeling thermal comfort

The energy management system needs to provide a comfortable indoor environment while minimizing the total energy costs. The external factors having impacts on the human body comfort include the ambient temperature, relative humidity, air velocity, and mean radiant temperature. The internal factors include the age, gender, physical strength, personal activity intensity and strength, clothing thickness and others [¹⁸]. In order to quantify the influence of the temperature on thermal comfort, the Fanger thermal comfort equations are developed and the PMV (predicted mean vote scale) indexes established to approximately estimate the thermal comfort in Ref. [16]. The PMV can be calculated by Eqs. (13)–(17) with the above factors taken into account. PMV= 0 means that the indoor thermal environment is at the most comfortable condition. The relationship between the PMV and thermal comfort is shown in Table 1.

(13)$$\begin{gathered} \mathrm{P}\mathrm{M}\mathrm{V}=(0.028+0.303{\mathrm{e}}^{-0.036M})\times \\ \{M-W-3.05\times {10}^{-3}\times [5733-6.99(M-W)-P_{\mathrm{a}}] \\ -0.42(M-W-58.15)-1.7\times {10}^{-5}M(5867-P_{\mathrm{a}}) \\ -0.0014M(34-{\mathit{\text{t}}}_{\mathrm{a}})-3.96\times {10}^{-8}{\mathit{\text{f}}}_{\mathrm{c}\mathrm{l}} \\ [{({\mathit{\text{t}}}_{\mathrm{c}\mathrm{l}}+273)}^4-{({\mathit{\text{t}}}_{\mathrm{r}}+273)}^4]-{\mathit{\text{f}}}_{\mathrm{c}\mathrm{l}}{h}_{\mathrm{c}}({\mathit{\text{t}}}_{\mathrm{c}\mathrm{l}}-{\mathit{\text{t}}}_{\mathrm{a}})\}, \end{gathered}$$

(14)f_{\mathrm{c}\mathrm{l}}=\{\begin{array}{l}\mathrm{1.00}+\mathrm{1.290}{I}_{\mathrm{c}\mathrm{l}}, I_{\mathrm{c}\mathrm{l}}\le \mathrm{0.078}\\ \mathrm{1.05}+\mathrm{0.645}{I}_{\mathrm{c}\mathrm{l}}, I_{\mathrm{c}\mathrm{l}}>\mathrm{0.078} \end{array},

(15)$$\begin{gathered} t_{\mathrm{c}\mathrm{l}}=35.7-0.028(M-W) \\ -{\mathit{\text{I}}}_{\mathrm{c}\mathrm{l}}\{3.96\times {10}^{-8}{\mathit{\text{f}}}_{\mathrm{c}\mathrm{l}}[{({\mathit{\text{t}}}_{\mathrm{c}\mathrm{l}}+273)}^4-{({\mathit{\text{t}}}_{\mathrm{r}}+273)}^4] \\ +{\mathit{\text{f}}}_{\mathrm{c}\mathrm{l}}{h}_{\mathrm{c}}({\mathit{\text{t}}}_{\mathrm{c}\mathrm{l}}-{\mathit{\text{t}}}_{\mathrm{a}})\}, \end{gathered}$$

(16)

(17)P_{\mathrm{a}}=\phi \times \exp [\mathrm{16.653}-\mathrm{4030.183}/(t_{\mathrm{a}}+\mathrm{235})],

where W is the rate of the mechanical work accomplished, M is the rate of metabolic heat production, t_a is the air temperature, P_a is the water vapour pressure, f_cl is the ratio of the clothed surface area over the DuBois surface area, t_cl is the average surface temperature of the clothed body, t_r is the mean radiant temperature, h_c is the convection heat transfer coefficient, I_cl is the effective thermal resistance of clothing, V_a is the air velocity, and j is the air relative humidity.

It is difficult to attain PMV by solving Eqs. (13)–(17), since these equations are highly nonlinear and very complicated. Of the factors stated above, the most intuitive feeling of people on indoor environment is the temperature [¹⁹]. Therefore, this paper focuses on the study of the impact of temperature on human bodies and the concept of thermal comfort is introduced to describe the satisfaction degree of the user to the external environment.

In Ref. [¹⁹], PMVs are calculated for different air temperatures t_a, by applying the single variable method, i.e., only one variable (t_a) is applied each time, given that M = 69.84 W/m², I_cl = 0.08 (m²∙K)/W, V_a = 0.25 m/s, t_r = t_a, and W = 0. The relationship between PMV and indoor air temperature can be formulated as

(18)

The minimum PMV is attained with the indoor temperature at 26°C. In ISO 7730 [²⁰], it is stated that PMV should be between –0.5 and 0.5, and hence the comfortable temperature interval is [24.8°C, 27.3°C].

Besides, people usually feel more comfortable when the indoor temperature and hot water temperature change slowly than the case when the temperature changes abruptly. This could be imposed with the following constraints:

(19)\left|{\theta }_{t+\mathrm{1}}^{\mathrm{i}\mathrm{n}}-{\theta }_t^{\mathrm{i}\mathrm{n}}\right|\le \mathrm{\Delta }{\theta }^{\mathrm{i}\mathrm{n}},

(20)\left|{\theta }_{t+\mathrm{1}}^{\mathrm{w}\mathrm{s}}-{\theta }_t^{\mathrm{w}\mathrm{s}}\right|\le \mathrm{\Delta }{\theta }^{\mathrm{w}\mathrm{s}},

where Dqⁱⁿ and Dq^ws are the given upper limits of the air and water temperature variations between t and t+1.

Optimal operating model of HEH considering thermal comfort

The HEH optimizes its operation with an objective of minimizing the total energy costs as formulated by

(21)\min C={\displaystyle \sum _{t=\mathrm{1}}^T[{\lambda }_t^{\mathrm{g}}{f}_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}\mathrm{\Delta }T+{\lambda }_t^{\mathrm{e}}{p}_t^{\mathrm{e}}\mathrm{\Delta }T]},

where T is the number of time slots and p_t^{\mathrm{e}} is the electricity purchased from or sold to the distribution system at time t when p_t^{\mathrm{e}} is positive or negative. The second part on the right hand of Eq. (21) represents the revenue from selling the redundant electricity to the distribution system.

Constraints

(1) Power balance in the HEH

(22)p_t^{\mathrm{e}}+p_t^{\mathrm{C}\mathrm{C}\mathrm{H}\mathrm{P}}=d_t+p_t^{\mathrm{E}\mathrm{C}},

where p_t^{\mathrm{E}\mathrm{C}} is the output heating power of the electric chiller at time t and d_t is the sum of electrical demands except the EV.

(2) Operating constraints of the mCCHP

When the mCCHP supplies heating, the heating power h_t^{\mathrm{ws}} and h_t^{\mathrm{wa}} from the mCCHP should respect the following constraint:

(23)h_t^{\mathrm{ws}}+h_t^{\mathrm{wa}}=h_t^{\mathrm{CCHP}}.

When the mCCHP supplies cooling, the balances inside the chillers and the mCCHP are formulated as

(24)h_t^{\mathrm{AC}}\cdot Z_{\mathrm{AC}}+p_t^{\mathrm{EC}}\cdot Z_{\mathrm{EC}}=h_t^{\mathrm{ca}},

(25)h_t^{\mathrm{ws}}+h_t^{\mathrm{AC}}=h_t^{\mathrm{CCHP}},

where h_t^{\mathrm{AC}} and p_t^{\mathrm{EC}} are respectively the output heating power by the absorption chiller and electric chiller.

The indoor temperature needs to be within a comfortable temperature interval, as defined by

(26){\theta }_{\min }^{\mathrm{in}}\le {\theta }_t^{\mathrm{in}}\le {\theta }_{\max }^{\mathrm{in}}.

The output limitations of the mCCHP are formulated as

(27)h_{\min }^{\mathrm{CCHP}}\le h_t^{\mathrm{CCHP}}\le h_{\max }^{\mathrm{CCHP}},

(28)p_{\min }^{\mathrm{CCHP}}\le p_t^{\mathrm{CCHP}}\le p_{\max }^{\mathrm{CCHP}},

where h_{\max }^{\mathrm{CCHP}} and h_{\min }^{\mathrm{CCHP}} are respectively the upper and lower bounds of heating power from the mCCHP while p_{\max }^{\mathrm{CCHP}} and p_{\min }^{\mathrm{CCHP}} are respectively the upper and lower bounds of electric power from the mCCHP.

Robust optimization model for HEH considering the uncertainties of electricity prices

The optimal operating model of HEH presented in Section 2.4 is based on the assumption that the electricity price is precisely known. In fact, since the real-time electricity price cannot be accurately predicted in advance, it is assumed that the actual value of the day-ahead electricity price is within a certain range around the predicted value, defined as

(29)\overline{{\lambda }_t^{\mathrm{e}}}-\epsilon \overline{{\lambda }_t^{\mathrm{e}}}\le {\lambda }_t^{\mathrm{e}}\le \overline{{\lambda }_t^{\mathrm{e}}}+\epsilon \overline{{\lambda }_t^{\mathrm{e}}} \forall t\in [\mathrm{1},T],

where \overline{{\lambda }_t^{\mathrm{e}}} denotes the predicted value while e represents the accuracy degree of the prediction.

Since it is difficult to attain the probability distribution of the electricity price, the well-established robust optimization theory [21] is a good choice to address this problem. In the objective function of the optimal operating model of HEH presented in Section 2.4, there are uncertain parameters. The corresponding robust optimization model could be obtained by employing the linear transformation and dual transformation, as detailed below.

(30)\mathrm{Min} w,

(31)\mathrm{s}.\mathrm{t}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{t=\mathrm{1}}^T{\lambda }_t^{\mathrm{g}}{f}_t^{\mathrm{CCHP}}\mathrm{\Delta }T+}{\displaystyle \sum _{t=\mathrm{1}}^T\overline{{\lambda }_t^{\mathrm{e}}}{p}_t^{\mathrm{e}}\mathrm{\Delta }T}+\Gamma z+{\displaystyle \sum _{t=\mathrm{1}}^{T}q(t)\le w},

(32)\mathrm{s}.\mathrm{t}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}z+q(t)\ge [\epsilon \overline{{\lambda }_t^{\mathrm{e}}}\mathrm{\Delta }T]y(t),

(33)\mathrm{s}.\mathrm{t}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-y(t)\le p_t^{\mathrm{e}}\le y(t),

(34)\mathrm{s}.\mathrm{t}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}z,q(t)\ge \mathrm{0},

\mathrm{s}.\mathrm{t}.\hspace{0.17em}\hspace{0.17em}\mathrm{Constraints}(1)–(\mathrm{29}).

This robust optimization model is a linear programming problem, in which w represents the optimization objective; G denotes the robust control coefficient; and z, q(t), and y(t) are the auxiliary variables introduced by dual transformation.

It is conceivable that as the real-time electricity price may take values different from the predicted ones [22], the constraints with uncertain parameters, and in this work Eq. (31), may be violated. In order to make the probability of violating constraint (31) (with uncertain parameters) lower than e, Ref. [²³] shows that the robust control coefficient needs to respect the following constraint:

(35)\Gamma \ge \mathrm{1}+{\Phi }^{-\mathrm{1}}(\mathrm{1}-\epsilon )\sqrt{n},

where n is the number of uncertainty parameters.

Case study

A typical winter day and a typical summer day of a household are employed to demonstrate the essential characteristics of the proposed method. The time-of-use tariffs, hot water demands, and demand curve of inflexible electricity loads are demonstrated in Figs. 2, 3 and 4 respectively, and the natural gas price is approximately $0.36/m³. The parameters of the flexible electricity loads and the mCCHP are listed in Tables 2 and 3 respectively. The parameters of the thermal resistance, temperature, specific heat capacity, water storage volume, and water density are presented in Table 4. The CPLEX solver based on the AMPL platform is used to solve the optimization model developed.

The energy management system optimizes its operation with an objective of minimizing the total energy costs. Two scenarios are examined:

Scenario 1: the energy management is implemented without considering the objective of minimizing total energy costs.

The energy management is implemented by controlling the indoor temperature at 26°C, the water temperature in the storage water tank at 45°C, and operating the flexible appliances randomly. The costs in a typical winter day and a typical summer day are respectively $3.352 and $2.664.

Scenario 2: the energy management is implemented considering the objective of minimizing total energy costs and a robust model included with electricity price uncertainty taken into account.

The energy management is implemented with the objective of minimizing total energy costs taking into account. When the electricity price uncertainty is not considered, the temperature control results, storage water tank heating power, air heating power, and output heating power of the mCCHP in a typical winter day are depicted in Fig. 5.

It can be noticed from Fig. 5 that the heating power from the mCCHP is closely related to the electricity price. As presented in Section 2.2, the HEH can be regarded as convertible loads between electricity and gas. When t∈[1, 7] and t∈[21, 24], the unit cost of generating electricity by the mCCHP is higher than that of purchasing electricity from the distribution system. Therefore, the energy management system dispatches the mCCHP to output less electricity and less heating power accordingly. Thus, when t∈[1, 7] and t∈[21, 24], the air heating power and water heating power from the mCCHP only maintain the indoor temperature and water temperature at a low level. The mCCHP does not work at the rated power, because increasing temperature means outputting more heating power and electricity correspondingly, and it is not economical to generate more electricity during these periods. But when t∈[8, 20], the unit cost of generating electricity by the mCCHP is lower than the cost of purchasing electricity from the distribution system. The home energy system dispatches the mCCHP to generate more electricity, and outputs more heating power accordingly. Therefore, the indoor temperature and hot water temperature increase until the allowable maximal temperatures are reached. Besides, at the end of the peak electricity price period, the indoor temperature and hot water temperature reach high levels (large values) in the comfortable temperature interval. In this process, the air and water play a role of energy storage, and the mCCHP outputs less electricity at flat and valley periods (e.g. t∈[1, 7], t∈[21, 24]) to avoid transmitting extra electricity to the distribution system.

Without considering uncertainties of the electricity price, the temperature control results, storage water tank heating power, air cooling power, and output heating power of the mCCHP in a typical summer day are shown in Fig. 6.

As analyzed in Section 2.2, the cooling demand can be regarded as a convertible load between electricity and gas. The cooling power is provided by the absorption chiller and the electric chiller. When t∈[8, 20], the unit cost of generating electricity by the mCCHP is lower than the cost of purchasing electricity from the distribution system. The home energy management system dispatches the mCCHP to generate more electricity, and outputs more heating power accordingly. The home energy management system dispatches the absorption chiller prior to electric chiller. During this period, the indoor temperature decreases and the hot water temperature increases until the allowable maximal temperature is reached, in order to output less heating power during other periods. When t∈[1, 7] and t∈[21, 24], the unit cost of generating electricity by the mCCHP is higher than that of purchasing electricity from the distribution system. Therefore, the home energy management system dispatches the mCCHP to output less electricity and less heating power accordingly, and dispatches the electric chiller prior to the absorption chiller. During these periods, the electricity from the distribution system is cheaper and, hence, the mCCHP outputs as less electricity and heating power as possible.

To consider the electricity price uncertainty, the robust optimization model developed is employed. By changing the robust coefficient G, the costs and the relative change of the optimization results with G specified at a certain value (but not 0) compared with the optimization result with G specified to be 0 in a typical winter day and a typical summer day are given in Table 5. From Table 5, it is found that the HEH can employ a cooperative scheduling strategy to attain a specified low economic risk level at the discretion of the HEH.

It is also observed from Table 5 that the energy management considering the objective of minimizing the cost of total energy costs less than the situation without considering this objective.

Concluding remarks

A mCCHP based comprehensive energy solution for households is proposed in this paper. First, the mathematical model of a HEH is presented, and the optimal operation of the HEH formulated as a mixed integer linear programming problem. Next, a robust optimization model for the HEH is developed with electricity price uncertainties taking into account. The simulation results with a case study show that the comprehensive energy solution proposed can not only satisfy demands for multiple forms of energy in the HEH, but also provide a comfortable indoor environment with the total energy costs minimized.

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