A CCP-Based Decentralized Optimization Approach for Electricity–Heat Integrated Energy Systems with Buildings

Abstract

With the widespread application of combined heat and power (CHP) units, the coupling between electricity and heat systems has become increasingly close. In response to the problem of low operational efficiency of electricity–heat integrated energy systems (EH-IESs) with buildings in uncertain environments, this paper proposes a chance-constrained programming (CCP)-based decentralized optimization method for EH-IESs with buildings. First, based on the thermal storage capacity of building envelopes and considering the operational constraints of an electrical system (ES) and thermal system (TS), a mathematical model of EH-IESs, accounting for building thermal inertia, was constructed. Considering the uncertainty of sunlight intensity and outdoor temperature, a CCP-based optimal scheduling strategy for EH-IESs is proposed to achieve a moderate trade-off between the optimal objective function and constraints. To address the disadvantages of high computational complexity and poor information privacy in centralized optimization, an accelerated asynchronous decentralized alternating direction method of multipliers (A-AD-ADMM) algorithm is proposed, which decomposes the original optimization problem into sub-problems of ES and TS for distributed solving, significantly improving solution efficiency. Finally, numerical simulations prove that the proposed strategy can fully utilize the thermal storage characteristics of building envelopes, improve the operational economics of the EH-IES under uncertain environments, and ensure both user temperature comfort and the information privacy of each subject.

1. Introduction

With the escalating tension between the fossil energy crisis and growing social energy demand, high-efficiency energy utilization has gradually become a research priority [1]. Integrated energy systems (IESs) that couple multiple energy forms can achieve coordinated planning, interactive response, and complementary coordination among heterogeneous energy subsystems, thereby significantly enhancing energy utilization efficiency, which has emerged as one of the most crucial development trends in future energy systems [2]. According to statistics, electricity–heat integrated energy systems (EH-IESs) can improve fuel efficiency by approximately 50% while reducing carbon emissions by 13% to 18%, establishing them as one of the most prevalent forms of IESs [3].

In 2021, 30% of total global energy consumption was accounted for by building energy use [4]. As the main consumer of diversified energy in cities, buildings have enormous regulation potential, which plays a critical role in EH-IES scheduling. A coordinated optimization scheduling method for buildings and IES source–load was proposed in [5], fully exploring the energy flexibility of building equipment such as air conditioning (AC) by analyzing building energy consumption behavior characteristics and utilizing demand response mechanisms, effectively improving energy utilization efficiency. On the basis of ensuring user comfort, Ref. [6] utilized the thermal dynamics of heating networks and buildings to increase renewable energy use, achieving the optimal economic operation of EH-IESs with buildings. An EH-IES model with building thermal load response was introduced in [7], improving the operational adaptability of the EH-IES by exploring the potential adaptability of AC, significantly reducing wind and solar curtailment rates. The above literature fully explored the cooling/heating flexibility of temperature control loads such as AC based on the building heat balance equation, which improves the operational economy of EH-IESs. However, the building model used is not accurate and cannot establish the mathematical formulation between the external environment and indoor temperature, resulting in an actual indoor temperature deviation from the set comfort range for users, failing to fully guarantee user comfort.

As the system scale increases and the operating environment becomes more complex, the centralized algorithm adopted above will face problems, such as high model complexity, heavy computational burden, and privacy leakage risk [8,9]. The distributed algorithm based on ADMM allows each subject to make decisions based solely on local information and to achieve optimal scheduling of the EH-IES with minimal information exchange, which can protect the privacy of different subjects. Ref. [10] adopted the ADMM algorithm to solve the optimal scheduling system of high-speed railways in a distributed manner, maintaining the security and interests of railway dispatch units. A low-carbon economic operation model for IES was constructed in [11], which used ADMM to solve the model, cutting alliance costs and safeguarding IES information privacy. In [10,11], the ADMM required that all subproblems be solved before proceeding to the next iteration, resulting in low computational efficiency, while the involvement of a central operator introduced potential privacy risks, making it difficult to fully ensure data confidentiality.

In addition, external environmental uncertainty can also affect the effectiveness of optimal scheduling strategies for EH-IESs, and the chance-constrained programming (CCP) method can accurately describe external uncertainty factors, achieving a better balance between the economy and safety of the strategy [12]. A CCP-based distributed planning method for EH-IESs was proposed in [13]. Accounting for wind output variability and probabilistic power balance, the method optimized EH-IES operation. Ref. [14] introduced a stochastic multi-objective scheduler for systems under renewable and load uncertainties, providing a safe, economical, and environmentally friendly scheduling scheme for the system. However, the above literature has not yet utilized the thermal storage characteristics of building envelope structures, and the flexible loads in buildings have not actively focused on optimizing EH-IES scheduling.

Overall, this article proposes a CCP-based decentralized optimization approach for EH-IESs with buildings; the contributions are outlined as follows:

(1) An optimization model for EH-IESs considering building thermal inertia was constructed. By characterizing the thermal dynamic change process of building envelopes, we fully tapped into the demand response potential of AC, significantly improving the energy utilization efficiency of EH-IESs while ensuring user thermal comfort.

(2) Considering the uncertainty of loads and external temperature, a CCP-based optimal scheduling method for EH-IESs, including buildings, is proposed, which balances the economy and security of the scheme well. In addition, by adopting a combined heating method that coordinates centralized heating and AC, the operation flexibility of EH-IESs in uncertain environments has been effectively improved.

(3) An accelerated asynchronous decentralized ADMM (A-AD-ADMM) algorithm is proposed, in which the electric system (ES) and thermal system (TS) can be solved in parallel while exchanging boundary information with each other at different iteration times, avoiding privacy leakage caused by the control center, and improving the algorithm’s solving efficiency. Additionally, the step size is dynamically adjusted in each iteration to enhance convergence performance.

2. Optimization Model of EH-IESs, Considering Building Thermal Inertia

A detailed model, a power flow model of the ES, and an operation model of the TS are constructed sequentially in this section. To improve the solving speed of these models, the second-order cone relaxation (SOCR) technology is used to process the above models.

2.1. Detailed Thermal Dynamic Model of Buildings

A detailed thermal dynamic model of the buildings is shown in Figure 1 [15]. This model consists of four wall nodes and one indoor node. Heat is transferred through thermal resistance and stored through thermal capacitance. The user’s temperature comfort is satisfied by the AC.

The thermal dynamic change process of the room can be described as follows [16]:

$$C_k^{\mathrm{room}}(T_{k,t+1}^{\mathrm{room}}-T_{k,t}^{\mathrm{room}})=\mathrm{\Delta }t[{\displaystyle \sum _{n=2}^5\frac{T_{k,n,t}^{\mathrm{wall}}-T_{k,t}^{\mathrm{room}}}{R_{k,n}^{\mathrm{wall}}}}+{\displaystyle \frac{T_{5,t}^{\mathrm{out}}-T_{k,t}^{\mathrm{room}}}{R_k^{\mathrm{win}}}}+{\mathcal{l}}_k^{\mathrm{win}}{S}_k^{\mathrm{win}}{L}_{k,t}^{\mathrm{int}}-P_{k,t}^{\mathrm{AC}}{\eta }^{\mathrm{AC}}]$$

(1)

where $C_k^{\mathrm{room}}$ denotes the indoor heat capacity; $R_{k,n}^{\mathrm{wall}}$ (n = 2, 3, 4, 5)/$R_k^{\mathrm{win}}$ denote the thermal resistance of walls and windows; $T_{k,t}^{\mathrm{room}}$ denotes the indoor temperature; $T_{k,n,t}^{\mathrm{wall}}$ denotes the temperature of walls; $\mathrm{\Delta }t$ denotes the scheduling time interval; $T_{5,t}^{\mathrm{out}}$ denotes the outdoor temperature adjacent to the windows; ${\mathcal{l}}_k^{\mathrm{win}}$ denotes the Boolean variable to determine whether the wall contains windows; $S_k^{\mathrm{win}}$ denotes the surface area of the windows; $L_{k,n,t}^{\mathrm{int}}$ denotes the intensity of sunlight received by the wall; $P_{k,t}^{\mathrm{AC}}$ denotes the AC operating power; and ${\eta }^{\mathrm{AC}}$ denotes the AC heating coefficient.

The thermal dynamic change process of the wall can be described as follows:

$$\{\begin{array}{c}{C}_{k,2}^{\mathrm{wall}}(T_{k,2,t}^{\mathrm{wall}}-T_{k,2,t-1}^{\mathrm{wall}})=\mathrm{\Delta }t({\displaystyle \frac{T_{k,t}^{\mathrm{room}}-T_{k,2,t}^{\mathrm{wall}}}{R_{k,2}^{\mathrm{wall}}}}+{\displaystyle \frac{T_{2,t}^{\mathrm{out}}-T_{k,2,t}^{\mathrm{wall}}}{R_{k,2}^{\mathrm{wall}}}}+{\alpha }^{k,2}{ƛ}^{k,2}{S}_{k,2}^{\mathrm{wall}}{L}_{k,2,t}^{\mathrm{int}})\\ C_{k,3}^{\mathrm{wall}}(T_{k,3,t}^{\mathrm{wall}}-T_{k,3,t-1}^{\mathrm{wall}})=\mathrm{\Delta }t({\displaystyle \frac{T_{k,t}^{\mathrm{room}}-T_{k,3,t}^{\mathrm{wall}}}{R_{k,3}^{\mathrm{wall}}}}+{\displaystyle \frac{T_{3,t}^{\mathrm{out}}-T_{k,3,t}^{\mathrm{wall}}}{R_{k,3}^{\mathrm{wall}}}}+{\alpha }^{k,3}{ƛ}^{k,3}{S}_{k,3}^{\mathrm{wall}}{L}_{k,3,t}^{\mathrm{int}})\\ C_{k,4}^{\mathrm{wall}}(T_{k,4,t}^{\mathrm{wall}}-T_{k,4,t-1}^{\mathrm{wall}})=\mathrm{\Delta }t({\displaystyle \frac{T_{k,t}^{\mathrm{room}}-T_{k,4,t}^{\mathrm{wall}}}{R_{k,4}^{\mathrm{wall}}}}+{\displaystyle \frac{T_{4,t}^{\mathrm{in}}-T_{k,4,t}^{\mathrm{wall}}}{R_{k,4}^{\mathrm{wall}}}}+{\alpha }^{k,4}{ƛ}^{k,4}{S}_{k,4}^{\mathrm{wall}}{L}_{k,4,t}^{\mathrm{int}})\\ C_{k,5}^{\mathrm{wall}}(T_{k,5,t}^{\mathrm{wall}}-T_{k,5,t-1}^{\mathrm{wall}})=\mathrm{\Delta }t({\displaystyle \frac{T_{k,t}^{\mathrm{room}}-T_{k,5,t}^{\mathrm{wall}}}{R_{k,5}^{\mathrm{wall}}}}+{\displaystyle \frac{T_{5,t}^{\mathrm{out}}-T_{k,5,t}^{\mathrm{wall}}}{R_{k,5}^{\mathrm{wall}}}}+{\alpha }^{k,5}{ƛ}^{k,5}{S}_{k,5}^{\mathrm{wall}}{L}_{k,5,t}^{\mathrm{int}})\end{array}$$

(2)

where $C_{k,n}^{\mathrm{wall}}$ denotes the heat capacity of the wall; $T_{n,t}^{\mathrm{out}}$ denotes the outdoor temperature; ${\alpha }^{k,n}$ denotes the Boolean variable used to determine whether sunlight is shining on the wall; ${ƛ}^{k,2}$ denotes the heat absorption coefficient of the wall; $S_{k,n}^{\mathrm{wall}}$ denotes the surface area of the wall; and $L_{k,n,t}^{\mathrm{int}}$ denotes the intensity of sunlight received by the wall.

During the operation of AC, the following constraints should be satisfied:

$$P_{\min }^{\mathrm{AC}}\le P_{k,t}^{\mathrm{AC}}\le P_{\max }^{\mathrm{AC}}$$

(3)

where $P_{\min }^{\mathrm{AC}}$ denotes the minimum operating power for AC systems; and $P_{\max }^{\mathrm{AC}}$ denotes the maximum operating power for AC systems.

The upper and lower bounds of users’ indoor temperature are constrained as follows:

$$T_{\min }^{\mathrm{room}}\le T_{k,t}^{\mathrm{room}}\le T_{\max }^{\mathrm{room}}$$

(4)

where $T_{\min }^{\mathrm{room}}$ and $T_{\max }^{\mathrm{room}}$ define the feasible range of indoor comfortable temperature.

It should be pointed out that the thermal comfort standards for users depend on environmental factors such as temperature, humidity, and airflow rate, as well as personal factors such as clothing and metabolic rates. Therefore, the acceptable indoor temperature setting range for users may vary in different scenarios. Airflow velocity and pressure have little effect due to their relatively constant nature, while temperature and humidity are the most important influencing factors, with temperature having the greatest impact. Humidity only modifies the effect of temperature [17]. Therefore, this study ignores the influence of factors such as air velocity and pressure, and takes into account the corrective effect of humidity on temperature. As described in reference [18], the thermal neutral temperature range for humans is between 18 °C and 28 °C. Based on this thermal neutral temperature, the comfortable temperature range for building users is set to 20 °C to 24 °C in this study.

Building energy consumption is usually composed of conventional loads and flexible loads (represented by AC):

$$P_t^{\mathrm{load}}=P_t^{\mathrm{AC}}+P_t^{\mathrm{el}}$$

(5)

where $P_t^{\mathrm{load}}$ denotes the total load in buildings; $P_t^{\mathrm{el}}$ denotes the power consumption for regular loads.

Renewable energy from photovoltaic (PV) systems can provide a portion of the energy supply for buildings, and the PV generation can be represented as follows:

$${\tilde{P}}_t^{\mathrm{PV}}=\kappa S^{\mathrm{PV}}{L}_t^{\mathrm{int}}$$

(6)

where ${\tilde{P}}_t^{\mathrm{PV}}$ denotes the furcating output power of PV; and $\kappa$ and $S^{\mathrm{PV}}$ denote the conversion efficiency and areas of PV.

Since both external temperature and sunlight intensity are predicted values, in order to describe the uncertainty of the external environment, this article represents outdoor temperature and sunlight intensity as the sum of their predicted values and prediction errors [19]:

$$T_t^{\mathrm{out}}={\tilde{T}}_t^{\mathrm{out}}(1+E_{\mathrm{out},t}^{\max }{R}_t)$$

(7)

$$L_t^{\mathrm{int}}={\tilde{L}}_t^{\mathrm{int}}(1+E_{\mathrm{int},t}^{\max }{R}_t)$$

(8)

where $T_t^{\mathrm{out}}$, ${\widehat{T}}_t^{\mathrm{out}}$ denote the actual and predicted values of outdoor temperature; $E_{\mathrm{out},t}^{\max }$ denotes the maximum prediction error percentage for outdoor temperature; $Q_t^{\mathrm{rad}}$, ${\widehat{Q}}_t^{\mathrm{rad}}$ denote the actual and predicted values of light intensity; $E_{\mathrm{rad},t}^{\max }$ denotes the maximum prediction error percentage for sunlight intensity; R_t denotes a random value that follows a U (−1,1) distribution, indicating the degree of uncertainty in the predicted results.

And the actual output power of PV $P_t^{\mathrm{PV}}$ can be calculated as follows:

$$P_t^{\mathrm{PV}}={\tilde{P}}_t^{\mathrm{PV}}(1+E_{\mathrm{int},t}^{\max }{R}_t)$$

(9)

2.2. Power Flow Model of the ES

The model of the ES after SOCR is represented as follows [20]:

$$v_{m,t}-v_{n,t}=2(r_{mn}{P}_{mn,t}+x_{mn}{Q}_{mn,t})-(r_{mn}^2+x_{mn}^2)i_{mn,t}$$

(10)

$${‖ {\left[2P_{mn,t}\hspace{1em}2Q_{mn,t}\hspace{1em}{i}_{mn,t}-v_{m,t}\right]}^{\mathrm{T}}‖}_2\le i_{mn,t}+v_{m,t}$$

(11)

$$P_{\mathrm{in},n,t}={\displaystyle \sum }_{l:n\to l}{P}_{nl,t}-\left(P_{mn,t}-i_{nm,t}{r}_{nm}\right)$$

(12)

$$Q_{\mathrm{in},n,t}={\displaystyle \sum }_{l:n\to l}{Q}_{nl,t}-\left(Q_{mn,t}-i_{mn,t}{x}_{mn}\right)$$

(13)

where $v_{m,t}$ and $v_{n,t}$ denote the squared voltage norms at nodes m and n, respectively; $i_{mn,t}$ is the squared voltage norms of the current in mn; $r_{nm}$ and $x_{mn}$ denote the resistance and reactance of mn; $l:n\to l$ means that l is the child node of n; $P_{mn,t}$/$Q_{mn,t}$ denote the active power and reactive power flowing through mn; $P_{\mathrm{in},n,t}$/$Q_{\mathrm{in},n,t}$ denote the active power and reactive power injected into node n.

The transformer substation, CHP units, and photovoltaics can provide the electricity supply for buildings [21]:

$$P_{n,t}^{\mathrm{tg}}=P_{n,t}^{\mathrm{eg}}+P_{n,t}^{\mathrm{CHP}}+P_{n,t}^{\mathrm{PV}}$$

(14)

where $P_{n,t}^{\mathrm{tg}}$ is the total power supply that the buildings can receive; $P_{n,t}^{\mathrm{eg}}$ is the power supplied by the transformer substation; and $P_{n,t}^{\mathrm{CHP}}$ is the power supplied by CHP units.

The operation constraints of the transformer substation are as follows [22]:

$$P_{\mathrm{eg}}^{\min }\le P_{n,t}^{\mathrm{eg}}\le P_{\mathrm{eg}}^{\max }$$

(15)

$$Q_{\mathrm{eg}}^{\min }\le Q_{n,t}^{\mathrm{eg}}\le Q_{\mathrm{eg}}^{\max }$$

(16)

where $Q_{\mathrm{eg},n,t}$ is the reactive power supplied by the transformer substation; $P_{\mathrm{eg}}^{\min }$\$Q_{\mathrm{eg}}^{\min }$ denote the minimum active and reactive power that substations can provide; $P_{\mathrm{eg}}^{\max }$\$Q_{\mathrm{eg}}^{\max }$ denote the maximum active and reactive power that substations can provide.

The voltage and current must meet the following constraints:

$$u_i^{\min }\le u_i\le u_i^{\max }$$

(17)

$$l_{ij}^{\min }\le l_{ij}\le l_{ij}^{\max }$$

(18)

where $u_i^{\min }$ and $u_i^{\max }$ are the feasible ranges of the square of the voltage; and $l_{ij}^{\min }$ and $l_{ij}^{\max }$ are the feasible ranges of the square of the current.

2.3. Operation Model of TS

2.3.1. The Model of CHP Units

This paper uses back-pressure CHP units as coupling devices, and their electric and thermal outputs can be expressed as follows:

$${\gamma }_{\min }{P}_{n,t}^{\mathrm{CHP}}\le H_{n,t}^{\mathrm{CHP}}\le {\gamma }_{\max }{P}_{n,t}^{\mathrm{CHP}}$$

(19)

$$H_{n,t}^{\mathrm{CHP}}=C_{\mathrm{w}}{m}_{n,t}^{\mathrm{w}}(T_{n,t}^{\mathrm{s}}-T_{n,t}^{\mathrm{r}})$$

(20)

where $H_{n,t}^{\mathrm{CHP}}$ is the heat supplied by CHP units; ${\gamma }_{\min }$/${\gamma }_{\max }$ represent the minimum and maximum heat-to-electric ratios; $C_{\mathrm{w}}$ denotes the specific heat capacity of the heat medium; $T_{n,t}^{\mathrm{s}}$/$T_{n,t}^{\mathrm{r}}$ denote the supply and return temperatures of the heat medium.

The operational limits of the CHP units are as follows:

$$P_{\min }^{\mathrm{CHP}}\le P_{n,t}^{\mathrm{CHP}}\le P_{\max }^{\mathrm{CHP}}$$

(21)

$$-\chi ·P_{\min }^{\mathrm{CHP}}\le P_{n,t}^{\mathrm{CHP}}-P_{n,t-1}^{\mathrm{CHP}}\le \chi ·P_{\min }^{\mathrm{CHP}}$$

(22)

where $P_{\min }^{\mathrm{CHP}}$/$P_{\max }^{\mathrm{CHP}}$ denote the feasible range of the CHP unit’s electrical output; $\chi$ denotes the climbing speed of the CHP units.

2.3.2. The Heating Network Model

The secondary network hydraulic model is constructed as follows [23]:

$${\mathit{M}}_{NP}{m}_t^{\mathrm{w}}=0$$

(23)

$${\displaystyle \sum _{i\in Z_{k,\mathrm{s}}}(m_{i,t}^{\mathrm{w}}{T}_{i,t}^{\mathrm{sout}})}=T_{m,t}^{\mathrm{s}}{\displaystyle \sum _{j\in Q_{k,s}}{m}_{j,t}^{\mathrm{w}}}$$

(24)

$${\displaystyle \sum _{i\in Z_{k,r}}(m_{i,t}^{\mathrm{w}}{T}_{i,t}^{\mathrm{rout}})}=T_{m,t}^r{\displaystyle \sum _{j\in Q_{k,r}}{m}_{j,t}^{\mathrm{w}}}$$

(25)

where Equation (23) denotes the mass flow rate conservation constraint. Equation (24) denotes the mixing temperature equation constraint; ${\mathit{M}}_{\mathrm{NP}}$ denotes the node-pipeline correlation matrix of the secondary heating network; $Z_{k,s}$/$Z_{k,r}$ denote the supply and return pipe sets, with k as the terminal node; $Q_{k,s}$/$Q_{k,r}$ denote the supply and return pipe sets, with k as the starting node; $T_{i,t}^{\mathrm{sout}}$/$T_{i,t}^{\mathrm{rout}}$ denote the outlet temperature of the supply and return pipe sets.

To ensure user comfort, the secondary heating network must also meet the following constraints during heating [24]:

$$m_{\min }^{\mathrm{w}}\le m_{i,t}^{\mathrm{w}}\le m_{\max }^{\mathrm{w}}$$

(26)

$$T_{m,\min }^{\mathrm{s}}\le T_{m,t}^{\mathrm{s}}\le T_{m,\max }^{\mathrm{s}}$$

(27)

$$T_{m,\min }^{\mathrm{r}}\le T_{m,t}^{\mathrm{r}}\le T_{m,\max }^{\mathrm{r}}$$

(28)

where $m_{\min }^{\mathrm{w}}$/$m_{\max }^{\mathrm{w}}$ denote the feasible range of the mass flow rate; $T_{m,\min }^{\mathrm{s}}$/$T_{m,\max }^{\mathrm{s}}$ denote the permissible range of supply temperature values; $T_{m,\min }^{\mathrm{r}}$/$T_{m,\max }^{\mathrm{r}}$ denote the permissible range of return temperature values.

3. A CCP-Based Optimization Approach for EH-IESs with Buildings

3.1. Objective Function

$$\min \left({\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N_{\mathrm{b}}}{F}_{n,t}^{\mathrm{eg}}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{CHP}}}{F}_{i,t}^{\mathrm{CHP}}}}\right)$$

(29)

where $F_{n,t}^{\mathrm{eg}}$ denotes the electricity purchase cost from the higher-level grid; $F_{i,t}^{\mathrm{CHP}}$ denotes the natural gas consumption cost of CHP units; T denotes the scheduling cycle; N_b, N_CHP denote the quantity of transformers and CHP units; and $F_{n,t}^{\mathrm{eg}}$ and $F_{i,t}^{\mathrm{CHP}}$ can be calculated as follows:

$$F_{n,t}^{\mathrm{eg}}=W_t·P_{n,t}^{\mathrm{eg}}$$

(30)

$$F_{\mathrm{CHP},i,t}=w_0+w_1{P}_{n,t}^{\mathrm{CHP}}+w_2{H}_{n,t}^{\mathrm{CHP}}+w_3{(P_{n,t}^{\mathrm{CHP}})}^2+w_4{(H_{n,t}^{\mathrm{CHP}})}^2+w_5{P}_{n,t}^{\mathrm{CHP}}{H}_{n,t}^{\mathrm{CHP}}$$

(31)

where $W_t$ denotes the electricity purchase cost of ES; $w_0$, $w_1$, $w_2$, $w_3$, $w_4$, $w_5$ denote the operational costs of CHP units.

3.2. Deterministic Transformation of CCP

A short-term, small-range deviation of indoor temperature exceeding the comfort zone will not cause significant discomfort to users. Therefore, this paper selects indoor temperature as a chance constraint, using its flexible adjustability to cope with the uncertainty of the external environment. And CCP is employed to transform the rigid constraint in (4) into a soft constraint, which allows the optimization scheme to violate the constraint to some extent, but the probability of the constraint holding must not be lower than a set confidence level. This achieves a moderate trade-off between the optimal objective function and constraints in an uncertain environment, enabling the optimization scheme to achieve a better balance [25].

$$\mathrm{Pr}\left\{\left.T_{k,\min }^{\mathrm{room}}\le T_{k,t}^{\mathrm{room}}\le T_{k,\max }^{\mathrm{room}}\right\}\right.\ge \epsilon$$

(32)

where $\mathrm{Pr}\left\{\left.-\right\}\right.$ denotes the probability of occurrence of $\left\{\left.-\right\}\right.$; and $\epsilon$ denotes the confidence level of CCP.

To integrate CCP into the optimization problem, it is necessary to introduce Boolean variables into the above expression and perform a deterministic transformation on the CCP shown in Equation (32), as follows:

$$T_{k,t,\min }^{\mathrm{room}}-\left(1-{\gamma }_{k,t}\right)\left(T_{k,t,\min }^{\mathrm{room}}-T_{k,t,\min \lim }^{\mathrm{room}}\right)\le T_{k,t}^{\mathrm{room}}\le T_{k,t,\max }^{\mathrm{room}}-\left(1-{\gamma }_{k,t}\right)\left(T_{k,t,\max }^{\mathrm{room}}-T_{k,t,\max \lim }^{\mathrm{room}}\right)$$

(33)

$${\displaystyle \sum _{t=1}^T{\gamma }_t\ge \epsilon ·T}$$

(34)

In the formula, ${\gamma }_{k,t}$ is a Boolean variable. When the variable does not satisfy $T_{k,t,\min }^{\mathrm{room}}\le T_{k,t}^{\mathrm{room}}\le T_{k,t,\max }^{\mathrm{room}}$, assign a value of 0; otherwise, assign a value of 1. Equations (33) and (34) indicate that the probability of the indoor temperature that does not exceed $T_{k,t,\max }^{\mathrm{room}}$ or fall below $T_{k,t,\min }^{\mathrm{room}}$ during the scheduling period is not less than $\epsilon$. Even when the CCP takes effect, the indoor temperature will not exceed $T_{k,t,\max \lim }^{\mathrm{room}}$ or $T_{k,t,\min \lim }^{\mathrm{room}}$.

4. An ADMM-Based Decentralized Solution Method for EH-IESs with Buildings

To protect information privacy, the coupling variable is chosen as the CHP unit’s power output, and ADMM is used to decouple the original problem into an ES subproblem and a TS subproblem for separate solutions:

• The subproblem for ES is as follows:

  $$\begin{array}{l}\min \left({\displaystyle \sum _{t=1}^T{\displaystyle \sum _{n=1}^{N_{\mathrm{b}}}{F}_{n,t}^{\mathrm{eg}}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{q\in S_{\mathrm{EH}}}\left({\mathit{\lambda }}_{\mathrm{e},q,t}^{(k)}\left(x_{\mathrm{E},q,t}^{(k+1)}-{\mathit{z}}_{q,t}^{(k)}\right)\right)+}}\right.\left.{\displaystyle \frac{\rho }{2}}{{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{q\in S_{\mathrm{EH}}}‖x_{\mathrm{E},q,t}^{(k+1)}-{\mathit{z}}_{q,t}^{(k)}‖}}}_2^2\right)\\ s.t.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(5)-(6),(9)-(18)\end{array}$$

  (35)

  where ${\mathit{\lambda }}_{\mathrm{e},q,t}^k$ is the vector composed of Lagrange multipliers on the ES side; ${\mathit{x}}_{\mathrm{E},q,t}^{k+1}$ is the vector composed of coupled variables; ${\mathit{z}}_{q,t}^k$ is the vector composed of global variables; $\rho$ is the iteration step size; and $S_{\mathrm{EH}}$ is the connecting line that couples two subsystems.
• The subproblem for TS is as follows:

  $$\begin{array}{l}\min \left({\displaystyle \sum _{t=1}^T{\displaystyle \sum _{i=1}^{N_{\mathrm{CHP}}}{F}_{n,t}^{\mathrm{CHP}}}}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{q\in S_{\mathrm{EH}}}\left({\mathit{\lambda }}_{\mathrm{h},q,t}^{(k)}\left({\mathit{x}}_{\mathrm{H},q,t}^{(k+1)}-{\mathit{z}}_{q,t}^{(k)}\right)\right)+}}\right.\left.{\displaystyle \frac{\rho }{2}}{{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{q\in S_{\mathrm{EH}}}‖{\mathit{x}}_{\mathrm{H},q,t}^{(k+1)}-{\mathit{z}}_{q,t}^{(k)}‖}}}_2^2\right)\\ s.t.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}(1)-(4),(7)-(8),(19)-(28)\end{array}$$

  (36)

  where ${\mathit{\lambda }}_{\mathrm{h},q,t}^k$ is the vector composed of Lagrange multipliers on the TS side; ${\mathit{x}}_{\mathrm{H},q,t}^{k+1}$ is the vector composed of coupled variables on the TS side.

When using the traditional ADMM algorithm to solve the above subproblems, the control center needs to participate in the solution, which poses a privacy leakage risk. The synchronous parallel ADMM (SP-ADMM) algorithm can complete the solution without the participation of the control center, thereby better protecting the privacy of each subject. However, the solution to each subsystem needs to be synchronized, and there is a lot of idle time; the solution time is shown in Figure 2a. An asynchronous decentralized ADMM (AD-ADMM) algorithm is a distributed optimization algorithm designed to address the resource waste and high privacy leakage risks caused by synchronous parallel mechanisms in traditional centralized optimization methods. Its core feature lies in the combination of an asynchronous parallel solving mechanism and a decentralized architecture. The solution time of AD-ADMM is shown in Figure 2b, and the solution process can be described as follows:

TS and ES perform local optimization separately. Once a subsystem completes its solution, it immediately uses the latest coupling information from the other subsystem to update variables. Taking the completion of ES solving as an example, the update process is as follows:

$${\mathit{z}}_{q,\mathrm{t}}^{(k+1)}={\displaystyle \frac{1}{2}}\left({\mathit{x}}_{\mathrm{E},q,t}^{(k+1)}+{\mathit{x}}_{\mathrm{H},q,t}^{(b)}\right)$$

(37)

$${\mathit{\lambda }}_{\mathrm{e},q,t}^{(k+1)}={\mathit{\lambda }}_{\mathrm{e},q,t}^{(k)}+\rho \left({\mathit{x}}_{\mathrm{E},q,t}^{(k+1)}-{\mathit{z}}_{q,t}^{(k+1)}\right)$$

(38)

The convergence of the iteration can be determined by the following equations:

$${‖{\mathit{r}}^{k+1}‖}_2^2={‖{\mathit{x}}_{\mathrm{E}/\mathrm{H},q,t}^{k+1}-{\mathit{z}}_{q,t}^{k+1}‖}_2^2\le {\epsilon }^{\mathrm{pri}}$$

(39)

$${‖{\mathit{s}}^{k+1}‖}_2^2={‖\left(-\rho \right)\left({\mathit{z}}_{q,t}^{k+1}-{\mathit{z}}_{q,t}^k\right)‖}_2^2\le {\epsilon }^{\mathrm{dual}}$$

(40)

Notably, the AD-ADMM algorithm requires adherence to a bounded delay constraint. Specifically, when the iteration count discrepancy between subsystems reaches the predefined threshold k, the faster one must halt progress until the slower one completes its current iteration.

In addition, this paper proposes an accelerated AD-ADMM (A-AD-ADMM) to dynamically adjust the iteration step size:

$${\rho }^{(k+1)}=\left\{\begin{array}{ll}{\rho }^{(k)}\left(1+\lg {\displaystyle \frac{‖{\mathit{r}}^{(k)}‖}{‖{\mathit{s}}^{(k)}‖}}\right)& \mathrm{if} {‖{\mathit{r}}^{(k)}‖}_2\ge \delta {‖{\mathit{s}}^{(k)}‖}_2\\ {\rho }^{(k)}& \mathrm{otherwise}\\ {\rho }^{(k)}/\left(1+\lg {\displaystyle \frac{‖{\mathit{r}}^{(k)}‖}{‖{\mathit{s}}^{(k)}‖}}\right)& \mathrm{if} {‖{\mathit{s}}^{(k)}‖}_2\ge \delta {‖{\mathit{r}}^{(k)}‖}_2\end{array}\right.$$

(41)

where $\delta$ is the set ratio of original and dual residuals.

5. Case Studies

5.1. Case Data

The subsequent section, the CCP-based decentralized optimization approach for EH-IESs with buildings, is validated using GUROBI within the MATLAB-YALMIP platform.

The IEEE 33-bus ES and a 12-node TS (E33T12), as shown in Figure 3, are selected as the test system. The system includes two CHP units to couple TS and ES, and their operating parameters are consistent with [26]. The four building clusters are simultaneously connected with ES and TS. The building clusters contain six buildings, with each user in the buildings equipped with an AC. The building envelope structure parameters and AC parameters are based on [27]. The PVs are located at E10, E24, and E27, and the related parameters of PV and sunlight intensity can be found in [28].

The optimization duration for this model is 24 h. Given the gradual thermodynamic changes of indoor temperatures, the time interval is set to 1 h. The optimal thermal comfort zone is defined as 20–24 °C. The energy efficiency of AC is 3. In the A-AD-ADMM algorithm, the initial iteration step is 1, and the convergence threshold of the residual is set to 10⁻³ [29]. The outdoor temperature and electricity price curves are shown in Figure 4 [30].

5.2. Case Results and Analysis

5.2.1. Impact Analysis of Heating Systems and User Comfort Levels on EH-IES Operation

When user comfort is not considered, the internal temperature of the building is set to remain constant at 22 °C. The optimized outputs of the heating equipment (AC and CHP unit) are shown in Figure 5.

In this scenario, the operating cost of the EH-IES is USD 5152.5. The indoor temperature is mainly maintained by AC and CHP units. CHP units generate heat by burning gas, while AC units consume electricity to generate heat. Therefore, when electricity prices are low, ACs operate at a higher power output and CHP units operate at a lower output. Conversely, when electricity prices are high, CHP unit output increases and AC output decreases, fully taking advantage of heat source synergies to boost the system’s overall energy utilization. However, because user temperature comfort is not considered, the indoor temperature remains constant, resulting in the operating power of the AC being greatly influenced by the outdoor temperature, which limits its flexibility within the EH-IES.

When user comfort is considered, the internal temperature of the building is set to 20–24 °C, and the output of the optimized heating equipment is shown in Figure 6.

Considering user comfort, AC and CHP units can fully utilize the thermal storage characteristics of the building envelope. The AC can store heat in advance before electricity prices rise, while CHP units further increase their output during peak electricity price periods, thereby reducing the operating power of the AC when electricity prices are high. This strategy further enhances the operational flexibility of the AC, with an EH-IES operating cost of USD 5002.9, significantly reducing the energy costs of buildings and the EH-IES.

5.2.2. Analysis of EH-IES Optimization Results Based on CCP

A short-term, small-range deviation of indoor temperature from the comfortable range will not cause significant discomfort to humans. Therefore, this section models indoor temperature as the chance constraint, setting the confidence levels of CCP to 100%, 90%, 80%, and 70%, respectively, to further improve the operational economy of the EH-IES. Considering certain errors in the prediction process of outdoor temperature and sunlight intensity, 100 Monte Carlo simulations were performed.

Figure 7 presents the cumulative distribution function (CDF) of indoor temperature under different confidence levels, averaged across 100 Monte Carlo runs. As the temperature increases from 18 °C to 26 °C, the cumulative probability transitions from 0 to 1. At full confidence, the building temperature lies entirely within the comfort range (20–24 °C). As the confidence level decreases, the range of indoor temperature variation gradually increases. When the confidence level is 90%, the indoor temperature remains bounded above by 24 °C, but its lower bound decreases to 18 °C. When the confidence level is 80%, both temperature bounds slightly exceed the comfort range, although the extent of the exceedance is not significant. When the confidence level drops to 70%, the upper and lower temperature limits further expand beyond the comfort range. A lower confidence level can provide greater flexibility in adjusting the indoor temperature, but it also increases the risk of the indoor temperature exceeding the comfort range. Therefore, a balance between the confidence level and user comfort needs to be considered when formulating scheduling strategies.

To analyze how confidence levels affect the operational economy of EH-IESs,

Table 1. Comparison of operating costs for EH-IESs at different confidence levels.

Confidence Level	Total Operating Cost /USD
100%	5002.9
90%	4959.6
80%	4878.2
70%	4756.24

compares the total operating costs of EH-IESs at different confidence levels based on the average results of 100 Monte Carlo simulations.

A decrease in the confidence level leads to improved economic performance of the EH-IES. Specifically, at 70% confidence, the total operating cost is 2.54% lower than that at 100%. Because the confidence level of the comfortable temperature decreases, the operational flexibility of AC and CHP units further improves, allowing the system to more reasonably utilize the complementary characteristics of heat sources, improve energy utilization efficiency, and enhance the operational economy of the EH-IES.

5.2.3. Convergence and Optimality Analysis of the A-AD-ADMM Algorithm

Figure 8 shows the convergence curves obtained using the A-AD-ADMM algorithm to solve the optimal scheduling strategy while considering user comfort. Both the primal residual and the dual residual reach the convergence threshold. Moreover, the operating cost of the EH-IES obtained by the distributed optimization method differs from the centralized result by only 0.15%, indicating the accuracy of the A-AD-ADMM used in this paper.

Table 2. Comparison of different ADMM algorithms.

Algorithm	Iterations	Solution Time/s
Traditional ADMM	35	743.62
SP-ADMM	35	269.64
AD-ADMM	63	133.55
A-AD-ADMM	47	89.34

shows the number of iterations and solution times required for different ADMM algorithms to solve this strategy. Although the AD-ADMM algorithm has more iterations than traditional methods, its final solution time is only 133.55 s due to the significantly shorter single iteration times, which are 82.04% and 50.47% faster than traditional ADMM and SP-ADMM, respectively. In addition, the use of the step size dynamic correction method can further shorten the solution time and the number of iterations, effectively improve the convergence performance of the algorithm, and have a greater advantage in distributed solving.

6. Conclusions

This paper proposes a CCP-based decentralized optimization method for EH-IESs with buildings; the main findings are as follows:

(1) Based on the thermal inertia of buildings, CHP units and AC can fully utilize the thermal storage characteristics of building envelope structures and the peak-valley difference of electricity prices, achieving flexible and economical operation of EH-IESs while ensuring the thermal comfort needs of users. Specifically, accounting for the thermal inertia of a building reduces operating costs by 2.91% compared to not considering building thermal inertia.

(2) The CCP-based stochastic optimization method for EH-IESs considers the uncertainty of sunlight intensity and outdoor temperature. By adopting lower confidence levels for user comfort temperature, the total operating cost of the EH-IES can be further reduced. Specifically, at a 70% confidence level, the total operating cost of the EH-IES decreases by 2.54% compared to that at the 100% confidence level.

(3) A-AD-ADMM adopts a decentralized solving algorithm to achieve global optimal scheduling of EH-IESs by exchanging a small number of boundary variables, effectively ensuring the information privacy of ES and TS. In addition, A-AD-ADMM significantly improves the convergence speed of the algorithm, with computational speed increases of 87.97%, 66.86%, and 33.10% compared to ADMM, SP-ADMM, and AD-ADMM, respectively.