A Cost Optimization Method Based on Adam Algorithm for Integrated Demand Response

Abstract

Demand response has gradually evolved into integrated demand response (IDR) as energy integration technology develops in integrated energy systems (IESs). The IES has a large amount of data interaction and an increasing concern for users’ privacy protection. Based on the combined cooling, heating, and power model, our study established an IDR management model considering demand-side energy coupling, focusing on cost optimization. In terms of privacy protection in the IDR management process, an optimization method based on the Adam algorithm was proposed. Only nonsensitive data, such as gradients, were transmitted during the processing of the Adam-based method by relying on a centralized iterative computing architecture similar to federated learning. Thus, privacy protection was achieved. The final simulation results proved that the proposed IDR management model had a cost reduction of more than 9% compared with a traditional power demand response. Further simulations based on this model showed that the efficiency and accuracy of the proposed Adam-based method are better than those of other distributed computing algorithms.

1. Introduction

Renewable energy and ever-changing energy demands have brought increasingly complex problems to power system dispatching; moreover, operation uncertainties have been increasing, making use of advanced tools and ideas crucial to ensuring the operation’s reliability [1]. Given the generation of massive user data and the diversification of user energy demands in multienergy systems, novel collaborative interaction models should be designed to achieve some traditional goals, such as peak clipping and valley filling or renewable energy consumption.

Demand response technology is generally applied in a single energy carrier system. However, most power users do not want to suffer power fluctuations, power interruptions, or delayed power consumption. Therefore, a single energy carrier system cannot fully utilize demand-side resources to implement demand response plans. IESs can achieve the coordination of multiple heterogeneous energy systems. Therefore, most users in IESs are multienergy consumers with complex load characteristics; this scenario brings new challenges to demand-side management [2]. The control of coupling energy is indispensable in establishing an IES. Thus, the energy dispatch process involves complex and changeable energy sources and cash flows from multiple parties. Therefore, we planned to use a widely researched energy hub [3,4] to model the CCHP model used in this study and summarize the energy interaction characteristics. This model can reflect the relationship between the input and output of an IES.

As a popular demand-side management measure, the IDR can fully tap the potential of multienergy consumers on the demand side to participate in demand response [5,6]. Heat and power cogeneration and integration technologies of multienergy carriers have developed in recent years, thereby promoting the evolution from the demand response of single smart grids to the IDR in IESs [5]. Similar to electricity demands, user energy demands are adjustable if stimulated appropriately under an IDR. A reasonable IDR mechanism has become important to encourage multienergy subject interaction, realize integrated energy management, and optimize resource allocation. In this study, an IDR management model based on CCHP technology was established. This modeling method was based on the performance of coupled energy in demand response to explore the impact of energy coupling on response costs.

The IDR based on IESs is a general form of peak clipping and valley filling, which are accompanied by a sharp increase in the amount of interaction information and the increasing importance attached to user privacy [7]. The need to protect privacy, such as energy usage behavior and energy usage habits, is important for energy users [8,9]. For instance, if users’ energy usage habits are maliciously exploited, it may lead to a reduction in power supply reliability or even a network-wide blackout. More importantly, failure to protect relevant privacy will lead to malicious quotations and unfair bidding by some other users. Therefore, a new privacy protection method specific to the cost optimization of IDR was investigated in the present study. In existing studies on IDR optimization problems, a virtual power plant scheduling framework considering demand response was proposed in Ref. [10] to develop IDR-related technologies in IESs. This framework enables heat and power cogeneration to participate in the IDR operation. In Ref. [11], a two-stage coordinated operation method was proposed to coordinate CCHP and flexible electric heating loads based on price-based demand responses. In Ref. [12], a two-layer coordinated power and gas network model was established. In these studies, the researchers believe that CCHP must cooperate with flexible loads to achieve optimization. Moreover, the possibility of CCHP reaching the IDR scheduling goal alone was not considered. Considering this situation, some scholars studied the application of CCHP in demand response and drew valuable conclusions [13,14,15,16]. In these studies, the demand response operation and economic optimization among different energy systems were well explored. However, transmitting the complete information of users is always adopted in the optimization process. For IDR, stakeholders are often in multilateral energy markets with incomplete information because of the different energy unit types. Thus, research on the application of privacy protection in IDR must be prioritized.

Among the existing methods for protecting the stakeholders’ privacy, the Bayesian game [17,18] and the ADMM [19,20] have been widely used. The Bayesian game is often used in the bidding field of the supply and demand sides of the power market under incomplete information [21]. Ref. [22] used Bayesian theory to optimize the energy consumption problems in demand response management. Ref. [21] applied Bayesian theory in a noncooperative game with incomplete information among IDR players for optimal bidding strategies. The Bayesian game can reach the Bayesian–Nash equilibrium point without spreading private information. However, participants in the Bayesian game must know other participants in advance to increase the decision-making time. Moreover, the joint probability distribution in the uncertain decision-making environment is difficult to determine. The ADMM algorithm has been widely used in consistency optimization under a fully distributed architecture. Ref. [23] used ADMM to protect end users’ privacy, and a two-stage robust microgrid-side energy management problem was solved. Ref. [24] investigated the joint coordination of demand response and AC optimal power flow to save costs under an ADMM-based method. Before the solution is reached, each demand response participant must construct the augmented Lagrangian function in advance. Moreover, many boundary conditions for the interaction between different participants exist, resulting in an increase in the extra calculation and frequent failure to converge to the optimal solution effectively. Therefore, a new computing architecture that does not need complex initialization in advance should be explored to realize flexible, reliable, and efficient IDR cost optimization on the basis of privacy protection.

The Adam algorithm is an efficient stochastic optimization algorithm that requires only a first-order gradient and has few memory requirements [25]. It combines the advantages of AdaGrad [26] and RMSProp [27]. Thus, it is suitable for sparse gradients, nonlinearities, and nonstationary scenes. The Adam algorithm has been successfully used in previous research [28,29]. In the studies listed above, the Adam algorithm ensured that the complete original data during the training process were not transmitted because of their gradient dependence. However, research on applying the Adam algorithm to cost optimization in IESs to achieve privacy protection is lacking. Inspired by this research gap, we intend to propose an IDR management method based on the Adam algorithm to achieve IDR cost optimization while considering privacy protection. Given that the Adam algorithm has few memory requirements, the Adam-based method proposed in this study can alleviate communication pressure to improve computing efficiency. This result was proved in the case study.

The main contributions of this study are summarized as follows:

• An IES internal coupling model based on CCHP technology is established. This model can clarify the relationship between the input and output of an IES. An IDR optimization model based on this coupling model is also established to explore the cost advantages of energy coupling in IDR management.
• Based on the proposed IDR optimization model, an optimization method that does not transmit energy users’ private information is proposed. This method is based on the Adam algorithm and can optimize costs generated by the IDR management of energy users under incomplete information markets.
• Compared with other distributed optimization algorithms, the computing framework based on the Adam algorithm proposed in this study can improve the efficiency and accuracy of the IDR optimization process.

The whole paper is organized as follows: In Section 2, a user-side multienergy demand model of an IES was established, and the IDR model was planned by controlling the CCHP energy distribution ratio and users’ energy use behaviors based on CCHP coupling and user-side multienergy complementary energy use behaviors. In Section 3, a new optimization algorithm based on Adam was established. This new algorithm integrated the Adam algorithm and IDR model to achieve optimal IDR policy planning without information privacy leakage. In Section 4, three simulation scenarios were set up to demonstrate the effectiveness and superiority of the IDR model and the optimization algorithm proposed in this study. Section 5 summarizes the study.

2. Distribution-Side IDR Model

2.1. CCHP Internal Coupling Relationship Modeling

Using the combination of a regional power system, a thermal system, and a cooling system based on CCHP, the IES investigated in this study provides consumers with electricity, heating, and refrigeration. This IES involves new energy-generating sets, natural gas companies, coupling equipment, and multienergy loads. The most significant difference between the IES and a traditional regional power system is that various energy resources have an interactive relationship between production and consumption.

Energy-generating sets can be divided into three categories: pure power-generating sets, pure heating sets, and heat and power cogenerating sets. The heat and power cogenerating sets usually refer to CCHP, in which power generation and heating are coupled according to the operation mode. In general, pure heating sets are gas furnaces, and pure power-generating sets are power supplies connected to transformers. In general, the coupling relationship at the supply side of the IES can be described using an EH model, in which a microgas turbine and a gas furnace are contained with their energy conversion relationship, as seen in (1).

$$\left[\begin{array}{c}{P}_{\mathrm{out}}^{\mathrm{e}}\\ P_{\mathrm{out}}^{\mathrm{h}}\end{array}\right]=\left[\begin{array}{cc}{\eta }_{\mathrm{T}}& {\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{e}}\\ 0& \left(1-{\omega }_1\right){\eta }_{\mathrm{F}}+{\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{h}}\end{array}\right]\left[\begin{array}{l}{P}_{\mathrm{in}}^{\mathrm{e}}\hfill \\ P_{\mathrm{in}}^{\mathrm{g}}\hfill \end{array}\right],$$

(1)

where ${\omega }_1$ is the ratio of the gas consumed by the microgas turbine; $\left(1-{\omega }_1\right)$ denotes the ratio of gas consumed by the gas furnace; $P_{\mathrm{out}}^{\mathrm{e}}$ and $P_{\mathrm{out}}^{\mathrm{h}}$ are the electric power and heating power outputs of EH, respectively; $P_{\mathrm{in}}^{\mathrm{e}}$ and $P_{\mathrm{in}}^{\mathrm{g}}$ are the electric power and natural gas power inputs of EH, respectively; ${\eta }_{\mathrm{T}}$ and ${\eta }_{\mathrm{F}}$ are the transformer efficiency and the gas furnace’s heating efficiency, respectively; and ${\eta }_{\mathrm{MT}}^{\mathrm{e}}$ and ${\eta }_{\mathrm{MT}}^{\mathrm{h}}$ represent the electrical efficiency and heating efficiency of the microgas turbine, respectively. Furthermore, an EH model, including an absorption chiller, is considered. If the system’s refrigeration demand can be satisfied by the adsorption chiller, the extended EH model can be modified into (2).

$$\left[\begin{array}{l}{P}_{\mathrm{out} }^{\mathrm{e}}\hfill \\ P_{\mathrm{out}}^{\mathrm{h}}\hfill \\ P_{\mathrm{out} }^{\mathrm{q}}\hfill \end{array}\right]=\left[\begin{array}{ll}{\eta }_{\mathrm{T}}\hfill & {\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{e}}\hfill \\ 0\hfill & \left(1-{\omega }_2\right)\left[\left(1-{\omega }_1\right){\eta }_{\mathrm{F}}+{\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{h}}\right]\hfill \\ 0\hfill & {\omega }_2{\sigma }_{\mathrm{hq}}\left[\left(1-{\omega }_1\right){\eta }_{\mathrm{F}}+{\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{h}}\right]\hfill \end{array}\right]\left[\begin{array}{l}{P}_{\mathrm{in}}^{\mathrm{e}}\hfill \\ P_{\mathrm{in}}^{\mathrm{g}}\hfill \end{array}\right],$$

(2)

where ${\omega }_2$ is the heat recovery ratio of the adsorption chiller, ${\sigma }_{\mathrm{hq}}$ represents the refrigeration efficiency, and $P_{\mathrm{out} }^{\mathrm{q}}$ is the cold output power of EH. Equation (2) expresses the double-port model of CCHP and displays the transfer relation between energy input and output; the energy output of CCHP can be controlled by controlling the energy distribution ratio $\left\{{\omega }_1,{\omega }_2\right\}$. The IES described by (2) is shown in Figure A1 in the Appendix A. In this study, each regional large load was presumably managed by an LSO. This LSO was responsible for the strategic implementation of IDR. It could also determine the dispatching plan and coordinate the interaction between energy users and CCHP.

2.2. IDR Design of IES

Given that some loads consume more than one kind of energy (e.g., electric power and heating power), the energy use behavior of these loads leads to an increase in various energy demands. On the contrary, a substitutive effect also exists between different kinds of energy. For example, the microgas turbine can directly replace electric boilers to supply heat when generating cheap heating power to reduce electrical loads. Therefore, the demand for another kind of energy changes when the demand for one kind of energy changes.

The above analysis indicates that an IDR mechanism based on general demand response can be designed. The designed IDR mechanism can reduce the power load directly or indirectly as follows: (1) The direct way of stimulating consumers to reduce part of the load by providing incentives is widely used. (2) In the indirect way, the mutual substitutive relationship between electricity demand and cold and heating demand is considered, and the system manager encourages the increase in cold and heating demand through multienergy coupling to reduce the electrical loads.

In the market mechanism, the LSO acquires auxiliary services like reserves through bidding. The LSO is controlled by the upper-layer ISO, which takes charge of the market clearing strategy of this IES. The specific process of IDR is as follows:

• The LSO carries out day-ahead load forecasting and reports it to the ISO for bidding to enter the power market.
• The LSO obtains the load reduction index from the market clearing results of the ISO.
• The LSO determines the dispatching plan by solving the IDR optimization model, informs consumers of the results, and controls the operation of CCHP;
• Consumers adjust their energy demand at the specified time. CCHP adjusts its energy production accordingly to balance the load, and the net load of the IES is reduced as planned.

Under this process, the information of the LSO and users is uploaded in the form of original data on the network, which have the potential to be attacked. Centralized clearing also causes communication network congestion when the amount of information is large. In Section 3, an FL algorithm is introduced based on this process to solve the network security and congestion problems. The load forecasting technology was not involved in the calculation of this study to highlight the key points. The daily load reduction under each time scale is initialized constantly.

2.3. IDR Management Model Based on Demand-Side Coupling

An LSO aims to minimize the sum of incentive, heating, and refrigeration costs (hereinafter collectively referred to as the operating cost). The demand-side IDR optimization model can be described as follows by taking 1 h as the time step size:

$$\underset{\Delta L_{i,t}^{\mathrm{ee}},\Delta L_{i,t}^{\mathrm{h}},\Delta L_{i,t}^{\mathrm{q}},{\omega }_1,{\omega }_2}{\min C_{\mathrm{sum}}}={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{t=1}^T\left[C_{i,t}^{\mathrm{IL}}+{\rho }_{\mathrm{h}}\Delta L_{i,t}^{\mathrm{h}}+{\rho }_{\mathrm{q}}\Delta L_{i,t}^{\mathrm{q}}\right]}},$$

(3)

where n is the number of LSOs, i labels a specific LSO, T is the total duration, t is the number of hours, and $C_{i,t}^{\mathrm{IL}}$ is the incentive cost; ${\rho }_{\mathrm{h}}$ and ${\rho }_{\mathrm{q}}$ represent the power–heat and power–cold cost coupling coefficients, which are generally negative and represent the unit cost of nonelectrical energy power; $\Delta L_{i,t}^h$ and $\Delta L_{i,t}^q$ are the variations in the heating and cooling loads generated by an IDR. The total cost of the IES depends on the total operating cost of all the LSOs. Thus, this optimization problem is actually the optimization of the sum of the costs of all the LSOs. Since $\Delta L_{i,t}^{\mathrm{ee}},\Delta L_{i,t}^{\mathrm{h}},\Delta L_{i,t}^{\mathrm{q}},{\omega }_1,{\omega }_2$ reflect the LSOs’ energy response characteristics and energy consumption characteristics, these decision variables are the private data to be protected in this study.

${\alpha }_i$ and ${\beta }_i$ are set as the user-side incentive cost coefficients, and $\Delta L_{i,t}^{\mathrm{ee}}$ is the electrical load reduction due to electric excitation. The incentive cost can be expressed by the following equation:

$$C_{i,t}^{\mathrm{IL}}={\alpha }_i\Delta L_{i,t}^{\mathrm{ee}2}+{\beta }_i\Delta L_{i,t}^{\mathrm{ee}}.$$

(4)

In addition, the variations in heating power, cooling power, and electric power are expressed as follows:

$$\begin{gathered} \Delta L_{i,t}^{\mathrm{h}}=\Delta L_{i,t}^{\mathrm{he}}+\Delta L_{i,t}^{\mathrm{hq}}, \\ \Delta L_{i,t}^{\mathrm{q}}=\Delta L_{i,t}^{\mathrm{qe}}+\Delta L_{i,t}^{\mathrm{qh}}, \\ \Delta L_{i,t}^{\mathrm{e}}=\Delta L_{i,t}^{\mathrm{ee}}+\Delta L_{i,t}^{\mathrm{eq}}+\Delta L_{i,t}^{\mathrm{eh}}, \end{gathered}$$

(5)

where $\Delta L_{i,t}^{\mathrm{he}}$ and $\Delta L_{i,t}^{\mathrm{hq}}$ represent the reduction in heating power due to the electrical power changing and the cooling power changing; $\Delta L_{i,t}^{\mathrm{qe}}, \Delta L_{i,t}^{\mathrm{qh}}$ represent the reduction in the cooling power due to the electrical power changing and the heating power changing; $\Delta L_{i,t}^{\mathrm{eq}}$, $\Delta L_{i,t}^{\mathrm{eh}}$ represent the reduction in the electrical power due to the cooling power changing and the heating power changing. These variables have the following relationships:

$$\begin{gathered} \Delta L_{i,t}^{\mathrm{he}}={\mu }_{\mathrm{he}}\Delta L_{i,t}^{\mathrm{e}}, \Delta L_{i,t}^{\mathrm{hq}}={\mu }_{\mathrm{hq}}\Delta L_{i,t}^{\mathrm{q}}, \\ \Delta L_{i,t}^{\mathrm{qe}}={\mu }_{\mathrm{qe}}\Delta L_{i,t}^{\mathrm{e}}, \Delta L_{i,t}^{\mathrm{qh}}={\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}, \\ \Delta L_{i,t}^{\mathrm{eh}}={\mu }_{\mathrm{eh}}\Delta L_{i,t}^{\mathrm{h}}, \Delta L_{i,t}^{\mathrm{eq}}={\mu }_{\mathrm{eq}}\Delta L_{i,t}^{\mathrm{q}}, \end{gathered}$$

(6)

where ${\mu }_{*}\le 0\left(*\in \left\{\mathrm{e},\mathrm{h},\mathrm{q}\right\}\right)$ is the product of the energy conversion efficiency and coupling coefficient, ${\mu }_{\mathrm{qh}}{\mu }_{\mathrm{hq}}=1$ and $t\in \left\{1,2,\dots ,T\right\}$. Equation (2) shows that the cooling power involved in this study is completely converted from thermal energy; thus, ${\mu }_{\mathrm{qe}}={\mu }_{\mathrm{eq}}=0$. In the case of coupling, any kind of energy can be represented by other energy sources, and (5) can be further changed to (7).

$$\begin{gathered} \Delta L_{i,t}^{\mathrm{h}}={\mu }_{\mathrm{he}}\Delta L_{i,t}^{\mathrm{e}}+{\mu }_{\mathrm{hq}}\Delta L_{i,t}^{\mathrm{q}}, \\ \Delta L_{i,t}^{\mathrm{q}}={\mu }_{\mathrm{qe}}\Delta L_{i,t}^{\mathrm{e}}+{\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}={\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}, \\ \Delta L_{i,t}^{\mathrm{e}}=\Delta L_{i,t}^{\mathrm{ee}}+{\mu }_{\mathrm{eh}}\Delta L_{i,t}^{\mathrm{h}}. \end{gathered}$$

(7)

Heating loads are constantly affected by controlling cooling loads. Furthermore, the number of electric, heating, and cooling loads in the IES is set to $n^{\mathrm{e}}$, $n^{\mathrm{h}}$, and $n^{\mathrm{q}}$, respectively; $P_{\mathrm{out},t}^{\mathrm{e}}$, $P_{\mathrm{out},t}^{\mathrm{h}}$, and $P_{\mathrm{out},t}^{\mathrm{q}}$ are the electric, heating, and cooling output power of CCHP, respectively; $P_{\mathrm{in}}^{\mathrm{e}}$ and $P_{\mathrm{in}}^{\mathrm{g}}$ represent the input power of electricity and natural gas of CCHP, respectively. Then, the constraints of the IDR optimization model are designed as follows by combining Equations (1), (2) and (7):

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{l}\begin{array}{c}\begin{array}{l}\Delta L_{i,t}^{\mathrm{e}}=\Delta L_{i,t}^{\mathrm{ee}}+{\mu }_{\mathrm{eh}}\Delta L_{i,t}^{\mathrm{h}}\\ \Delta L_{i,t}^{\mathrm{q}}={\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}\\ \Delta L_{i,t}^{\mathrm{h}}={\mu }_{\mathrm{he}}\Delta L_{i,t}^{\mathrm{e}}+{\mu }_{\mathrm{hq}}\Delta L_{i,t}^{\mathrm{q}}\\ \begin{array}{c}\Delta L_{i,\min }^{\mathrm{ee}}\le \Delta L_{i,t}^{\mathrm{ee}}\le \Delta L_{i,\max }^{\mathrm{ee}}\\ \Delta L_{i,\min }^{\mathrm{h}}\le \Delta L_{i,t}^{\mathrm{h}}\le \Delta L_{i,\max }^{\mathrm{h}}\\ \Delta L_{i,\min }^{\mathrm{q}}\le \Delta L_{i,t}^{\mathrm{q}}\le \Delta L_{i,\max }^{\mathrm{q}}\\ P_{in,\min }^{\mathrm{g}}\le \Delta P_{in,t}^{\mathrm{g}}\le P_{in,\max }^{\mathrm{g}}\end{array}\\ P_{in,\min }^{\mathrm{e}}\le \Delta P_{in,t}^{\mathrm{e}}\le P_{in,\max }^{\mathrm{e}}\end{array}\\ P_{\mathrm{out},t}^{\mathrm{e}}={\displaystyle \sum _{i=1}^{n^{\mathrm{e}}}\left(\Delta L_{i,t}^{\mathrm{e}}\right)}\\ P_{\mathrm{out},t}^{\mathrm{h}}={\displaystyle \sum _{i=1}^{n^{\mathrm{h}}}\left(\Delta L_{i,t}^{\mathrm{h}}\right)}\\ P_{\mathrm{out},t}^{\mathrm{q}}={\displaystyle \sum _{i=1}^{n^{\mathrm{q}}}\left(\Delta L_{i,t}^{\mathrm{q}}\right)}\end{array}\\ \left[\begin{array}{l}{P}_{\mathrm{out} }^{\mathrm{e}}\hfill \\ P_{\mathrm{out}}^{\mathrm{h}}\hfill \\ P_{\mathrm{out} }^{\mathrm{q}}\hfill \end{array}\right]=\left[\begin{array}{ll}{\eta }_{\mathrm{T}}\hfill & {\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{e}}\hfill \\ 0\hfill & \left(1-{\omega }_2\right)\left[\left(1-{\omega }_1\right){\eta }_{\mathrm{F}}+{\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{g}}\right]\hfill \\ 0\hfill & {\omega }_2{\sigma }_{\mathrm{hq}}\left[\left(1-{\omega }_1\right){\eta }_{\mathrm{F}}+{\omega }_1{\eta }_{\mathrm{MT}}^{\mathrm{g}}\right]\hfill \end{array}\right]\left[\begin{array}{l}{P}_{\mathrm{in}}^{\mathrm{e}}\hfill \\ P_{\mathrm{in}}^{\mathrm{g}}\hfill \end{array}\right].\end{array}$$

(8)

The above inequality constraints limit the maximum and minimum values of the cooling, heating, and power loads and the maximum and minimum values of the purchased energy power. The equality constraints describe the load coupling balance constraints, energy supply and demand balance constraints, and CCHP dual-port power balance constraints.

3. IDR Management Based on Adam

3.1. Overview of the Adam Algorithm

This section takes the application of the Adam algorithm to FL as an example to introduce the Adam algorithm.

FL can save the training data locally in client devices while performing global training. The client devices participating in the training update the local model according to their original data and then send it to the central parameter server (CPS) to update the global model. The CPS cannot control or modify the original data of the client. Moreover, it is not responsible for maintaining and protecting the original data of the client. After receiving the data, the CPS aggregates the encrypted data and then returns the result to each client. Each client updates its local model. This process is consistent with the pursuit of data security and easing the pressure on information channels in this study.

A typical FL operation mechanism between the client and CPS can be summarized as shown in Figure 1.

The Adam algorithm, with good convergence characteristics and computational efficiency, has often been used to solve FL problems. Compared with the steepest descent method, the Adam algorithm increases the first-order momentum and the second-order momentum. Thus, the model can reflect spatial–temporal characteristics and effectively suppress oscillations in the convergence process. In the FL training process, the CPS usually initializes a global model and sends it to all low-level users. Each user uses a local data set to train the model. After training, the trained model is uploaded to the CPS. After the model is integrated and processed by the CPS, it is issued again. The iteration is repeated several times until convergence. In the Adam-based FL training process, the parameter vector, first-order momentum vector, second-order momentum vector, two hyperparameters, and step size must be initialized in advance. Then, the loop iteratively updates each part (the number of iterations, gradient, first-order momentum vector, second-order momentum vector, and parameter vector) until the parameter converges. Moreover, Adam aims to find a set of parameters that minimize the mean squared error. Adam does not require a fixed target; it uses sparse gradients and naturally performs some form of step size annealing [30].

3.2. A New Cost Optimization Method Based on Adam

In this section, a new nonlinear optimization algorithm based on Adam was proposed to optimize the operation cost of IDR. This algorithm was based on the nonlinear programming problem. The gradient descent method used in the Adam algorithm was applied to the optimal solution of IDR management from a cost perspective. This Adam-based method is a new nonlinear optimization algorithm with good privacy protection characteristics. In the IES designed in this study, the CPS was responsible for parameter aggregation and was located at the ISO. It also took charge of collecting the parameter sets of the LSO.

The training process of FL with Adam is displayed in Algorithm 1 as an example.

In Algorithm 1, n is the iteration round of the FL algorithm; ${\alpha }_1=0.9$ and ${\alpha }_2$ = 0.99 are the hyperparameters that determine the gradient descent training speed of the Adam algorithm; Weight_i is the importance weight of each user client i; $\eta$ is the learning rate; $\ell (\ast )$ is the activation function form of $f(\ast )$; and $m_i^{n-1}$ is the gradient change of the Adam algorithm at the (n − 1)-th iteration of client i, which follows the iteration rule shown in (9).

$$m_i^n=\frac{\eta \left({\alpha }_1{m}_i^{n-1}+\left(1-{\alpha }_1\right)\nabla \ell \left({\vartheta }_i^n\right)\right)}{\sqrt{{\alpha }_2\left({\displaystyle {\sum }_{\tau =1}^n\nabla \ell {\left({\vartheta }_i^{\tau }\right)}^2}\right)+\left(1-{\alpha }_2\right)\nabla \ell {\left({\vartheta }_i^n\right)}^2}}$$

(9)

The upper part of the fraction in (9) indicates that the learning rate is multiplied by the Adam gradient, which consists of the gradient of the previous iteration and the current gradient. The lower part of the fraction is the root of Adam’s secondary momentum, comprising the sum of historical Adam’s secondary momentum and current Adam’s secondary momentum.

Algorithm 1 Training process of FL with Adam

Server executes:  Initialize ${\vartheta }^0$  ${\vartheta }_k^0\leftarrow {\vartheta }^0$  for each round n = 0,1,2,… do for each client i do      ${\vartheta }_i^n\leftarrow {\vartheta }^n$      ${\vartheta }_i^{n+1}\leftarrow$ Client Update $\left({\vartheta }_i^n\right)$   for end   ${\vartheta }_{}^{n+1}\leftarrow {\displaystyle {\sum }_{i=1}^u\frac{Weigh{t}_i}{TotalWeight}{\vartheta }_i^{n+1}}$  for end Server execution end

Clientkexecutes:  Client Update $\left({\vartheta }_i^n\right)$: ${\vartheta }_i^{n+1}\leftarrow {\vartheta }_i^n-\frac{\eta \left({\alpha }_1{m}_i^{n-1}+\left(1-{\alpha }_1\right)\nabla \ell \left({\vartheta }_i^n\right)\right)}{\sqrt{{\alpha }_2\left({\displaystyle {\sum }_{\tau =1}^n\nabla \ell {\left({\vartheta }_i^{\tau }\right)}^2}\right)+\left(1-{\alpha }_2\right)\nabla \ell {\left({\vartheta }_i^n\right)}^2}}$  return ${\vartheta }_i^{n+1}$ to server Client k execution end

For the IDR nonlinear programming problem mentioned in this section, the pseudocode of the new cost optimization method based on Adam is shown in Algorithm 2. This method combines the Adam algorithm with the IDR nonlinear programming problem by considering privacy protection and showing good convergence performance. Its main difference from the traditional FL algorithm mentioned in Algorithm 1 is that the proposed optimization method does not need to define the training data set in advance. Moreover, the convergence of the algorithm is guaranteed by the convergence of the nonlinear programming problem and the convergence of the gradient descent method used in the Adam algorithm.

In Algorithm 2, $x_k^0={\left[E_k^{\mathrm{e},0},E_k^{\mathrm{c},0},E_k^{\mathrm{h},0}\right]}^{\mathrm{T}}$ is the initial value of the demand-side decision variable, $\epsilon$ denotes the convergence precision, $x_k^i$ is the decision variable of user k in the i-th iteration, and $g\left(x_k^i\right)={-W_{\mathrm{LA},k}|}_{x_k^i}$ represents the demand-side objective function. Similar to Algorithm 1, Algorithm 2 indicates that only the gradient data are transmitted in the communication line, whereas the other data of practical significance are not. Thus, privacy protection is realized. Then, Algorithm 1 is further compared with Algorithm 2.

Algorithm 2 New cost optimization method based on the Adam algorithm

Server executes:  Initialize $x_k^0={\left[E_k^{\mathrm{e},0},E_k^{\mathrm{c},0},E_k^{\mathrm{h},0}\right]}^{\mathrm{T}}$ and $m_{}^0=0$   for each round i = 0,1,2,… do  for each client k do   $m_k^i\leftarrow m^i$   $\left(x_k^{i+1},m_k^{i+1}\right)\leftarrow$ Client Update $\left(x_k^i,m_k^i\right)$    for end    $m_{}^{i+1}\leftarrow {\displaystyle {\sum }_{k=1}^K\frac{Weigh{t}_k}{TotalWeight}{m}_k^{i+1}}$    if $m_{}^{i+1}-m_{}^i<\epsilon$     Loop end    if end   for end Server execution end

Client k executes:  Client Update $\left(x_k^i,m_k^i\right)$:     $m_k^i=\frac{\left({\alpha }_1{m}_k^{i-1}+\left(1-{\alpha }_1\right)\nabla g\left(x_k^i\right)\right)}{\sqrt{{\alpha }_2\left({\displaystyle {\sum }_{\tau =1}^i\nabla g{\left(x_k^i\right)}^2}\right)+\left(1-{\alpha }_2\right)\nabla g{\left(x_k^i\right)}^2}}$     $x_k^{i+1}\leftarrow x_k^i-\eta m_k^i$   return $m_k^i$ to server Client k execution end

(a) In terms of the calculation principle, the gradient descent method is adopted for Algorithms 1 and 2. However, Algorithm 1 optimizes the weight of the loss function, whereas Algorithm 2 directly faces nonlinear programming problems and optimizes decision variables. In addition, Algorithm 1 averages the weights on the server side, whereas Algorithm 2 averages the gradients on the server side, thereby ensuring that in Algorithm 2, every user has the same model at the beginning of each iteration and that the results can reflect the localization characteristics of different clients in subsequent iterations.

(b) For the computing structure, Algorithm 1 is implemented based on the machine learning ability of the client. Thus, the landing process based on Algorithm 1 becomes relatively tedious, and the required function fails within a short amount of time. The client only needs simple numerical computing power to implement Algorithm 1. Moreover, the landing process based on Algorithm 2 is relatively simple. Therefore, the Adam-based optimization method proposed in this study is a privacy protection solution with favorable extensibility.

4. Example Analysis

The running environment is an Intel Core i7-10870H CPU 2.21 GHz processor with 32 GB of RAM. In this study, the IES of a region in Qingdao, China, in winter was used to prove the effectiveness of the proposed method. This region has one CCHP and eight large-scale flexible load bodies (or load buses). The CCHP system is managed by an ISO. Each large-scale flexible load body is managed by an LSO, and eight LSOs are controlled by this ISO in a centralized way.

The following three scenarios were set in this study:

(1)

Scenario I: ${\mu }_{\mathrm{he}}=0$ was set by emphasizing the investigation of the supporting effect of nonelectrical resources on the IDR of electrical resources. The new optimization method based on Adam was used for this calculation example.

(2)

Scenario II: ${\mu }_{\mathrm{he}}\ne 0$ was set by emphasizing the investigation of the influence of demand-side coupling on the supply of nonelectrical resources (with heating power supply as an example). The new optimization method based on Adam was also applied to this scenario.

(3)

Scenario III: Under Scenario II, the performances of the proposed Adam-based method in this study, the D-PPDS method mentioned in Ref. [31], and the ADMM algorithm used in Ref. [32] were compared to prove the good convergence and high efficiency of the proposed method.

4.1. Scenario I

A schematic diagram of the energy supply network of the IES studied in this calculation example is shown in Figure A1 in Appendix A. The maximum limit, $L_{i,\max }^{\mathrm{ee}}$, of the power load reduction for each LSO is 15 MW. The maximum limits, $L_{i,\max }^{\mathrm{h}}$ and $L_{i,\max }^{\mathrm{q}}$, of both the heating power and cooling power load reduction are 10 MW, and the minimum limit of the electric, cooling, and heating load reduction is 0. The electric power response value $\Delta L_{i,t}^{\mathrm{e}}$ of all LSOs is set in

Table A1. Incentive parameters and power load response values of eight buses at a certain moment.

LSO Index i	$\mathbf{\Delta }{\mathit{L}}_{\mathit{i}}^{\mathbf{e}}$ (MW)	$\mathit{\alpha }$ (USD2/MW)	$\mathit{\beta }$
1	8.52	0.96	90
2	8.06	1.00	88
3	8.01	0.91	92
4	7.13	1.1	90
5	7.41	0.92	94
6	6.31	0.98	96
7	7.14	0.97	92
8	7.67	1.1	98

. The consumer incentive parameters are also listed in Table A1 in Appendix A. For all buses, ${\mu }_{\mathrm{eq}}=3$ and ${\mu }_{\mathrm{eh}}=1$. The first-order momentum parameter is ${\alpha }_1=0.9$, and the second-order momentum parameter is ${\alpha }_2=0.99$.

In addition, the research emphasis of the example is on the substitutive effect of the nonelectrical energy in the IDR of electrical energy. Thus, the influence of the change in the heating load on the change in the electrical load can be neglected. Hence, ${\mu }_{\mathrm{he}}=0$ is set in Equation (6). Then, Equation (7) can be further rewritten as Equation (10).

$$\begin{gathered} \Delta L_{i,t}^{\mathrm{h}}={\mu }_{\mathrm{he}}\Delta L_{i,t}^{\mathrm{e}}+{\mu }_{\mathrm{hq}}\Delta L_{i,t}^{\mathrm{q}}={\mu }_{\mathrm{hq}}\Delta L_{i,t}^{\mathrm{q}}, \\ \Delta L_{i,t}^{\mathrm{q}}={\mu }_{\mathrm{qe}}\Delta L_{i,t}^{\mathrm{e}}+{\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}={\mu }_{\mathrm{qh}}\Delta L_{i,t}^{\mathrm{h}}, \\ \Delta L_{i,t}^{\mathrm{e}}=\Delta L_{i,t}^{\mathrm{ee}}+{\mu }_{\mathrm{eh}}\Delta L_{i,t}^{\mathrm{h}}+{\mu }_{\mathrm{eq}}\Delta L_{i,t}^{\mathrm{q}}. \end{gathered}$$

(10)

Figure 2 shows the calculation convergence process under an Adam-based method of the LSO operating cost on eight load buses in the same period, and Figure 3 exhibits the calculation convergence process of ${\omega }_1$ and ${\omega }_2$ in this period. Given the different initial parameters of the different LSOs, the convergence results and convergence speed of the different LSO costs are slightly different. However, the final results can converge completely after 60 iterations (iteration time is 0.15 s).

In this calculation, the gas ratio consumed by the microgas turbine, ${\omega }_1$, and the heat recovery ratio of the absorption chiller, ${\omega }_2$, converged to 0.56 and 0.50, respectively. This result means that the IDR can achieve cost optimization when the gas is evenly distributed to microgas turbine and gas boiler and half of the thermal power output is used for cooling.

Table 1. Cooling, heating, and power load reduction values of eight buses at a certain moment.

LSO Index i	$\mathbf{\Delta }{\mathit{L}}_{\mathit{i},\mathit{t}}^{\mathbf{ee}}$ (MW)	$\mathbf{\Delta }{\mathit{L}}_{\mathit{i},\mathit{t}}^{\mathbf{h}}$ (MW)	$\mathbf{\Delta }{\mathit{L}}_{\mathit{i},\mathit{t}}^{\mathbf{q}}$ (MW)
1	6.73	−1.79	−0.35
2	7.90	−0.15	−0.03
3	5.52	−2.48	−0.49
4	6.40	−0.72	−0.14
5	4.96	−2.45	−0.49
6	3.41	−2.89	−0.57
7	5.38	−1.75	−0.35
8	2.52	−5.15	−1.03

shows that under the coupling of cooling, heating, and electricity, the electric load response is shared by the electrical loads and heating loads. Given that the cooling efficiency of the absorption chiller is approximately 0.18, the change in the cooling loads significantly impacts the heating loads. Thus, although an increase in cooling load cannot directly affect the electrical loads, it can support the electrical demand response by acting on the heating loads.

The Adam-based method proposed in this study was used to calculate the demand response cost without considering the coupling.

Table 2. Comparison of objective function values with and without demand-side coupling.

	With Coupling (USD)	Without Coupling (USD)	Improvement (%)
Total cost:	4385.12	5351.73	18%

shows that the value of the objective function that considers the coupling (approximately USD 4385.12) is significantly lower than that without considering the coupling (approximately USD 5351.73) on the demand side. The total cost reduction of 18% shows that the multienergy coupling on the demand side significantly reduces the cost of the demand response. This finding also reflects the feasibility of the calculation method proposed in this study.

4.2. Scenario II

The example simulation was carried out on a certain day (24 h) to verify the effectiveness of the optimization model proposed in this study within a long time scale when ${\mu }_{\mathrm{he}}\ne 0$. In this scenario, the total load demand curve of electricity, heating, and cooling on that day is shown in Figure A2 in Appendix A. The energy demand shown in Figure A2 was randomly assigned to eight LSOs, and the simulation was repeated 100 times. The subsequent conclusions are the average of the sum of these experimental results. In addition, the external electricity price is shown in

Table A2. Electricity prices at different periods of the day.

Period (h)	0–8	8–14, 17–19, 22–24	14–17, 19–22
Electricity prices (USD/MWh)	64	71	79

in Appendix A.

The influence of multienergy coupling on the total load reduction can be analyzed by solving the optimization model. Figure 4 shows that compared with the heat supply in the case without considering the coupling, the heat supply in the case considering coupling significantly decreases. The analysis shows that one part of the electric power is converted into heat, and traditional thermal equipment such as boilers can meet users’ heating power demand in the case of a low power output. Thus, the IDR based on a multienergy coupling proposed in this study can achieve low-carbon goals and provide a possibility for accessing clean energy to some extent.

The cooling, heating, and power loads reduced by the IDR in 1 day are shown in Figure 5. The cost curve of this method was compared with the cost curve of using the power demand response in the absence of a cooling–heating coupling. During the daytime (8:00–16:00), many cooling, heating, and power load values must be reduced, and the corresponding two cost curves have high values. However, problems, such as network congestion and energy waste, were avoided because of the multienergy coupling. Thus, the curve that considers coupling is lower than the curve that does not consider coupling. The coupling model established in this study was advantageous in terms of the demand response cost (the total cost was reduced by 9%).

4.3. Scenario III

The performance of the proposed method was compared with that of the D-PPDS [31] and ADMM algorithms [32] under the same simulation model in the following example to demonstrate the efficiency and feasibility of the Adam-based method proposed in this study.

ADMM, D-PPDS, and the Adam-based method all use the gradient descent method (or dual ascent method) to obtain a global solution, and they all need many iterations. The difference is that ADMM and D-PPDS rely on topology and healthy distributed communication networks, and the data transmitted between units are multiple auxiliary variables that change with iterations. However, the Adam-based method uses centralized communication, and the data transmitted between units and servers contain gradient information. The curves shown in Figure 6 are the $C_{\mathrm{sum}}$ values of the three algorithms under the same model (Scenario II). Although the curves corresponding to the different methods are almost the same, the Adam-based method has a lower response cost and smoother curves than D-PPDS at times because of its historical characteristics and second-order momentum characteristics. Thus, the unnecessary switching actions and the unit start–stop in the dispatching plan can be avoided.

Table 3. Comparison of computing performance between D-PPDS and Adam-based method under different system models.

Method	Single Iteration Time (ms)	Convergence Iterations	Total Time Spent (ms)	Convergence Error (%)
D-PPDS (8 buses)	4.83	52	251	0.65
ADMM (8 buses)	6.27	79	495	0.93
Adam-based (8 buses)	5.93	46	273	0.34–0.61
D-PPDS (33 buses)	4.99	76	379	0.91
ADMM (33 buses)	6.44	101	650	1.18
Adam-based (33 buses)	5.94	48	285	0.23–0.55

shows the iteration time, convergence times, and accuracy (convergence error) of the three algorithms under different scales of models. The eight-bus model is the model proposed in Figure A3 in Appendix A, and the 33-bus model uses the topology structure of the standard IEEE-33-bus model, where each bus is an LSO load bus. The initialization method for the energy demand values of these LSOs is the same as that in Scenario II. Under the same model (eight-bus model), the performance of the Adam-based method is close to that of the D-PPDS algorithm, and it takes a long time to converge to the exact solution because ADMM cannot deal with the global inequality constraints with incomplete information. Given the centralized computing characteristics of the Adam-based method, the CPS needs data acquisition and parameter integration. Thus, the calculation time of the Adam-based method for a single iteration is longer than that of D-PPDS.

The comparison of the computing performance of the Adam-based method in the eight-bus model with that in the IEEE-33-bus model indicates that the single iteration time and the iteration times are almost unaffected by the system scale because each bus in the Adam-based method can grasp the global gradient information at any time and perform the convergence task together. However, the D-PPDS and ADMM algorithms need numerous iterations in large-scale systems to guarantee convergence. Compared with the eight-bus model, the IEEE-33-bus model has large-scale data, which effectively ensures the accuracy of the Adam-based method. Therefore, the Adam-based method proposed in this study can realize privacy protection and has good feasibility and high computational efficiency when solving IDR problems.

5. Discussion

The Adam algorithm is widely used in machine learning and has shown good training performance and convergence efficiency. The protection principle of Adam is to achieve global optimization by transmitting gradient information from sensitive data. This principle not only saves communication costs but also achieves good privacy protection. However, with the advancement in attack technology [33,34], selfish energy users (such as LSOs) are still unwilling to take risks to contribute their own private data without good economic incentives [35]. Therefore, subsequent research should quantify the privacy costs of energy users and study the economic incentive mechanism based on Adam to fully protect the privacy benefits of energy users.

In addition to the research on IDR, the future research on Adam can also be extended to the application of Adam in Digital Twin modeling, load forecasting, and fault recognition of electric power systems, thereby laying a good foundation for realizing global energy interconnection.

6. Conclusions

In this study, the advanced application of the Adam algorithm in distribution networks with multiple energy carriers was preliminarily explored. Based on the beneficial impact of multienergy coupling on the IDR cost in IESs, the Adam-based method proposed in this study was verified in IDR management. It shows better computing performance and computing effects than other mature distributed algorithms considering privacy preservation, such as ADMM. In addition, the Adam-based method only transmits gradient information. Thus, compared with other data protection methods, the Adam-based method can complete all the work with very minor data interactions. This feature also manifests the unlimited potential of the Adam algorithm in intelligent power distribution. Moreover, the first-order momentum and the second-order momentum used by the Adam algorithm can prevent the occurrence of unreasonable data, thereby accelerating the convergence speed and improving the convergence accuracy of IDR management. The Adam-based method is not sensitive to the system scale. Thus, it has a huge advantage when the demand-side coupling deepens and the scale and information complexity of the IES increase.