**1. Introduction**

Thermal energy storage is considered as one of the advanced energy technologies [1]. Electric energy can be stored in the form of heat during off-peak demand periods and used for heating of rooms during peak demand periods. The improvement of thermal storage is useful to reasonably arrange the electricity consumption of thermal storage loads and promote the thermal storage peak shaving incentive mechanism realization. Therefore, solid electric thermal storage (SETS) has become one of the most promising solutions as a flexible demand response (DR) in demand-side management (DSM) [2]. Consequently, SETS prediction is an important precondition for peak shaving in DSM. Due to the electric–thermal time shift characteristics of SETS, the electricity prediction can provide multiple options for peak shaving and dispatching of the power system. However, there are few studies on the SETS prediction, and the prediction accuracy needs to be improved. Therefore, if the SETS prediction is applied to the actual operation of the power system, it is necessary to enhance the prediction accuracy of SETS based on advanced algorithms.

There are few studies on the electricity prediction of those devices. A physical model (PM) of SETS is integrated into an energy management system for isolated microgrids [3]. SETS is used to accommodate wind power to supply heat load in isolated microgrids [4]. A PM for residential forced-air electric furnaces is built to predict the thermal energy storage, which is an early application

of SETS [5]. This work does not take into account the effect of continuously changing the ambient temperature on thermal energy storage. A PM is integrated into the TRNSYS calculation tool to evaluate the optimal thermal energy storage of forced-air electric furnaces with changing ambient temperature [6]. Therefore, it is necessary to improve the basis of the existing methods, and enhance the prediction accuracy. However, the customers' behavior characteristics of SETS are not considered in the above-mentioned PMs, which is very important to improve the accuracy of prediction.

In [7], a sparse continuous conditional random fields method was proposed to predict electric load with the identification of behavior. The data from advanced metering infrastructure is used to understand the power consumption patterns to improve the load forecasting accuracy in [8]. The prediction accuracy would be significantly enhanced with the consideration of behavior. However, the working mode of SETS are completely different from those of conventional electric loads [9]. SETS is charged by the off-peak electricity, and its thermal energy is released all-day. During the off-peak hours, usually from 21:00 to 6:00, the heating elements quickly heat the dense bricks to a high temperature owning to its cheaper electricity prices. During the peak period from 6:00 to 21:00, the heating elements are switched off, and SETS continues to release its thermal energy to warm the rooms. Many behavior characteristics of SETS directly affect the heat load demand, such as all-day continuous work (e.g., convenience store), and holiday and non-holiday period (e.g., star hotel), which need to be considered. The conventional models of predicting electric load are not adequate for SETS. Due to the wide geographical distribution of SETS installation, the PM cannot consider all the situations comprehensively. The prediction accuracy of the PM still needs to be improved. Under this motivation, this paper considers the behavior characteristics into the PM to enhance the accuracy of SETS.

Another prediction approach of electric load is based on cyber models (CMs), such as auto-regression algorithm [10,11], fuzzy algorithm [12], support vector machine [13], extreme learning machine [14], stochastic methods [15], and multi-stages estimators of nonlinear additive models [16]. Among the existing methods, machine learning (neural network) is commonly used in heat load prediction. A data-driven approach with machine learning is presented to predict the heat load in the rooms [17], and a bi-directional long short-term memory recurrent neural network is proposed to combine the correlation between past information and future information to predict the thermal storage time in [18]. A linear regression model with the ambient temperature is proposed to predict heat load in [19]. These prediction methods are based on historical data combined with multiple machine learning methods. However, it is difficult to predict a sudden increase and decrease of heat load for the above CMs.

With the comprehensive consideration, the PM is sensitive to the sudden increase and decrease of heat load by adding the restriction conditions of electric behaviour, but it cannot consider all the influencing factors. On the other hand, the CM can make good use of historical data to reflect the influencing factors on the SETS, which lacks the guidance of the PM. Modern smart grids have applied cyber–physical systems (CPS) to energy systems including modeling energy systems, energy efficiency, energy resource management, and energy control [20]. Therefore, this paper used a method that combines model-based and data-driven [21], that integrates the PM of SETS and the CM.

The difficulty in using the cyber–physical model (CPM) is how to integrate the cyber components (including influencing factors) and physical components (including behavior characteristics and average power consumption) of SETS. In this paper, we develop a PM of SETS to predict daily average power consumption based on the ambient temperature. The integration of K-means and one-versus-one support vector machine is applied to extract the behavior of SETS. The error levels of the PM and influencing factors are considered as the input of the Back Propagation (BP) neural network-based CM to further enhance the prediction accuracy of electricity consumption. Our contributions are summarized as follows:

• To the best of the authors' knowledge, this is the first work to use the cyber–physical approach to predict the SETS' load change. The physical and cyber components of SETS are integrated.


The rest of this paper is organized as follows: The PM of SETS, including its structure, principle, formula derivation, the customers' behavior characteristics of SETS, and influencing factors are presented in Section 2. The cyber–physical approach combining the PM and CM of SETS is proposed in Section 3. Simulations are conducted to validate the effectiveness of the cyber–physical approach in Section 4. The conclusion is drawn in Section 5.
