## 3 Results
They can also be connected at both the input and the output of various energy supply channels. The flow diagram for one of the energy supply channels is shown in Figure 5.  
In the case of energy accumulation at the input and energy supply from the storage device at the output, the equations of input and output energy flow and a converter of one energy type into another will be  
\[\widetilde{p}_{\alpha}=p_{\alpha}-q_{\alpha}; \tag{13}\]  
\[\widetilde{l}_{\beta}=l_{\beta}+m_{\beta}. \tag{14}\]  
Given all inputs and outputs of the converter of one energy type into another, the vector of input and output will be  
\[\tilde{L}=C\tilde{P}. \tag{15}\]  
Based on Equations (7) and (8), we can write  
\[L+M=C[P-Q]. \tag{16}\]  
Figure 4: A simplified flow diagram of energy storage device. Figure 5: Flow diagram of an energy supply channel with energy storage devices. The concept of an equivalent vector of energy flow of a storage device \(M^{eq}\) is introduced to factor in storage devices; then, the vector of energy flows at the energy hub output is  
\[L=C[P-Q]-M=CP-M^{eq}. \tag{17}\]  
In turn,  
\[M^{eq}=CQ+M. \tag{18}\]  
The matrix form, given coefficient \(\dot{E}\), is  
\[\underbrace{ \begin{vmatrix}m_{a}^{eq}\\ m_{\beta}^{eq}\\ \vdots\\ m_{\omega}^{eq}\end{vmatrix} }_{M^{eq}}=\underbrace{ \begin{vmatrix}s_{aa}&s_{ ga}&\cdots&s_{\omega u}\\ s_{\beta\beta}&s_{\beta\beta}&\cdots&s_{\omega\beta}\\ \vdots&\vdots&\ddots&\vdots\\ s_{\alpha\omega}&s_{\beta\omega}&\cdots&s_{\omega\omega}\\ \end{vmatrix} }_{S}\frac{ \begin{vmatrix}\dot{e}_{\alpha}\\ \dot{e}_{\beta}\\ \vdots\\ \dot{e}_{\omega}\\ \end{vmatrix} }_{\dot{E}} \tag{19}\]  
The matrix of the energy hub outputs, with the storage devices taken into account, is  
\[L=CP-S\dot{E}=[C-S]\bigg{|} \begin{vmatrix}P\\ \dot{E}\end{vmatrix} . \tag{20}\]  
This mathematical description is the basis for studying the operating conditions of the energy hub. As a result, we have systems of linear algebraic equations with constant coefficients. This description is optimal for investigating the energy hub to optimize its functioning depending on the objective function. At present, there is no specific implementation of mathematical models. Even within the framework of one technical object, different channels are monitored using different software implementations [30; 31]. The monitoring data are stored in different databases and are not related to each other by time stamps. This circumstance does not allow accumulating information, that is, for example, using complex neural networks as control systems. Hence, the conclusion that the major issue of the widespread use of MES is the absence of simulation modeling systems enabling an analysis of the structure and behavior of an object in the design stage. At the same time, the approach that has recently become widespread in studying the operating conditions of energy objects suggests the use of simulation models. The data obtained from modeling are the basis for solving optimization problems or for designing various analysis and control systems (big data, digital shadow, forecasting, and control systems based on the use of neural networks) [32]. In [33], two possible approaches to model multi-energy systems are considered. These are an integrated approach with all elements of a multi-energy system modeled on a single software platform and an approach that involves harmonizing simulation models of individual elements with the aid of a single program module that matches the inputs and outputs of separate modules with each other. Moreover, the individual components have their dedicated models, which are characteristic of various energy supply channels [34]. The process of developing approaches to model multi-energy systems should take account of some fundamental factors, one of which is the application area of the concept.