# Optimal Configuration of Integrated Energy System Based on Energy-Conversion Interface
## 2 Energy-Conversion Interface Model
### Complex Input-Output Coupling Model  
The energy flow of a complex ECI is shown in Figure 4. Compared to a simple ECI, there is an intermediate energy-flow process between the devices. The above simple input-output coupling model is no longer applicable. The input-output model for a complex ECI is derived below. For convenience of representation, this paper proposes a device-grading model. A device that directly receives energy from the energy-supply side and does not have another input from another device is defined as a primary conversion device. A device that is powered by a primary conversion device and does not have a higher-level input is a secondary conversion device. By this analogy, a device that is powered by an \(S\)-1 device and does not have a higher-level device is defined as an \(S\)-class conversion device. In a complex ECI, there are \(m\) types of input energy, \(n\) types of output energy, \(k\) types of energy-conversion devices, and \(S\) energy-conversion levels, as shown in Figure 5. Different energy form collection other than the energy-supply side and the demand side occurs during the energy-transfer process. Assuming that there are \(p\) types of intermediate energy forms, the same energy forms existing on the energy-supply side and the demand side are treated as different energy forms. If \(q\) represents the number of same energy forms on both sides of supply and demand, there should be \(q\) virtual-conversion devices whose conversion factors is \(1\). Totally, there are \(m+n+p\) energy forms and \(k+q\) energy-conversion devices in the system. Figure 4: Energy-flow diagram of a complex energy-conversion interface. Figure 5: Schematic diagram of the hierarchical structure of a complex ECI. Energy form property matrix \(\mathbf{D}\) is defined to represent the correspondence between energy-output port J, energy-input port K, and the energy form in the system. In this matrix, \(d_{ij}\) represents the correspondence between the \(i\)-th input/output port and the \(j\)-th energy form, that is, the value of \(d_{ij}\) is 1 when the \(i\)-th port is about the \(j\)-th energy form, and 0 otherwise. \(\mathbf{D}1\sim\mathbf{D}4\) are the corresponding relationship between power-supply-side output port J1, device output port J2, demand-side input port K1, device input port K2, and the energy forms. The devices are arranged in order of level. According to the above complex model, the number of columns of the four matrices is \(m+n+p\), and the numbers of rows of the four matrices are \(m\), \(\sum\limits_{i=1}^{k+q}N_{i}\), \(n\), \(\sum\limits_{i=1}^{k+q}M_{i}\) respectively, where \(M_{i}\) and \(N_{i}\) are the number of input ports and the number of output ports of the \(i\)-th energy-conversion device, respectively. We define branch connection characteristic matrix \(\mathbf{F}\) to indicate the connection relationship between each energy-output port and the energy-input port. Block matrices \(\mathbf{F}1\sim\mathbf{F}4\) represent the connection relationship between J1 and K1, J1 and K2, J2 and K1, J2 and K2, respectively. Then, the relationship between matrix \(\mathbf{F}\) and energy form characteristic matrix \(\mathbf{D}\) can be expressed as:  
\[\mathbf{F}=\left[ \begin{array}{cc}\mathbf{D}3\cdot\mathbf{D}1^{\text{T}}& \mathbf{D}4\cdot\mathbf{D}1^{\text{T}}\\ \mathbf{D}3\cdot\mathbf{D}2^{\text{T}}&\mathbf{D}4\cdot\mathbf{D}2^{\text{T}} \end{array} \right]=\left[ \begin{array}{cc}\mathbf{F}1&\mathbf{F}2\\ \mathbf{F}3&\mathbf{F}4\end{array} \right] \tag{4}\]  
The distribution matrix corresponds to the branch connection characteristic matrix. \"1\" in the branch connection characteristic matrices corresponds to the corresponding distribution coefficient, and \"0\" in the branch connection characteristic matrices corresponds to \"0\" in the corresponding distribution matrices. That is, the sparsity of the two characteristic matrices is the same, and the matrices are both highly sparse matrices.