# Energy Bus-Based Matrix Modeling and Optimal Scheduling for Integrated Energy Systems
## 2 Matrix-Modeling Method
### Steady-State Model  
If the IES is in the static state, the inputs and outputs of each device are unchanged. The energy conversion process of the devices can be expressed as  
\[\mathbf{e}_{i,\text{out}}=\mathbf{H}_{i}\mathbf{e}_{i,\text{in}} \tag{4}\]  
where \(\mathbf{e}_{i,\text{out}}\) and \(\mathbf{e}_{i,\text{in}}\) are the output and input of the device \(i\), and \(\mathbf{H}_{i}\) is the energy conversion matrix of the device \(i\) and is defined as  
\[\mathbf{H}_{i}= \begin{bmatrix}\eta_{i,\text{G2G}}&\eta_{i,\text{E2G}}&\eta_{i, \text{C2G}}&\eta_{i,\text{Q2G}}\\ \eta_{i,\text{G2E}}&\eta_{i,\text{E2E}}&\eta_{i,\text{C2E}}&\eta_{i,\text{Q2E }}\\ \eta_{i,\text{G2C}}&\eta_{i,\text{E2C}}&\eta_{i,\text{C2C}}&\eta_{i,\text{Q2C }}\\ \eta_{i,\text{G2Q}}&\eta_{i,\text{E2Q}}&\eta_{i,\text{C2Q}}&\eta_{i,\text{Q2Q }}\end{bmatrix}  \tag{5}\]  
where \(\eta_{i,\text{Z2y}}\) represents the conversion efficiency of the energy \(x\) to \(y\) of the device \(i\). Regarding the PGU, its input is gas and its outputs are electricity and heat. It has two efficiency coefficients: the conversion efficiency of gas to electricity \(\eta_{\text{Pgu,e}}\) and the conversion efficiency of gas to heat \(\eta_{\text{Pgu,h}}\). Hence, \(\mathbf{H}_{\text{Pgu}}\) is  
\[\mathbf{H}_{\text{Pgu}}= \begin{bmatrix}\eta_{\text{Pgu,e}}&0&0&0\\ 0&0&0&0\\ \eta_{\text{Pgu,h}}&0&0&0\end{bmatrix}  \tag{6}\]  
and \(\mathbf{e}_{\text{Pgu,in}}\) is  
\[\mathbf{e}_{\text{Pgu,in}}=[\alpha_{\text{Pgu}}\quad 0\quad 0]^{\text{T}} \tag{7}\]  
where \(\alpha_{\text{Pgu}}\) is the gas input of the PGU. According to the energy conversion mechanism of the gas boiler, absorption chiller, and electric chiller, their energy conversion matrices are defined as  
\[\mathbf{H}_{\text{gb}}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \eta_{\text{gb}}&0&0&0\end{bmatrix}  \tag{8}\]  
\[\mathbf{H}_{\text{ac}}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&COP_{\text{ac}}\\ 0&0&0&0\end{bmatrix}  \tag{9}\]  
\[\mathbf{H}_{\text{ec}}= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&COP_{\text{ec}}&0&0\\ 0&0&0&0\end{bmatrix}  \tag{10}\]  
where \(COP_{\text{ac}}\) and \(COP_{\text{ec}}\) are the coefficients of performance of the absorption chiller and electric chiller, respectively. \(\eta_{\text{Pgu,e}}\), \(\eta_{\text{Pgu,h}}\), \(\eta_{\text{gb}}\), \(COP_{\text{ac}}\), and \(COP_{\text{ec}}\) can be fitted with the operating data of the corresponding devices. They can be represented by constants or polynomials related to the part-load ratio [33]. This means that the models can be linear or nonlinear. Since most of the conversion devices and components have only one type of input, \(\mathbf{e}_{i,\text{out}}\) can be simplified as  
\[\mathbf{e}_{i,\text{out}}=\mathbf{h}_{i}\alpha_{i} \tag{11}\]  
where \(\mathbf{h}_{i}\) is the nonzero column of \(\mathbf{H}_{i}\), and \(\alpha_{i}\) is the input of the device. If purchases of electricity and gas are allowed, \(\mathbf{e}_{\text{grid}}\) can be defined as  
\[\mathbf{e}_{\text{grid}}=\left[\alpha_{\text{grid},\text{g}}\quad\alpha_{\text{grid },\text{e}}\quad 0\quad 0\right]^{\text{T}} \tag{12}\]  
Electricity, chilled water, and hot water storage units are employed in the IES as \(\mathbf{e}_{\text{s}}\) and are defined as  
\[\mathbf{e}_{\text{s}}=\left[0\quad\alpha_{\text{s},\text{e}}\quad\alpha_{\text{s},\text{c}}\quad\alpha_{\text{s},\text{h}}\right]^{\text{T}} \tag{13}\]  
Hence, the steady-state balance of the energy flow of the IES can be simplified as  
\[\left[ \begin{array}{c}1\\ 0\\ 0\\ 0\end{array} \right]\alpha_{\text{grid},\text{g}}+\left[ \begin{array}{c}0\\ 1\\ 0\\ 0\end{array} \right]\alpha_{\text{grid},\text{e}}+\left[ \begin{array}{c}-1\\ 0\\ 0\\ \eta_{\text{pgu},\text{rh}}\end{array} \right]\alpha_{\text{pgu}}+\left[ \begin{array}{c}-1\\ 0\\ 0\\ \eta_{\text{gb}}\end{array} \right]\alpha_{\text{gb}}+\left[ \begin{array}{c}0 \\ 0\\ COP_{\text{ac}}\\ -1\end{array} \right]\alpha_{\text{ac}}+\left[ \begin{array}{c}0\\ -1\\ 0\end{array} \right]\alpha_{\text{ec}} \tag{14}\] \[+\left[