## 3 Distributed Optimized Scheduling of EGHS
### PCCP-ADMM-FE Algorithm Steps  
Based on the above algorithms, a convex distributed optimization framework for internal and external co-ordination is established in this paper. The specific steps are as follows:  
(1) Initialize the parameters of the EGHS system, ignore Equation (28), and use the second-order cone program model to obtain a good initial iteration point. Initialize PCCP parameters and enter the inner layer calculation. (2) Relax the equality constraint to the objective function, then yield an augmented Lagrange function:  
\[ \begin{split}& L(x,y,z,\lambda)=C_{sum}+\lambda_{P2G}^{T}(\mathbf{Y}_{ \text{P2G}}-\mathbf{Y}_{\text{P2G}})+\lambda_{G2P}^{T}(\mathbf{X}_{\text{G2P}}-\mathbf{Y}_ {\text{G2P}})+\lambda_{CHPxyz}^{T}(\mathbf{X}_{\text{CHP}}-\mathbf{Y}_{\text{CHP}})\\ &+\lambda_{CHPxyz}^{T}(\mathbf{Z}_{\text{CHP}}-\mathbf{Y}_{\text{CHP}})+ \lambda_{CHPxyz}^{T}(\mathbf{X}_{\text{CHP}}-\mathbf{Z}_{\text{CHP}})+\rho/2\|\mathbf{X}_{ \text{P2G}}-\mathbf{Y}_{\text{P2G}}\|^{2}+\rho/2\|\mathbf{X}_{\text{G2P}}-\mathbf{Y}_{ \text{G2P}}\|^{2}\\ &+\rho/2\|\mathbf{Z}_{\text{CHP}}-\mathbf{Y}_{\text{CHP}}\|^{2}+\rho/2\| \mathbf{X}_{\text{CHP}}-\mathbf{Y}_{\text{CHP}}\|^{2}+\rho/2\|\mathbf{X}_{\text{CHP}}-\mathbf{ Z}_{\text{CHP}}\|^{2}\end{split}  \tag{38}\]  
In the above, \(\mathbf{X}_{\text{P2G}}\), \(\mathbf{X}_{\text{G2P}}\), and \(\mathbf{X}_{\text{CHP}}\) are the P2G, G2P, and CHP variables in the power grid, respectively. \(\mathbf{Y}_{\text{P2G}}\), \(\mathbf{Y}_{\text{G2P}}\), and \(\mathbf{Y}_{\text{CHP}}\) are P2G, G2P, and CHP variables in the gas network, respectively. \(\mathbf{Z}_{\text{P2G}}\) is the CHP variable in the heating network. The iteration format of ADMM-FE is as follows:  
\[\mathbf{X}^{(k+1)}=\underset{\mathbf{X}}{\text{argmin}}\bigg{\{}C_{e}+\mathbf{\lambda}_{ xy,k}^{T}(\mathbf{X}-\mathbf{Y}^{(k)})+\rho/2\|\mathbf{X}-\mathbf{Y}^{(k)}\|^{2}+\mathbf{\lambda}_{xz,k}^{T} (\mathbf{X}-\mathbf{Z}^{(k)})+\rho/2\|\mathbf{X}-\mathbf{Z}^{(k)}\|^{2}\bigg{\}} \tag{39}\]  
\[ \begin{split}\mathbf{Y}^{(k+1)}=\underset{Y}{\text{argmin}}\Big{\{}C_ {g}+\mathbf{\lambda}_{xy,k}^{T}(\mathbf{X}^{(k+1)}-\mathbf{Y})+\rho/2\|\mathbf{X}^{(k+1)}-\mathbf{ Y}\|^{2}\mathbf{\lambda}_{zy,k}^{T}(\mathbf{Z}^{(k)}-\mathbf{Y})+\rho/2\|\mathbf{Z}^{(k)}-\mathbf{Y}\|^{2} \\ +\tau\beta/2\|\mathbf{Y}-\mathbf{Y}^{(k)}\|^{2}+\rho^{(k)}\sum\limits_{k \in\Gamma}\sum\limits_{(i,j)\in\Gamma_{\text{pipe}}}\sum\limits_{d\in\Gamma_{ d}}\delta_{ij,d}^{t}\bigg{\}}\\ \mathbf{Z}^{(k+1)}=\underset{\mathbf{Z}}{\text{argmin}}\bigg{\{}C_{h}+ \mathbf{\lambda}_{zy,k}^{T}(\mathbf{Z}-\mathbf{Y}^{(k)})+\rho/2\|\mathbf{Z}-\mathbf{Y}^{(k)}\|^{2} +\mathbf{\lambda}_{xz,k}^{T}(\mathbf{X}-\mathbf{Z}^{(k)})+\rho/2\|\mathbf{X}-\mathbf{Z}^{(k)}\|^{2} \\ +\tau\beta/2\|\mathbf{(Z}-\mathbf{Z}^{(k)})\|^{2}\bigg{\}}\end{split}  \tag{41}\]  
In the above, \(C_{e}\), \(C_{g}\), and \(C_{h}\) are the power grid, gas network, and heat grid costs separated in the objective function, respectively. \(\lambda_{xy,k}\) is the Lagrange multiplier of the grid and gas network coupling variables. \(\lambda_{zy,k}\) is the Lagrange multiplier of the coupled variable of the heating network and the gas network. \(\lambda_{xz,k}\) is the Lagrange multiplier of the coupled variable of the grid and the heating network. \[\left\{ \begin{array}{l}\lambda_{\text{P2G,}k+1}=\lambda_{\text{P2G,}k}+\beta \Big{(}\mathbf{\lambda}_{\text{P2G}}^{(k+1)}-\mathbf{\gamma}_{\text{P2G}}^{(k+1)} \Big{)}\\ \lambda_{\text{G2P,}k+1}=\lambda_{\text{G2P,}k}+\beta\Big{(}\mathbf{\lambda}_{\text{ G2P}}^{(k+1)}-\mathbf{\gamma}_{\text{G2P}}^{(k+1)}\Big{)}\\ \lambda_{\text{CHPxyz,}k+1}=\lambda_{\text{CHPxyz,}k}+\beta\Big{(}\mathbf{\lambda}_ {1}\mathbf{\lambda}_{\text{CHP}}^{(k+1)}-\mathbf{\gamma}_{\text{CHP}}^{(k+1)}\Big{)}\\ \lambda_{\text{CHPzz,}k+1}=\lambda_{\text{CHPzz,}k}+\beta\Big{(}\mathbf{\lambda}_ {\text{G}}\mathbf{\lambda}_{\text{CHP}}^{(k+1)}-\mathbf{Z}_{\text{CHP}}^{(k+1)}\Big{)} \end{array} \right. \tag{42}\]  
In the above, \(\rho\), \(\tau\), and \(\beta\) are penalty parameters.