## Abbreviations
\begin{tabular}{l l} \(K_{\rm h}\) & Pipeline resistance coefficient vector \\ \(\mathbf{\Phi_{\rm n}}/\mathbf{\Phi_{\rm s}}\) & Consumed heat power vector of non-source/source nodes \\ \(C_{\rm p}\) & Specific heat capacity of water \\ \(T_{\rm n}^{\rm s}/T_{\rm s}^{\rm R}\) & Outflow temperature vector of non-source/source nodes \\ \(T_{\rm sp}^{\rm s}/T_{\rm sp}^{\rm R}\) & Specified inflow temperature vector of supply/return nodes \\ \(T_{\rm outk}^{\rm p}/T_{\rm ink}^{\rm p}\) & Outlet/inlet temperature of heat pipeline \(k\) \\ \(T_{a,k}\) & Ambient temperature of heat pipeline \(k\) \\ \(\lambda_{\rm h,k}\) & Heat transfer coefficient of heat pipeline \(k\) \\ \(L_{\rm h,k}\) & Length of heat pipeline \(k\) \\ \(m_{k}\) & Mass flow of heat pipeline \(k\) \\ \(\mathbb{H}^{\rm pipe}\) & Set of heat pipelines \\ \(T_{\rm outk}^{\rm p^{\prime}}/T_{\rm ink}^{\rm p^{\prime}}\), & Equivalent outlet/inlet temperature of heat pipeline \(k\) \\ \(T_{\rm inj,k}^{\rm N^{\prime}}/T_{\rm outj}^{\rm N^{\prime}}\), & Inflow/outflow temperature of node \(i/j\) \\ \(A_{\rm hd}\) & Directional node-branch incidence matrix of DHN \\ \(S_{\rm h}\) & Directional factor vector of heat pipelines \\ \(m_{\rm out}/m_{\rm in}\) & Mass flow leaving/entering a node \\ \(\mathbb{H}^{\rm node}\) & Set of DHN nodes \\ \(\mathbb{H}^{\rm out,\prime}/\mathbb{H}^{\rm inj}\) & Set of heat pipelines with Node \(i\) as the inlet/heat pipelines with node \(i\) as the outlet \\ \(T_{\rm in}^{\rm N^{\prime}}/T_{\rm out}^{\rm N^{\prime}}\) & Nodal inflow/outflow temperature vector \\ \(D_{1}^{\rm S}/D_{1}^{\rm R}\) & Weighted mass flow leaving into pipelines from supply/return nodes \\ \(D_{2}^{\rm S}/D_{2}^{\rm R}\) & Weighted mass flow leaving into loads from supply/return nodes \\ \(D_{3}^{\rm S}/D_{3}^{\rm R}\) & Weighted mass flow entering from pipelines from supply/return nodes \\ \(f_{\rm P}\) & Gas pipeline flow vector \\ \(S_{\rm g}\) & Directional factor vector of gas pipelines \\ \(K_{\rm P}\) & Gas pipeline coefficient vector \\ \(\Pi_{\rm s}/\Pi_{\rm e}\) & Squared pressure vector of gas pipeline start/end node \\ \(M_{\rm P}\) & Constant part in \(K_{\rm P}\) \\ \(Z_{\rm P}\) & Gas pipeline compressibility factor vector \\ \(\rho\) & Specific gravity vector \\ \(\varphi_{\rm p}/L_{\rm p}/D_{\rm P}\) & Friction factor/length/diameter vector of gas pipeline \\ \(C_{\rm p}/T_{\rm g}\) & Constant parameter/gas temperature of gas pipeline \\ \(\chi/\kappa/\mu\) & Flow exponent/diameter exponent/specific gravity exponent \\ \(A_{\rm g}\) & Node-pipeline incidence matrix of DGN \\ \(A_{\rm gd}\) & Directional node-pipeline incidence matrix of DGN \\ \(H_{\rm C}/B_{\rm C}\) & Adiabatic horsepower/power coefficient vector of compressor \\ \(f_{\rm C}/\theta_{\rm C}\) & Gas flow/compression exponent vector of compressor \\ \(\pi_{\rm i}/\pi_{\rm o}\) & Inlet/outlet pressure vector of compressor \\ \(C_{\rm C}\) & Compressor characteristic constant \\ \(P_{\rm st}/T_{\rm st}\) & Standard temperature/pressure \\ \(T_{\rm g,in}\) & Gas temperature at the compressor inlet \\ \(Z_{\rm C,in}\) & Gas compressibility factor vector at the compressor inlet \\ \(\mathbf{\omega}\) & Gas polytropic coefficient vector \\ \(\mathbf{\tau}_{\rm C}\) & Consumed gas flow vector of compressor \\ \(\mathbf{\tau}_{\rm C,st}\) & Equivalent energy consumption vector of the extracted gas under standard GCV \\ \(q_{\rm C}\) & GCV vector at the extraction node \\ \(\alpha_{\rm C}/B_{\rm C}/\gamma_{\rm C}\) & Efficiency coefficient vector \\ \(B_{\rm g}\) & Node-equipment incidence matrix of DGN \\ \(B_{\rm g}^{\rm S}/B_{\rm g}^{\rm C}\) & Node-slack source incidence matrix/node-compressor incidence matrix \\ \(f_{\rm E}/f_{\rm S}\) & Equipment gas flow/slack source flow vector \\ \(f_{\rm L}/f_{\rm D}/f_{1}\) & Net load/volume demand/non-slack source flow vector \\ \(E_{\rm D}\) & Load energy demand vector \\ \(q\) & Nodal GCV vector \\ \end{tabular}