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PDEPartial differential equation. SASSemi-analytical solution. SOE Second-order explicit scheme. TVD Total variation decreasing. B. Operators diag(•)Transform a column vector into a diagonal matrix. ⊗ Convolution. 𝑿 (0 : 𝑘) Linear combination of 𝑿 (0), 𝑿 (1), • • • , 𝑿 (𝑘). C. VariablesThe capitalized/bold forms of these variables denote the DT/vector of these variables respectively.𝑚/ 𝑚 in Pipe/node injection mass flow rate. 𝜏 s/r Node supply/return temperature. 𝜏 in/out,s/r Inlet/outlet temperature of supply/return pipes. 𝜏 amb Ambient Temperature. 𝜙 Node power. 𝑒/ 𝑓 Real/imaginary part of bus voltage. 𝑝/𝑞 Active/reactive power. D. Coefficients 𝛾 Cross-sectional area. 𝜌 Density. 𝐻 Length of pipe. 𝐶 p Thermal capacity of mass flow. 𝜆 Overall heat transfer coefficient. 𝐾 Resistance coefficient. 𝑍 Trade-off between heat supply and electric power. 𝑐 m1 Heat-to-power ratio. 𝜂 e Electrical efficiency. 𝐹 in Fuel input rate of steam turbine. Δ𝑥/Δ𝑡 Spatial/temporal step sizes. 𝜃 Parameter that balances the dissipative/dispersive errors in PDE solutions. 𝑉/𝐿 Node/loop incidence matrix. 𝐺/𝐵 Conductance/susceptance matrix. E. Superscripts/Subscripts D Diagonal matrix generated by diag(•).