Physical models

monee uses steady-state models for each energy carrier. Select a tab to see the governing equations and key variables for that domain.

Electricity

AC power flow

The standard nonlinear AC power-flow equations are used, with voltage magnitude V and voltage angle δ as the node decision variables.

Active and reactive power injected at bus i:

\[P_i = \sum_j V_i V_j \left( G_{ij} \cos(\delta_i - \delta_j) + B_{ij} \sin(\delta_i - \delta_j) \right)\]

\[Q_i = \sum_j V_i V_j \left( G_{ij} \sin(\delta_i - \delta_j) - B_{ij} \cos(\delta_i - \delta_j) \right)\]

where G and B are the conductance and susceptance entries from the nodal admittance matrix Y.

Variable	Symbol	Unit
Voltage magnitude	V	per unit
Voltage angle	δ	radians
Active power injection	P	MW
Reactive power injection	Q	Mvar

The MISOCP relaxation (MISOCP_NETWORK_FORMULATION) replaces the bilinear voltage products with lifted variables and second-order cone constraints, yielding a convex problem solvable globally by a MIQCP solver such as Gurobi or HiGHS.

Gas

Weymouth pipe flow

Pressure drops along a gas pipe are governed by the Weymouth equation. Pressure is modelled as pressure-squared p² to keep the equation quadratic:

\[\left(p_i^2 - p_j^2\right) \cdot C^2 = \dot{m}_{ij}^2 - \dot{m}_{ji}^2\]

The pipe constant C encodes the physical properties of the pipe and gas:

\[C = f\!\left(D,\, L,\, T_\text{gas},\, Z\right)\]

where D is the inner diameter, L is the pipe length, T_gas is the gas temperature, and Z is the compressibility factor.

The Swamee–Jain approximation is used for the Darcy–Weisbach friction factor within the Weymouth constant. Gas density along each pipe is computed from the ideal-gas law using the average pressure between the two endpoints.

Variable	Symbol	Unit
Pressure squared	p²	Pa²
Mass flow	ṁ	kg/s

Note: Using p² as the primary variable avoids the square root that would otherwise appear in a direct pressure formulation, keeping the problem structure quadratic and easier for NLP solvers.

Water / Heat

Darcy–Weisbach hydraulic flow

Hydraulic pressure drops along water pipes follow the Darcy–Weisbach equation:

\[p_i - p_j = R_m \left( \dot{m}_{ij}^2 - \dot{m}_{ji}^2 \right)\]

where the hydraulic resistance R_m is:

\[R_m = f \cdot \frac{L}{D} \cdot \frac{1}{2 \rho A^2}\]

with f the friction factor (laminar approximation f = 64 / Re), L the pipe length, D the inner diameter, ρ the fluid density, and A the cross-sectional area.

Temperature propagation

The outlet temperature accounts for insulation losses and the ambient temperature:

\[T_\text{out} = T_\text{ext} + \alpha \left( T_\text{in} - T_\text{ext} \right)\]

The attenuation factor α depends on the UA product of the insulation layer and the specific heat capacity of water:

\[\alpha = \exp\!\left( -\frac{UA}{\dot{m}\, c_p} \right)\]

Variable	Symbol	Unit
Node pressure	p	Pa
Mass flow	ṁ	kg/s
Inlet / outlet temperature	T_in, T_out	K
External temperature	T_ext	K