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User energy model

This section describes the mathematical model used to represent the energy system beyond power in EnergyCommunity.jl. The model integrates:

  • Heat Pumps (HP): devices providing heating and cooling
  • Thermal Storage (TES): storage tanks with losses and capacity limits for heat and/or cool
  • Boilers: fuel-fired boilers
  • Thermal Loads: time-varying heating/cooling demand

Heat Pumps

Each heat pump $h \in A^{HP}_j$ operated by user $j$ consumes electricity $P^{HP,el}_{j,h,t}$ to provide thermal power $P^{HP}_{j,h,t}$, where $x_{j,h}$ is the variable of the nominal electrical capacity of the heat pump.

\[0 \le P^{HP,el}_{j,h,t} \le x_{j,h}\]

The heat pump can operate in either heating or cooling mode, determined by the variable mode: when mode is greater than or equal to +0.5, the heat pump operates in heating mode; when mode is less than or equal to –0.5, it operates in cooling mode.

  • Heating power when mode ≥ +0.5
  • Cooling power when mode ≤ –0.5

Accordingly, the thermal output $P^{HP}_{j,h,t}$ of the heat pump is defined based on the mode of operation using temperature-dependent COP/EER values:

\[P^{HP}_{j,h,t} = \begin{cases} \;\; P^{HP,el}_{j,h,t} \cdot COP_{j,h,t}, & \text{(heating mode)}\\ -\, P^{HP,el}_{j,h,t} \cdot EER_{j,h,t}, & \text{(cooling mode)} \end{cases}\]

The performances of heat pump are modelled using a second-law efficiency multiplied by the Carnot efficiency:

\[COP_{j,h,t} = \eta^{II,heat}_{j,h,t} \cdot \frac{T_{sink}}{T_{sink} - T_{source}(t)}\]

\[EER_{j,h,t} = \eta^{II,cool}_{j,h,t} \cdot \frac{T_{source}(t)}{T_{sink} - T_{source}(t)}\]

where:

  • $\eta^{II,heat}_{j,h,t}$ and $\eta^{II,cool}_{j,h,t}$ are the second-law efficiencies for heating and cooling, respectively.
  • $T_{sink}$ is the temperature of the heat sink (e.g., indoor temperature for heating, outdoor temperature for cooling).
  • $T_{source}(t)$ is the time-varying temperature of the heat source (e.g., outdoor temperature).

Thermal Storage (TES)

Each thermal energy storage $s \in A^{TES}_j$ has volumetric capacity $x_{j,s}$ and supports storing heat and/or cool depending on the input parameters and operation $mode$. The thermal energy $E^{TES}_{j,s,t}$ stored in the storage $s$ at time step $t$ is modelled depending on the specific heat capacity of the fluid ${cp}_{j,s}$, the reference temperature of the fluid in the storage in heating/cooling mode $T^{ref,heat/cool}_{j,s,t}$, and the input temperature of the fluid $T^{in,heat/cool}_{j,s,t}$:

When in heating mode (mode ≥ +0.5):

\[0 \le E^{TES}_{j,s,t} \le {cp}_{j,s} x_{j,s} \left(T^{ref,heat}_{j,s,t} - T^{in,heat}_{j,s,t} \right)\]

When in cooling mode (mode ≤ –0.5):

\[{cp}_{j,s} x_{j,s} \left(T^{in,cool}_{j,s,t} - T^{ref,cool}_{j,s,t} \right) \le E^{TES}_{j,s,t} \le 0\]

TES loses heat proportionally by factor $k_{j,s}$ to stored energy and to the temperature difference between the reference heating or cooling temperature of the fluid $T^{ref,heat/cool}_{j,s,t}$, in the storage and the unheated-zone temperature $T^U_{j,s,t}$:

When in heating mode (mode ≥ +0.5):

\[L^{TES}_{j,s,t} = k_{j,s} E^{TES}_{j,s,t-1} \left(T^{ref,heat}_{j,s,t} - T^U_{j,s,t}\right)\]

When in cooling mode (mode ≤ –0.5):

\[L^{TES}_{j,s,t} = k_{j,s} E^{TES}_{j,s,t-1} \left(T^U_{j,s,t} - T^{ref,cool}_{j,s,t}\right)\]

The effective temperature $T^U_{j,s,t}$ is modelled as a fraction of the indoor–outdoor temperature gradient:

\[T^U_{j,s,t} = T^{int}_{j,s,t} - b^{tr}_{j,s} \left( T^{int}_{j,s,t} - T^{ext}_{j,s,t} \right)\]

where:

  • $T^{int}_{j,s,t}$ = indoor reference temperature (user profile "T_int")
  • $T^{ext}_{j,s,t}$ = external ambient temperature (user profile "T_ext")
  • $b^{tr}_{j,s}$ = thermal transmittance factor ("b_tr_x") controlling how much of the indoor–outdoor gradient affects the TES

Boilers

Boilers $o \in A^{BOIL}_j$ are fuel-fired heating units producing thermal power $P^{boil}_{j,o,t}$. Their nominal capacity is represented by variable $x_{j,o}$ and limits the thermal power output:

\[0 \le P^{boil}_{j,o,t} \le x_{j,o}\]

Thermal Energy Balance

For each user $j$ at every time step $t$, the thermal balance equation ensures that thermal demand for each thermal load $l$ is met by thermal technologies that feed to it:

\[\sum_{s \in A^{TES}_j(l)} \left[ E^{TES}_{j,s,t} - E^{TES}_{j,s,t-1} + L^{TES}_{j,s,t} \right] + P^{th}_{j,l,t} \Delta_t = \sum_{h \in A^{HP}_j(l)} P^{HP}_{j,h,t} \Delta t + \sum_{o \in A^{BOIL}_j(l)} P^{boil}_{j,o,t}\Delta t , \quad \forall l \in A^{TL}_j\]

where:

  • $A^{TL}_j$ is the set of thermal loads for user $j$
  • $A^{TES/HP/BOIL}_j$ is the set of thermal storages, heat pumps and boilers for user $j$ that feed to the thermal load $l$
  • $P^{th}_{j,l,t}$ is the thermal power demand of load $l$ at time $t$ (positive for heating, negative for cooling)
  • $\Delta_t$ is the time step duration
  • $L^{TES}_{j,s,t}$ are the thermal losses from TES
  • $P^{HP}_{j,h,t}$ is the thermal power provided by heat pumps
  • $P^{boil}_{j,o,t}$ is the thermal power provided by boilers
  • $E^{TES}_{j,s,t}$ is the energy stored in TES at time $t$