Model

The energy hub concept is used to model an energy community where multi-energy carriers can supply diverse end use demands through building-level equipment and district-level infrastructure optimally interconnected and operated. For a delimited perimeter of buildings, REHO selects the optimal energy system configuration minimizing the specified objective function. All the energy flows at building-level and district-level are then fully characterized by the model decision variables.

Fig. 1 District-level energy hub model in REHO

Energy demands considered by the model are: thermal comfort (space heating and cooling), domestic hot water (DHW), domestic electricity, and mobility needs. Domestic electricity, DHW, and mobility needs are generated using standardized profiles according to norms or measurements in a pre-processing step. In contrast, the thermal comfort demand is modeled within the framework itself in order to include the control strategy of the energy management system and the possibility of a thermal renovation of the building. This heating or cooling demand is impacted by factors such as the conductive heat losses through the building envelope, the heat capacity of the building and the heat gains from occupants, electric appliances and solar irradiation. Furthermore, the energy demand associated with thermal comfort is characterized by the desired comfort temperature of the rooms, the nominal return and supply temperature of the heat distribution system and the control strategy of latter.

Heating and cooling requirements can be satisfied by energy conversion technologies (such as a heat pump, an electrical heater, a fuel cell, a gas boiler, an air conditioner…) or directly from a district heating infrastructure. Energy can be stored in installed equipment (such as a battery or a water tank), or in the form of building thermal inertia. Photovoltaic panels act as a renewable energy source. The building-level energy system is interconnected to the energy distribution infrastructure of the district (electrical grid, natural gas grid, …).

Fig. 2 REHO model architecture

Fig. 3 Building-level energy hub in REHO

List of symbols

Note

In the following, all decision variables of the model are denoted with bold letters to distinguish them from the parameters.

Variables

\(\boldsymbol{C}\)	cost	\(\text{currency}\)
\(\boldsymbol{E}\)	electricity	\(kW(h)\)
\(\boldsymbol{G}\)	global warming potential	\(kg_{CO_2, eq}\)
\(\boldsymbol{H}\)	natural gas or fresh water	\(kW(h)\)
\(\boldsymbol{Q}\)	thermal energy	\(kWh\)
\(\boldsymbol{R}\)	residual heat	\(kWh\)
\(\boldsymbol{T}\)	temperature	\(K\)
\(\boldsymbol{f}\)	sizing variable	\(\diamondsuit\)
\(\boldsymbol{\lambda}\)	decomposition decision variable, master problem	\(-\)
\(\boldsymbol{y}\)	decision variable, binary	\(-\)

Parameters

\(A\)	area	\(m^2\)
\(C\)	heat capacity coefficient	\(kW/m^2K\)
\(F\)	bound of validity range of unit sizes	\(\diamondsuit\)
\(\Phi\)	specific heat gain	\(kW/m^2\)
\(Q\)	thermal power	\(kW\)
\(T\)	temperature	\(K\)
\(U\)	heat transfer coefficient	\(kW/m^2K\)
\(V\)	volume	\(m^3\)
\(\alpha\)	azimuth angle	\(^{\circ}\)
\(\beta\)	limiting angle	\(^{\circ}\)
\(c\)	energy tariff	\(\text{currency}/kWh\)
\(c_p\)	specific heat capacity	\(kJ/(kgK)\)
\(d\)	distance	\(m\)
\(d_p\)	frequency of periods per year	\(d/yr\)
\(d_t\)	frequency of timesteps per period	\(h/d\)
\(e\)	electric power	\(kW/m^2\)
\(\epsilon\)	elevation angle	\(^{\circ}\)
\(\nu\)	efficiency	\(-\)
\(f_{b,r}\)	spatial fraction of a room in a building	\(-\)
\(f^s\)	solar factor	\(-\)
\(fû\)	usage factor	\(-\)
\(g\)	global warming potential streams	\(kg_{CO_2, eq}/kWh\)
\(\gamma\)	tilt angle	\(^{\circ}\)
\(g^{glass}\)	ratio of glass per facades	\(-\)
\(h\)	height	\(m\)
\(i\)	interest rate	\(-\)
\(i^{cl}\)	fixed investment cost	\(\text{currency}\)
\(i^{c2}\)	continuous investment cost	\(\text{currency}/\diamondsuit\)
\(i^{g1}\)	fixed impact factor	\(kg_{CO_2, eq}\)
\(i^{g2}\)	continuous impact factor	\(kg_{CO_2, eq}/ \diamondsuit\)
\(irr\)	irradiation density	\(kWh/m^2\)
\(l\)	lifetime	\(yr\)
\(m\)	mass	\(kg\)
\(n\)	project horizon	\(yr\)
\(pd\)	period duration	\(h\)
\(\phi\)	solar gain fraction	\(kW/m^2\)
\(q\)	thermal power	\(kW/m^2\)
\(\rho\)	density	\(kg/m^3\)
\(s\)	shading factor	\(-\)
\(x\)	coordinate, pointing east	\(-\)
\(y\)	coordinate, pointing north	\(-\)
\(z\)	coordinate, pointing to zenith	\(-\)

Dual variables

\([\beta]\)	epsilon constraint for multi objective optimization
\([\mu]\)	incentive to change design proposal
\([\pi]\)	cost or global warming potential of electricity

Superscripts

A	appliances
B	building
L	light
P	people
bat	bateobatle
bes	bes
cap	cap
chp	chp
cw	cw
\(-\)	demand
dhw	domestic hot water
el	electricity
ERA	enery reference area
ext	external
gain	heat gain
ghi	global horizontal irradiation
gr	grid
hp	heat pump
int	internal
inv	investment
irr	irradiation
max	maximum
min	minimum
net	netto
ng	natural gas
op	operation
pv	photovoltaic panel
r	return
ref	reference
rep	replacement
s	supply
SH	space heating
stat	static
\(+\)	supply
tot	total
TR	transformer

Indexes

0	nominal state
II	ref. to 1st law of thermodynamics
II	ref. to 2nd law of thermodynamics
\(b\)	building
\(f\)	facades
\(i\)	iteration
\(k\)	temperature interval
\(l\)	linearization interval
\(p\)	period
\(pt\)	patch
\(r\)	replacement
\(t\)	timestep
\(u\)	unit

Sets

\(A\)	azimuth angles
\(B\)	buildings
\(F\)	facades
\(I\)	iterations
\(K\)	temperature levels
\(L\)	linearization intervals
\(O\)	orientations
\(P\)	typical periods
\(R\)	roofs
\(S\)	skydome patches
\(T\)	timesteps
\(U\)	units
\(U_r\)	units that need replacements
\(Y\)	tilt angles

Inputs

For the application of REHO, the energy hub description needs to contain - as highlighted by REHO model architecture :

• the End Use Demands (EUDs), from the meteorological data and the buildings characteristics,

• the resources to which it has access to provide those EUDs, namely the grids,

• the equipments that can be used to convert those resources into the required services.

End use demand profiles

Middelhauve² - Section 1.2

The EUDs profiles to be determined are:

• The demand profile for domestic hot water

• The demand profile for domestic electricity

• The demand profile for space heating computed with:

  • The internal heat gains from occupancy,

  • The internal heat gains from electric appliances,

  • The heat exchange with the exterior,

  • The solar gains from the irradiance,

• The demand profile for mobility.

Statistical profiles

When real data is not available, the profiles can be estimated using statistical data.

In the case of REHO, the consumption profiles are computed from statistical data on buildings characteristics, combined with weather data.

Buildings characteristics

The buildings are defined by their usage type, their morphology, and their heating performance.

Usage

Usage is defined by the building category (I to XII) from SIA 380/1:2016. It defines, combined with SIA 2024:2015, the statistical profiles for each category in terms of occupation, lighting and hot water demand.

These profiles are generally specific to each room type and usage.

List of SIA 380/1 categories

I	Collective housing
II	Individual housing
III	Administrative
IV	School
V	Commercial
VI	Restaurant
VII	Gathering places
VIII	Hospital
IX	Industry
X	Shed, warehouse
XI	Sports facilities
XII	Covered swimming-pool
XIII	Other

Morphology

• Energy reference area (ERA) \(A_{ERA} [m^2]\)

• Roof surfaces \(A_{roofs} [m^2]\)

• Facades surfaces \(A_{facades} [m^2]\)

• Glass fraction \(g^{glass} [-]\)

Heating performance

• Year of construction or renovation

• Quality of thermal envelope

  • Overall heat transfer coefficient \(U_{h} [kW/K/m^2]\)

  • Heat capacity coefficient \(C_{h} [Wh/K/m^2]\)

• Temperatures of supply and return for heating system \(T_{h,supply}-T_{h,return} [°C]\)

• Temperatures of supply and return for cooling system \(T_{c,supply}-T_{c,return} [°C]\)

• Reference indoor temperature \(T_{in} [°C]\)

The heating technique is maily measured in degrees Celsius. In building we have heating and cooling system. They include supply and return temperatures for both heating and cooling. The supply and return temperatures for cooling are captured by temperature_cooling_supply_C and temperature_cooling_return_C, respectively. Similarly, the parameters temperature_heating_supply_C and temperature_heating_return_C represent the corresponding temperatures for the heating system. The target temperature to be reached inside the building is defined by the parameter temperature_interior_C.

Weather data

To calculate energy demand profiles the outdoor ambient temperature global irradiation for the region in study are necessary.

• Outdoor ambient temperature (yearly profile) \(T_{out} [°C]\)

• Global horizontal irradiation (yearly profile) \(Irr_{out} [°C]\)

Data reduction

The hourly timesteps of a typical annual profile, leads to 8760 data points per year. This leads, together with the complexity of the model, to computationally untraceable models. Reducing the size of the data representing the energy demand of the renewable energy hub and weather conditions is required. The aggregation of timeseries to typical periods is specifically popular, as patterns occur naturally in the supply and demand of energy, which arise in the time dimension through hourly, daily and seasonal cycles. The k-medoids clustering algorithm is used in REHO. Typical days are identified based on two variables: global irradiation and ambient temperature.

NB: Extreme periods are also considered, but only for the design of the capacities.

Grids

In the REHO model, a grid is characterized by the energy carrier it transports and its specifications.

Energy layers

Several energy carriers are considered in REHO, namely:

• Electricity,

• Natural gas,

• Oil,

• Wood,

• District heat,

• Biomethane,

• Hydrogen,

• Carbon dioxide,

• Fossil fuel,

• Mobility (service expressed in pkm),

• Data (ICT service).

These layers are modeled through parameters that can be changed in the model:

• Import and export tariffs,

• Carbon content,

• Environmental impact.

They can be set as constant through the year or specified at an hourly resolution.

Specifications

Cost, environmental impact, maximal capacity for district imports and exports.

Equipments

The model has to choose between several energy conversion and energy storage technologies that can be installed to answer the EUDs.

The units are parametrized by:

• Specific cost (fixed and variable costs, valid for a limited range \(f_{min}\) - \(f_{max}\))

• Environmental impact (= grey energy encompassing the manufacturing of the unit, and distributed over the lifetime of the unit)

• Thermodynamics properties (efficiency, temperature of operation)

Building-level units

District-level units

The units cannot be used at the building-scale.

Technology	Input stream	Output stream	Reference unit
Energy conversion technologies
Gas boiler	natural gas	heat	$$kW_{th}$$
Geothermal heat pump	ambient heat, electricity	heat	$$kW_{th}$$
District heating network	heat	heat	$$kW_{th}$$
Cogeneration	natural gas	electricity, heat	$$kW_{e}$$
Electricity storage technologies
EV charger	electricity*	electricity*	$$kWh$$
Electrical vehicle	electricity*	electricity*, mobility	$$kWh$$
ICE vehicle (internal combustion engine)	fossil fuel	mobility	$$unit$$
Bike		mobility	$$unit$$
Electric bike	electricity	mobility	$$unit$$
Battery	electricity	electricity	$$kWh$$

Note

EVs are not directly connected to the Layer electricity. Rather, intermediate variables representing the exchanges between EVs and charging stations are used, and the import of electricity from the Grid to charge the vehicles can be observed through the EV charger demand \(\boldsymbol{\sum_{u \in EVcharger}\dot{E}_{u,p,t}^{-}}\) (see annex).

Model

Objective functions

Middelhauve² - Section 1.2.4

REHO can optimize energy hubs considering economic indicators (minimizing operational expenses, capital expenses, total expenses) or environmental indicators (global warming potential).

As objectives can be generally competing, the problem can be approached using a Multi-Objective Optimization (MOO) approach. MOO is implemented using the \(\epsilon\)-constraint method to generate Pareto curves.

Annual operating expenses

\[\boldsymbol{C^{op}_b} = \sum_{l \in \text{L}} \sum_{p \in \text{P}} \sum_{t \in \text{T}} \left( c^{l, +}_{p,t} \cdot \boldsymbol{ \dot{E}^{gr,+}_{b,l,p,t} } - c^{l,-}_{p,t}\cdot \boldsymbol{ \dot{E}^{gr,-}_{b,l,p,t} } \right) \cdot d_t \cdot d_p \quad \forall b \in \text{B}\]

Annual capital expenses

\[\begin{split}\begin{align} \boldsymbol{C^{cap}_b} &= \frac{i(1+i)}{(1+i)^n -1} \cdot \left(\boldsymbol{C^{inv}_b } + \boldsymbol{C^{rep}_b } \right) \label{eq_ch1:Ccap}\\ \boldsymbol{C^{inv}_b }&= \sum_{u \in \text{U}} b_{u} \cdot \left( i^{c1}_{u} \cdot \boldsymbol{y_{b,u}} + i^{c2}_{u} \cdot \boldsymbol{f_{b,u}} \right) \label{eq_ch1:Cinv}\\ \boldsymbol{C^{rep}_b} &= \sum_{u \in \text{U}} \sum_{r \in \text{R}} \frac{1}{\left( 1 + i \right)^{r \cdot l_u}} \cdot \left( i^{c1}_{u} \cdot \boldsymbol{y_{b,u}} + i^{c2}_{u} \cdot \boldsymbol{f_{b,u}} \right) \quad \forall b \in \text{B} \label{eq_ch1:Crep} \end{align}\end{split}\]

Annual total expenses

\[\boldsymbol{C^{tot}_b} = \boldsymbol{C^{cap}_b} + \boldsymbol{C^{op}_b} \quad \forall b \in \text{B}\]

Global warming potential

\[\boldsymbol{G^{op}_b} = \sum_{l \in \text{L}} \sum_{p \in \text{P}} \sum_{t\in \text{T}} \left( g^{l,+}_{p,t} \cdot \boldsymbol{\dot{E}^{gr,+}_{b,l,p,t}} - g^{l,-}_{p,t} \cdot \boldsymbol{\dot{E}^{gr,-}_{b,l,p,t}} \right) \cdot d_p \cdot d_t \quad \forall b \in \text{B}\]

\[\boldsymbol{G^{bes}_b }= \sum_{u \in \text{U}} \frac{1}{l_u}\cdot \left( i^{g1}_u \cdot \boldsymbol{y_{b,u}} + i^{g2}_u\cdot \boldsymbol{f_{b,u}} \right) \quad \forall b \in \text{B}\]

\[\boldsymbol{G^{tot}_b} = \boldsymbol{G^{bes}_b} + \boldsymbol{G^{op}_b} \quad \forall b \in \text{B}\]

Building-level constraints

Sizing constraints

Upper and lower bounds for unit installations are necessary for identifying the validity range for the linearization of the cost function of the unit.

The main equation for sizing and scheduling problem units are described by:

\[\begin{split}\begin{align} \boldsymbol{y_{b,u}} \cdot F_u^{min} &\leq \boldsymbol{f_{b,u}} \leq \boldsymbol{y_{b,u}} \cdot F_u^{max} \\ \boldsymbol{f_{b,u,p,t}} &\leq \boldsymbol{f_{b,u}}\\ \boldsymbol{y_{b,u,p,t}} &\leq \boldsymbol{y_{b,u}}\\ & \quad \forall b \in \text{B} \quad \forall u \in \text{U} \quad \forall p \in \text{P} \quad \forall t\in \text{T} \nonumber \end{align}\end{split}\]

Energy balance

The energy system of the building includes all the different unit technologies that are used to fulfil the building’s energy demand.

\[\begin{split}\begin{align} \boldsymbol{\dot{E}_{b,p,t}^{gr,+}} + \sum_{u \in \text{U}} \boldsymbol{ \dot{E}_{b,u,p,t}^{+}} &= \boldsymbol{\dot{E}_{b,p,t}^{gr,-}}+ \sum_{u \in \text{U}} \boldsymbol{\dot{E}_{b,u,p,t}^{-}} + \dot{E}_{b, p, t}^{B,-} \label{eq_ch1:Ebalance} \\ \boldsymbol{\dot{H}_{b,p,t}^{gr,+}} &= \sum_{u \in \text{U}} \boldsymbol{\dot{H}_{b,u,p,t}^{-}} \qquad \qquad \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T} \label{eq_ch1:Hbalance} \end{align}\end{split}\]

Heat cascade

\[\begin{split}\begin{align} \boldsymbol{\dot{R}_{k,b,p,t} }- \boldsymbol{ \dot{R}_{k+1,b,p,t}} &= \sum_{u_h \in \text{S}_h} \boldsymbol{\dot{Q}_{u_h,k,b,p,t}^{-}}- \sum_{u_c \in \text{S}_c} \boldsymbol{\dot{Q}_{u_c,k,b,p,t}^{+}} \label{eq_ch1:heatK1}\\ \boldsymbol{\dot{R}_{1,b,p,t}}&= \boldsymbol{\dot{R}_{n_k+1,b,p,t}} = 0 \qquad \qquad \forall k \in \text{K} \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T} \label{eq_ch1:heatK2} \end{align}\end{split}\]

Thermal comfort

The general form of the SH demand can be expressed by the first order dynamic model of buildings:

\[\boldsymbol{\dot{Q}_{b,p,t}^{SH}} = \dot{Q}_{b,p,t}^{gain} - U_{b}^{h} \cdot A^{ERA}_b \cdot (\boldsymbol{T^{int}_{b,p,t}} - T^{ext}_{p,t}) - C^h_b \cdot A^{ERA}_b \cdot (\boldsymbol{T^{int}_{b,p,t+1}} - \boldsymbol{T^{int}_{b,p,t}}) \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T}\]

Where heat gains are constituted by:

\[\dot{Q}^{gain}_{b,p,t} = \dot{Q}^{int}_{b,p,t} + \dot{Q}^{irr}_{b,p,t}\quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T}\]

With internal heat gains calculated based on SIA 2024:2015 and include the rooms usage:

\[\dot{Q}^{int}_{b,p,t} = A^{net}_b \cdot \sum_{r \in Rooms} f_{b,r} \cdot f^{u}_{r,p} \cdot (\Phi^{P}_{r,p,t} + \Phi^{A+L}_{r,p,t}) \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T}\]

And solar heat gains proportional to the global irradiation, through a solar gain coefficient:

Middelhauve² - Section 3.2.4 Solar heat gains

\[\dot{Q}^{irr}_{b,p,t} = A^{ERA}_b \cdot \phi^{irr} \cdot \dot{irr}^{ghi}_{b,p,t} \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T}\]

Note

The internal building temperature \(T_{int}\) is considered as a variable to be optimized. This allows the building heat capacity to work as an additional, free thermal storage for the building energy system, thus making it possible to use available surplus electricity, which was generated onsite.

Penalty costs

Clearly, comfort should also be taken into account: this is achieved through the introduction of a penalty cost in the optimization problem objective at each hour when the indoor temperature exceeds pre-defined bounds. These penalty costs are deduced in a post-computing step.

Domestic hot water

\[{Q}^{dhw,-}_{b} = A^{net}_b \cdot \sum_{r \in Rooms} f_{b,r}\cdot f^{u}_{r,p} \cdot V^{dhw,ref}_{r} \cdot \frac{n^{ref}}{A^{net}_r}\cdot c_p^{dhw} \cdot \rho^{dhw} ( T^{dhw} - T^{cw}) \quad \forall b \in \text{B}\]

Domestic electricity

\[\dot{E}^{B}_{b,p,t} = A^{net}_b \cdot \sum_{r \in Rooms} f_{b,r} \cdot f^{u}_{r,p} \cdot \dot{e}^{A+L}_{r,p,t} \quad \forall b \in \text{B} \quad \forall p \in \text{P} \quad \forall t\in \text{T}\]

Storage

Cyclic constraints are imposed both on the indoor temperature and on thermal and electrical energy storage systems, to ensure the the state is reset to its initial status at the end of each period.

A tank for domestic hot water is mandatory, and one for space heating is possible – generally helps to increase the self-consumption of PV + HP combination.

District-level constraints

Decomposition algorithm (Dantzig-Wolfe) to break down the energy community into a master problem (transformer perspective) and one subproblem for each building ones. The obtained solution is an approximation of the compact formulation (= solving all the buildings simultaneously, exponential computational complexity) but has a linear computational complexity.

Configuration selection

\[\begin{split}\begin{align} 0 \leq \boldsymbol{\lambda_{i,b}} & \leq 1 \quad \forall i \in \text{I}, \quad \forall b \in \text{B} \label{eq_ch4:convex_1}\\ \sum_{i \in \text{I}} \boldsymbol{\lambda_{i,b}} &= 1 \quad \forall b \in \text{B} \quad \backsim [\mu_b] \label{eq_ch4:convex_2}\ \end{align}\end{split}\]

\[\sum_{i \in \text{I}} \sum_{b \in \text{B}} \boldsymbol{\lambda_{i,b}} \cdot \left( \dot{E}^{gr,+}_{i,b,p,t} - \dot{E}^{gr,-}_{i,b,p,t} \right) \cdot d_p \cdot d_t = \boldsymbol{E^{TR,+}_{p,t}} - \boldsymbol{ E^{TR,-}_{p,t} }\quad \forall p \in \text{P}, \quad \forall t \in \text{T} \quad \backsim [\pi_{p,t}]\]

\[\boldsymbol{C^{el}} = \sum_{p \in \text{P}} \sum_{t \in \text{T}} \left( c^{el, +}_{p,t} \cdot \boldsymbol{E^{TR,+}_{p,t}} - c^{el,-}_{p,t}\cdot \boldsymbol{ E^{TR,-}_{p,t}} \right)\]

\[\boldsymbol{G^{el}} = \sum_{p \in \text{P}} \sum_{t\in \text{T}} \left( g^{el}_{p,t} \cdot \boldsymbol{E^{TR,+}_{p,t}} - g^{el}_{p,t} \cdot \boldsymbol{E^{TR,-}_{p,t}} \right)\]

\[\begin{split}\begin{align} \boldsymbol{C^{op}} &= \boldsymbol{C^{el}} + \sum_{i \in \text{I}} \sum_{b \in \text{B}} \boldsymbol{\lambda_{i,b}} \cdot C^{gas}_{i,b} \label{eq_ch4:opex} \\ \boldsymbol{C^{cap}} &= \sum_{i \in \text{I}} \sum_{b \in \text{B}} \boldsymbol{\lambda_{i,b}} \cdot C^{cap}_{i,b} \label{eq_ch4:capex} \\ \boldsymbol{C^{tot}} &= \boldsymbol{C^{cap}} + \boldsymbol{C^{op}} \label{eq_ch4:totex}\\ \boldsymbol{G^{tot}} &= \boldsymbol{G^{el}} + \sum_{i \in \text{I}} \sum_{b \in \text{B}} \boldsymbol{\lambda_{i,b}} \cdot \left(G^{gas}_{i,b} + G^{bes}_{i,b} \right) \label{eq_ch4:GWP} \end{align}\end{split}\]

\[\begin{split}\begin{align} &\boldsymbol{TOTEX} = \boldsymbol{OPEX} + \boldsymbol{CAPEX} \label{totex}\\ &\boldsymbol{OPEX} = \sum_{\substack{l\in L}} c^+_l \cdot \boldsymbol{E^{net, +}_l} -c^-_l \cdot \boldsymbol{E^{net, -}_l} \label{opex}\\ &\boldsymbol{CAPEX} = \frac{i(1+i)}{(1+i)^n-1}(\boldsymbol{C^{inv}}+\boldsymbol{C^{rep}}) \label{capex}\\ &\boldsymbol{C^{inv}} = \sum_{\substack{u\in U}}b_u\cdot(i^{c1}_u\cdot \boldsymbol{y_u}+i^{c2}_u\cdot \boldsymbol{f_u}) \label{cinv}\\ &\boldsymbol{C^{rep}} = \sum_{\substack{u\in U}}\sum_{\substack{r\in R}}\frac{1}{(1+i)^{r\cdot l_u}}\cdot(i^{c1}_u\cdot \boldsymbol{y_u}+i^{c2}_u\cdot \boldsymbol{f_u}) \label{crep} \end{align}\end{split}\]

Grid capacity

The maximum capacity of the local low-voltage transformer is considered. The electricity export and the import is constrained within the feasibility range of the transformer.

Fig. 4 Energy flows and network constraints in REHO

Energy flows and network constraints in REHO distinguishes the:

• Grid = energy flows within the district boundary

• Network = exchanges with the district exterior, through the interface (transformer perspective)

\[\begin{split}\begin{align} &\sum_{b \in \text{B}} (\boldsymbol{\dot{E}^{gr,+}_{b,l,p,t}} - \boldsymbol{\dot{E}^{gr,-}_{b,l,p,t}}) \cdot d_p \cdot d_t = \boldsymbol{E^{net,+}_{l,p,t}} - \boldsymbol{ E^{net,-}_{l,p,t} } \qquad \forall l, p, t \in \text{L, P, T} \label{grid constraints}\\ &\boldsymbol{\dot{E}^{net,\pm}_{l,p,t}} \leq \dot{E}^{net, max}_l \qquad \forall l, p, t \in \text{L, P, T} \label{Transformer max} \end{align}\end{split}\]

Outputs

Decision variables

• Installed capacities for building-level and district-level units

• Operation time throughout a year

These fully characterize the energy flows at building-level and district-level, as well as the financial flows (investments + operational costs).

Key performance indicators

The KPIs are divided in four subgroups: Environmental, economical, technical and security indicators. For more information on how to calculate the KPIs presented below, please refer to Middelhauve² - Section 1.2.5 Key performance indicators.