As it has been discussed in section 3.1.1, the most appropriate time resolution for studying the integration of renewable energies is the hourly one. In a first effort to run the model with hourly time steps, the annual share of lighting end-uses (%l i g ht i ng), the annual share of SH end-uses (%sh),
and the period capacity factor (cp,t) are changed from a monthly to a hourly resolution. The time
periods duration (top) is modified to 1 hour in the model.
However it was not possible to start the optimization problem due to memory issues. The attempts were done with a desktop computer with an Intel(R) Xeon(R) CPU E3-1240 V2 3.40 GHz processor, 8.00 GB of installed memory (RAM) and having the virtual memory managed by the system. The computation process was stopped because the system was running out of memory when loading eq. 17 of the model in [34]. This equation defines the operation strategy for the decentralized heating technologies. It warranties that the relative use of each technology in each period is constant, except for the solar thermal, which has priority since it has zero operational cost.
Hence we propose a modified formulation of the model in order to avoid the use of Eq. 17 in [34] while maintaining an equivalent operation strategy for the decentralised heating technologies, and the hourly time resolution. In order to facilitate the better understanding of the new formulation, we would like to recommend the reader to skim through the first section of the SI in [34], where the equations, variables and parameters of the original MILP model are explained.
As mentioned the first modification consist on deleting Eq. 17. To warranty that solar thermal has priority over the other decentralized heating technologies (e.g. decentralised boilers), its heat supply is discounted to EndUses(HeatLowTDec) (Eq. 3.1). Furthermore, it has been assumed that solar thermal technologies can be also installed in buildings connected to district heating networks (DHNs). The calculation of EndUses(HeatLowTDHN) (Eq. 3.2) is modified to apply this new assumption. In order to respect the linearity of Eq. 3.1 and Eq. 3.2, the ratio centralized over total low-temperature heating (%Dhn) is defined as parameter, which is explicitly calculated in Eq. 3.3.
EndUses(Heat LowT Dec, t )= (EndUsesInput(HeatLowT HW )/
t∈Ttop(t )+ EndUsesInput(Heat LowT SH )· %sh(t )/top(t )− Ft(DecSolar, t ))· (1 − %Dhn)
∀t ∈ T (3.1) EndUses(Heat LowT D H N , t )= (EndUsesInput(HeatLowT HW )/
t∈Ttop(t )+ EndUsesInput(Heat LowT SH )· %sh(t )/top(t )− Ft(DecSolar, t ))· %Dhn
∀t ∈ T (3.2) %Dhn=
i∈TECH OF EUT(HeatLowTDHN)fmin(i )
j∈TECH OF EUC(HeatLowT)fmin( j )
(3.3)
A parameter (FfixSolar) and an equation (Eq. 3.4) are added to fix the decentralised solar thermal
installed capacity. A second equation defines the operation in each period (Eq. 3.5). Since solar ther- mal is treated differently in comparison to the other heating and CHP technologies, Eq. (3.6,3.7,3.8) do not apply to it anymore. The parameter fmin(Solar) is thus always equal to zero.
F(DecSolar)= F f i xSolar (3.4)
Ft(DecSolar, t )= F(DecSolar)cp,t(DecSolar, t ) t∈ T (3.5)
Ft( j , t )≤ F(j)cp,t( j , t ) k= DecSolar,∀j ∈ TECH \ {k},∀t ∈ T (3.6)
t∈T
Ft( j , t )top(t )≤ F(j)cp( j )
t∈T
top(t ) k= DecSolar,∀j ∈ TECH \ {k} (3.7)
i∈RES∪TECHX\(STO∪DecSolar) f (i , l )Ft(i , t ) +
j∈STO
(Stoout( j , l , t )− Stoin( j , l , t ))− EndUses(l,t) = 0
∀l ∈ L,∀t ∈ T (3.8) Eq. 3.9 defines the operation strategy for the heating and transport technologies. The relative use of each technology in each period should be constant. This equation warranties the operation strategy that was defined by the problematic constraint Eq. 17 in [34], and makes dispensable the use of Eq. (18,19,20) from [34]. Hence they are removed together with the variable YSolar.
fmin,%( j ) j ∈TECH OF EUC(euc)\DecSolar Ft( j , t )≤ Ft( j , t )≤ fmax,%( j ) j ∈TECH OF EUC(euc)\DecSolar Ft( j , t )
k= Electr ici t y,∀euc ∈ EUC \ k,∀j ∈ TECH OF EUC(euc),∀t ∈ T (3.9) However, if because of the operation strategy imposed by Eq. 3.9, CHP systems are active during periods with electricity excess, fuel will be used to produce electricity that will find no consumer. Thus this operation strategy may lead to non-optimal operation of the heating system from an energy and economic point of view. In that case, boilers should provide the heating service. Eq. (3.10,3.11,3.12,3.13) are implemented to avoid that to happen. They are based on the idea that
decentralized CHP systems are installed in combination with auxiliary boilers. In other words, a building with a CHP system also have a peak boiler. These boilers can be operated when the use of CHP systems induces excess electricity production. This use of auxiliary boilers combined with CHP system is introduced in [92]. Only a percentage (%CogenBoiler) of the installed capacity of boilers
is assumed to be able to work as auxiliary boilers. They do not represent an extra capacity to the existing boilers, hence their use as auxiliary systems is limited by Eq. (3.11,3.12,3.13).
i∈AUX REPLACED(j)
Ft(i , t )≤ F(j)%CogenBoiler ∀j ∈ REPL ACED,∀t ∈ T (3.10)
t∈T ( i∈AUX REPLACES(j) Ft(i , t )+ Ft( j , t ))top(t )≤ F(j)cp( j ) t∈T top(t ) ∀j ∈ REPL ACES (3.11) Ft( j , t )+ i∈AUX REPLACES(j) Ft(i , t )≤ F(j) ∀j ∈ REPL ACES,∀t ∈ T (3.12) i∈AUX REPLACES(j)
Ft(i , t )≤ F(j) ∀j ∈ REPL ACED,∀t ∈ T (3.13)
Eq. (3.14,3.15) are added into the model to avoid the optimiser to increase the installed capacity of the CHP technologies aiming to have more capacity for the auxiliary boilers. Eq. 3.15 applies to both DHN and decentralized heating, hence Eq. 24 in [34] is deleted. The parameter %PeakDHNin [34], which defines the ratio peak/maximum monthly average DHN heat demand is substituted by
%PeakHeatLowT, fixing the ration peak/maximum average hourly low temperature heat demand.
HeatLowTmax= max
t∈T
EndUses(Heat LowT D H N , t )+ EndUses(HeatLowT Dec,t)
∀t ∈ T (3.14) F( j )= HeatLowTmaxfmin( j )%PeakHeatLowT
k= DecSolar,∀j ∈ TECH OF EUC(HeatLowT) \ {k} (3.15)
The inputs from and outputs to the layers (f ) of an auxiliary boiler are calculated as the opposite to the ones of the CHP technology that is substituting, plus the inputs and outputs of the boiler itself (see table 3.2 for an example). In this way, the Ftof the CHP technology does not change, hence Eq. 3.9 is respected. The replacement is only done at the layer balances.
Table 3.2 – Calculation fo the layer input and output for an auxiliary decentralised NG boiler that can replace decentralised NG CHP systems.
Technology Inputs from and outputs to the layers
a
ELECTRICITY NG HEAT LOW T DECEN
-(Dec CHP NG) -0.957 2.174 -1
Dec Boiler NG 0 -1.111 1
Aux Boiler -0.957 1.063 0
aA negative value represents an input from the layer (consumption). A positive values is represents an output to the
layer (production).
REPLACED set includes the CHP technologies in TECH OF EUC (HEAT HIGH T) and TECH OF EUC (HEAT LOW T) sets (see Figure 1 in [34]). REPLACES set contains the boilers in the same two sets. The auxiliary technologies are parellely listed in two sets (AUX REPLACED and AUX REPLACES). The content of AUX REPLACED and AUX REPLACES is the same, the difference is the way the technologies are grouped in subsets. AUX REPLACED (replaced) subset groups the auxiliary technologies acting as auxiliary technology of the replaced CHP system in REPLACED. For example, AUX REPLACED (DecCHPng) contains DecCHPng-AUXBOILERng, DecCHPng-AUXBOILERwood and DecCHPng-AUXBOILERoil. AUX REPLACES (replaces) contains the auxiliary technologies using the replaces boiler. For example, AUX REPLACES (DecBOILERng) has DecCHPgas-AUXBOILERng, DecCHPoil-AUXBOILERng, AdvCHPng-AUXBOILERng and AdvCHPh2-AUXBOILERng. TECHX set is equivalent to the union of the technologies in TECH set merged with the auxiliary technologies, i.e. technologies in AUX REPLACES or AUX REPLACED.
This current version of the model requires about 20 minutes to be solved with the desktop computer whose characteristics have been reported earlier in this section.