\section{Uncertainty on decay heat}
The envelope decay heat in the core model at equilibrium was discussed and estimated in Sect. \ref{coreDH}. Fuel assemblies with different burnup convey the largest value at different decay time and the contribution of the nuclide species more relevant for decay heat is established as function of time.\\
In this section the uncertainty on the decay heat as function of decay time is estimated. The sources of uncertainty considered in this analysis are those introduced in the previous section. The objective here is to comprehend how the uncertainties of individual nuclides combine to form the overall uncertainty of an integrated quantity like decay heat.\\

\textit{Uncertainty on decay heat evolution provided by most irradiated assembly}\\
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{images/plots/FA/uncertainty/decayHeat_unc.png}
\caption{Decay heat uncertainty evolution from cross section and fission yield uncertainty propagation. A decay time up to 70 years is considered.}\label{unc_dh}
\end{figure}
A first result presented in Fig. \ref{unc_dh} shows the evolution of decay heat uncertainty for a fuel assembly being irradiated for 18 cycles, that gives the largest contribution in the long term. The \acrshort{FP}s relevant for decay heat are not affected by cross section uncertainties but they have large uncertainty coming from \acrshort{NFPY} since they are produced directly by fission or by primary fission products decay . This phenomenon is demonstrated by the plot since the decay heat uncertainty is dominated by \acrshort{NFPY} uncertainty in the short term, when fission products are relevant. For longer decay time, the major contributors are actinides, $^{242}$Cm up to 2 years followed by $^{238}$Pu and $^{241}$Am. The cross section uncertainties of these nuclides explain the evolution of decay heat uncertainty determined by \acrshort{XS} propagated from different \acrshort{NDL}, that were already commented in the previous section. Conservative estimations are provided by JENDL-4.0u that gives larger uncertainty estimation of $^{242}$Cm, $^{238}$Pu and $^{241}$Am concentrations. JEFF-3.3 gives the largest uncertainty in the last part of the transient. The peak uncertainty estimated is around 1.75\% between 5 and 200 days after discharge and it is provided propagating the cross sections from JENDL-4.0u.\\

The total decay heat uncertainty quantification is performed propagating cross section and \acrshort{NFPY} at the same time and considering both the least and most irradiated assembly that convey the largest decay heat respectively for short and long decay periods. JEFF-3.3 sampled nuclear data are employed to generate perturbed input files. \\
\begin{comment}
The evolution of the uncertainty of envelope decay heat is depicted in Fig. \ref{} and it reflects the plot showing the separate effects even if in that case only the most irradiated assembly was taken into account.\\
\end{comment}
The decay heat uncertainty of the most irradiated assembly is expected to be the quadratic sum of the two effects (cross section and \acrshort{NFPY}) considered separately if the two mechanisms are independent; the mathematical formulation is written down in eq. \eqref{uncertaintyProp}.
\begin{equation}
    u_{global}=\sqrt{u_{xs}^2 \ + u_{nfpy}^2}
    \label{uncertaintyProp}
\end{equation}
\begin{comment}
interazione tra fy e xs nel modello,
non correlazione tra gli input (xs,fy)
superposition principle works
\end{comment}
The two approaches do not provide a significant difference on the uncertainty evolution, therefore the methodologies are equivalent and there is negligible interaction between the uncertainties conveyed by cross sections and \acrshort{NFPY}, so that the effects on the decay heat are independent.\\

The uncertainty quantification of the decay heat of the fuel assembly irradiated for one cycle is performed, since it provides the envelope decay heat in the short decay period after discharge. The uncertainty propagation is carried out propagating both cross sections and fission yields from JEFF-3.3 and the outcome is shown in Fig. \ref{unc_dh_corr}.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{images/plots/FA/uncertainty/dh_unc_cycles.png}
\caption{Decay heat uncertainty evolution of a fuel assembly irradiated for 1 and 18 cycles. Brown line shows the correlation between the two cases.}\label{unc_dh_corr}
\end{figure}
In the short term in the assembly irradiated one cycle shows the largest uncertainty as well as the largest decay heat. For longer time scales the uncertainty of the 18 cycles irradiated FA is larger. This phenomenon is related to the uncertainty of the main contributors that can vary with burnup, as for plutonium isotopes (Fig. \ref{unc_pu}) and to the different sensitivity of the decay heat to certain nuclides. The uncertainty will increase if a nuclide with large uncertainty is more relevant for the decay heat at a specific decay time.
The correlation evolution, the dashed curve in Fig. \ref{unc_dh_corr}, gives an indication on how much the decay heat uncertainty in the two cases comes from the same source .
For example in the long term decay heat is given by $^{238}$Pu and $^{241}$Am: this takes place both in the fuel assembly irradiated for 1 and 18 cycle and this implies high correlation, although at the same time the resulting uncertainty is quite different since it increases with burnup for the two nuclides. Therefore the resulting uncertainty after 1 cycle is much lower.\\
Similar conclusions can be drawn on the high correlation from the beginning of the transient up to one day of decay. In this interval of time, decay heat is dominated by the short lived nuclides produced in the same way in the first and last irradiation cycle (even if in different amount). The phenomenon brings to high level of correlation in the time interval. Lower correlation is observed in the intermediate decay time and it suggests that uncertainty is associated to different nuclides. The analysis could be expanded to consider all fuel assembly batches with intermediate burnup level. Even though it's expected that they will exhibit intermediate behavior compared to the least and most irradiated fuel assemblies, a comprehensive analysis of these intermediate cases can provide valuable insights into the how the various factors affect the decay heat uncertainty.\\

\begin{center}
\fbox{
\begin{minipage}{0.98\textwidth}
The uncertainty on decay heat coming from nuclear data is established for a fuel assembly irradiated for one and 18 cycles, that provide the largest decay heat in the short and long cooling time, respectively. The uncertainty is larger, up to 2.5\%, for the FA irradiated for one cycle in the short term and then it reduces to almost zero. The uncertainty in the most irradiated fuel assembly is between 1 and 1.5\% for the entire transient considered.  
\end{minipage}}
\end{center}\\