## 2 The Proposed Methodology
### Energy Retrofit Results and Discussion  
In this case study, we first give the analysis of this case in a single-period setting, and then extend the single-period model to a multi-period model considering different cases of analysis until \(K=30\) years. The single-period energy retrofit planning problem (4) presents 48 integer variables, 96 bounding constraints, and 1 inequality constraint, whilst the multi-period energy retrofit planning problem (15) presents 96 \(K\) integer variables, 192 \(K\) bounding constraints, 48\(K\) equality constraints, and \(K\) inequality constraints. The analysis of the widest planning horizon (i.e., \(K=30\)) implies solving a problem with 2880 variables, 5760 bounding constraints, 1440 equality constraints, and 30 inequality constraints. In both settings, the problem is implemented and solved in the MATLAB environment integrated with IBM ILOG CPLEX Optimization Studio [34]. All the presented computations are executed on a PC equipped with a 4.0 GHz Intel Core i7 CPU and 16 GB RAM. The computational times are less than 1 minute in all the cases. Figure 1 compactly reports the results of the planning problem in terms of retrofit actions in the case of initial investment budget \(b=30\),000.00 (\(\xi\)) for both single- and multi-period (with \(K=20\) years) setting. Figure 1a shows the quantities of actions to be deployed normalized with respect to the application potentials. Conversely, Figure 1b shows the cumulative quantities of actions to be deployed, normalized with respect to the application potentials over the planning horizon. As expected, from Figure 1 it is evident that using a multi-period planning approach is more convenient than the single-period approach. In fact, after the initial investment, the subsequent savings in energy expenditures are used to accumulate cash to invest later in further energy retrofit actions, generating a beneficial snowball effect. More in detail, the single-period retrofit plan consists in implementing a limited set of actions (namely, retrofit actions \(j\in\{11,15,18,19,20,21,22,28\}\)) in Table 2, whilst the multi-period retrofit plan allows implementing a wider set of actions (namely, in addition retrofit actions \(j\in\{1,6,26,27,31,32,36,37,42,45\}\)) in Table 2. Specifically, the obtained multi-period retrofit planning consists of replacing a total of 379 luminaries with compatible LED lamps and installing a total of 926 EHMs, whilst the single-period retrofit planning consists of replacing 34 luminaries with compatible LED lamps and installing 1 EHM only. To compare results in terms of overall achieved energy savings, and ensure the equality of conditions in the single- and multi-period setting, we consider an identical time window in both cases. Hence, we compare the total energy savings obtained by the multi-period optimization over the planning horizon \(\mathcal{K}\) and the cumulative energy savings obtained by extrapolating the energy savings achieved by the single-period optimization in the initial period throughout the whole analysis time window \(\mathcal{K}\). Denoting \(\mathbf{x}_{1}^{*},\ldots,\mathbf{x}_{k}^{*},\ldots,\mathbf{x}_{K}^{*}\) as the optimal retrofit plan computed by the multi-period approach--i.e., a solution of optimization problem (12)--the total energy savings achieved over the planning horizon \(\mathcal{K}\) is expressed as:  
\[ES_{K}^{m}=\sum_{k=1}^{K}\sum_{j=1}^{J}p_{kj}\sum_{i=1}^{k}x_{ij}^{*} \tag{18}\]  
where \(\mathbf{p}_{k}\) (\(k=1,\ldots,K\)) are the unitary payoff vectors estimated in accordance with (16). On the other hand, in case of single-period planning, denoting \(\dot{\mathbf{x}_{1}}\)--i.e., an optimizer of (4)--the total energy saving achieved at the end of time period \(K\) is computed as follows:  
\[ES_{K}^{*}=\sum_{k=1}^{K}\mathbf{p}_{k}^{T}\dot{\mathbf{x}_{1}}. \tag{19}\]  
Figure 2 shows the energy savings over the analysis time window achieved by the single- and multi-period retrofit planning for different initial investment budgets.