## 2 The Proposed Methodology
### Energy Retrofit Results and Discussion
It is apparent that the multi-period planning produces significantly higher energy savings for the same initial investment budget. For instance, the initial budget \(b=\xi\) 10,000.00 yields a total energy saving of 0.7 MWh in \(K=30\) years by using the single-period retrofit planning. This equals about one twentieth of the total energy savings achieved in the same time window for the same initial budget by using the multi-periodretrofit planning (16 MWh). Finally, Figure 2 shows (as expected) that the deviation between results in the single- and multi-period planning is more marked for longer planning windows and lower initial investment budgets. In order to further highlight the advantages of using a multi-period retrofit optimization approach over a single-period method, we also compare the net present values (NPVs) of the financial gains achieved when single- and multi-period retrofit planning techniques are used to maximize the energy savings. Indeed, NPV is used in the budgeting and investment planning phase as one of the most common profitability indexes for investment projects [35]. It is defined as the difference between the present value of cash inflows and the present value of cash outflows over a period of time [36]. Figure 1: The cumulative energy savings over time intervals achieved by single- (**a**) and multi-period (**b**) retrofit planning for an initial investment budget \(b=\epsilon\) 30,000.00. Therefore, in the two cases of single- and multiple-time periods, NPVs are expressed as follows, respectively:  
\[NPV_{K}^{s}=\left(b-c^{T}\dot{x_{1}}\right)+\sum_{k=1}^{K}\frac{s_{k}^{T}\dot{x_{ 1}}}{\prod_{i=1}^{k}(1+r_{i})}-b \tag{20}\]  
\[NPV_{K}^{m}=\frac{\left(b_{K}-c_{K}^{T}x_{K}\right)(1+r_{K})+s_{K}^{T}z_{K}}{ \prod_{k=1}^{K}(1+r_{k})}-b. \tag{21}\]  
where \(c_{k}\) and \(s_{k}\) (\(k=1,\ldots,K\)) are the unitary cost and cost energy savings vectors estimated by the corresponding models (9) and (13), and \(r_{k}\) (\(k=1,\ldots,K\)) denotes the interest rate at time interval \(k\). Figure 2: The cumulative energy savings over time intervals achieved by single- and multi-period retrofit planning for different initial investment budgets: (**a**) \(b=\xi\) 10,000.00, (**b**) \(b=\xi\) 30,000.00, and (**c**) \(b=\xi\) 50,000.00. Figure 3 shows the NPVs achieved by the single- and multi-period retrofit planning for different initial investment budgets. It is apparent that, considering a short planning horizon \(K\) (i.e., less than 20, 24, and 26 years for an initial investment budget \(b\) equal to \(\xi\)10,000.00, \(\xi\)30,000.00, and \(\xi\)50.000.00, respectively), the single-period approach has a higher NPV than the multi-period planning. This is caused by the fact that in the multi-period planning approach, the energy cost savings are invested in further retrofit actions. Conversely, substantial savings are obtained in the remaining part of the planning horizon in the multi-period approach. Hence, exceeding the above identified break point, the multi-period planning produces significantly higher NPVs for the same initial investment. For instance, the initial budget \(b=\xi\) 10,000.00 yields NPVs equal to \(\xi\) 75,000.00 considering a time analysis window of \(K=30\) years by using the single-period retrofit planning. This results about half the NPVs that are achieved considering the same time window for the same initial budget by using the multi-period retrofit planning (\(\xi\) 150,000.00). Finally, Figure 3 shows that the deviation between results obtained in the single- and multi-period planning is more marked for longer planning windows and lower initial investment budgets. The above numerical experiments carried out by varying the time horizon demonstrate that, while staging the interventions on the system may be a valuable strategy which allows savings to be obtained in several cases to (either energetical or economical), it might not be the best approach when the time horizon decreases. Hence, a proper assessment of the right strategy needs to be performed on a case by case basis.