## 2 The Proposed Methodology
### Optimization Problem Formulation
#### 2.2.2 Retrofit Planning in a Multi-Period Setting
The first one (i.e., \(\Big{(}b_{k-1}-c_{k-1}^{T}x_{k-1}\Big{)}(1+r_{k-1})\)) representsthe budget portion that has not been used in the preceding period (\(k-1\)) and is properly revalued based on the corresponding interest rate \(r_{k-1}\). The second contribution in (12) (i.e., \(\mathbf{s}_{k-1}^{T}\mathbf{z}_{k-1}\)) represents how monetary savings in energy expenditure in preceding periods are accumulated as financial resources to be subsequently invested in further retrofit interventions. Note that, denoting by \(\mathbf{e}=(e_{1},\ldots,e_{k},\ldots,e_{K})\) the column vector of unitary energy prices over the planning horizon, \(\bm  
around 2 GWh [25]. Table 1 reports the outcome of the diagnosis phase: 24 types of lamps are totally installed, classified into the following five categories: high-pressure sodium vapor lamp (HPS); metal halide lamps (MH); low-pressure mercury vapor lamps (Hg_LP); mercury mixed-light lamps (Hg_ML); metal halide projectors (MH_pr); high-pressure sodium vapor floodlights (HPS_pr). As a further outcome of the decision design phase, we identify two class of retrofitting actions in the compliance with road lighting norms and standards [26; 27; 28]: replacement of old lamps with well-suited LED lamps and installation of energy harvesting modules (EHM) compatible to lighting unit characteristics. Several street lighting LED luminaires are nowadays available on the market, given remarkable advantages of LED technologies in terms of reduction in electrical energy consumption due to their high efficiency and in maintenance costs due to their long lifetime [29]. Hence, for each lighting unit type in the existing street lighting system, we select a new LED luminaire having the same photometric features, but characterized by a lower installed power. Similarly, energy savings could be realized by equipping street lighting units with EHMs (systems that store the energy from natural resources--typically photovoltaic panels-in rechargeable batteries) [30]. Consequently, for each lighting unit type we select a compatible EHM. For all the candidate actions, the compliance with the existing light management system is verified. Table 2 reports the complete list of retrofitting actions (in total \(J=48\) actions, i.e., 24 actions for each class), including the current energy savings (in kWh/year) and costs (in \(\xi\)), whilst application potentials are straightforwardly inferable from Table 1. Estimates of current energy savings and costs in Table 2 are based on data sheets for urban lighting products [31] and on updated prices for public works in the Bari city area [32]. As regards the energy savings variation over the planning horizon, we assume that payoffs do not significantly change, i.e., the model function in (8) corresponds to the identity mapping:  
\[\mathbf{p}_{1}=\mathbf{p},\ \ \mathbf{p}_{k}=\mathbf{p}_{k-1},\ \ \ k=2,\ldots,K. \tag{16}\]  
As for the costs variation over planning horizon, we assume that costs grow at a constant inflation rate, i.e., the model function in (9) corresponds to the revaluation by a fixed rate \(\varphi=2\%\):  
\[z_{kj}=\sum_{i=1}^{k}x_{ij},\;k=1,\ldots,K,j=1,\ldots,J. \tag{6}\]0  
Finally, we set the current unitary energy price to be equal to the electricity price (taxes included) for non-household medium size consumers (i.e., annual consumption within the range of [500, 2.000] MWh) in the first half 2018 in Italy [33]: \(e_{1}=0.1642\) (\(\xi\)). Furthermore, we assume that the variation of unitary energy price over planning horizon is negligible, as occurred to the average electricity price (taxes included) for non-household medium size consumers in Europe since first half of 2016 [33]. Therefore, we set \(e_{k}=e_{1},\ k=2,\ldots,K\). Finally, we assume that the interest rate is constant, specifically \(r_{k}=2\%,k=1,\ldots,K\).