## 2 The Proposed Methodology
### Optimization Problem Formulation
#### 2.2.2 Retrofit Planning in a Multi-Period Setting  
The multi-period decision model aims at identifying the optimal retrofitting interventions to be separately implemented at the beginning of the \(K\) time periods. To this aim, \(K\) column vectors ofnon-negative integer decision variables \(x_{k}=\left(x_{k1},\ldots,x_{kj},\ldots,x_{kj}\right)\) (\(k=1,\ldots,K\)) are defined to reflect the quantitative choices on the retrofitting interventions implementation over the given planning horizon. For instance, if action \(j\) denotes the deployment of energy harvesting modules, \(x_{kj}\) represents the quantity of modules to be deployed at the beginning of time interval \(k\). All the decision variables are lower bounded by zero (i.e., no intervention) and are upper bounded by the application potential of the given intervention:  
\[x_{k}\in\chi,k=1,\ldots,K. \tag{5}\]  
In addition, \(K\) column vectors of supporting integer variables \(\mathbf{z}_{k}=\left(z_{k1},\ldots,z_{kj},\ldots,z_{kj}\right)\) (\(k=1,\ldots,K\)) are introduced to define the cumulative quantities of retrofitting interventions implemented along the time horizon. For instance, \(z_{kj}\) represents the quantity of action \(j\) deployed up to the time interval \(k\):  
\[z_{kj}=\sum_{i=1}^{k}x_{ij},\;k=1,\ldots,K,j=1,\ldots,J. \tag{6}\]  
It is apparent that also the above defined supporting variables are lower bounded by zero (i.e., no intervention) and are upper bounded by the application potential of the given interventions:  
\[z_{k}\in\chi,k=1,\ldots,K. \tag{7}\]  
Furthermore, differently from the single-period setting, the multi-period planning implies considering time-varying pattern for energy savings and retrofit action costs [22]. As for energy savings, we assume that the performance of the energy efficiency retrofits can deteriorate subject to usage of the retrofitted items. Denoting \(\mathbf{p}_{k}=\left(p_{k1},\ldots,p_{kj},\ldots,p_{kj}\right)\) (\(k=1,\ldots,K\)) as column vectors of unitary payoffs over the planning horizon, we model the payoff dynamics as follows:  
\[\mathbf{p}_{1}=\mathbf{p},p_{kj}=g^{(p)}\Big{(}p_{(k-1)j}\Big{)},k=2,\ldots,K,\;j=1, \ldots,J \tag{8}\]  
where \(g^{(p)}(\cdot)\) is the payoff variation model function. As for retrofit action costs, denoting \(\mathbf{c}_{k}=\left(c_{k1},\ldots,c_{kj},\ldots,c_{kj}\right)\) (\(k=1,\ldots,K\)) as column vectors of unitary costs related to retrofit actions as implemented along the planning horizon, we model the cost dynamics as follows:  
\[\mathbf{c}_{1}=\mathbf{c},\mathbf{c}_{k}=g^{(c)}(\mathbf{c}_{k-1}),k=2,\ldots,K \tag{9}\]  
where \(g^{(c)}(\cdot)\) is the cost variation model function. The choice of the decision variables over the given planning horizon is constrained by the financial resources limitation. Hence, the following set of inequalities must be satisfied:  
\[\sum_{j=1}^{J}c_{kj}x_{kj}=c_{k}^{T}x_{k}\leq b_{k},\;k=1,\ldots,K \tag{10}\]  
where \(b_{k}\) denotes the budget allocated to the retrofit actions plan of the whole system at the beginning of time period \(k\). At the first time interval, the entire initial investment budget can be used (i.e., \(b_{1}=b\)), whilst in the succeeding time intervals the available budget is determined in accordance with cash-flow dynamics:  
\[b_{1}=b \tag{11}\]  
\[b_{k}=\Big{(}b_{k-1}-c_{k-1}^{T}x_{k-1}\Big{)}(1+r_{k-1})+s_{k-1}^{T}z_{k-1}, k=2,\ldots,K. \tag{12}\]  
where \(r_{k}\) denotes the interest rate at time interval \(k\), and \(\mathbf{s}_{k}=\left(s_{k1},\ldots,s_{kj},\ldots,s_{kj}\right)\) (\(k=1,\ldots,K\)) are column vectors collecting all the energy cost savings achieved during time interval \(k\) for each retrofit action (specifically, \(s_{kj}\) is the energy cost saving achieved during time period \(k\) in case retrofit action \(j\) is implemented no later than the beginning of time period \(k\)). Note that budget \(b_{k}\) in Equation (12) is obtained by summing two contributions.