## 5 The Agent-Based Model
### Simulation Phase
Each factor in Equation (1) was weighted by a class-dependent parameter; these weights are \(w_{P}(c_{v})\), \(w_{B}(c_{v})\), \(w_{N}(c_{v})\) and \(w_{E}(c_{v})\)--the notation serves to express their dependency on the agent class \(c_{v}\). Each agent had its set of specific weights; agents of the same social class do not share the same weights (at least not necessarily--this still could happen as a byproduct of the parameters tuning procedure). A key aspect of the proposed approach was assigning correct values to these weights: this is the crucial operation where the self-reported behaviour obtained with questionnaires and interviews (which guided the definition of the agents behaviour) was merged with the observed behaviour (historical data of installed PV power). This passage will be described in detail in Section 6. The first factor in Equation (1) regards the payback period, \(pp\). To obtain a balanced influence of all factors in the utility functions, all factors where normalized in a [0,1] range. In the case of the payback time, the normalization takes advantage of the bounds on the minimum payback period \(min(pp)\) (assumed to be equal to one year) and on the maximum payback period \(max(pp)\), assumed to be 21 years since the expected useful life for PV systems is 20 years. Hence, the payback influence for agent \(v\) is computed following [28] and expressed by this equation:  
\[u_{P}(v)=\frac{max(pp)-pp(v)}{max(pp)-min(pp)}=\frac{21-pp(v)}{20} \tag{2}\]  
where \(pp(v)\) is the payback period for the initial investment. Its value is computed using the net present value (NPV) of the PV system--the NPV typically starts with negative values (due to the initial cost of the investment) and it gradually gets closer to zero, while the initial cost is offset by yearly gains due to electrical bill savings and sale of own-produced energy. When the NPV turned from negative to positive it indicated the point when the investment became profitable. The computation of the NPV was based on the yearly cash flows--each agent measures its expenses and gains (taking into account also national and regional incentives) and computes its yearly NPV accordingly. The household budget \(u_{B}(v)\) is given by:  
\[u_{B}(v)=\frac{e^{p_{budget}}}{v_{equity}} \tag{3}\]  
with the initial investment \(v_{equity}\) computed as the PV panel installation cost minus any applicable incentive. \(v_{B}\) is the disposable budget of the household. The third factor contributing to the agents' decision is the environmental benefit that can be gained by adopting PV technology, instead of consuming electrical energy coming from non-renewable resources. These benefits are measured in terms of oil saved, which is in turn correlated with an overall decrease in CO2 production. The Italian Regulatory Authority for Electricity and Gas provides a factor to convert the produced energy (expressed in MWh) to the equivalent in tonnes of oil (TOE) (A TOE is defined as the amount of energy released by burning one tonne of oil, or 0.187 TOE for each MWh produced). Thanks to this conversion, the ecological benefits can be computed with the following equation:  
\[u_{E}(v)=\frac{1}{e^{oil_{intConsumed}-oil_{consumed}}} \tag{4}\]  
The final component of the desire function (Equation (1)) is the influence of the other members of the social network of the agent. This factor is identified by \(u_{N}(v)\) and it encapsulates the importance of the neighbours' choices in shaping the household behaviour. As previously mentioned, the agent's neighbours are the nodes (other households) with a shared links; the vicinity of two nodes dependson geographical proximity and social class similarity. The neighbourhood influence contribution is computed with the following equation:  
\[u_{N}(v)=\frac{1}{1+e^{\frac{1}{2}L_{v,tot}L_{v,adopter}}} \tag{5}\]  
with \(L_{v,tot}\) being the total number of links of agent \(v\) and \(L_{v,adopter}\) the number of links shared with adopters.