## 6 Problem Formulation
### Constraints
#### 6.2.1 Operating Constraints for Micro-Turbines  
For \(\forall i,t\), the following constraints are considered:  
\[C_{i,t,s}^{opv}=C_{i,t,s}^{gen}+C_{i,t,s}^{start}+C_{i,t,s}^{en} \tag{9}\]  
\[C_{i,t,s}^{gen}=a_{i}V_{i,t,s}+\Delta T\sum_{m=1}^{N_{i}}\lambda_{i,m}P_{i,m,t,s} \tag{10}\]  
Figure 6: Model implementation process.  
\[P_{i,1,t,s}\leq P_{i,1}^{\max}-P_{i}^{\min} \tag{11}\]  
\[P_{i,m,t,s}\leq P_{i,m}^{\max}-P_{i,m-1}^{\max} \tag{12}\]  
\[P_{i,NL_{i},t,s}\leq P_{i}^{\max}-P_{i,NL_{i}-1}^{\max} \tag{13}\]  
\[P_{i,t,s}=P_{i}^{\min}V_{i,t,s}+\sum_{m=1}^{N_{i}}P_{i,m,t,s} \tag{14}\]  
\[C_{i,t,s}^{start}=k_{start}on_{i,t,s} \tag{15}\]  
\[C_{i,t,s}^{em}=C_{CO_{2}}CO_{2,i,t}P_{i,t,s} \tag{16}\]  
where \(C_{i,t,s}^{epe}\) is the operation cost of micro-turbine \(i\) in scenario \(s\) at time \(t\) and consists of generation, startup and emission costs as (9). \(C_{i,t,s}^{gen}\) is the generation cost of micro-turbines \(i\) in scenario \(s\) at time \(t\) and is modeled in (10)-(14) by piecewise linear function [47]. \(a_{i}\) is the generation cost of micro-turbine \(i\) at its minimum generation. \(m\) is the segments index in the cost function of micro-turbine \(i\). \(N_{i}\) and \(\lambda_{i,m}\) are the number of segments in the cost function of micro-turbine \(i\) and the marginal cost of segment \(m\) in the cost function of micro-turbine \(i\) ($/kWh), respectively. \(P_{i,m}^{max}\) is the maximum limit of power generation in the \(m\)-th segment of micro-turbine \(i\) cost function (kW). \(P_{i,m,t,s}^{max}\) is the power generation of micro-turbine \(i\) from the \(m\)-th segment in scenario \(s\) at time \(t\). Furthermore, Start-up cost and emission cost are given in (15) and (16), respectively. \(k_{start}\) is startup constant cost. \(CO_{2,i,t}\) is carbon dioxide pollutant of micro-turbine \(i\) at time \(t\) and \(C_{CO_{2}}\) is externality cost of generation carbon dioxide ($/kg).  
\[P_{i}^{\min}V_{i,t}\leq P_{i,t,s}\leq P_{i}^{\max}V_{i,t} \tag{17}\]  
\[P_{i,t,s}-P_{i,t-1,s}\leq RU_{i}(1-on_{i,t,s})+P_{i}^{\min}on_{i,t,s} \tag{18}\]  
\[P_{i,t-1,s}-P_{i,t,s}\leq RD_{i}(1-off_{i,t,s})+P_{i}^{\min}off_{i,t,s} \tag{19}\]  
\[\sum_{\ell=t}^{t+MUT_{i}-1}V_{i,\ell}\geq MUT_{i}on_{i,t,s} \tag{20}\]  
\[\sum_{\ell=t}^{t+MDT_{i}-1}(1-V_{i,\ell})\geq MDT_{i}off_{i,t,s} \tag{21}\]  
\[on_{i,t,s}-off_{i,t,s}=V_{i,t,s}-V_{i,t-1,s} \tag{22}\]  
\[on_{i,t,s}+off_{i,t,s}\leq 1 \tag{23}\]  
In (17), \(P_{i,t,s}\) is the generating power of micro-turbine \(i\) in scenario \(s\) at time \(t\) and is limited to their bounded maximum and minimum capacity, i.e., \(P_{i}^{max}\) and \(P_{i}^{min}\). \(RU_{i}\) and \(RD_{i}\) are the ramping up and down rates of micro-turbine \(i\) and are limited in (18) and (19), respectively. The (20) and (21) impose the minimum up/down time constraints for micro-turbines. \(MUT_{i}\) and \(MDT_{i}\) are the minimum up and down times of micro-turbine \(i\). Finally, the binary variables relationships are formulated as (22) and (23). \(on_{i,t,s}\), \(off_{i,t,s}\) and \(V_{i,t,s}\) are the startup, shutdown and commitment indicators of micro-turbine \(i\)[48; 49; 50].