## 3 Proposed Method
### Point Estimation Method  
Once the states of micro-turbine generators and elastic loads have been determined on the master level, the remaining problem becomes a stochastic optimization problem, which is solved on the slave level. The uncertain PV generation is estimated using the point estimation method [39; 40], as follows. Let the expected values, standard deviation and skewness coefficient of each uncertain PV generation (i.e., \(\widetilde{PV_{j}}\)) be \(\mu_{PV_{j}}\), \(\sigma_{PV_{j}}\) and, \(\lambda_{PV_{j}}\), respectively, where \(j=1,2,\ldots,G_{2}\).  
Step C1: Set the first and second moments of the \(\widetilde{P}_{i}(t)\) at the \(i\)-th generator, \(\widetilde{P}_{p}(t)\) and \(\widetilde{P}_{s}(t)\) to zero where \(i=1,2,\ldots,G_{1}\).  
\[E\left(\widetilde{P_{i}^{1}}(t)\right)=0,\;E\left(\widetilde{P_{i}^{2}}(t) \right)=0,\;E\left(\widetilde{P_{p}^{1}}(t)\right)=0,\;E\left(\widetilde{P_{ p}^{2}}(t)\right)=0,\;E\left(\widetilde{P_{S}^{1}}(t)\right)=0,\;E\left(\widetilde{P_{S}^{2}}(t) \right)=0 \tag{8}\]  
where the symbol \(E\) stands for the expectation operator.  
Step C2: Compute the two perturbations:  
\[\xi_{j,1}(t)=\frac{\lambda_{PVj}(t)}{2}+\sqrt{G_{2}+\left(\frac{\lambda_{PVj}( t)}{2}\right)^{2}},\;j=1,2,\cdots,G_{2};\;t=1,2,...,T \tag{9}\]  
Figure 6: (**a**) Insertion of a pair of slots; and (**b**) trimming values in slots to make the sum 24.  
Figure 7: Mutation operation for elastic load encoding.  
\[\xi_{j,2}(t)=\frac{\lambda_{PVj}(t)}{2}-\sqrt{G_{2}+\left(\frac{\lambda_{PVj}(t)}{2 }\right)^{2}},\;j=1,2,\cdots,G_{2};\;t=1,2,...,T \tag{10}\]  
In addition, compute the two weighting factors:  
\[\omega_{j,1}(t)=-\frac{\xi_{j,2}(t)}{G_{2}\big{(}\xi_{j,1}(t)-\xi_{j,2}(t) \big{)}} \tag{11}\]  
\[\omega_{j,2}(t)=\;\frac{\xi_{j,1}(t)}{G_{2}\big{(}\xi_{j,1}(t)-\xi_{j,2}(t) \big{)}} \tag{12}\]  
\[\sum_{j\;=\;1}^{G_{2}}\omega_{j,1}+\omega_{j,2}=1 \tag{13}\]  
Step C3: Estimate the two location parameters:  
\[PV_{j,k}(t)=\mu_{PV_{j}}(t)+\xi_{j,k}(t)\cdot\sigma_{PV_{j}}(t),\;k=1,2;\;j=1, 2,\cdots,G_{2};\;t=1,2,...,T \tag{14}\]  
Step C4: Let \(m\) be 1 where \(m\) denotes a moving index given for the PV arrays.  
Step C5: Let \(k\) be 1. Start to apply the positive perturbation using Equation (9) for the \(m\)-th PV array.  
Step C6: Find the optimal \(P_{i}(t),\;P_{p}(t)\) and \(P_{s}(t)\) where \(t=1,2,\ldots,T\).  
\[\text{Min}f=\sum_{t\;=\;1}^{T}\sum_{i\in\text{CI}(t)}[F_{i}(P_{i,mk}(t))+S_{i} (t)]+p_{p}(t)\cdot P_{p,mk}(t)-p_{s}(t)\cdot P_{s,mk}(t) \tag{15}\]  
subject to  
\[P_{p,mk}(t)-P_{s,mk}(t) +\sum_{i\in\text{CI}(t)}P_{i,mk}(t)+\sum_{j\in\text{G}_{2},j\
eq m }\mu_{PV_{j}}(t)+PV_{m,k}(t)=P_{in}(t)+\sum_{\ell\in\text{C2}(t)}P_{el}^{\ell} (t) \tag{16}\] \[t=1,\;2,\;\ldots,\;T\]  
\[P_{i}^{min}\leq P_{i,mk}(t)\leq P_{i}^{max},\;i\in\text{C1}(t) \tag{17}\]  
and the constraints on ramp rate that are expressed in Equations (5) and (6). The symbols \(C1(t)\) and \(C2(t)\) are the set of generators and elastic loads in the on state.  
Step C7: Let \(k=k+1\). If \(k=2\), then start to conduct the negative perturbation using Equation (10) for the \(m\)-th PV array and go to Step C6; else go to Step C8.  
Step C8: \(m=m+1\). If \(m>\) G2, then go to Step C9; else go to Step C5.  
Step C9: Compute the mean and standard deviation of \(P_{i}(t)\), \(i=1,2,\;\ldots,\;G_{2}\), \(t=1,2,\;\ldots,\;T\).  
\[E\big{(}P_{i}(t)\big{)}=\sum_{m\;=\;1}^{G_{2}}\sum_{k\;=\;1}^{2}\omega_{j,k}( t)\times P_{i,mk}(t) \tag{18}\]  
\[\sigma_{P_{i}(t)}=\sqrt{E\big{(}P_{i}^{2}(t)\big{)}-\big{(}E\big{(}P_{i}(t) \big{)}\big{)}^{2}} \tag{19}\]  
The calculations of the means and standard deviations of \(\widetilde{P}_{p}(t)\), \(\widetilde{P}_{s}(t)\) and cost are similar and omitted here.  
Solving the problem in Step C6 becomes easy because it is a quadratic programming problem. This work employs the interior point method in MATLAB to solve this problem [41].