## Appendix C  
The change curve of wind power, PV power, and load can be simulated by Weibull distribution and normal distribution. In the intraday stage, the wind farm, solar station, and load have random characteristics. Therefore, the two-point estimation method is used to deal with this uncertainty, in which each uncertainty variable is replaced with the deterministic value on both sides of the node mean, and the deterministic power flow is calculated. In form, this calculation can be regarded as a multivariate nonlinear function:  
\[Z=h(X) \tag{10}\]  
where \(X\) is a random input vector, which is used to describe the wind power output, photovoltaic power output, and load, and \(Z\) is the output variable, which represents the intraday adjustment cost of thermal power unit relative to the day-ahead scheduling plan.  
\[X=[X_{1},X_{2},X_{3},\cdots,X_{M}] \tag{11}\]  
where \(M\) is the dimension of a random variable, which is determined by the number of wind power units, photovoltaic units, and loads. The estimated information of each random variable consists of a group \((x_{l,K},p_{l,K})\), \(I\) = 1,2...,\(M\), \(K\) = 1,2, and \(x_{l,K}\) is estimated point, and the corresponding estimate vector is \([\mu_{X,1},\mu_{X,2},\cdots x_{l,K},\cdots,\mu_{X,M}]\), \(p_{l,K}\) is the corresponding weight. The variables \(Z(I,K)=h(\mu_{X,1},\mu_{X,2},\cdots x_{l,K},\cdots,\mu_{X,M})\) are evaluated by the optimization algorithm and all the estimated points are calculated successively. Finally, the statistical information of the desired variables can be obtained.  
Given the probability distribution of random variables, the two-point estimation method is used to evaluate the probability distribution function of average adjustment cost \(Z\). The specific steps are as follows:  
Step 1: input system data, including branch parameters, transformer ratio, and other deterministic data, as well as the expectation and generation-load distribution function, and determine the number \(M\) of uncertain variables.  
Step 2: initialize \(I=1\);\(E(Z)=0\);\(E\!\left(Z^{2}\right)=0\).  
Step 3: select the random variable of node injection \(X_{I}\), and determine the position coefficient \(\xi_{I,1}\) and \(\xi_{I,2}\), and probability concentration \(p_{I,1}\) and \(p_{I,2}\).  
\[\left\{\begin{array}{l}\xi_{I,1}=\sqrt{m}\\ \xi_{I,2}=-\sqrt{m}\\ p_{I,1}=p_{I,2}=\frac{1}{2m}\end{array}\right. \tag{12}\]  
Step 4: calculate the \(X_{I,1}\) and \(X_{I,2}\) of estimated distribution points on both sides of the mean value;  
\[\left\{\begin{array}{l}X_{I,1}=\mu_{X,I}+\xi_{I,1}\sigma_{X,I}\\ X_{I,2}=\mu_{X,I}+\xi_{I,2}\sigma_{X,I}\end{array}\right. \tag{13}\]  
where \(\mu_{X,I}\) and \(\sigma_{X,I}\) are the mean and standard deviation of random quantities respectively.  
Step 5: the improved bat algorithm is adopted to calculate deterministic power flow optimization for random variables \(X_{I,K}\)(\(K\) = 1,2), and the average adjustment cost in each period of each estimation point can be obtained.  
\[X=[\mu_{X,1},\mu_{X,2},\cdots,x_{l,K},\cdots,\mu_{X,M}] \tag{14}\]  
Step 7: update \(E(Z)\) and \(E\!\left(Z^{2}\right)\).  
\[E(Z)\approx\sum_{I=1}^{M}\sum_{K=1}^{2}(P_{I,K}h([\mu_{X,1},\mu_{X,2},\cdots,X _{l,K},\cdots,\mu_{X,M}])) \tag{15}\]  
\[E\!\left(Z^{2}\right)\approx\sum_{I=1}^{M}\sum_{K=1}^{2}\!\left(P_{I,K}h([\mu _{X,1},\mu_{X,2},\cdots,X_{l,K},\cdots,\mu_{X,M}])^{2}\right) \tag{16}\]  
Step 7: repeat from step 3 to step 6, \(I=I+1\) and loop over the variables.  
Step 8: calculate the mean and standard deviation:  
\[\mu_{Z}=E(Z)\] (A15)  
\[\sigma_{Z}=\sqrt{E(Z^{2})-\mu_{Z}^{2}}\] (A16)