## 4 Case Studies
### Probabilistic Forecasting of ALSC
#### 4.2.3 The Impact of Multi-Slack Buses on Probabilistic ALSC  
In this study, two types of power imbalance allocations are performed at multi-slack buses. In order to study the impact of multi-slack based on probabilistic ALSC, according to the different number of times and principles of power imbalance allocation, four different methods are considered, which are shown in Table 11. Probabilistic evaluations of ALSC are performed with four different methods, and the results are shown in Table 12. More specifically, the influence of multi-slack buses on probabilistic ALSC is analyzed from two perspectives: (1) the impact of the principle of the first power imbalance allocation; (2) the impact of the principle of the second power imbalance allocation. First, the influence of the first power imbalance allocation on the results of probabilistic ALSC is discussed. By comparing the results of Method 1 and Method 3 (i.e., the first and the third rows in Table 12) and the results of Method 2 and Method 4 (i.e., the second and the fourth rows in Table 12), it can be found that the probabilistic results of ALSC of Method 3 and Method 4 containing the first power imbalance allocation are different from those of Method 1 and Method 2 without the first power imbalance allocation, demonstrating the first part of power imbalance caused by forecasting errors can affect the probabilistic results of ALSC. The formation mechanism of the first power imbalance is as follows. Due to the forecasting errors, wind power, photovoltaic power and load power are all random variables. Probabilistic ALSC calculation needs samples from these random variables and then performs deterministic ALSC calculation. The sampling values are different each time,  
Furthermore, it can be seen that the \(S_{\rm{ALSC}}\) of Method 3 is larger than that of Method 1 and the \(S_{\rm{ALSC}}\) of Method 4 is larger than that of Method 2, indicating that the consideration of the first power imbalance allocation increases the \(S_{\rm{ALSC}}\). The reason lies in that the first power imbalance allocation brings a larger difference of ALSC calculation results with different sampling values. To be more specific, the cause of the \(S_{\rm{ALSC}}\) is that the different sampling values of wind power, photovoltaic power and load power lead to different results of deterministic ALSC calculation. When the sampling values of wind power and photovoltaic power are large and the sampling value of load power is small (i.e., the first part of power imbalance \(\Delta P_{1}>\)0), the first power imbalance allocation will decrease the power of traditional generators at slack buses, while when the sampling values of wind power and photovoltaic power are small and the sampling value of load power is large (i.e., the first part of power imbalance \(\Delta P_{1}<\)0), the first power imbalance allocation will increase the power of traditional generators at slack buses. If the first power imbalance allocation is not required, the initial power (i.e., the power before the load growth and the second power imbalance allocation) of the traditional generator is constant. With the first power imbalance allocation, different sampling values of wind power and photovoltaic power and load power will cause the initial power of the traditional generators to change, which magnifies the difference between the different sampling values and hence increases the \(S_{\rm{ALSC}}\). Next, the influence of the principles of the second power imbalance allocation on the results of probabilistic ALSC is discussed. Two different allocation principles of the second power imbalance allocation are considered herein: (1) Method 1 and Method 3 with the ratio of the current power of the generators; (2) Method 2 and Method 4 with the ratio of reciprocal of each generator cost. According to the results in Table 11, it is obvious that the \(E_{\rm{ALSC}}\) calculated by the method with the second allocation principle is smaller.