## 3 Probabilistic Forecasting of ALSC
### The Repeated Power Flow Considering Multi-Slack Buses  
RPF is a common method for solving the deterministic ALSC problem, whose basic principle is that, starting from the basic condition, the load is gradually increased according to the given load growing pattern and power flow is calculated repeatedly until any electrical constraint is violated. The maximum load growing percentage \(\lambda\) is defined as  
\[\lambda=\frac{S_{Li}-S_{Li0}}{S_{Li0}} \tag{22}\]  
where \(S_{Li}\) is the increased load of bus \(i\) and \(S_{Li0}\) is the basic load of bus \(i\). In the process of RPF, the load will continue to increase. If all the growing load is borne solely by the slack bus, the generator at the slack bus have a high probability of exceeding its maximum output capacity, which does not conform to the actual situation and affects the accurate evaluation of ALSC. Therefore, it is necessary to set multi-slack buses during the RPF calculation. The main task of multi-slack buses is to allocate the power imbalance of the system. In this study, apart from the growing load, the forecasting errors of wind power, photovoltaic power and load power in the basic condition are another source of power imbalance. For two different power imbalance sources, two types of allocations are proposed and performed amongst multiple slack buses herein. 1. The first power imbalance allocation: the power imbalance source is the power forecasting errors. More specifically, the power forecasting errors are modeled as having a Gaussian distribution in Section 2.3. When LHS is used to sample from the distribution of random variables, each sampling value is different, which corresponds to the fluctuation of renewable energy generation and load power. This will lead to the power imbalance, which is relatively small. Therefore, the generator power adjustment process is similar to the secondary frequency regulation in the power system. For this kind of power imbalance, power allocation is performed according to the ratio of the remaining capacity of each generator amongst multi-slack buses. 2. The second power imbalance allocation: the power imbalance source arises from the growing load power. During the RPF calculation, loads need to be increased continuously to find the maximum load point, which can cause a significant power imbalance. This part of the power imbalance is relatively large and the load change is regular. Therefore, the generator power adjustment process is similar to the tertiary frequency regulation. The tertiary frequency regulation is essentially a problem of economic dispatch. Therefore, the allocation principle of power imbalance is selected according to the ratio of generator economy, which can be represented by the reciprocal of the cost of each generator. The calculation steps of solving the deterministic ALSC problem by RPF are introduced as follows:  
1. Obtain the reference value of power system loss \(P_{loss}\) by a power flow calculation under the mean value of all random input variables. The active power equation of power flow equations is as follows:\[P_{Gi}-P_{Di}-U_{i}\sum\limits_{j=1}^{N}U_{j}(G_{ij}\cos\delta_{ij}+B_{ij}\sin \delta_{ij})=0\] (23) where \(N\) is the number of buses; \(P_{Gi}\) is the active power of generator at bus \(i\); \(P_{Di}\) is the active power of load at bus \(i\); \(U_{i}\) is the voltage amplitude of bus \(i\); \(\delta_{ij}\) is the phase angle difference between bus \(i\) and bus \(j\); and \(G_{ij}\) and \(B_{ij}\) are the real part and imaginary parts of the bus admittance matrix, respectively. 2. Sample from the forecasting distribution of load, wind power and photovoltaic power at the current moment \(t\) and get the sampling points.