## 3 Reliability Criteria for IBG Units in MPPT Mode  
When IBG under MPPT control does not participate in frequency regulation, the uncertainties of the renewable generation and load are two critical factors related to reserve determination. Since the distribution of their prediction errors does not affect the proposed method, for simplicity, it can be assumed that the prediction errors obey a zero-mean Gaussian distribution. At the time period \(t\), the forecast error of RG is denoted by \(\Delta P_{t}^{\text{R}}\), of which the probability density function is \(f(\Delta P_{t}^{\text{R}})\sim N(0,{r_{t}}^{2})\). Similarly, the forecast error of demand is denoted by \(\Delta P_{t}^{\text{D}}\), of which the probability density function is \(f(\Delta P_{t}^{\text{D}})\sim N(0,{d_{t}}^{2})\). Furthermore, the net load forecast error can be defined as \(\Delta P_{t}^{\text{N}}\), which is defined as:  
\[\Delta P_{t}^{\text{N}}=\Delta P_{t}^{\text{D}}-\Delta P_{t}^{\text{R}} \tag{7}\]  
of which the probability density function is \(f(\Delta P_{t}^{\text{N}})\sim N(0,{r_{t}}^{2}+{d_{t}}^{2})\), since renewable generation and load are independent of each other. It is noted that for a power system with zero penetration of renewables, the net load forecast error only contains the error from the demand forecast. Additionally, the forced outage rate (FOR) of traditional generators is denoted by \(p^{\text{FO}}\).  
In this paper, the forecast error of the renewable generation and the demand, as well as the outage of traditional generators, are three factors that cause frequency events that lead to load shedding or renewable spillage incidents. Therefore, two indexes, LOLP and RSP, are proposed as two reliability criteria to evaluate the system.  
The frequency of the power system is directly coupled with the power balance between generation and demand. The under-frequency event may happen when the generation cannot meet the demand. Since the probability of outage of two or more generators is very low [35], in this paper, the scenarios of no generation outage and single generator outage are considered. The LOLP at a specific period \(t\) is denoted by \(p_{t}^{LOLP}\), which can be expressed as:  
\[\begin{array}{l}p_{t}^{LOLP}=\prod\limits_{i=1}^{N}(1-p_{i}^{\text{FO}}) \cdot p\Big{\{}(\Delta P_{t}^{\text{N}}-R_{t}^{u}  
Over-frequency events may happen when the generation exceeds the demand, which the downward reserve is unable to compensate. The calculation of RSP in period \(t\) is denoted by \(p_{t}^{RSP}\), which can be expressed by  
\[p_{t}^{RSP}=p\Big{\{}(\Delta P_{t}^{N}+R_{t}^{d})<0\Big{\}} \tag{9}\]  
where \(R_{t}^{d}\) is the downward reserve at period \(t\). If the rated LOLP and RSP are denoted by \(p_{rated}^{LOLP}\) and \(p_{rated}^{RSP}\), then the reliability constraints can be expressed as:  
\[p_{t}^{LOLP}\leq p_{rated}^{LOLP} \tag{10}\]  
\[p_{t}^{RSP}\leq p_{rated}^{RSP} \tag{11}\]  
The values of \(p_{rated}^{LOLP}\) and \(p_{rated}^{RSP}\) are set by the dispatch system operators, which are usually between 0 and 0.05. The solutions of the upward and downward reserve \(R_{t}^{u}\) and \(R_{t}^{d}\) satisfying the desired reliability criteria can be found by searching the LOLP and RSP solution spaces, respectively.