## 3 PSSS Assessment Using Developed Efficient Probabilistic RES Model  
### Development of PVG Probability Function and WTG Output Power  
PVG output power increases with solar radiation until it reaches standard solar radiation (around 1000 W/m\({}^{2}\)), beyond which it remains constant at its rated capacity, as shown in Figure 2[28]. Moreover, output power from a solar farm is zero at night time. These two situations cause discontinuities in the PDF of PVG output power when it is computed using the general procedure, and are usually modeled as impulse functions, as shown in Figure 2[2; 3]. To avoid these discontinuities, researchers [8; 9; 10] commonly only use the linear region (where solar radiation is greater than zero and less than standard radiation) to compute the PDF of the PVG output power, which results in a continuous function and is much more convenient to use. However, the realistic probabilistic model of PVG output power should be a mixed random variable with two impulse functions, and a continuous function between them. However, this model is difficult to use with analytical methods, and a more efficient method based on power-forecast error to construct the probability density of PVG output power was utilized in our study. Figure 1: Proposed method to coordinate power-system-stabilizer (PSS) and battery energy-storage-system (BESS) controllers under renewable-energy-source (RES) integration. Both solar-power and wind-power forecast errors are commonly modeled as normal distribution, and vary with feed-in power as well as the time of day [29; 30]. Power-forecast error \(\Delta P_{m,pv}\) for an \(m^{th}\) solar farm is defined as  
\[\Delta P_{m,pv}=P_{m,pv}-\bar{P}_{m,pv} \tag{1}\]  
where \(P_{m,pv}\) is actual output power for solar farm \(m^{th}\), and \(\bar{P}_{m,pv}\) is the forecast output power of the solar farm, which is deterministic (average solar-farm output power [29] in this paper). The probability function of the power-forecast error can be computed as:  
\[f(\Delta P_{m,pv})=\frac{1}{\sigma_{\Delta P_{m,pv}}\sqrt{2\pi}}exp-\frac{[ \Delta P_{m,pv}-\mu_{\Delta P_{m,pv}}]^{2}}{2\sigma_{\Delta P_{m,pv}}^{2}} \tag{2}\]  
where \(\mu_{\Delta P_{m,pv}}\) and \(\sigma_{\Delta P_{m,pv}}\) are the mean and standard deviation of solar-power-forecast error, respectively. Thus, the output-power PDF for the \(m^{th}\) solar farm can be derived using Equations (1) and (2) with the help of a transformation:  
\[f(P_{m,pv})=\frac{1}{\sigma_{\Delta P_{m,pv}}\sqrt{2\pi}}exp-\frac{[P_{m,pv}- (\bar{P}_{m,pv}+\mu_{\Delta P_{m,pv}})]^{2}}{2\sigma_{\Delta P_{m,pv}}^{2}} \tag{3}\]  
As wind-power-forecast error is also commonly modeled as normal distribution [5; 29; 30], the output-power PDF for the \(n^{th}\) wind farm can also be derived in a similar manner to PDF computation of PVG output power as:  
\[f(P_{n,w})=\frac{1}{\sigma_{\Delta P_{n,w}}\sqrt{2\pi}}exp-\frac{[P_{n,w}-( \bar{P}_{n,w}+\mu_{\Delta P_{n,w}})]^{2}}{2\sigma_{\Delta P_{n,w}}^{2}} \tag{4}\]  
where \(\bar{P}_{n,w}\), \(\mu_{\Delta P_{n,w}}\) and \(\sigma_{\Delta P_{n,w}}\) are the forecast output power of the wind farm, which is deterministic (average wind-farm output power [29] in this paper), and mean and standard deviation of wind-power-forecast error respectively. Both Equations (3) and (4) are continuous and very convenient to use with analytical methods. The graphical illustration of PDF computation of PVG output power using solar-power-forecast error is shown in Figure 3. Beta distribution is commonly used to characterize solar radiation and has some value of shape parameters \(a\) and \(b\) depending on location [31]. These shape parameters are then used to estimate average solar radiation \(\bar{S}\), which is further used to compute the average PVG output power with the help of solar-farm power characteristics [28] in our work. Finally, the PDF of PVG output power is obtained by substituting the value of computed average PVG output power into Equation (3).