## 2 Problem Outlines
### The Objective Function
#### 2.3.9 Load Forecast Uncertainty  
Load forecasting is a critical item in many planning applications in power systems. Load forecasting can also be performed in different intervals for various purposes. Load forecasts are dependent on weather conditions (temperature, humidity, air brightness, wind speed). In addition, each day of the week has its own load curve. Load consumption curves on holidays and non-holidays are also different from each other. In different seasons of the year, according to the factors specific to each season, such as day length, the load consumption pattern will be different. Short-term load forecasting is a pivotal data needed for running a day-ahead market. The results of STLF are employed by GENCOs to discover prices with regard to the way other market participants may act. Price forecast signals are also applied to bidding strategy techniques to maximize profitability. The system operator will clear the market for the next-day delivery according to the forecasted values along with the bids from GENCOs. ISO is responsible for balancing the market between generation level and the forecasted demand at each interval. However, with regard to the inaccuracy of forecasts, there is always a violation that must be redressed by storage units or other controlling actions such as load shedding. There are multiple approaches to deal with load forecasting. The most common methods are the time series method, regression methods, and intelligent methods. Intelligent methods are also classified into several categories such as artificial neural networks, fuzzy logic-based methods, also ANFIS models, wavelet transformation methods, as well as support vector machines. In order to deal with uncertainties, probabilistic models can be integrated into the employed load forecasting paradigm. In this study, the Gaussian process quantile regression (GPQR) is employed to deal with uncertainties [38]. Due to the stochastic nature of load patterns as well as various external factors such as weather conditions, calendar effects, and seasonal factors, the power demand signals exhibit intermittent and volatile characteristics. Hence, a prediction scheme is needed to provide the most probable distribution of power demand patterns rather than a crisp value. A Gaussian process model is a set of random variables, in which any member corresponds with a probability distribution that can be represented by the following function:  
\[f(l)\sim GP\big{(}\mu(l),COV\big{(}l,l^{\prime}\big{)}\big{)} \tag{23}\]  
In this equation, \(\mu(l)\) and \(\text{COV}(l,l^{\prime})\) denote the mean and covariance functions that can be calculated as follows:  
\[\mu(l)=E\big{\{}f(l)\big{\}} \tag{24}\]  
\[COV\big{(}l,l^{\prime}\big{)}=E\Big{\{}f(l)-\mu(l)]\times\big{[}f(l^{\prime} )-\mu(l^{\prime})\big{]}^{T}\Big{\}} \tag{25}\]  
The covariance function illustrates the similarity between data points. One of the most widely-used functions for describing covariance is the squared exponential (SE) model that can be represented as follows:  
\[COV_{SE}\big{(}l,l^{\prime}\big{)}=\theta_{f}^{2}\exp\left(\frac{\|l-l^{ \prime}\|}{\theta_{length}^{2}}\right) \tag{26}\]  
The parameters \(\theta_{f}\) and \(\theta_{l}\) control the scale and length. With regard to the differentiability of this covariance, which implies that the Gaussian process is very smooth, a simplified alternative, known as Matern covariance, can be replaced because, in practice, no phenomena do not reflect such a strong smoothness. \[COV_{Mat}\big{(}l-l^{\prime}\big{)}=\sigma^{2}\frac{2^{1-v}}{\Gamma(v)}\bigg{(} \sqrt{2v}\frac{l-l^{\prime}}{i}\bigg{)}^{v}B_{v}\bigg{(}\sqrt{2v}\frac{l-l^{ \prime}}{i}\bigg{)} \tag{27}\]  
In the above, \(B_{v}\) represents the modified Bessel function, in which \(v\) and \(i\) are both positive hyperparameters. To simplify this covariance function, the value of \(v\) is supposed to be \(v=p+1/2\). In this parameter, \(p\) is a non-negative integer.