## 3 Modeling of Power Consumption Profiles for Residential, Commercial, and Industrial Consumers
### Influence of Temperature Sensitivity on Power Consumption of Consumer  
The data of humidity and temperature is gathered from the meteorological department and incorporated TS influence on power consumption by using Pearson correlation [39] given as  
\[\rho=\sum_{i=1}^{n}\frac{(x_{i}-\overline{x})(y_{i}-\overline{y})}{S_{x}S_{y}(n- 1)} \tag{8}\]  
where \(n\) is the number of samples, \(\overline{x}\) and \(\overline{y}\) are the mean, while \(S_{x}\) and \(S_{y}\) are the standard deviations of \(x\) and \(y\), respectively.  
The above correlation helps to obtain the correlation coefficient of power consumption, humidity, and temperature for commercial, industrial, and residential consumers [40]. A high value of correlation coefficient 0.88 is calculated for temperature versus power consumption, which indicates that the rise in temperature will significantly increase the power consumption. On the the rainy days, a higher level of humidity with low temperature is observed, which is also determined by the correlation coefficient having a negative value for power consumption versus humidity and temperature. Therefore, humidity in rainy weather influences less as compared to high temperature in dry weather. The power consumption of all three classes of consumers as a function of humidity and temperature is calculated by multiple regression analysis by  
\[P(T_{n},\,H_{n})=1.2-1.48T_{n}-2.53H_{n}+0.89T_{n}^{2}+1.7T_{n}H_{n}+1.3H_{n} \tag{9}\]  
where \(T_{n}\) is the normalized temperature, which is a ratio of actual temperature \(T\) and mean temperature \(T_{mean}\). Similarly, \(H_{n}\) is normalized humidity, which is a ratio of actual humidity \(H\) and mean humidity \(H_{mean}\). The influence of temperature on power consumption profile of consumers is given as  
\[P_{n}=\alpha+\beta T_{n}+\gamma T_{n}^{2} \tag{10}\]  
where \(P_{n}\) is the normalized power consumption of consumers, which is the ratio of actual power \(P\) and mean power \(P_{mean}\). The coefficients of regression model are denoted by \(\alpha\), \(\beta\), and \(\gamma\). The discussion on the influence of temperature and humidity on power consumption plots is presented in the results and discussion section of paper. The upper and lower bounds of 96% confidence intervals are given by  
\[P=P_{n}\pm t_{\alpha/2}s_{P_{n}} \tag{11}\]  
where the standard deviation of \(P_{n}\) is denoted by \(s_{P_{n}}\) and \(t_{\alpha/2}\) represent the confidence interval. High fitness of regression model is obtained for temperature and power consumption, with a deterministic coefficient greater than 0.81, thus the \(t\) and \(F\) tests of \(\alpha\), \(\beta\), and \(\gamma\) are verified.  
The first order derivative for power consumption and temperature effect is applied to determine the TS of power consumption by  
\[\text{TS}=\frac{\partial P_{n}}{\partial T_{n}}=\beta+2\gamma T_{n} \tag{12}\]  
The power consumption with 1 degree centigrade rise in temperature is given as  
\[\Delta P_{n}=\text{TS}\times P_{mean}\times\frac{\Delta T_{n}}{T_{mean}} \tag{13}\]  
where \(\Delta T_{n}\) is the change in normalized temperature.  
Cumulative Effect of Residential, Commercial, and Industrial Consumers due to Temperature Sensitivity  
The temperature rise has a different effect on different classes of consumers, each of which has different load profiles and power consumption criteria on various time instantsof 24 h, week, month, and year. Based on power consumption profiles and TS analysis, the power consumption of all three consumer classes are solved by integrating them, given by  
\[\Delta P=(TS_{R}\times P_{R}+\text{TS}_{C}\times P_{C}+\text{TS}_{I}\times P_{I })\times\frac{\Delta T}{Tmean} \tag{14}\]  
where \(P_{R}\), \(P_{C}\), and \(P_{I}\) represent the base power demands of residential, commercial, and industrial consumers, respectively, whereas \(\Delta T\) is the change in actual temperature.