## 2 S-L Analysis Method
### S-L Analysis Method of Single-Supply System  
#### 2.3.1 Conventional Single-Supply System  
(1) Primary Frequency Regulation  
Using the basic ideas above, an S-L analysis method for a general single-supply system was obtained (see Figure 4). Figure 3: Diagram of S-L analysis method. The proposed S-L analysis method is an evolution of the one introduced in Reference [20]. As we can see from Figure 4, the load absorbed power and its frequency characteristic lies in the first quadrant, which means the corresponding power value is positive. Conversely, the power source output power has a frequency characteristic in the second quadrant. Straight lines \(\rm L_{G}\) and \(\rm L_{L}^{\ 0}\) are the frequency characteristics of the power source and the total load in the initial state, respectively. The intersections of the straight line o with the vertical axis, \(\rm L_{G}\) and \(\rm L_{L}^{\ 0}\) are the points x, a and b respectively, which satisfy \(\rm ax\ =\ bx\), \(P_{G}{}^{0}\)=\(\rm P_{L}^{\ 0}\). Therefore, in the initial state, the system's operation line is a straight line o. The coordinate at point x is \(f_{\rm N}\) and the system is run at nominal frequency. After the load is increased, the frequency characteristic of the load would move to the right and become the straight line \(\rm L_{L}^{\ 1}\) in Figure 4. Instantaneously, the load increases from \(\rm P_{L}^{\ 0}\) to \(\rm P_{L-}^{\ 0}\) while the power delivered by the generator remains the same. This results in insufficient power and a decrease in frequency. As the figure shows, in the process of decreasing frequency, the generator generates additional power and the load absorbs less power. Under the frequency regulation of the power supply and the load, when the operation line is reduced to the straight line o\({}^{\prime}\), there is \(\rm dy\ =\ ey\), which reaches a new power balance of \(P_{G}{}^{\ 1}\ =\ P_{L}^{\ 1}\) and the frequency drops to \(f_{1}\) at that time. In the following, the rationality of S-L analysis method is deduced from a mathematical relation. In Figure 4, the initial power balance corresponding to the operation line o is obtained as the equation below. \[P_{G}{}^{\ 0}\ =\ P_{L}^{\ 0} \tag{8}\]  
From the final power balance corresponding to the operation line o\({}^{\prime}\), there is:  
\[P_{G}{}^{\ 1}\ =\ P_{L}^{\ 1} \tag{9}\]  
Considering the transitional process from operation line o to operation line o\({}^{\prime}\) for the output power, there is  
\[P_{G}{}^{\ 0}+\Delta P_{G}\ =\ P_{G}{}^{\ 1} \tag{10}\]  
The load, first sudden increases, then reduces for self-regulation. \[P_{L}{}^{\ 0}+\Delta P_{L}{}^{\ 0}-\Delta P_{L}{}^{\ 1}\ =\ P_{L}{}^{\ 1} \tag{11}\]  
Using Equations (8)-(11), we can deduce Equation (12) below. \[\Delta P_{G}\ =\ \Delta P_{L}{}^{\ 0}-\Delta P_{L}{}^{\ 1} \tag{12}\]  
Equation (13) is easily obtained from the mathematical relationship in Figure 4.  
\[\Delta P_{G}\cdot R_{G}\ =\ \Delta P_{L}{}^{\ 1}\cdot R_{L}\ =\ |\Delta f| \tag{13}\]  
Figure 4: S-L analysis method of single-supply system. where, \(R_{\rm G}\), \(R_{\rm L}\) are the magnitude of the slope of the power-frequency characteristic of power supply and load. It can be seen that Equation (12) is consistent with Equation (3) and Equation (13) is consistent with Equation (7). Therefore, the S-L analysis method meets the two basic relations of frequency regulation process, which is not contrary to the law of power-frequency proven to be feasible. (2) Secondary Frequency Regulation  
If the system frequency does not satisfy the requirements after frequency regulation, there is a need for secondary frequency regulation. For convenience, take the realization of the frequency zero error adjustment as an example to illustrate the process (see Figure 5). For ease of reading, the information of primary frequency regulation was omitted and the second frequency information was shown in green. After primary frequency regulation, the frequency is \(f_{\rm 1}\), with cy \(=\) dy, that is \({P_{\rm G}}^{\rm 1}\)\(=\)\({P_{\rm L}}^{\rm 1}\).