## 2 S-L Analysis Method  
### Overview of Power System Frequency Regulation  
When the power system is operating in a steady state, the power of the generator is balanced with the power of the load. \[\sum_{i\ =\ 1}^{m}{P_{\text{G}i}}^{0}=\sum_{j\ =\ 1}^{n}{P_{\text{L}j}}^{0} \tag{1}\]  
where \(m\) is the number of generators in the system, \(P_{\text{G}i}\) is the active power from the generator, \(i\)(\(i\) = 1,2,\(\cdots\),\(m\)), \(n\) is the number of motors in the system, and \(P_{\text{L}j}\) is the active power absorbed by the load \(j\)(\(i\) = 1,2,\(\cdots\),\(n\)). Frequency changes occur because system load varies randomly throughout the day so that an exact forecast of real power demand cannot be assured. The imbalance between real power generation and load demand (plus losses) throughout the daily load cycle causes kinetic energy of rotation to be added to or taken from the on-line generating units. Frequency throughout the interconnected system varies as a result. Then, the system will reach a new balance:  
\[\sum_{i\ =\ 1}^{m}{P_{\text{G}i}}^{1}=\sum_{j\ =\ 1}^{n}{P_{\text{L}j}}^{1} \tag{2}\]If we subtract Equation (2) from Equation (1), we have the following equation. \[\sum_{i\ =\ 1}^{m}\Delta P_{\text{G}i}\ =\ \Delta P_{\text{L}}{}^{0}-\sum_{j\ =\ 1}^{n}\Delta P_{\text{L}j}{}^{1} \tag{3}\]  
where \(\Delta P_{\text{L}}{}^{0}\) is the total load change, \(\Delta P_{\text{L}j}{}^{1}\) is the amount of power change caused by the load \(j\) due to its self-regulation, and \(\Delta P_{\text{G}i}\) is the amount of power change caused by the generator \(i\) due to its self-regulation. Equation (3) is one of the two basic relations of the frequency regulation process. Next, the specific frequency regulation process of the generator set was analyzed. Take boiler-turbine-generator combination of a thermal generating unit as an example to help understand the control actions at the power plants. Most steam turbogenerators are equipped with turbine speed governors. The function of the speed governor is to continuously monitor the turbine-generator speed and to control the throttle values, which adjust steam flow into the turbine in response to changes in system frequency. To permit parallel operation of generating units, the frequency-versus-power output governing characteristic of each unit has a droop, which means that a decrease in speed should accompany an increase in load, as depicted by the straight line in Figure 1. The per-unit droop or speed regulation \(r_{\text{u}}\) of the generating unit is defined as the magnitude of the change in steady-state speed and expressed in per unit of rated speed when the output of the unit is gradually reduced from 1.00 per-unit rated power to 0. Therefore, per-unit regulation is simply the magnitude of the slope of the frequency-versus-power output characteristic when the frequency axis and the power-output axis are each scaled in per unit of their respective rated values. From Figure 1 it follows that per-unit regulation is given by Equation (4). \[r_{\text{u}}\ =\ -\frac{\Delta f(\%)}{\Delta P(\%)}\ =\ \frac{(f_{2}-f_{1})/f_{ \text{R}}}{P_{\text{GR}}/S_{\text{R}}} \tag{4}\]  
where, \(f_{2}\) is the frequency (in Hz) at no load, \(f_{1}\) is frequency (in Hz) at rated megawatt output \(P_{\text{GR}}\), \(f_{\text{R}}\) is the rated frequency (in Hz) of the unit, and \(S_{\text{R}}\) is the megawatt base. Multiplying each side of Equation (4) by \(f_{\text{R}}/S_{\text{R}}\) gives Equation (5). \[R\ =\ r_{\text{u}}\frac{f_{\text{R}}}{S_{\text{R}}}\ =\ \frac{f_{2}-f_{1}}{P_{ \text{GR}}}\ \text{Hz}/\text{MW} \tag{5}\]  
where \(R\) is the magnitude of the slope of the power-frequency characteristic (in Hz/MW). Suppose that the unit is supplying output power \(P_{\text{GO}}\) at frequency \(f_{0}\) when the load is increased to \(P_{\text{G}}\ =\ P_{\text{GO}}+\Delta P_{\text{G}}\)  
Figure 1: Per-unit regulation characteristic of a generating unit.