## 3 Mathematical Model of the System
### Power Electronics Systems Controlling the Flow of Power  
From the point of view of the executed algorithm (point 2.1), power electronics systems related to DC electrical energy conversion into AC and vice versa (in the kinetic storage structure) and control of the energy flow between the turbine, storage and power grid according to Equation (1), are all important elements of the system presented in Figure 1. The efficiency of the abovementioned systems is a parameter that greatly affects the effectiveness of power conversion. A typical relationship of a power electronics converter \(\tau_{\mathrm{INV}}\) as a function of load \(\eta_{\mathrm{INV}}=f\) (\(P/P_{\mathrm{INVn}}\)) is used in the acquired model, where \(P\) is converted power, and \(P_{\mathrm{INVn}}\) is the rated power of the converter assembly. Figure 4 presents the assumed efficiency characteristics of power electronics systems. Indicators of Electrical Energy Supplies from a Wind Farm--Kinetic Energy Storage System to the Power Grid  
An assessment of electrical energy generation reliability for the assumed period involves the identification of the extent to which power demand \(P_{x}(t)\) is covered by the capacity of subsystem \(P_{\mathrm{w}}(t)\). Information about reliability established for future periods (e.g., following changes in the structure of the interesting part of the grid) includes load connection, additional unstable sources etc. A model of power generation reliability, as a difference in the value of stochastic processes \(P_{x}(t)\) and \(P_{w}(t)\)  
Figure 4: Changes in the efficiency of power electronics systems used in the analysed system as a function of a relative value of converted power (\(P/P_{\mathrm{INVn}}\)). is also defined as a stochastic process D(\(t\)) which is a power deficit. This is described by the general relationship [29]:  
\[\text{D}(t)=\left\{ \begin{array}{c}P_{x}(t)-P_{\text{w}}(t)\text{ for }P_{x}(t)>P_{\text{w}}(t)\\ 0\text{ for }P_{x}(t)\leq P_{\text{w}}(t)\end{array} \right. \tag{7}\]  
Indicators related to the duration of power deficit, power not supplied to customers and frequency and duration of occurrence, loss of energy expectation (LOEE), frequency and duration indices (F&D), expected capacity deficiency (ECD), probability of capacity deficiency (PCD), expected loss of load (XLOL) etc.--are used as quantitative parameters describing a stochastic process of a power deficit D(\(t\)) [29; 30; 31; 32; 33]. A method based on an analysis of the changes in the level of the output power P\({}_{3}\) for a one-year period was used to compare the reliability of electrical energy supplies to the power grid from a wind farm--kinetic energy storage system with various structures. The application of an energy storage system helps to shape the changes in the power supplied to the system as compared to the power level resulting from changes in the wind energy. The applied indicators are similar to the parameters defined for a power deficit stochastic process D(t) but they still take into consideration the acquired power flow control algorithm (Section 2.1). Furthermore, they are identified based on historical data.