## 2 Methods
The capacitance of the SC increased nonlinearly in a given voltage range at its terminals. Given this dependence, the total capacitance of the SC can be approximated, with sufficient accuracy, as the sum of the fixed capacitance \(C_{1}\) = const and capacitance \(C_{\rm v}(U)=k\ |U|\), which linearly depends on the voltage \(U\). The coefficient \(k\) = const and has dimension [F/V]. According to the physics of the double electric layer [2; 3; 18; 20], the total capacitance of supercapacitors is determined by the expression:  
\[C(U)=C_{1}+k\ |U|. \tag{3}\]  
Taking into account expression (3) for branching the modes of operation of the SC, in accordance with works [2; 3; 8; 9; 10], it is necessary to use branched substitution schemes. In order to take into account the distributed electrical parameters of supercapacitors, their branched substitution schemes with different numbers of parallel branches containing concentrated linear and nonlinear elements \(C\), \(R\) and \(L\) were proposed. The schemes can be improved taking into account the duration of energy processes. This paper investigates the change in the energy and dynamic characteristics of the SC when the voltage at its terminals changes when charged by the battery. The optimal solution of this problem is to use an equivalent scheme of substitution of SC with a variable number of parallel branches with different values of time constants \(\tau\) = \(RC\)[20]. With a duration of transient processes to \(\tau\) \(\leq\) 30 min, the scheme of replacement of SC is shown in Figure 1, which accurately reflects the energy processes in the supercapacitor. The parameters of this substitution scheme were determined experimentally [20]. The first branch (instantaneous branch) is represented by a capacitor, the capacitance of which depends on the voltage [6; 18] according to expression (3). The branch consists of elements \(C_{1}\) and \(R_{1}\), the values of which do not change, and the element \(C_{\rm v}(U_{1})\), the value of which depends on the applied voltage to the terminals of the SC. This branch has a low time constant and its capacitors are charged in the second time range. The second branch (delayed branch), with constant parameters \(C_{2}\) and \(R_{2}\), operates in the minute time range. The third (long-term) branch has the longest time constant and operates in the range of 10 min (parameters \(C_{3}\) and \(R_{3}\) are also constant). To take into account the self-discharge of the SC, a substitution circuit is added in parallel resistor \(R_{4}\). The battery is represented by a DC voltage source \(E\) with internal resistance \(R_{AB}\). In refs. [3; 18] it was experimentally confirmed that the equivalent circuit substitution scheme accurately reflects its energy and dynamic characteristics in charge-discharge processes. Voltages \(U_{R_{AB}}(t),U_{R_{1}}(t)\), respectively, on the resistors \(R_{AB},R_{1}\) of Figure 1, are expressed by the formulas:  
\[U_{R_{AB}}(t)=R_{AB}\cdot i(t); \tag{4}\]  
\[U_{R_{1}}(t)=R_{1}\cdot i(t), \tag{5}\]  
After combining the resistances of the charging circuit \(R_{\Sigma}=R_{AB}+R_{1}\), according to Formula (3), we can write:  
\[U_{R_{\Sigma}}(t)=R_{\Sigma}\cdot i(t)=R_{\Sigma}\cdot\left((C_{1}+2\cdot k \cdot|U_{SC}(t)|)\cdot\left(\frac{dU_{SC}(t)}{dt}\right)\right) \tag{6}\]  
Coefficient \(k\) = const and has the dimension (F/V).