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Telegraph equations are derived from the equations of transmission line theory. They describe the relationships between the currents and voltages on a portion of an electric line as a function of the linear constants of the conductor (resistance, conductance, inductance, capacitance). Their resolution makes it possible to determine the variation of the current and the voltage as a function of time at each point of the line. By adopting a general sinusoidal form, we propose a new exact solution to the telegraphers' partial differential equations. Different simulations have been carried out considering the parameter of the 12/20 (24) kV Medium Voltage Cable NF C 33,220. The curves of the obtained solution better fit the real voltage curves observed in the electrical networks in operation.

Residential photovoltaic (PV) systems connected to the grid are used for selfconsumption. Any surplus production is fed into the grid and contributes to improving the voltage. Several techniques are developed to model their connection. However, studies on methods of injecting energy production into the Low Voltage (LV) network are nowadays a problem. This paper proposes a mathematical model to determine the current to be injected and calculate each node's voltage. The current equation is a recurrence relation with an initial condition. This initial condition is for the case of a single PV system connected to the LV grid. The equation can also be written in matrix form. Similarly, the voltage solution is a recurrence relation. It also has an initial condition for the first node. Both mathematical formulae with the proposed initial conditions are consistent and can be used for the determination of the current and voltage of the different nodes in the grid.

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