The ampacity — a term that refers to the maximum current in amperes — of existing overhead lines can be increased using Dynamic Line Rating (DLR) systems. Based on ambient online conditions and electrical power flow measurements, together with a forecast of future ambient conditions, the ampacity of overhead lines can be dynamically adjusted. In essence, the ampacity of an overhead line is determined by the operating temperature of the conductor that still guarantees the required safe distances to the objects under the overhead line. This paper presents a calculation algorithm for the sag of a conductor that addresses an unequal temperature of the conductor along the line as a response to an expected problem in the future, when increasing numbers of DLR sensors will be installed on or even near overhead lines. A novel approach to the sag-temperature calculation useful for a DLR is developed based on the linearization of cable equations. As a result, the computational costs are greatly reduced, which is an important factor when the DLR system includes a large number of overhead lines and processes them simultaneously. The presented methods are open and can be adapted for an arbitrary DLR system.