Operating method of a laundry washing machine (1) comprising: an internal water softening device (13) which is crossed by the fresh water directed towards the washing tub (3) and/or the detergent dispenser (10), and which is filled with a water softening agent capable of reducing the hardness degree of the fresh water flowing through the same water softening device (13); a regeneration-agent container (21) containing a given amount of consumable salt or other regeneration agent; a first water-supply line (19) for selectively channelling a flow of fresh water into said regeneration-agent container (21) to form a brine capable of regenerating the water softening properties of said water softening agent; and a storage tank (35) which is fluidically connected to said regeneration-agent container (21) for receiving and accumulating the brine formed into said regeneration-agent container (21), and to said water softening device (13) for delivering said brine to the water softening device (13); the operating method
