An air-conditioning system for a building includes a hot water distribution circuit, a cold water distribution circuit, and several terminal air conditioning units. Each of the terminal air conditioning units includes a fan blowing air in a space of the building, a heating coil connected to the hot water distribution circuit and/or a cooling coil connected to the cold water distribution circuit. At least one ambient temperature control system allows to control heating power of the heating coils and cooling power of the cooling coils. The system further includes a calorific energy management system with a heat pump for transferring calorific energy: a) from the cold water distribution system to the hot water distribution system; b) from the cold water distribution system to atmosphere; and c) from the atmosphere to the hot water distribution system. The calorific energy management system can manage the calorific energy transfers with a three level control system to optimize energy consumption.