A multi-circuit heating or cooling system having a heating source circuit, HK, a buffer circuit, PK, and a mixed heating circuit, MK, with - at least one heat source, which is part of said heating source circuit, HK, with a heating source circuit flow, HK-VL, with an HK-VL temperature and a heating source circuit return, HK-RL, with an HK-RL temperature, and at least two heat sinks (103a, 105a), wherein - a first heat sink is part of said buffer circuit, PK, with a buffer circuit flow, PK-VL, with a PK-VL temperature and a first buffer circuit return, PK-RL1, with a PK-RL temperature and - a second heat sink is part of said mixed heating circuit, MK, with a mixed heating circuit flow, MK-VL, with an MK-VL temperature and a mixed heating circuit return, MK-RL, with an MK-RL temperature, wherein - said first heat sink is formed as a buffer store for the energy-transporting medium, wherein said buffer circuit has a second buffer circuit return, PK-RL2, with a PK-RL2 temperature, wherein - a distributor device with hydraulic means for distributing an energy-transporting medium is provided, characterised in that - said distribution device comprises a multiple-way mixing valve with at least five ports (A, B, C, D, E) - and said heating source circuit, said buffer circuit and said mixed heating circuit are connected to at least one of said ports (A, B, C, D, E) of said multiple-way mixing valve, wherein - said HK-VL is connected to a first port of said multiple-way mixing valve, - said HK-RL is connected both to said MK-RL and in parallel to said PK-RL2, - said PK-RL1 is connected to a second port of said multiple-way mixing valve, - said PK-VL is connected to a third port of said multiple-way mixing valve, - said MK-RL is connected to a fourth port of said multiple-way mixing valve and preferably also in parallel to said second port, - said PK-RL2 is connected to said MK-RL, - said MK-VL is connected to a fifth port of said multiple-way mixing valve so as to implement priority control for the filling of said buffer store by said heat source in an operating state of said multi-circuit heating or cooling system, wherein only said buffer circuit is supplied with heat from said heating source heating circuit. - at least one heat source, which is part of said heating source circuit, HK, with a heating source circuit flow, HK-VL, with an HK-VL temperature and a heating source circuit return, HK-RL, with an HK-RL temperature, and at least two heat sinks (103a, 105a), wherein - a first heat sink is part of said buffer circuit, PK, with a buffer circuit flow, PK-VL, with a PK-VL temperature and a first buffer circuit return, PK-RL1, with a PK-RL temperature and - a second heat sink is part of said mixed heating circuit, MK, with a mixed heating circuit flow, MK-VL, with an MK-VL temperature and a mixed heating circuit return, MK-RL, with an MK-RL temperature, wherein - said first heat sink is formed as a buffer store for the energy-transporting medium, wherein said buffer circuit has a second buffer circuit return, PK-RL2, with a PK-RL2 temperature, wherein - a distributor device with hydraulic means for distributing an energy-transporting medium is provided, characterised in that - said distribution device comprises a multiple-way mixing valve with at least five ports (A, B, C, D, E) - and said heating source circuit, said buffer circuit and said mixed heating circuit are connected to at least one of said ports (A, B, C, D, E) of said multiple-way mixing valve, wherein - said HK-VL is connected to a first port of said multiple-way mixing valve, - said HK-RL is connected both to said MK-RL and in parallel to said PK-RL2, - said PK-RL1 is connected to a second port of said multiple-way mixing valve, - said PK-VL is connected to a third port of said multiple-way mixing valve, - said MK-RL is connected to a fourth port of said multiple-way mixing valve and preferably also in parallel to said second port, - said PK-RL2 is connected to said MK-RL, - said MK-VL is connected to a fifth port of said multiple-way mixing valve so as to implement priority control for the filling of said buffer store by said heat source in an operating state of said multi-circuit heating or cooling system, wherein only said buffer circuit is supplied with heat from said heating source heating circuit.