Integrating energy systems have emerged and developed rapidly to tackle climate change and promote green and low-carbon transition. However

the coupling of heterogeneous energy carriers and the random fluctuation of wind

PV and load bring new challenges to the planning and operation of IES. Therefore

a probabilistic multi-energy flow solution method based on semi-invariant computing framework and principle of maximum entropy is proposed to reflect the operation of integrated electricity-gas-heat system with uncertainties fully. Firstly

the steady-state modeling and solution of heterogeneous energy carriers and coupling components are carried out to obtain the distribution of state variables in multi-energy coupling networks. Secondly

according to the numerical characteristics of uncertain input variables

a probabilistic multi-energy flow calculation model based on semi-invariant computing framework is constructed

and the probability density of output state variables is obtained by combining the maximum entropy rule. Then

based on the probability distribution results of state variables under the maximum entropy distribution

the weak nodes in heterogeneous energy systems are identified. Finally

the IES 34-20-14 and IES 57-20-32 bus systems are analyzed and compared with various state variable distribution fitting methods. The results show that this method can obtain accurate probability distribution under the maximum uncertainty

and realize the quick solution of IES probability energy flow.