Soft open points (SOPs) are power electronic devices which can replace conventional normally open points in distribution networks. SOPs enable full control of active power flow between the interconnected feeders and can inject reactive power at each node to which they are connected. SOPs integrated with energy storage (ES) have been recently proposed to realize both spatial and temporal flexibility in active distribution networks. The flexibility provided by integrated ES-SOP devices will allow network operators to run their networks closer to their limits, but only if there is appropriate management of the uncertainty arising from demand and renewable generation. The only existing model of an ES-SOP uses nonconvex nonlinear equations, neglects uncertainty, and represents converter losses in an oversimplistic manner. This paper presents a robust mixedinteger convex model for the optimal scheduling of integrated ESSOPs to ensure a zero probability of constraint violation. Losses of the subsystems comprising the ES-SOP are modelled using a proposed binary-polynomial model, enabling efficient scheduling of the energization state of subsystems to reduce no-load losses. The ES-SOP is considered in this paper to be owned by the network operator to: 1) manage power flow constraints, 2) minimize cost of losses, and 3) maximize arbitrage profit.