"run according to plan" is correct and usable in written English.
You can use it to express that something is going as expected. For example, "The event was well organized and it ran according to plan."
But such efforts don't always run according to plan, here are a few examples of university marketing gone wrong: Rebranding a university is always a delicate business – how do you create a modern, eye-catching image, without alienating current staff and students?
Back in 2007 Roy Keane paid Hearts £9m for the Scotland international, then a British record for a goalkeeper, but two broken arms plus a serious knee injury subsequently ensured things have not run according to plan at the Stadium of Light.
It's just a question of the tournament running according to plan".
If Benson's film is saying anything, it is that the script rarely runs according to plan.
Down in the main ITV studio, it was the politicians' turn to explain to Tom Bradby why everything was still running according to plan.
Unadorned in the way she dresses and decorates, Mrs. Bush is nevertheless obsessed with detail and expected to ensure that even the humblest White House occasion runs according to plan.
You're going to need them, because if all runs according to plan you will spend the second half of the evening tossing the movie the impact and the substance of it back and forth.
If the "coronation" runs according to plan, Mr. Landry, now deputy premier, will become premier in time to welcome President Bush and 33 other heads of government to Quebec City, the provincial capital, for a conference called the Summit of the Americas in mid-April.
But not every night on ecstasy runs according to plan.
The brigade plan was now a day behind schedule, although with the unexpected ease experienced by the Canadians, overall, the divisional attack was still running according to plan.
Very little time at the Laeken Summit had to be spent on the euro, because at the moment everything is running according to plan without any significant problems and, therefore, the introduction of the euro on 1 January 2002 will be able to take place.
