You can try to cast it to Cart and set it there. 

var implementation = cart as Cart;

if (implementation != null)

{

  implementation.ProviderId = ...;

}

//Save the cart

Thank you Quan,

I have one doubt.we have using Icart only for all operations. If I cast Icart as cart only set the provider id. But I am saving the Icart only. It will be saved or else.

in the snippet above, you can call either

_orderRepo.Save(cart);

or 

_orderRepo.Save(implementation);

They both work the same

Thank you Quan,

It worked for me. I used Icart to save.

But why providerId not available in Icart?

I don't know, there might be some reasons, or it might be just an oversight. 

Thanks Quan,