In Part I of this post, we have seen that the optimal power flow (OPF) problem in electricity networks is much more difficult than congestion control on the Internet, because OPF is nonconvex. In Part II, I will explain where the nonconvexity comes from, and how to deal with it.
Source of nonconvexity
Let’s again start with congestion control, which is a convex problem.
As mentioned in Part I, corresponding to each congestion control protocol is an optimization problem, called network utility maximization. It takes the form of maximizing a utility function over sending rates subject to network capacity constraints. The utility function is determined by the congestion control protocol: a different design to adapt the sending rate of a computer to congestion implies a different utility function that the protocol implicitly maximizes. The utility function is always increasing in the sending rates, and therefore, a congestion control protocol tends to push the sending rates up in order to maximize utility, but not to exceed network capacity. The key feature that makes congestion control simple is that the utility functions underlying all of the congestion control protocols that people have proposed are concave functions. More importantly, and in contrast to OPF, the network capacity constraint is linear in the sending rates. This means that network utility maximization is a convex problem.
I have discussed in a previous post that digitization (the representation of information by zeros and ones, and its physical implementation and manipulation) and layering have allowed us to confine the complexity of physics to the physical layer and insulate high-level functionalities from this complexity, greatly simplifying the design and operation of communication networks. For instance, routing, congestion control, search, and ad markets, etc. do not need to deal with the nonlinearity of an optical fiber or a copper wire; in fact, they don’t even know what the underlying physical medium is.
This is not the case for power networks.
The lack of an analogous concept of digitization in power means that we have been unable to decouple the physics (Kirchhoff’s laws) of power flow from high-level functionalities. For instance, we need to deal with power flows not only in deciding which generators should generate electricity when and how much, but also in optimizing network topology, scheduling the charging of electric vehicles, pricing electricity, and mitigating the market power of providers. That is, while the physics of the transmission medium is confined in a single layer in a cyber network, it permeates through the entire infrastructure in a cyber-physical network, and cannot be designed away.
How difficult is it to deal with power flows?
This post (and the one that follows) will illustrate some of these challenges by contrasting the problem of congestion control on the Internet and that of optimal power flow (OPF) in electricity.