"Free will" as broadly defined is "a particular sort of capacity of rational agents to choose a course of action from among various alternatives" (SEP). Although there are extreme views, such as Descartes' where "the will is by its nature so free that it can never be constrained," most people accept that our will can at the very least be influenced by factors external to ourselves, if not wholly controlled by it. The challenge, as you point out, is finding how to explain a notion of freedom through non-causation.
In a strict sense, the answer to your question is no. There have been no successful logical explanations for free will as traditionally defined. The problem lies with the fact that to be free requires non-causation, or indeterminacy. Proponents of free will often invoke abstruse theories about the quantum indeterminacy or the Heisenberg Uncertainty principle as examples of where the secret of free will lies, but the major problem which undercuts each of these theories is a lack of an explanation of how free will can be obtained through seemingly random processes.
Concordantly, I think it is more appropriate at this time—given our current understanding of the universe—to relax our notion of free will instead of trying to stretch it to afford an as of yet an indescribable freedom. To this end, the notion of free will in a deterministic system can be explained as an illusory sense of control that occurs when it appears there is more than one possible future before us.
The distinct feeling of freedom of action is not foreign to us, and even if we would call ourselves determinists we all still act under the assumption that we actually can control our own decisions and actions. We still talk and move around assuming our bodies operates under our own volition – if we did not we may as well simply collapse under the futility of it all. It is readily apparent that despite our understanding of determinism, our perceived control over our own actions completely permeates every moment of our lives — and thus the temptation to embrace free will. But closer inspection brings forth the actual degree of control we actually have, which is in my view broken into two distinct types of control: universal control and human control. Universal control is the notion that we are able, through whatever mechanisms, to break ourselves free of physical causation; this is impossible. In other words, it’s just an affirmation of standard causal determinism: that “every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences” (Van Inwagen, 1983). This type of control we do not have. However, we do have human control; that is, we have the ability to predict (or rather, inability to perfectly predict) the outcome of events which provides us with the illusion that there is a “possibility” between choices. Note that it is not the freedom to “decide” whether or not we predict or not; it is simply the distinct feeling of comfort and peace that comes from a sense of control, i.e., from one’s sense that he or she understands the future well enough such that his or her desires are favorably in-line with the likely future. Stated differently, we do not have control in the sense that we are agents free from causation, but we do have control in the sense that, given our inability to calculate the precise outcome of all the causal factors leading to a particular event, we are able to—with a fair degree of accuracy, especially for common events—predict at least a set of outcomes that will likely occur. While only one outcome is truly inevitable for any event, our limited intellects can often only narrow down the factors to several “possible” outcomes. The control we feel is the degree to which our predicted outcome is expected to match the inevitable outcome. Dennett (1984), illustrates this best with his remote-control airplane example:
When you control your plane perfectly, you don’t do it by controlling
all the causes that influence it. The weather, the density of the air,
and the force of gravity, for instance, are all beyond your control,
and they are the largest forces that act on your plane. The fact that
your plane is constantly under the influence of gravity does not
prevent you from controlling it–in fact in some regards gravity helps
you, just so long as you know its effects on the plane. But a sudden
and unanticipated gust of wind may upset your control, either
temporarily or permanently. What is important about the difference
between the gust of wind and gravity is not the steadiness of the
latter and the suddenness of the former, but the (relative)
unexpectability of the former. … Foreknowledge is what permits
–Dennett, Daniel (1984). Elbow Room: The Varieties of Free Will Worth Having. Cambridge. MA: MIT Press.
Thus, the nature of human control is not “real” control; it is the measure of our desires against our prediction of the future. At best, this leaves us with the illusion of free will. In this way (regarding the OP's question), free will can be quantified in a deterministic system without breaking any rules of causation.
In an effort to avoid some of the inevitable pro-"free will" comment backlash this post will generate, I just to want to add that I understand that people don't want to give up the idea of freedom of action; they don't want to be reduced purely physical automata. What they don't realize is that—if this is the way we really are—nothing really has to change. It's not like the slight alteration of our conceptual understanding of the free will flips our entire world upside down. At the end of the day, we still get up in the morning, we still take showers, eat breakfast and go to work. We still hang out with our friends when we want and watch our favorite TV shows as we please. Ice cream and brownies still taste delicious, the night sky full of stars still fills us with awe and wonder, and the pain of losing a loved one is still real to us. If determinism is true, no one has to give up anything.