I am not sure if or how Koltygin's answer: "To determine parameters in different operating modes" answers your question, but let me try a different way. The way in which you have posed your question is open to a number of interpretations, at least, in my mind. "How has the equations for the check valves been formulated to be QL=KL(PD-PL) and QP=KP(PD-PP)?" Everything appears to depend on what you mean by the word 'formulated'.
One possible interpretation is how the check valve equations are realized in the simulation below your mechanical drawing. For example, the equation for QL is realized, in the simulation, by applying PD to the plus input, and PL to the minus input of a summing junction; taking difference PD-PL and applying it to one input of a comparator, C, while applying a signal 0.5 to the other input of the comparator; and using the comparator output to close a switch allowing the difference PD-PL>0.5 to be applied to the multiplication block KL to yield an output KL(PD-PL).
Another possible interpretation of what you mean by the word 'formulated' might be explaining how a check value can be represented by the equation QL=KL(PD-PL).
A third possible interpretation might be explaining why your hydraulic system requires check valves.