It's great that this issue is getting some attention, i wanted to work on it myself but unfortunately did not find the time.

My main issue outlined in #1320 is that the value of s_nom_max affects the losses and hence the solution. The reason is that the tangents depend on s_nom_max (both slope and offset). This PR improves the situation by using s_nom_opt in suitable situations during iterative expansion modeling. However, in other situations it still relies on s_nom_max, so unfortunately i does not resolve #1320 completly.

I think the losses can be made independent of s_nom_max, because the shape of the loss parabola is independent of s_nom_max. In the loss constraint s_nom_max is only really needed to provide an upper bound on the losses (and should only be relevant for the model in negative price hours, if I understand the theory right). So my suggestion is to choose the approximation independently of s_nom_max, only relying on a specified loss_approximation_tolerance config, which gives the maximum error between the approximation and the parabola. The approach here - using an L2 approximation - seems to be one promising for constructing such a tighter approximation. I can't judge however, if there are reasons why an outer envelope would be more desirable.

It's great that this issue is getting some attention, i wanted to work on it myself but unfortunately did not find the time.

My main issue outlined in #1320 is that the value of s_nom_max affects the losses and hence the solution. The reason is that the tangents depend on s_nom_max (both slope and offset). This PR improves the situation by using s_nom_opt in suitable situations during iterative expansion modeling. However, in other situations it still relies on s_nom_max, so unfortunately i does not resolve #1320 completly.

I think the losses can be made independent of s_nom_max, because the shape of the loss parabola is independent of s_nom_max. In the loss constraint s_nom_max is only really needed to provide an upper bound on the losses (and should only be relevant for the model in negative price hours, if I understand the theory right). So my suggestion is to choose the approximation independently of s_nom_max, only relying on a specified loss_approximation_tolerance config, which gives the maximum error between the approximation and the parabola. The approach here - using an L2 approximation - seems to be one promising for constructing such a tighter approximation. I can't judge however, if there are reasons why an outer envelope would be more desirable.

I believe that even if we implement the loss approximation using the idea of setting a loss_approximation_tolerance, the concept of the maximum s is still relevant, because we need to define the range of the power over which the approximation is applied. In other words, the number of line segments is a function of both the maximum s and the tolerance. For example, if we only aim to achieve an approximation error below 0.01 within the interval [-1, 1], we may only need 3 segments; however, if we want to maintain an error below 0.01 over the entire range [-∞, +∞], we would in fact need an infinite number of segments.

Great, I will try to look into it soon! For now, just one question: In your image, it looks like the first tangent cuts through the red line. Shouldn't it be on the outside of the red line?

In your image, it looks like the first tangent cuts through the red line. Shouldn't it be on the outside of the red line?

Hi @fneum, this modification is intended to generate line segments that minimize approximation error without restricting the segments to be tangents.

Hey @WeiAi-Energy, @lindnemi proposed a secant model in #1495. What do you think of that? Would you see your PR as a third mode or would you say it's unnecessary if we go forward with #1495?

Hey @WeiAi-Energy, @lindnemi proposed a secant model in #1495. What do you think of that? Would you see your PR as a third mode or would you say it's unnecessary if we go forward with #1495?

Hi @fneum, I think the error control proposed in #1495 is a great idea. My only concern is that it still requires a s_nom_max, but in practice, it's often hard to assign an appropriate s_nom_max for every single line, which may increase unnecessary complexity in the model. For example, if s_nom_max is set to 10 GW but the optimal solution is 0.1 GW, then most of the constraint segments won’t be active. This can be avoided in the iterative case in this PR, so I would suggest combining the two PRs.

Hi @fneum, I think the error control proposed in #1495 is a great idea. My only concern is that it still requires a s_nom_max, but in practice, it's often hard to assign an appropriate s_nom_max for every single line, which may increase unnecessary complexity in the model. For example, if s_nom_max is set to 10 GW but the optimal solution is 0.1 GW, then most of the constraint segments won’t be active. This can be avoided in the iterative case in this PR, so I would suggest combining the two PRs.

I totally agree that it's undesirable to have many inactive constraints. Using s_nom instead of s_nom_max in #1495 when solving iteratively would reduce the number of constraints, however, it would also mean that the approximation error can no longer be guaranteed to be below the specified error tolerances for flows larger than s_nom. Because of this I don't think using s_nom works well in #1495, but we could add a further mode of loss approximation which trades the strict error control for sparser constraints.

@WeiAi-Energy thanks for the pr. After tackling it partially in #1495 (as it seems), I am wondering if is makes sense to pull this in. the code changes are not a lot and don't seem to be invasive. so I am happy to give this a go and review properly. otherwise we can close this. also tagging @lindnemi