Feature/planning/soft site power capacities

[victorgarcia98]

Work in progress.

Distinguish interpretation of soc-max/min from soc-maxima/minima, and site-consumption/production-capacity from site-power-capacity, representing hard and soft constraints

[Flix6x]

I'll think about naming.

[Flix6x]

This PR only addresses site level power constraints at the moment and not any device level constraints. As it stands, I feel that soft constraints for site level power would be more cleanly modelled as site level commitments (to the DSO) based on the contracted capacities, with some (assumed, or even known) penalty. That should mostly happen outside of the device scheduler.

If we do address device-level soft constraints, we should think about them too in the context of introducing device-level commitments (two birds, one stone).

[victorgarcia98]

I'm not sure we can model the mechanism that these PR introduces via commitments as they are defined at the moment.

---

Let's say we want to add two soft limits at 90 kW to a site of 100 kW.

We would add two commitments:

• Commitment = 90 kW, $Price_{Up} = 100$, $Price_{Down} = 0$
• Commitment = -90 kW, $Price_{Up} = 0$, $Price_{Down} = 100$

This would be equivalent to have a single commitment:

• Commitment = 0 kW, $Price_{Up} = 100$, $Price_{Down} = 100$

This applies in general. For example, if we have two commitments, the EMS-commitment couples eq. is:

$$ \sum_{d\in Devices} Power(d, j) = Commitment(1) + Commitment(2) + Down(1, j) + Down(2, j) - Up(1, j) - Up(2, j) $$

which is, inf fact, equivalent of having an single equivalent commitment defined as:

$$ Commitment(3) = Commitment(1) + Commitment(2) $$

with the equivalent prices:

$$ Price_{Up}(3) = Price_{Up}(1) + Price_{Up}(2) $$

$$ Price_{Down}(3) = Price_{Down}(1) + Price_{Down}(2) $$

---

How is it modeled in this PR?

Parameters

• ems_derivative_soft_{min,max}: upper/lower EMS soft limits
• ems_flow_soft_relaxation_cost: cost of exceeding the soft limits.

Variables

• ems_power_margin_{upper, lower}: how much the power is over/under the soft limit.

Constraints

• ems_derivative_soft_margin_{upper, lower}: ensures that the variable ems_power_margin_{upper, lower} is not higher than the difference between the hard and soft limits.
• ems_derivative_soft_lower_bound: makes the "margin" variables to track the difference between the soft limit and the EMS power.

---

Commitments model

The commitment appear in two equations:

    def ems_flow_commitment_equalities(m, j):
        """Couple EMS flows (sum over devices) to commitments."""
        return (
            0,
            sum(m.commitment_quantity[:, j])
            + sum(m.commitment_downwards_deviation[:, j])
            + sum(m.commitment_upwards_deviation[:, j])
            - sum(m.ems_power[:, j]),
            0,
        )

Which is equivalent to

$$ \sum_{d\in Devices} Power(d, j) = \sum_{c \in Commitments} [Commitment(c, j) + Down(c, j) - Up(c,j)] $$

and the objective functions:

        for j in m.j:
            for c in m.c:
                costs += m.commitment_downwards_deviation[c, j] * m.down_price[c, j]
                costs += m.commitment_upwards_deviation[c, j] * m.up_price[c, j]

[Flix6x]

We would add two commitments:

Commitment = 90 kW, $Price_{Up} = 100$, $Price_{Down} = 0$
Commitment = -90 kW, $Price_{Up} = 0$, $Price_{Down} = 100$

This would be equivalent to having a single commitment:

Commitment = 0 kW, $Price_{Up} = 100$, $Price_{Down} = 100$

This applies in general.

Counterexample: if the aggregate power flow is 50 kW, then in your first example the costs are 0, but in your second example the costs are 5$/h (assuming your prices are in $/MWh).

[victorgarcia98]

We would add two commitments:
Commitment = 90 kW, PriceUp=100, PriceDown=0
Commitment = -90 kW, PriceUp=0, PriceDown=100
This would be equivalent to having a single commitment:
Commitment = 0 kW, PriceUp=100, PriceDown=100
This applies in general.

Counterexample: if the aggregate power flow is 50 kW, then in your first example the costs are 0, but in your second example the costs are 5$/h (assuming your prices are in $/MWh).

True, good point.

Case A)

• Constraint: $50 = -90 + 90 + Down(1) + Down(2) + Up(1) + Up(2) = Down(1) + Down(2) + Up(1) + Up(2)$
• Objective: $100 \cdot Up(1) + 0\cdot Up(2) + 0\cdot Down(1) + 100 \cdot Up(2) = 100 \cdot [Up(1) + Down(2)]$

Case B)

• Constraint: $50 = 0 + Up(1) + Down(1) = Up(1) + Down(1) $
• Objective: $100 \cdot [Up(1) + Down(1)]$

Wouldn't case A) be unbounded?

Let's assume $Down(2)$, which appears in the objective function, is any large value, e.g. 10 MW. This will make 100 EUR/MWh * 10 MW = 1000 EUR/h. We can compensate $Down(2)$ with $Up(2)$ which doesn't appear in the objective but appears on the constraint.

In general, we can set $Up(1)$ and $Down(2)$ to be any value which drive up/down the objective function and find values for $Up(2)$ and $Down(1)$ to fulfill the constraint.