Dispatch Optimisation Formulation

Decision variables

\( q_{r,a,c,ts} \), where q represents c commodity flow in region r, to/from asset a, in time slice ts. Negative values are flows into the asset and positive values are flows from the asset; q must be ≤0 for input flows and ≥0 for output flows. Note that q is a quantity flow (e.g. energy) as opposed to an intensity (e.g. power).

where

r = region

a = asset

c = commodity

ts = time slice

Objective function

$$ min. \sum_{r}{\sum_{a}{\sum_{c}{\sum_{ts}}}} cost_{r,a,c,ts} * q_{r,a,c,ts} $$

Where cost is a vector of cost coefficients representing the cost of each commodity flow.

$$ cost_{r,a,c,ts} = var\_ opex_{r,a,pacs} + flow\_ cost_{r,a,c} + commodity\_ cost_{r,c,ts} $$

var_opex is the variable operating cost for a PAC. If the commodity is not a PAC, this value is zero.

flow_cost is the cost per unit flow.

commodity_cost is the exogenous (user-defined) cost for a commodity. If none is defined for this combination of parameters, this value is zero.

NOTE: If the commodity flow is an input (i.e. flow <0), then the value of cost should be multiplied by −1 so that the impact on the objective function is positive.

Constraints

Issue 1: It would reduce the size of the optimisation problem if all assets of the same type that are in the same region are grouped together in constraints (to reduce the number of constraints). However, this approach would also complicate pre- and post-optimisation processing which would need to unpick grouped assets and allocate back to their agent owners.

Asset-level input-output commodity balances

Non-flexible assets

Assets where ratio between output/s and input/s is strictly proportional. Energy commodity asset inputs and outputs are proportional to first-listed primary activity commodity at a time slice level defined for each commodity. Input/output ratio is a fixed value.

For each r, a, ts, c:

$$ \frac{q_{r,a,c,ts}}{flow_{r,a,c}} - \frac{q_{r,a,pac1,ts}}{flow_{r,a,pac1}} = 0 $$

for all commodity flows that the process has (except pac1). Where pac1 is the first listed primary activity commodity for the asset (i.e. all input and output flows are made proportional to pac1 flow).

TBD - cases where time slice level of the commodity is seasonal or annual.

Commodity-flexible assets

Assets where ratio of input/s to output/s can vary for selected commodities, subject to user-defined ratios between input and output.

Energy commodity asset inputs and outputs are constrained such that total inputs to total outputs of selected commodities is limited to user-defined ratios. Furthermore, each commodity input or output can be limited to be within a range, relative to other commodities.

For each r, a, c, ts:

(TBD)

for all c that are flexible commodities. “in” refers to input flow commodities (i.e. with a negative sign), and “out” refers to output flow commodities (i.e. with a positive sign).

Asset-level capacity and availability constraints

Primary activity commodity/ies output must not exceed asset capacity or any other limit as defined by availability factor constraint user inputs.

For the capacity limits, for each r, a, c, ts. The sum of all PACs must be less than the assets' capacity:

$$ \sum_{pacs} \frac{q_{r,a,c,ts}}{capacity\_ a_{a} * time\_ slice\_ length_{ts}} \leq 1 $$

For the availability constraints, for each r, a, c, ts:

$$ \sum_{pacs} \frac{q_{r,a,c,ts}}{capacity\_ a_{a} * time\_ slice\_ length_{ts}} \leq process.availability.value(up)_{r,a,ts} $$

$$ \sum_{pacs} \frac{q_{r,a,c,ts}}{capacity\_ a_{a} * time\_ slice\_ length_{ts}} \geq process.availability.value(lo)_{r,a,ts} $$

$$ \sum_{pacs} \frac{q_{r,a,c,ts}}{capacity\_ a_{a} * time\_ slice\_ length_{ts}} = process.availability.value(fx)_{r,a,ts} $$

The sum of all PACs must be within the assets' availability bounds. Similar constraints also limit output of PACs to respect the availability constraints at time slice, seasonal or annual levels. With appropriate selection of q on the LHS to match RHS temporal granularity.

Note: Where availability is specified for a process at daynight time slice level, it supersedes the capacity limit constraint (i.e. you don't need both).

Commodity balance constraints

Commodity supply-demand balance for a whole system (or for a single region or set of regions). For each internal commodity that requires a strict balance (supply == demand, SED), it is an equality constraint with just “1” for each relevant commodity and RHS equals 0. Note there is also a special case where the commodity is a service demand (e.g. Mt steel produced), where net sum of output must be equal to the demand.

For supply-demand balance commodities. For each r and each c:

$$\sum_{a,ts} q_{r,a,c,ts} = 0$$

For a service demand, for each c, within a single region:

$$\sum_{a,ts} q_{r,a,c,ts} = cr\_ net\_ fx$$

Where c is a service demand commodity and cr_net_fx is the exogenous (user-defined) demand for the given time slice selection. Note that the ts to be summed over will differ depending on the specified time slice level for a given commodity. If the time slice level is annual, it will be every time slice, if it's season then there will be separate constraints for each season and if it's time_slice then there will be separate constraints for every individual time slice.

TBD – commodities that are consumed (so sum of q can be a negative value). E.g. oil reserves.
TBD – trade between regions.

Asset-level commodity flow share constraints for flexible assets

Restricts share of flow amongst a set of specified flexible commodities. Constraints can be constructed for input side of processes or output side of processes, or both.

$$ q_{r,a,c,ts} \leq process.commodity.constraint.value(up)_{r,a,c,ts} * \left( \sum_{flexible\ c} q_{r,a,c,ts} \right) $$

$$ q_{r,a,c,ts} \geq process.commodity.constraint.value(lo)_{r,a,c,ts} * \left( \sum_{flexible\ c} q_{r,a,c,ts} \right) $$

$$ q_{r,a,c,ts} = process.commodity.constraint.value(fx)_{r,a,c,ts} * \left( \sum_{flexible\ c} q_{r,a,c,ts} \right) $$

Could be used to define flow limits on specific commodities in a flexible process. E.g. a refinery that is flexible and can produce gasoline, diesel or jet fuel, but for a given crude oil input only a limited amount of jet fuel can be produced and remainder of production must be either diesel or gasoline (for example).

Other net and absolute commodity volume constraints

Net constraint: There might be a net CO₂ emissions limit of zero in 2050, or even a negative value. Constraint applied on both outputs and inputs of the commodity, sum must less then (or equal to or more than) a user-specified value. For system-wide net commodity production constraint, for each c, sum over regions, assets, time slices.

$$\sum_{r,a,ts} q_{r,a,c,ts} \leq commodity.constraint.rhs\_ value(up)$$

$$\sum_{r,a,ts} q_{r,a,c,ts} \geq commodity.constraint.rhs\_ value(lo)$$

$$\sum_{r,a,ts} q_{r,a,c,ts} = commodity.constraint.rhs\_ value(fx)$$

Similar constraints can be constructed for net commodity volume over specific regions or sets of regions.

Production or consumption constraint: Likewise similar constraints can be constructed to limit absolute production or absolute consumption. In these cases selective choice of q focused on process inputs (consumption) or process outputs (production) can be applied.