Market

To allow for various representations of price-elastic production and demand with marginal cost/value \( C_{market} \) that are not covered by the specific categories otherwise defined, generic market steps are introduced through the variable \( market \), measured in GWh:

\[ \forall \enspace m \in \textrm{MarketSteps},\ k \in \textrm{Timesteps} \]

\[ 0 \leq \textrm{market}(s,k) \leq \textrm{MarketCapacity}(m,k) \qquad \space, \textrm{if } \textrm{MarketCapacity}(m,k) > 0 \]

\[ \textrm{MarketCapacity}(m,k) \leq \textrm{market}(m,k) \leq 0 \qquad , \textrm{if } \textrm{MarketCapacity}(m,k) < 0 \]

The additional cost component:

\[ Cost: += \sum_{m,k}C_{market}(m,k)*\textrm{market}(m,k)\]

Note that \( \textrm{MarketCapacity}(m,k) \) is negative for sale steps and positive for purchase steps.

Dynamically adjusted market steps

In this section, we detail a mode of simulating market steps in which the purchase/sale is adjusted dynamically according to previous decisions in the purchase/sale variables. Whereas the formulation above might lead to an abrupt adjustment of purchase/sale in response to changing prices, it might be more realistic to enable a more gradual adjustement of purchase/sales which smoothens out variations in the case of shorter but transient price peaks.

In the dynamically adjusted market step functionality, we separate the total purchase/sale capacity into three segments:

Part	Description
\( \textrm{FlexibleMarketCapacity} \)	Part of the total purchase capacity that can be used based on the current power price.
\( \textrm{InflexibleMarketCapacity} \)	Part of the total purchase capacity that is used no matter how high/low the price becomes. ( We distinguish between purchase and sales.)
\( \textrm{InflexibleUnusedMarketCapacity} \)	Part of the total purchase capacity that is not used no matter how low/high the price becomes.

The flexible capacity constrains the purchase and sales variables as follows

\[ \forall \enspace m \in \textrm{Steps},\ k \in \textrm{Timesteps} \]

\[ 0 \leq \textrm{purchase} (m,k) \leq \textrm{FlexibleMarketCapacity}(m,k) \]

\[ \textrm{FlexibleMarketCapacity}(m, k) \leq \textrm{sales} (m,k) \leq 0, \]

and the inflexible market capacity is subtracted from the right hand side of the power balance of area a:

\[ \textrm{POWBAL}(a,k) [RHS]: -= \textrm{InflexibleMarketCapacity}(m,k). \]

The objective function is updated with cost and capacities as follows:

\[ Cost: += \sum_{m,k} C_m^+*\textrm{purchase}(m,k) + \sum_{m,k} C_m^-*\textrm{sales}(m,k) \]

where \( purchase(m,k) \) or \( sales(m,k) \) is the variable representing how much energy is purchased/sold by the flexible capacity, depending on whether the market step is a purchase or sales option and \( C_m^{\pm} \) their respective costs.

The three response types differ in how the three capacity parts are calculated. Before showing the equations we define the total consumption of a step to be either

\[ \textrm{TotalConsumption}(k) = \textrm{InflexibleMarketCapacity}(k) + \textrm{purchase}(k) \]

or

\[ \textrm{TotalConsumption}(k) = \textrm{InflexibleMarketCapacity}(k) + \textrm{sales}(k) \]

and \( \textrm{TotalConsumption} \) will henceforth be understood to be the total amount purchased or total amount sold depending on the step type.

Momentary

The purchase/sale changes immediately according to price level with

\[\textrm{InflexibleMarketCapacity}(k) = 0\]

\[\textrm{InflexibleUnusedMarketCapacity}(k) = 0\]

\[\textrm{FlexibleMarketCapacity}(k) = \textrm{MarketCapacity(k)},\]

which is equivalent to the formulation in the previous section.

Asymptotic

The purchase (sale) approaches zero (max capacity) asymptotically when prices stay high (low) over time.

\[ \textrm{InflexibleMarketCapacity}(k) = \alpha(k) \textrm{TotalConsumption}(k - \Delta k) \]

\[ \textrm{InflexibleUnusedMarketCapacity}(k) = \alpha(k) (\textrm{MarketCapacity}(k) - \textrm{TotalConsumption}(k - \Delta k) )\]

\[ \textrm{FlexibleMarketCapacity}(k) = \textrm{MarketCapacity}(k) (1-\alpha(k)) \]

where \( \alpha(k) \in [0,1] \) is an inertia parameter and \( \Delta k > \text{DecisionProblemLength} \) is how far back in time to find the previous consumption.

If \( \textrm{InflexibleUnusedMarketCapacity}(k) < 0 \) and the market step is a purchase option or \( \textrm{InflexibleUnusedMarketCapacity}(k) < 0 \) and the step is a sales option we replace part of the above equations with

\[ \textrm{InflexibleUnusedMarketCapacity}(k) = 0 \]

\[ \textrm{InflexibleUsedMarketCapacity}(k) = \alpha(k) \textrm{MarketCapacity}(k) \]

Linear

The purchase (sale) approaches zero (max capacity) linearly when prices stay high (low) over time.

\[ \textrm{InflexibleMarketCapacity}(k) = \textrm{TotalConsumption}(k - \Delta k) - (1 - \beta(k)) \textrm{MarketCapacity(k)} \]

\[ \textrm{FlexibleMarketCapacity}(k) = 2(1-\beta(k)) \textrm{MarketCapacity}(k) \]

where \( \beta(k) \in [0,1] \) is an inertia parameter and \( \Delta k > \text{DecisionProblemLength} \) denotes how far back in time we need to go in order to find the previous consumption.

If \( \textrm{InflexibleMarketCapacity(k)} < 0 \) and the market step is a purchase option or \( \textrm{InflexibleMarketCapacity(k)} > 0 \) and the step is a sales option, we replace the above equations with

\[ \textrm{InflexibleMarketCapacity}(k) = 0 \]

\[ \textrm{FlexibleMarketCapacity}(k) = (1 - \beta(k)) \textrm{MarketCapacity}. \]

If \( \textrm{TotalConsumption}(k) > \textrm{MarketCapacity} \) and the market step is a purchase option or \( \textrm{TotalConsumption}(k) < \textrm{MarketCapacity} \) and the step is a sales option we use

\[ \textrm{FlexibleMarketCapacity}(k) = \textrm{MarketCapacity(k)} - \textrm{InflexibleMarketCapacity}(k). \]