# 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 

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