The cost#

One of the 24 fragments of examples/pypsa.yaml: the expected operating cost and its tail, weighted by scenario. It adds a term to total_cost. It reads CVaR_omega, scenario_opex, scenario_weight under given.

dimensions:
  scenario:
    description: the futures dispatch is chosen in, each with a weight

parameters:
  CVaR_alpha:
    description: PyPSA's `risk_preference['alpha']` — the confidence level; the tail holds the other `1 - alpha` of the probability
    dims: []

variables:
  CVaR_a:
    description: "`CVaR-a` — how far a scenario's operating cost exceeds the tail's start; nothing where it does not"
    dims: [scenario]
    bounds:
      lower: 0
  CVaR_theta:
    description: "`CVaR-theta` — where the tail starts, the value at risk"
    dims: []
  CVaR:
    description: "`CVaR` — the tail's average cost, what the objective prices at `omega`"
    dims: []

given:
  parameters:
    scenario_weight: { dims: [scenario] }
    CVaR_omega: { dims: [] }
  expressions:
    scenario_opex: { dims: [scenario] }
    total_cost: { dims: [] }

expressions:
  risk_weighted_opex:
    expression: (1 - CVaR_omega) * sum(scenario_weight * scenario_opex, over=scenario) + CVaR_omega * CVaR
    adds_to: total_cost

constraints:
  CVaR_excess:
    description: "`CVaR-excess-{s}` — a scenario's operating cost beyond the tail's start is its excess; PyPSA names one row per scenario"
    dims: [scenario]
    where: CVaR_omega > 0
    expression: CVaR_a - scenario_opex + CVaR_theta >= 0
  CVaR_def:
    description: "`CVaR-def` — the tail's average is at least where it starts plus the expected excess over the tail's probability"
    dims: []
    where: CVaR_omega > 0
    expression: CVaR_theta + 1 / (1 - CVaR_alpha) * sum(scenario_weight * CVaR_a, over=scenario) <= CVaR

Sets#

Symbol	Meaning
\(\Xi\)	index \(\xi\) — `scenario` — the futures dispatch is chosen in, each with a weight

Parameters#

Symbol	Meaning
\(\alpha\)	`CVaR_alpha` (scalar) — PyPSA's `risk_preference['alpha']` — the confidence level; the tail holds the other `1 - alpha` of the probability

Variables#

Symbol	Meaning
\(a\)	`CVaR_a` over \(\Xi\) — `CVaR-a` — how far a scenario's operating cost exceeds the tail's start; nothing where it does not
\(\theta\)	`CVaR_theta` (scalar) — `CVaR-theta` — where the tail starts, the value at risk
\(CVaR\)	`CVaR` (scalar) — `CVaR` — the tail's average cost, what the objective prices at `omega`

Given#

Symbol	Meaning
\(\pi\)	`scenario_weight` over \(\Xi\), data another file declares
\(\omega\)	`CVaR_omega` (scalar), data another file declares
\(\mathit{scenario\_opex}\)	`scenario_opex` over \(\Xi\), an expression another file defines
\(\mathit{total\_cost}\)	`total_cost` (scalar), an expression this file adds `risk_weighted_opex` to

Definitions#

Symbol	Meaning
\(\mathit{risk\_weighted\_opex}\)	`risk_weighted_opex` (scalar)

Subject to#

CVaR_excess

\[ a_{\xi} - \mathit{scenario\_opex}_{\xi} + \theta \ge 0 \qquad \forall\, \xi \in \Xi \,:\, \omega > 0 \]

CVaR_def

\[ \theta + \frac{1}{1 - \alpha} \cdot \left( \sum_{\xi \in \Xi} \pi_{\xi} \cdot a_{\xi} \right) \le CVaR \qquad \text{where } \omega > 0 \]

Definitions#

risk_weighted_opex

\[ \mathit{risk\_weighted\_opex} = \left( 1 - \omega \right) \cdot \left( \sum_{\xi \in \Xi} \pi_{\xi} \cdot \mathit{scenario\_opex}_{\xi} \right) + \omega \cdot CVaR \]

Variable domains#

CVaR_a

\[ a_{\xi} \ge 0 \qquad \forall\, \xi \in \Xi \]

CVaR_theta

\[ \theta \in \mathbb{R} \]

CVaR

\[ CVaR \in \mathbb{R} \]