Flow-based market coupling#

This file states PyPSA's flow-based domains. It is one of the 25 whole specs of one topic each (fragments) that merge composes into the spec of examples/pypsa.yaml. It adds a term to the sum Bus_injection. FlowBased_net_position is what a zone bus exports to its domain. GlobalConstraint_flow_based_ub holds each critical network element to its remaining available margin, and FlowBased_balance sums each domain's net positions to zero through a sum by a relation.

given:
  parameters:
    GlobalConstraint_type: { dims: [global_constraint], dtype: str }
    GlobalConstraint_sense: { dims: [scenario, global_constraint], dtype: str }
    GlobalConstraint_counts_snapshot: { dims: [scenario, global_constraint, snapshot], dtype: bool }
  variables:
    Link_p: { dims: [scenario, snapshot, link] }
  expressions:
    Bus_injection: { dims: [bus, scenario, snapshot] }

dimensions:
  scenario:
    description: the futures dispatch is chosen in, each with a weight
  snapshot:
    description: dispatch periods
    dtype: datetime
    ordered: true
  bus:
    description: network nodes
  link:
    description: controllable connections, each from one bus to the buses it delivers to
  global_constraint:
    description: PyPSA's `GlobalConstraint` rows, one label per declared limit
  flow_based_domain:
    description: >-
      the flow-based domains, each with its own zero-sum balance of net
      positions (PyPSA's `domain` of a `flow_based` row), data prep. A run whose
      flow-based rows name no domain has one. A run without them has none

maps:
  bus:
    flow_based_domain:
      description: >-
        the flow-based domain a zone bus belongs to: the domain whose rows set
        the bus's `ptdf_<bus>` column, data prep. A bus that is no zone has no
        row
      refers_to: flow_based_domain
      missing: absent

parameters:
  GlobalConstraint_ram:
    description: >-
      the remaining available margin of a `flow_based` row, one critical
      network element (CNEC). It may differ by snapshot. No value drops the
      row in that snapshot (`flow_based.py:201`)
    dims: [scenario, snapshot, global_constraint]
    missing: absent
  GlobalConstraint_zonal_ptdf:
    description: >-
      the zonal power transfer distribution factor of a `flow_based` row for a
      zone bus: PyPSA's `ptdf_<bus>` column read long, data prep. It may differ
      by snapshot. No value is zero, as PyPSA fills it
    dims: [scenario, snapshot, global_constraint, bus]
    missing: 0
  GlobalConstraint_link_ptdf:
    description: >-
      the factor of a `flow_based` row for a corridor link, whose flow enters
      the row: PyPSA's `ptdf_<link>` column read long, data prep. No value is
      zero
    dims: [scenario, snapshot, global_constraint, link]
    missing: 0
  Bus_flow_based_zone:
    description: >-
      whether a bus is a zone of a flow-based domain: some `flow_based` row
      has its `ptdf_<bus>` column, data prep
    dims: [bus]
    dtype: bool

variables:
  FlowBased_net_position:
    description: >-
      `FlowBased-net_position` — what a zone bus exports to the flow-based
      domain, unbounded. It leaves the zone's nodal balance
      (`flow_based.py:207-234`). A bus that is no zone has none
    dims: [scenario, snapshot, bus]
    where: Bus_flow_based_zone
    missing: 0

expressions:
  FlowBased_injection:
    description: what a zone bus exports to its flow-based domain leaves the bus (`constraints.py:1578`)
    dims: [scenario, snapshot, bus]
    expression: -FlowBased_net_position
    adds_to: Bus_injection

constraints:
  GlobalConstraint_flow_based_ub:
    description: >-
      `flow_based` — a CNEC's flow, the zones' net positions and the corridor
      links' flows each times its factor, is at most its remaining available
      margin. PyPSA names the rows `GlobalConstraint-ram`, one per CNEC and
      snapshot, and reads no `constant` (`flow_based.py:262-293`)
    dims: [scenario, snapshot, global_constraint]
    where: >-
      GlobalConstraint_type == 'flow_based' AND GlobalConstraint_sense == '<='
      AND GlobalConstraint_ram AND GlobalConstraint_counts_snapshot
    expression: >-
      sum(GlobalConstraint_zonal_ptdf * FlowBased_net_position, over=bus)
      + sum(GlobalConstraint_link_ptdf * Link_p, over=link)
      <= GlobalConstraint_ram
  FlowBased_balance:
    description: "`FlowBased-balance` — the net positions of each domain's zones sum to zero"
    dims: [scenario, snapshot, flow_based_domain]
    expression: sum(FlowBased_net_position, over=bus, by=bus[flow_based_domain]) == 0

assumptions:
  GlobalConstraint_flow_based_sense_is_le:
    holds: "GlobalConstraint_sense == '<='"
    where: "GlobalConstraint_type == 'flow_based'"
    description: >-
      PyPSA refuses a `flow_based` row whose sense is not `<=`
      (`flow_based.py:109-114`)

Given#

Symbol	Meaning
\(\mathrm{in}\)	`GlobalConstraint_counts_snapshot` over \(\mathcal{G} \times \Xi \times \mathcal{T}\), data another file declares
\(\mathrm{sense}\)	`GlobalConstraint_sense` over \(\mathcal{G} \times \Xi\), data another file declares
\(\mathrm{type}\)	`GlobalConstraint_type` over \(\mathcal{G}\), data another file declares
\(f\)	`Link_p` over \(\mathcal{L} \times \Xi \times \mathcal{T}\)
\(\mathit{Bus\_injection}\)	`Bus_injection` over \(\mathcal{N} \times \Xi \times \mathcal{T}\), an expression this file adds `FlowBased_injection` to

Sets#

Symbol	Meaning
\(\mathcal{N}\)	index \(n\) — `bus` with \(\mathrm{flow\_based\_domain}: \mathcal{N} \to \mathcal{L}\), `bus[flow_based_domain]` is `absent` where the data gives no label — network nodes
\(\mathcal{L}\)	index \(l\) — `flow_based_domain` with \(\mathrm{flow\_based\_domain}: \mathcal{N} \to \mathcal{L}\), `bus[flow_based_domain]` is `absent` where the data gives no label — the flow-based domains, each with its own zero-sum balance of net positions (PyPSA's `domain` of a `flow_based` row), data prep. A run whose flow-based rows name no domain has one. A run without them has none
\(\mathcal{G}\)	index \(g\) — `global_constraint` — PyPSA's `GlobalConstraint` rows, one label per declared limit
\(\mathcal{L}\)	index \(l\) — `link` — controllable connections, each from one bus to the buses it delivers to
\(\Xi\)	index \(\xi\) — `scenario` — the futures dispatch is chosen in, each with a weight
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods

Parameters#

Symbol	Meaning
\(\mathrm{zone}\)	`Bus_flow_based_zone` over \(\mathcal{N}\) — whether a bus is a zone of a flow-based domain: some `flow_based` row has its `ptdf_<bus>` column, data prep
\(\mathrm{PTDF}^{f}\)	`GlobalConstraint_link_ptdf` over \(\mathcal{G} \times \mathcal{L} \times \Xi \times \mathcal{T}\), \(0\) where the data has no row — the factor of a `flow_based` row for a corridor link, whose flow enters the row: PyPSA's `ptdf_<link>` column read long, data prep. No value is zero
\(\mathrm{RAM}\)	`GlobalConstraint_ram` over \(\mathcal{G} \times \Xi \times \mathcal{T}\), `absent` where the data has no row — the remaining available margin of a `flow_based` row, one critical network element (CNEC). It may differ by snapshot. No value drops the row in that snapshot (`flow_based.py:201`)
\(\mathrm{PTDF}^{z}\)	`GlobalConstraint_zonal_ptdf` over \(\mathcal{N} \times \mathcal{G} \times \Xi \times \mathcal{T}\), \(0\) where the data has no row — the zonal power transfer distribution factor of a `flow_based` row for a zone bus: PyPSA's `ptdf_<bus>` column read long, data prep. It may differ by snapshot. No value is zero, as PyPSA fills it

Variables#

Symbol	Meaning
\(\mathit{NP}\)	`FlowBased_net_position` over \(\mathcal{N} \times \Xi \times \mathcal{T}\), \(0\) where the mask leaves it out — `FlowBased-net_position` — what a zone bus exports to the flow-based domain, unbounded. It leaves the zone's nodal balance (`flow_based.py:207-234`). A bus that is no zone has none

Definitions#

Symbol	Meaning
\(\mathit{NP}^{\mathrm{inj}}\)	`FlowBased_injection` over \(\mathcal{N} \times \Xi \times \mathcal{T}\) — what a zone bus exports to its flow-based domain leaves the bus (`constraints.py:1578`)

Variable domains#

FlowBased_net_position

\[ \mathit{NP}_{n,\xi,t} \in \mathbb{R} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{zone}_{n} \]

Definitions#

FlowBased_injection

\[ \mathit{NP}^{\mathrm{inj}}_{n,\xi,t} = -\mathit{NP}_{n,\xi,t} \qquad \forall\, n \in \mathcal{N},\ \xi \in \Xi,\ t \in \mathcal{T} \]

Subject to#

FlowBased_balance

\[ \sum_{n \in \mathcal{N} \,:\, \mathrm{flow\_based\_domain}(n) = l} \mathit{NP}_{n,\xi,t} = 0 \qquad \forall\, l \in \mathcal{L},\ \xi \in \Xi,\ t \in \mathcal{T} \]

GlobalConstraint_flow_based_ub

\[ \sum_{n \in \mathcal{N}} \mathrm{PTDF}^{z}_{n,g,\xi,t} \cdot \mathit{NP}_{n,\xi,t} + \sum_{l \in \mathcal{L}} \mathrm{PTDF}^{f}_{g,l,\xi,t} \cdot f_{l,\xi,t} \le \mathrm{RAM}_{g,\xi,t} \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi,\ t \in \mathcal{T} \,:\, \mathrm{type}_{g} = \text{'}\mathrm{flow\_based}\text{'} \wedge \mathrm{sense}_{g,\xi} = \text{'}\mathrm{<=}\text{'} \wedge \mathrm{RAM}_{g,\xi,t} \text{ is defined} \wedge \mathrm{in}_{g,\xi,t} \]

Assumptions#

GlobalConstraint_flow_based_sense_is_le

\[ \mathrm{sense}_{g,\xi} = \text{'}\mathrm{<=}\text{'} \qquad \forall\, g \in \mathcal{G},\ \xi \in \Xi \,:\, \mathrm{type}_{g} = \text{'}\mathrm{flow\_based}\text{'} \]