The composed spec#

This page shows the spec that the surface, generators and loads make together:

import mathspec as ms

spec = ms.merge(['surface.yaml', 'generator.yaml', 'load.yaml'])

The file below is spec, the spec that merge returns, written as YAML with every default spelled out. No fragment holds this file, and the repository does not commit it. Each component fragment read Port_p under given:, and merge folded those readings into the entry of the surface, so Port_p is one entry here.

The objective comes from the surface, and it reads total_cost. The generator is the only fragment with a cost, so Generator_cost is the only term of that sum. A second fragment with a cost would add its own term to total_cost.

The math under the file has a tab per formulation: As composed is the spec above, and With commitment lays variants/commitment.yaml over it with override, which makes the generator a committed unit:

committed = ms.override(spec, ['variants/commitment.yaml'])

to_spec refuses a patch on its own, because the patch edits entries that it does not declare. So the patch has math only on the spec it lands on, and the tab prints the patch beside that math.

version: 0
dimensions:
  snapshot: {dtype: datetime, description: dispatch periods}
  bus: {dtype: str, description: network nodes}
  port: {dtype: str, description: 'the connections components make, one label per connection'}
  generator: {dtype: str, description: 'generating units, each on one port'}
  load: {dtype: str, description: 'demands, each on one port'}
relations:
  Port_bus: {key: port, values: bus}
  Generator_port: {key: generator, values: port}
  Load_port: {key: load, values: port}
parameters:
  Generator_p_nom:
    dims: [generator]
    dtype: float
    description: nominal power
  Generator_marginal_cost:
    dims: [generator]
    dtype: float
    description: cost of one unit of output
  Load_p_set:
    dims: [snapshot, load]
    dtype: float
    description: '`Load-p_set` — what a load takes in a snapshot'
variables:
  Port_p:
    dims: [snapshot, port]
    domain: continuous
    missing: absent
    description: what a port puts into its bus in a snapshot, negative for a withdrawal
  Generator_p:
    dims: [snapshot, generator]
    bounds: {lower: 0.0, upper: Generator_p_nom}
    domain: continuous
    missing: absent
    description: '`Generator-p` — what a generator produces in a snapshot'
expressions:
  Generator_cost: sum(Generator_p * Generator_marginal_cost)
  total_cost:
    dims: []
    expression: Generator_cost
    description: what running the system costs
objective: {sense: minimize, expression: total_cost}
constraints:
  Bus_nodal_balance:
    dims: [snapshot, bus]
    expression: sum(Port_p, over=port, by=Port_bus[bus]) == 0
    description: '`Bus-nodal_balance` — what the ports on a bus put in nets to nothing'
  Generator_injection:
    dims: [snapshot, generator]
    expression: at(Port_p, by=Generator_port[port]) == Generator_p
    description: 'what a generator produces is what its port injects. No PyPSA row stands for this: PyPSA
      writes the generator into the balance instead'
  Load_withdrawal:
    dims: [snapshot, load]
    expression: at(Port_p, by=Load_port[port]) == -Load_p_set
    description: 'what a load takes is what its port withdraws. No PyPSA row stands for this: PyPSA writes
      the load into the balance instead'

As composedWith commitment

Sets#

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{N}\)	index \(n\) — `bus` with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N}\) — network nodes
\(\mathcal{J}\)	index \(j\) — `port` with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N},\ \mathrm{Generator\_port}: \mathcal{G} \to \mathcal{J},\ \mathrm{Load\_port}: \mathcal{D} \to \mathcal{J}\) — the connections components make, one label per connection
\(\mathcal{G}\)	index \(g\) — `generator` with \(\mathrm{Generator\_port}: \mathcal{G} \to \mathcal{J}\) — generating units, each on one port
\(\mathcal{D}\)	index \(d\) — `load` with \(\mathrm{Load\_port}: \mathcal{D} \to \mathcal{J}\) — demands, each on one port

Parameters#

Symbol	Meaning
\(\mathrm{p}^{\mathrm{nom}}\)	`Generator_p_nom` over \(\mathcal{G}\) — nominal power
\(\mathrm{c}\)	`Generator_marginal_cost` over \(\mathcal{G}\) — cost of one unit of output
\(\mathrm{load}\)	`Load_p_set` over \(\mathcal{T} \times \mathcal{D}\) — `Load-p_set` — what a load takes in a snapshot

Variables#

Symbol	Meaning
\(f\)	`Port_p` over \(\mathcal{T} \times \mathcal{J}\) — what a port puts into its bus in a snapshot, negative for a withdrawal
\(p\)	`Generator_p` over \(\mathcal{T} \times \mathcal{G}\) — `Generator-p` — what a generator produces in a snapshot

Definitions#

Symbol	Meaning
\(\mathit{Generator\_cost}\)	`Generator_cost` (scalar)
\(\mathit{total\_cost}\)	`total_cost` (scalar) — what running the system costs

Variable domains#

Port_p

\[ f_{t,j} \in \mathbb{R} \qquad \forall\, t \in \mathcal{T},\ j \in \mathcal{J} \]

Generator_p

\[ 0 \le p_{t,g} \le \mathrm{p}^{\mathrm{nom}}_{g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Definitions#

Generator_cost

\[ \mathit{Generator\_cost} = \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot \mathrm{c}_{g} \]

total_cost

\[ \mathit{total\_cost} = \mathit{Generator\_cost} \]

Objective#

\[ \min \mathit{total\_cost} \]

Subject to#

Bus_nodal_balance

\[ \sum_{j \in \mathcal{J} \,:\, \mathrm{Port\_bus}(j) = n} f_{t,j} = 0 \qquad \forall\, t \in \mathcal{T},\ n \in \mathcal{N} \]

Generator_injection

\[ f_{t,\mathrm{Generator\_port}(g)} = p_{t,g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Load_withdrawal

\[ f_{t,\mathrm{Load\_port}(d)} = -\mathrm{load}_{t,d} \qquad \forall\, t \in \mathcal{T},\ d \in \mathcal{D} \]

variants/commitment.yaml

parameters:
  Generator_p_min_pu: { dims: [generator], description: "least output, per unit of nominal power" }
variables:
  Generator_status:
    dims: [snapshot, generator]
    domain: binary
    description: "`Generator-status` — whether a unit is on in a snapshot"
  Generator_p: { bounds: { upper: null } }
constraints:
  Generator_com_p_upper:
    description: "`Generator-com-p-upper` — a committed unit outputs at most its nominal power; off, at most nothing"
    dims: [snapshot, generator]
    expression: Generator_p <= Generator_p_nom * Generator_status
  Generator_com_p_lower:
    description: "`Generator-com-p-lower` — a committed unit outputs at least its minimum; off, at least nothing"
    dims: [snapshot, generator]
    expression: Generator_p >= Generator_p_min_pu * Generator_p_nom * Generator_status

Sets#

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) — `snapshot` — dispatch periods
\(\mathcal{N}\)	index \(n\) — `bus` with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N}\) — network nodes
\(\mathcal{J}\)	index \(j\) — `port` with \(\mathrm{Port\_bus}: \mathcal{J} \to \mathcal{N},\ \mathrm{Generator\_port}: \mathcal{G} \to \mathcal{J},\ \mathrm{Load\_port}: \mathcal{D} \to \mathcal{J}\) — the connections components make, one label per connection
\(\mathcal{G}\)	index \(g\) — `generator` with \(\mathrm{Generator\_port}: \mathcal{G} \to \mathcal{J}\) — generating units, each on one port
\(\mathcal{D}\)	index \(d\) — `load` with \(\mathrm{Load\_port}: \mathcal{D} \to \mathcal{J}\) — demands, each on one port

Parameters#

Symbol	Meaning
\(\mathrm{p}^{\mathrm{nom}}\)	`Generator_p_nom` over \(\mathcal{G}\) — nominal power
\(\mathrm{c}\)	`Generator_marginal_cost` over \(\mathcal{G}\) — cost of one unit of output
\(\mathrm{load}\)	`Load_p_set` over \(\mathcal{T} \times \mathcal{D}\) — `Load-p_set` — what a load takes in a snapshot
\(\underline{\mathrm{p}}\)	`Generator_p_min_pu` over \(\mathcal{G}\) — least output, per unit of nominal power

Variables#

Symbol	Meaning
\(f\)	`Port_p` over \(\mathcal{T} \times \mathcal{J}\) — what a port puts into its bus in a snapshot, negative for a withdrawal
\(p\)	`Generator_p` over \(\mathcal{T} \times \mathcal{G}\) — `Generator-p` — what a generator produces in a snapshot
\(u\)	`Generator_status` over \(\mathcal{T} \times \mathcal{G}\) — `Generator-status` — whether a unit is on in a snapshot

Definitions#

Symbol	Meaning
\(\mathit{Generator\_cost}\)	`Generator_cost` (scalar)
\(\mathit{total\_cost}\)	`total_cost` (scalar) — what running the system costs

Variable domains#

Port_p

\[ f_{t,j} \in \mathbb{R} \qquad \forall\, t \in \mathcal{T},\ j \in \mathcal{J} \]

Generator_p

\[ p_{t,g} \ge 0 \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_status

\[ u_{t,g} \in \{0, 1\} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Definitions#

Generator_cost

\[ \mathit{Generator\_cost} = \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot \mathrm{c}_{g} \]

total_cost

\[ \mathit{total\_cost} = \mathit{Generator\_cost} \]

Objective#

\[ \min \mathit{total\_cost} \]

Subject to#

Bus_nodal_balance

\[ \sum_{j \in \mathcal{J} \,:\, \mathrm{Port\_bus}(j) = n} f_{t,j} = 0 \qquad \forall\, t \in \mathcal{T},\ n \in \mathcal{N} \]

Generator_injection

\[ f_{t,\mathrm{Generator\_port}(g)} = p_{t,g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Load_withdrawal

\[ f_{t,\mathrm{Load\_port}(d)} = -\mathrm{load}_{t,d} \qquad \forall\, t \in \mathcal{T},\ d \in \mathcal{D} \]

Generator_com_p_upper

\[ p_{t,g} \le \mathrm{p}^{\mathrm{nom}}_{g} \cdot u_{t,g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]

Generator_com_p_lower

\[ p_{t,g} \ge \underline{\mathrm{p}}_{g} \cdot \mathrm{p}^{\mathrm{nom}}_{g} \cdot u_{t,g} \qquad \forall\, t \in \mathcal{T},\ g \in \mathcal{G} \]