The composed spec#

What 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 merge returns, written as YAML with every default spelled out. No fragment holds it, and nothing in the repository commits it. Port_p is one declaration here. Each component fragment read it under given:, and merging folded those readings into the surface's own declaration.

The objective is the generator's, carried as it was written, since no other fragment prices anything. A second priced fragment would add its term to this one, each term in parentheses.

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

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

A patch is refused on its own, since it edits declarations it does not declare. So the spec it lands on is the only place its math exists, 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
    absence: undefined
    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
    absence: undefined
    description: '`Generator-p` — what a generator produces in a snapshot'
constraints:
  Bus_nodal_balance:
    dims: [snapshot, bus]
    expression: sum(Port_p, by=Port_bus, over=port, into=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, over=port, into=generator) == 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, over=port, into=load) == -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'
objective: {sense: minimize, expression: sum(Generator_p * Generator_marginal_cost)}

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

Objective#

\[ \min \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot \mathrm{c}_{g} \]

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

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

Variable domains#

Port_p

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

Generator_p

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

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

Objective#

\[ \min \sum_{t \in \mathcal{T},\ g \in \mathcal{G}} p_{t,g} \cdot \mathrm{c}_{g} \]

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

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

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

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