Notation

This page explains the notation used in this mathematical formulation section of the documentation, using the following example:

\[\begin{split}& \forall \, c, h, \ l \notin L^{\text{RES}} \cup L^{\text{REN}} \cup L^{\text{STOR}}, n = n^D = n^O, y = y^A: \\& \sum_{h m n^L t ; y^V \leq y}{\left( \output \cdot \text{duration_time_rel}_{h^1=h^D;h^2=h} \cdot \text{ACT}_{h m n^L t y^A y^V} \right)} \\& - \sum_{h m n^L t; y^V \leq y}{\left( \text{input}_{c h h^O l m n^L n^O t y^A y^V} \cdot \text{duration_time_rel}_{h^1=h^O;h^2=h} \cdot \text{ACT}_{h m n^L t y^A y^V} \right)} \\& + \text{STOCK_CHG}_{c h l n y} \\& + \sum_{s}{\left( \left( \text{land_output}_{c h l n s y} - \text{land_input}_{c h l n s y} \right) \cdot \text{LAND}_{n s y} \right)} \\& - \text{demand_fixed}_{c h l n y} \\& = \\& \text{COMMODITY_BALANCE}_{c h l n y}\end{split}\]

  • The equality (=) or inequality (< or >) relation is on a line by itself. This allows to distinguish the left- and right-hand sides of the expression.

  • The first line begins with the “for all …” symbol (\(\forall\)) and gives:

    • The dimensionality of the equation or inequality, via a list of indices (in alphabetical order). In some cases, this is equivalent to the dimensionality of a single term on either side. In the example above, this is COMMODITY_BALANCE. Where both sides are more complicated, the first line informs about the dimensionality.

    • Any conditions or restrictions on the set members to which the indices apply. Implicitly, indices without condition may take any value from their corresponding set. In the example above:

      • Index \(l\) must be in the union of certain other sets.

      • The indices \(y\) (for parameters/variables with a dimension named “year”) must always align with indices \(y^A\) (for parameters/variables with a dimension named “year_active”).

      • Likewise, \(n\), \(n^D\) (node_dest), and \(n^O\) (node_origin) must all be aligned

      • The other indices are unconstrained, so implicitly \(c \in C\) and \(h \in H\).

  • References to items (exogenous parameters or endogenous variables) include:

    • The name in roman (upright) text. Parameter names are always lower case, for instance ‘output’. Variables names are always upper case, for instance ‘ACT’.

    • Its dimensionality as a right subscript with an alphabetical list of indices. These:

      • Always exactly restate the dimensionality of the item given elsewhere in this documentation. That is, the indices do indicate aggregation or other operations.

      • May express an alignment or restriction of indices, if and only if this is particular to one reference of the parameter/variable. Otherwise, these are expressed in the first line, as described above. In the above example, ‘duration_time_rel’ has indices \(h^1, h^2\), but these are aligned differently in the two places it is used in the equation.

      • Facilitate understanding of alignment and broadcasting between elements of parameters/variables with different dimensionality.

  • Indices are always exactly one character, for instance ‘y’. If a parameter or variable is indexed more than once by the same set, all but 0 or 1 occurrence of the same base has a right superscript, for instance \(y^A\) and \(y^V\). These correspond to different dimension- or index names given in the “sets and parameters” page of the documentation; for instance \(y^A\) is always the dimension/index name “year_active”, referring to a member of \(Y^A \subseteq Y\).

  • In both the left- and right-hand sides:

    • Individual additive terms are shown on separate lines.

    • Operands for operations like summation, if including multiple terms, are enclosed in parentheses.

LateX source

The following describes the usage of LaTeX markup to write equations. Users who are only reading the documentation to aid usage of message_ix do not need to read these instructions.

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