# Years, periods, and time slices

This page describes how time-related concepts are represented in the MESSAGEix framework and stored in ix modeling platform.

## Years and periods

The year set—in a message_ix.Scenario and in the MESSAGEix model formulation—is used to index parameter dimensions with the names “year”, “year_act”, “year_vtg”, etc. Elements in this set are representative years within a period of time. MESSAGEix treats periods as contiguous, but year elements need not be consecutive. The representative year is always the final year of the corresponding period.

Example 1

In a Scenario with consecutive year elements [1984, 1985, 1986, …], the element 1985 refers to the period from 1985-01-01 to 1985-12-31, inclusive.

Example 2

In a Scenario with non-consecutive year elements [1000, 1010, 1020, …]:

• the element 1000 refers to a period that ends 1000-12-31, i.e. at the end of the calendar year 1000.

• the element 1010 refers to a period that ends 1010-12-31, i.e. at the end of the calendar year 1010.

• the element 1010 therefore refers to the period from 1001-01-01 to 1010-12-31, inclusive.

The parameter duration_period stores the duration of each period, measured in years, indexed by elements from the year set. MESSAGEix only supports year set elements that can be represented as integers, and therefore periods with durations that are a whole number of years.

Other parameters may be aligned to the start or end of periods.

Example 3

A Scenario has year elements [1000, 1010, 1020, 1030, …], and a technology parameterized with technical_lifetime value of 20 years for the key year_vtg=1010.

• Capacity associated with this technology is constructed at the beginning of the period denoted by 1010, i.e. as of 1001-01-01.

• By the end of this period, 1010-12-31, this technology has operated for 10 years.

• The next period, labelled 1020, ends in 1020-12-31. By this date, the technology has operated for 20 years, equal to its technical lifetime.

• The following period, labelled 1030, begins on 1021-01-01. The capacity created in year_vtg=1010 is beyond its technical lifetime, and unavailable in this period.

## Time slices

The time set is used to index parameter dimensions with the names “time”, “time_origin”, “time_dest”, etc. These are variously referred to as “(sub-annual) time slices”, “time steps”, or other names. Elements in this set are labels for portions of a single year. The special value 'year' represents the entire year.

Since a year element refers to the representative, final year within a period, using year and time together denotes a portion of that specific year.

Example 4

In a Scenario with year elements [2000, 2002, 2004] and time elements [summer, winter]:

• The year element 2002 refers to the period from 2001-01-01 to 2002-12-31 inclusive, which has a duration_period of 2 years.

• The time element ‘summer’, used alone, refers to a portion of any year.

• In a MESSAGEix parameter indexed by (year, time, …), values with the key (2002, ‘summer’, …) refer to the ‘summer’ portion of the final year (2002-01-01 to 2002-12-31) within the entire period (2001-01-01 to 2002-12-31) denoted by 2002.

### Duration of sub-annual time slices

The duration of each sub-annual time slice should be defined relative to the whole year, with a value between 0 and 1, using parameter duration_time. For example, in a model with four seasons with the same length, duration_time of each season will be 0.25. Please note that the duration of time slices does not need to be equal to each other. This information is needed to calculate capacity of a technology that is active in different time slices. Time slices can be represented at different temporal levels, using sets lvl_temporal and map_temporal_hierarchy. This helps introducing a flexible temporal resolution, e.g., by representing some technologies at finer time resolution while others at year. When there are more than one temporal levels, e.g., “year”, “season”, “month”, “day”, etc., duration_time is defined for time slices at each temporal level separately. The sum of duration_time of time slices at each temporal level must be equal to 1. For example, in a model with 4 time slices as “season” and 10 time slices as “day” under each “season”, duration_time of each “season” and “day” can be specified as 0.25 and 0.025, respectively.

By default, the unit of ACT is treated per year in the GAMS formulation for different time slices. This means values reported in time slice “year” and “month” both have the same unit (e.g., GWa). However, the user can report the values across parameters and variables with different units relative to the length of the full year. For example, the user can report ACT in units of “GWa” and “GWh” for time slices of “year” and “hour”, respectively, in the same model. To activate this feature, the parent time slice for which the relative units are desired should be specified by set time_relative. This will ensure that parameter duration_time_rel is effective. Otherwise, this parameter is filled by value of 1, meaning that the units will be treated uniformly across different sub-annual time slices.

## Discounting

The interest_rate in MESSAGEix is defined for a period of one year, therefore, for periods of more than a year, the discounting is performed in a cumulative manner.

Example 5

Using the same setup as Example 2:

• Discounting for the element 1010 involves discounting for years 1001, 1002, … , 1010.

• Using the standard PV formula, we have that, for the year 1001 the discount factor would be $$(1 + interest_rate)^(1000 - 1001)$$, for the year 1002 the discount factor would be $$(1 + interest_rate)^(1000 - 1002)$$, and so on.

• Therefore, the period discount factor for the element 1010 is $$df_1010 = (1 + interest_rate)^(1000 - 1001) + (1 + interest_rate)^(1000 - 1002) + ... + (1 + interest_rate)^(1000 - 1010)$$

• Analogously, the period discount factor for the element 1020 is $$df_1020 = (1 + interest_rate)^(1000 - 1011) + (1 + interest_rate)^(1000 - 1012) + ... + (1 + interest_rate)^(1000 - 1020)$$

• So, if we have a cost of K_1010 for the element 1010, its discounted value would be df_1010 * K_1010, which means, all the years in element 1010 have a representative cost of K_1010 that is discounted up to the initial year of the setup, namely, the year 1000.

In practice, since the representative year of a period is always its final year, the actual calculation of the period discount factor within the model is performed backwards, i.e., starting from the final year of the period until the initial year.