According to the articles below, Russia’s main party United Russia used Approval Voting on 22 May 2016 for its primary elections. The articles lack details, and so far I've been unable to get confirmation. You can run the articles through Google Translate for a pretty good translation. Most of the content was irrelevant, but I’ve included here the relevant excerpts.
Elections for its own: will there be real competition in the main party's primaries?
Выборы для своих: будет ли реальная борьба на праймериз главной партии
After the primaries procedure are those who expect to get into the Duma on the list, and those who are running in single-mandate constituencies. Accordingly, this Sunday, voters will receive two ballots - to select the single-mandate and spisochnikov. The vote will be rated, which means that each will be able to choose more than one candidate, and will have the chance to vote for anyone who gives him sympathy. This is one of the fundamental differences between Russian primaries from US, where the vote for one candidate, says political analyst Yevgeny Minchenko.
Experts explained the difference between United Russia's primaries and western elections
Эксперты объяснили отличия праймериз «Единой России» от западных выборов
Primaries "United Russia" in many ways unique and not like the Western models, the experts stated, "Minchenko Consulting". For example, in Russia one voter is allowed to vote for more candidates
In Russia, analysts say, the ruling party during the primaries, used unusual for the world of practice system - a vote for an unlimited number of candidates (in the West used the principle of "one voter - one vote").
United Russia’s website has a couple more articles about the primary elections:
but they lack mention of the voting method.