Convergence bidding is a mechanism in two-settlement electricity markets to reduce the price gap between the day-ahead market (DAM) and the real-time market (RTM). It also provides opportunities for market participants to arbitrage on the difference between the DAM locational marginal prices (LMPs) and the RTM LMPs. Two technical subjects related to convergence bids (CBs) are studied in this dissertation: 1) Given the fact that CBs have a significant impact on the operation of electricity markets, it is important to understand how market participants strategically select their CBs in real-world. This open problem is addressed in this dissertation by developing a data-driven reverse engineering method which results in identifying three main clusters of CB strategies in the California electricity market. Based on the lessons learned from the existing real-world strategies, a new CB strategy is proposed. We show through case studies that the performance of the proposed new strategy outperforms the most lucrative market participants in the California electricity market. 2) The operation and impact of CBs during blackouts are also investigated in this dissertation. The amount of load shedding is modeled as a function of the amount of the cleared CBs. The sign of the slope of this function is proposed as a metric to determine if a CB exacerbates or heals the power outages. It is proved mathematically that, when there is no congestion in the DAM, the metric is always positive. Using numerical case studies, we show that, not only when there is no congestion, but also most often when there is congestion, the introduced metric is positive. The engineering implications of these results are discussed. Furthermore, the impact of load shedding on the profit of CBs is also analyzed to draw a complete picture in this analysis. It is shown that load shedding usually creates advantage for supply CBs and disadvantage for demand CBs in terms of their profit. The real-world California market data during the blackouts in August 2020 is also analyzed. It is shown that the decision to suspend CBs during this event matches the results that are obtained in this dissertation.