Balancing power markets involve complex procurement auction mechanisms that are challenging both to design for auctioneers and to take part in for suppliers. Recent empirical work indicates that auction results from Austria and Germany do not match game-theoretic predictions. Instead, suppliers adjust their bids to previous auctions results and do not reveal their actual costs within their bids. Therefore, this work focuses on bidding strategies of suppliers in the Austrian and German automatically-activated Frequency Restoration Reserve auctions. First, the operating principle of the auctions is analyzed and the cost and profit structures are illustrated. Then, a theoretic approach for the derivation of optimal bidding strategies is presented, that allows the integration of price expectations based on historical market data. We validate our approach by a numerical application of the bidder's calculus. Finally, our theoretic results are confronted with Austrian and German auction outcomes. We find evidence that the identified bidding strategies are applied by the suppliers.