An Auction Story: How Simple Bids Struggle with Uncertainty

Short-term electricity markets are key to an efficient production by generation units. We develop a two-period model to assess different bidding formats to determine for each bidding format the optimal bidding strategy of competitive generators facing price-uncertainty. We compare the results for simple bidding, block bidding and multi-part bidding and find that even under optimal simple and block bidding generators face the risk of ex-post suboptimal solutions, whereas in multi-part bidding these do not occur. This points to efficiency gains of multi-part bidding in the presence of uncertainty in electricity markets.


Introduction
Short-term electricity markets, especially day-ahead and intraday markets, have the goal to efficiently match generation and demand of electricity. A key challenge for market operators and participants is to account for inter-temporal linkages between time, which is caused by the technical characteristics of generation units connected to the electricity grid (Elmaghraby and Oren, 1999). An example of these intertemporal technical limitations are start-up costs that occur once when a power plant is started up, ramping constraints that limit the rate of change of production, the maximum time a load can be shed without impacting its service quality (for This topic is broadly linked to the general literature on auction theory. However, while most literature deals with auctions for single goods, the literature on multi-unit auctions is not as well developed (Klemperer, 2004). The most closely related strand are combinatorial auctions, which allow bidders to express the willingness to buy packages of goods (Cramton et al., 2007). In this paper, we concentrate on the specific case of multi-unit auctions which allow expressing complementaries via bidding multiple parameters, which exist in the praxis of electricity markets.
Various bidding designs exist to coordinate production of generators, usually distinguishing between simple bidding (single time period with one price component), block bidding (multitime period with one price component), and multi-part bidding (often in the form of three-part bids), where generators submit bids describing their variable and start-up costs (and/or other technical and financial parameters).
In simple bidding, generators need to account for startup costs in individually accepted bids, to exclude inefficient solutions, while not excluding too many profitable combinations where the price is low in one period, but so high in the other as to make the combination of the two profitable. Due to this inherent trade-off in bidding, ex-post several inefficiencies in production are possible for simple bidding: the unit is accepted for one or two periods, although the price is not high enough to recover costs, and only one period is accepted, when it would have been efficient to dispatch the unit in two periods. For block bidding, the unit can only be accepted jointly, while under low prices below variable cost in one period and high prices in the other, it would be more efficient to let the unit run for a single time period. To capture these inefficiencies, we built a simple two-period model of a price taker's optimal bids under three different bidding formats: simple, block and multi-part bidding.
The advantages and disadvantages of these market designs have been discussed previously in literature. Elmaghraby and Oren (1999) compare energy only hourly auctions (termed vertical auctions) with horizontal auctions (defined by load slices) in an analytical model under certainty and conclude that, even under perfect information, no efficient outcome can be achieved under energy only hourly auctions. Sioshansi et al. (2010) compare multi-part bids with simple energy bids in a deterministic simulation study and find that energy-only simple bidding results in inefficiencies due to ignoring the inter-temporal linkages and non-convex generation costs, while multi-part bidding may result in some equity and incentive issues (which can partly be mitigated by uplift payments). Sioshansi and Nicholson (2011) explicitly look at market power issues with regard to multi-part (here two-part) and simple bidding in an analytic framework of a duopoly under deterministic demand levels (with a numerical quantification). The authors find that under both formats strategic bidding is taking place, leading in many cases to mixed-strategy equilibria. Finally, they find that energy only markets with simple bidding yield higher prices. O'Neill et al. (2005) emphasise that in markets with non-convexities market-equilibria without multi-part bids might not exist, and present clearing mechanisms and how to interpret prices in mixed-integer problems to arrive at market clearing prices. Reguant (2014) analyses the usage of complementary bidding mechanisms (i.e. multipart bids in the terminology of this paper) in the Spanish electricity market using a structural model and how they can be used to better explain apparent mark-ups that in part exist due to start-up costs; however, she does not address the question of what drives firms to use one or the other bidding format. In the corresponding thesis chapter (Reguant, 2011), an additional computational analysis is presented, which compares the welfare effects of simple bidding and multi-part bidding using a computational counter-factual analysis. Here, multi-part bidding leads to efficiency gains, even in the presence of market power, as well as lower prices. This is caused by increased volatility in the market if simple bids only are allowed, and the reduced coordination of start-up decisions.
While these papers discuss inefficiencies arising in equilibrium with and without market power, uncertainty may lead to further inefficiencies due to possible coordination issues. Cramton (2017) discusses qualitatively that the internalisation of start-up and other costs based on market expectations would often turn out to be wrong and distort the energy offer curve.
Engineering literature on the other hand, uses stochastic optimisation to derive optimal bidding strategies under price uncertainty (Conejo et al., 2002). To our knowledge, however, the role uncertainty explicitly plays in reducing the efficiency of simple bidding as compared to multipart bidding has not been discussed in literature using an analytical framework. In contrast, this paper concentrates on the effect that price uncertainty has in isolation from other effects on the bidding strategies of generators in the presence of start-up cost. While this inefficiency is similar in nature to the principal coordination issue of simple marginal prices if there are indivisibilities and other non-convexities (O'Neill et al., 2005), in our model the inefficiency would cease to exist in case that there was no uncertainty. This is a relevant problem especially for short-term markets since in intra-day trading price uncertainty has very significant levels (Neuhoff et al., 2016), and may increase in future with higher levels of variable renewable energy sources.

Bidding format description & optimal bidding strategies
We are investigating bidding formats using a simplified analytical framework of a single, pricetaking actor with a power plant exposed to uncertain power prices in two time periods. The power plant is participating in a uniform price auction in a large competitive market. The model consists of two stages, which are presented in Figure 1. In Table 1 (Appendix A), all parameters and variables of the model can be found.

Bid
Clearing & production Price uncertainty In the first stage, depending on the bidding format, the actor chooses his bids in order to maximize his expected profit over both trading periods, given the price expectations, as well as the clearing rules of the bidding format.
In the second stage, the market is cleared according to the bids of the actor, the clearing rule of the bidding formats (described in detail below) and the stochastic realisations of the prices in the two consecutive periods. The prices are independently uniformly distributed U (0, p max ). As a result of market clearing, bids of the actor are either accepted or not. If any of the bids are accepted the power plant needs to produce power and its owner receives the revenue of the accepted period(s), as well as incurs production costs. The power plant has fixed, positive start-up costs c s , which occur if the power plant produces in one or two periods.
Secondly, the plant has positive variable production costs of c v , which occur in every period that the plant is producing. For simplicity, we assume that the capacity K of the power plant, as well as the length l of one trading period are 1, so prices and costs can be simply added to result in the profit. We assume, that there is no possibility for the power plant to not produce, by, for example, paying an imbalance price instead, or by retrading electricity in a consecutive market. We relax the assumption of a single trading round, as well as the power plant size and period length in Section 4. In the following, the three bidding formats and clearing rules of the auction are described.

Simple bids
In the simple bid format, the power plant can bid for each period separately with the bids b 1 and b 2 . If the realised prices p 1 and p 2 are equal or larger than the respective bid, the bid is accepted and the unit needs to produce with the associated costs. Bids can be individually accepted, so that a unit may produce in one period, but not the other.
Depending on the prices p 1 and p 2 the profit for the actor is thus: The payoffs for an example can be found in Figure 2, where the profit for the actor is depicted depending on realisations of p 1 and p 2 and for a bid b = b 1 = b 2 (marked with dotted lines in the figure, and corresponding to the optimal bid derived below). To the right of the vertical dotted line period 1 is accepted as p 1 >= b 1 , and above the horizontal dotted line the bid in period 2 is accepted as p 2 >= b 2 . Within the fourth quadrant in the top right, bids in both periods are accepted. In this example, the actor faces a risk of losing money for low levels of p 1 and p 2 that are just above the bid level b.
Expected profits & optimal bidding: For simple bidding, since the two distributions are assumed to be identical, bidding in the two time periods is symmetric (a full derivation to show that the bids are identical can be found in the appendix B.1) and we can simplify the derivation to an identical decision variable b for both periods. Given the probability of having a bid accepted (1 − b p max ) and the expected profit conditional on the acceptance of a bid ( p max +b 2 ), the expected profit for a given bid level b is: .
Deriving for b and checking for sufficient and necessary optimality conditions (see Appendix B.1), the optimal bidding strategy (with b * = p max equivalent to no bidding) is: Thus the optimal strategy is to adjust the bid above marginal short-term cost, in the presence of start-up costs (in case of zero start-up costs, it is optimal to bid at variable costs). Two points are notable about the solution: First, the optimal bid can only be equal to or lower than the simple sum of variable and fixed costs. This means that in case of start-up costs the actor is willing to run the risk of losing money in case only one time period is accepted rather than two, as the probability of profits by being accepted in two periods is outweighing the risk of only being accepted in a single period. Second, the solution is not equal to the simple heuristic of bidding variable costs plus the distributed fixed cost over the two time periods. The optimal bid may be both lower and higher than this heuristic.
With the optimal bids, the expected profit can be derived: As anticipated the expected profit under optimal bidding is decreasing both with start-up costs c s and variable costs c v .

Block bids
In block bidding, the actor couples two bids, so that the two periods are either jointly accepted or rejected. Hence, if the block bid is accepted, the unit is producing in both periods. Therefore the profit function is: The payoffs for realisations of p 1 and p 2 for an example can be found in Figure 3, which corresponds to the optimal bidding strategy derived below.
Expected profits & optimal Bidding: For block bidding the expected profit is: The expected profit for block bidding is: Two cases can be distinguished. For 0 ≤ b b < p max the integral needs to distinguish two areas: For 2p max > b b ≥ p max , only one area needs to be considered: As it is not profitable to produce if the cost exceed the maximum revenue over two periods, the expected profit is: Differentiating the expected profit, and checking that the optimality conditions are fulfilled (see appendix) we find the optimal bid b b to be: No bid otherwise (11) This is also intuitively correct, as the actor, as a price taker under uniform bidding, will bid his true cost for a joint delivery of two periods. In contrast to simple bidding, there is no trade-off between risking potential losses for a higher expected profit: as a joint bid for two periods is submitted there is no risk involved in only having one bid accepted. However, there are situations in which one period has a very high price covering both variable and fixed costs in a single period, and the other period has a very low price not covering variable costs (as will be discussed further in the Section 3).
Inserting the optimal bidding strategy into the expected profit function of block bidding yields the expected profit under optimal bidding: The expected profit under optimal bidding differentiates for two cases. In the first case, the bid (which equals the cost of producing in two periods) could theoretically be covered by the prices in one period. The second case represents the setting where the bid is larger than the highest reachable price in the market and the revenues from both periods are necessary.

Multi-part bidding
For multi-part bidding, the actor can bid start-up cost b s and variable cost b v that are valid for both periods. Dependent on market prices the actor will be accepted in one, two, or no periods for which the profit function is: The payoffs for realisations of p 1 and p 2 for an example can be found in Figure 4. Visually the case differentiation between the dispatch of one and two units can be seen to occur as kinks in the iso-lines at variable cost levels.
Expected profits & optimal bidding: For multi-part bidding the expected profit is: 2 )dp 1 dp 2 As the multi-part algorithm is defined in such a way that units are only accepted in the auction if the prices exceed the variable costs in the respective period and if startup costs are covered over either one or two periods (according to bids) potential profit are foregone if the actor bids either above or below the true costs. It is thus optimal to bid the truthful variable and start-up costs (i.e. b * s = c s and b * v = c v . A formal derivation of this result can be found in the Appendix B.3).
By distinguishing three cases defined by combinations of c s and c v , that determine whether only jointly accepted bids or also individually accepted are possible solutions, we find the expected profit under optimal bidding to be: The case differentiation for the expected profit (at c s + c v = p max ) occurs at the combination of start-up and variable costs where the dispatch in a single time period becomes inefficient, as start-up and variable cost exceed the maximum price level in the period. In that case, the expected profits of block and multi-part bidding are identical.

Inefficiencies & comparison of expected profits
We In block bidding these interdependencies are considered, however acceptance in single time periods is ruled out.
It is important to note that this inefficiency is not identical (but similar in nature) to the principal coordination issue of simple marginal prices if there are indivisibilities and other non-convexities (O'Neill et al., 2005). In contrast, in our model the inefficiency would cease to exist in case that there was no uncertainty in at least one of the two periods, as the bid for the uncertain period could be adjusted taking into account the price of the certain period.
The example in Figure 5 shows three types of inefficiency of simple as compared to multipart bidding and two types of inefficiencies of block bidding as compared to multi-part bidding.
1 Discrete start-up choices and other non-convexities can also lead to market results, where prices indicate profitable production opportunities while a unit is not dispatched and vice versa (leading to uplift payments or paradoxically rejected bids). However, the effect here is not due to the discreteness of the start-up decisions, but due to the uncertainty over several periods, making the ex-post losses (or unrealised profits) an inefficiency.
In areas B we depict the inefficiency of an ex-post underproduction where the plant produces in only one time period, whereas a production in two time periods is taking place under multipart bidding. The difference between the profit functions of simple and multi-part bidding can be expressed as 2 As the profit under simple bidding is smaller than under multi-part bidding this represents an inefficiency.
In area C we see the inefficiency of an ex-post overproduction in two periods, where the plant is dispatched in two time slots as compared to no dispatch at all in the efficient case (and as is the case under multi-part bidding).
The example in Figure 5 shows two inefficiencies of block as compared to multi-part bidding.
In the areas D the plant is not dispatched at all, although a dispatch in one period would be efficient as is taking place under multi-part bidding, representing a missed profit: In the area E the plant is accepted for two periods under block bidding, although it would be more efficient to produce in a single period, as the revenue in the additional period exceeds the variable cost.
As a result for all combinations of variable costs c v and start-up costs c s with non-zero profits, multi-part expected profits are larger than either single bid or block bid profits (for proof see appendix C). Whether block or simple bidding is more profitable depends on the combination of c v and c s as compared to the price distribution.
In   As sometimes markets allow both block and simple bids, market participants will choose the optimal bid type according to their power plant type. Figure 8 shows the expected profits if the optimal choice (in expectation) is made between block and simple bids (Subfigure a).
Where the isolines have a kink, block and simple bidding yield the same profits. Subfigure b shows the ratio between the optimal simple-block choice and multi-part bidding. Given the chosen period length and price distributions the biggest inefficiencies occur for OCGT plants and CCGT plants that are half-way cooled down.
The results in this section are example quantifications for the specific case of a dispatch in two consecutive hours under high price uncertainty. It is important to stress that results  We do not asses the profitability of a combination of simple and block bids against multipart bids due to the following reason: A combination of block bids and simple bids would lead to the risk of double selling the production volumes as soon as the simple bid and the block bid is accepted. Therefore, either an additional buy offer in form of a block bid or the simple bid could be used to offset the double buying (trading in several time periods is outside of the scope of this model).

Model extension: retrading for simple bidding
So far we did not model a second trading round where agents can readjust their position. This represents a limitation of the model, as a second trading round (and further trading rounds) can potentially reduce the inefficiencies identified in the paper for simple bidding: • Case A: When the unit is accepted in one of two time periods, but ex-post it would have been more efficient to be accepted in none, the power plant operator aims to buy power back in order not to produce at all.
• Case B: When the unit is accepted in one of two time periods, but ex-post it would have been more efficient to be accepted in both periods, the actor aims to sell in the additional period.
• Case C: When the unit is accepted in both periods, but ex-post it would have been more efficient to be accepted in none, the actor tries to buy power back for both periods. However, several effects may limit the effectiveness of sequential markets to reduce the discussed inefficiencies.
• As the time of delivery gets closer fewer flexibility options can be utilized, which leads to a steeper merit order curve (Henriot, 2014) and the missed utilisation of more efficient units.
• Secondly, in the case of continuous markets at a given point, the market usually only has limited liquidity, leading to price effects for relatively small volumes of traded energy and increased uncertainty for actors that have open positions (Neuhoff et al., 2016).
Therefore, in order to approximate these effects, we consider the effect of the traded capacity on the price in a second trading round.

Case A: Ex-post over supply
This case describes the inefficiency area A, shown in Figure 5, as in this area, the power plant is dispatched in one time period, although a dispatch is not efficient giving the prices of the first auction round. Therefore, the power plant operator could buy the power back in a second auction round, in order not to need to produce.
Equation (21) shows the inequality for which production would result in lower costs than retrading in a second round, and uses the price sensitivity m (in Eur/(MW h · MW )), which depends on the traded capacity K. As long as this inequality holds, the actor will decide not to retrade.
In order for this inequality to hold, m 1 must be bigger than a certain threshold for the price sensitivity.
To assess what the impact of different variable costs and start-up costs is on retrading, we look at the situation of a just accepted bid (p α 1 = b * ), which we hold constant, while varying either c v or c s . Reformulating equation (22) (see appendix for full documentation) towards c v will lead to: Equation (23) shows that for an increasing share of c v in the production costs, the necessary m 1 for retrading to be unprofitable is decreasing. This means that for power plants with relatively high variable costs compared to their start-up costs (e.g. fast starting gas power plants) retrading is only an option when the price elasticity of the market is very low. Fast starting power plants with high variable cost are normally used at the upper end of the merit-order where a higher price elasticity can be observed. Reformulating equation (22) (see appendix for full documentation) towards c s will lead us to: This means that for power plants with relatively high startup costs compared to their variable costs (e.g. lignite or nuclear power plants) retrading is an option even when the price elasticity of the market is high. However, these power plants are normally very large (hence a big K) which will reduce m 1 again.

Case B: Ex-post under supply
This case describes the inefficiency area B, shown in Figure 5, as in this area the plant is dispatched in only one time period, although a dispatch in two time periods would be efficient giving the prices of the first auction round. Therefore, the power plant operator could have an intention to sell the electricity for the second time period in a second auction round.
Similarly to the case before, setting the price p α 2 = b * , and holding b * constant while varying either c v or c s yields: This shows that for increasing shares of c v in the total production cost, the necessary m 2 for retrading to be unprofitable is decreasing. Hence, power plants with a high share of variable costs have no intention to retrade as already a small price elasticity makes it unprofitable.
Power plants with a low share of variable cost and higher shares of start-up costs, on the other hand, will be willing to retrade also during times of high price elasticity. However, these power plants are normally very large which will then reduce necessary m 2 again.
Case C: Ex-post over supply in two periods In order to compare the effect of c v and c s on the necessary price elasticity for the model to be correct, we reformulate equation (28) by assuming m 1 and m 2 to be equal and the bids to be just accepted in the two periods at p α 1 = p α 2 = b * . Secondly, as done previously, we fix b * , and vary only one of the two parameters at a time (see appendix for full documentation).
If b is fixed, and we vary c v , we need to substitute c s , which results in the following condition: Thus, as in the previous cases, with an increasing variable cost share in the bidding price, the necessary price elasticity for retrading to be unprofitable is decreasing.
If b is fixed, and we vary c s , we need to substitute c v , which results in the following condition: Thus, as previously, with increasing fixed costs, the necessary price elasticity in the market needs to be higher for electricity producers preferring not to retrade.

Reflections on assumptions
The analytic model is based on a set of simplifications necessary for tractability and clarity of economic effects. Only two time slots, rather than 24 hourly or even more quarterly products are considered. Usually commitment to longer periods is possible, which reduces the significance of start-up costs; however, it also implies additional interactions across multiple hours that could increase the relevance of the effect. Secondly, we model in a sensitivity analysis (cf. Section 4) a second auction round and could identify a mitigation of some of the effects. Additional auction rounds thus might further mitigate inefficiency of simple and block bidding as compared to multi-part bidding.
However, a second effect might partially off-set this opportunity. Market parties may be find it increasingly costly to close an open positions in subsequent markets, as these usually have limited depth, (Neuhoff et al., 2016) and exhibit a steepening merit order curve as fewer units can respond closer to real-time.
Finally, we do not model a portfolio of generators and the possibility for generators to make a generic bid and then dispatch those of their units which can fulfil the bids at the lowest cost (portfolio based bidding is still common in Western European power markets). This may also reduce the inefficiencies of simple bids discussed in this paper for those market players that have a large portfolio, whereas smaller market participants would be more exposed to this sort of inefficiency.

Conclusions
Several bidding formats exist to clear electricity markets, with simple bids and block bidding on the one side and multi-part bidding, which explicitly lets generators state cost components and technical constraints in their bids on the other side. While these bidding formats have been discussed in the literature, this discussion has mostly focused on the direct coordination problem due to indivisibility, and other non-convexities which exists both in deterministic and uncertain settings. The suitability of different bidding formats to coordinate the clearing of electricity markets in the presence of uncertainty and complementarities across several hours has been less well analysed in economic literature.
We derive optimal bidding strategies for each of the bidding formats simple bidding, block bidding and multi-part bidding using a two-period model of a price taker in a simultaneously cleared uniform price auction. We show in an analytic model that in the presence of start-up costs and price uncertainty in electricity markets, generators face ex-post suboptimal market outcomes under simple and block bidding, while this doesn't occur in multi-part bidding. The question, whether simple or block bidding is more efficient depends on whether start-up costs are high as compared to variable costs, in which case block bids are more profitable.
With uncertainty about auction clearing prices and simple or block bidding formats do not allow producers to specify their willingness to produce as function of the outcome in the adjacent hour for which they face complementarities in their production function (for example due to start-up costs). Using numerical examples of different power plant types we show that this effect could occur at significant levels given common electricity market prices and power plant characteristics.
Our results point to inefficiencies in electricity markets in the absence of multi-part bids.
These may be mitigated to the extent that further trading rounds or continuous trading allows generators to re-adjust their production schedules. In an extension of the model, we assess how a further trading can reduce the inefficiency. While subsequent markets may help to alleviate the problem, this might be offset by declining liquidity and depth, as well as the steepening merit order curve over time.   Figure 1 and add the shaded areas in Figure 5.