Design and regulation of balancing power auctions: an integrated market model approach

We present an integrated market model which considers the dependencies between the wholesale market and the highly regulated balancing power markets. This fosters the understanding of the mechanisms of these markets and, thus, allows the evaluation of the designs of these markets and their interplay. In contrast to existing literature, in our model the prices on the different markets are interdependent and endogenously determined, which also applies to the switch from inframarginal suppliers to extramarginal suppliers. Linked to this, the implementation of a specific assignment of the suppliers to the different markets is according to their production costs and their ability to provide balancing power. We prove the existence of a market equilibrium, analyze its outcome and contrast this with German market data. Based on this model, we assess design changes, partly stipulated by recent European regulation. This includes uniform pricing as a common settlement rule (effect: no truthful bidding in general), standardized prequalification criteria (promising measure for cost reduction), market flexibilization via “free energy bids” (no increased competition) and the alternative score “mixed-price rule” (no effect on the equilibrium).


Introduction
To ensure a proper working of the electricity system, ancillary services are used. The most important short-term service is balancing power (henceforth abbreviated as BP, see Appendix 1 ): it balances demand and supply deviations in real-time. This ensures a stable grid frequency in alternating current grids. BP procurement is carried out by the system operator entity, e.g. transmission system operators (TSO) in Europe (ENTSO-E 2020). The mechanism on the BP markets are procurement auctions, i.e., prequalified suppliers compete for BP provision (Ocker et al. 2016).
Prequalified suppliers can act on different electricity markets (wholesale market and BP markets), and the suppliers, who provide BP, must run their plant at a minimal load and sell this electricity on the wholesale market. This causes interdependencies between the wholesale market and the BP markets (Just and Weber 2008). This is a major focus of our research. We provide a stylized integrated market model to identify and examine the characteristics of the different markets and their interdependencies. This approach does not only foster the understanding of the interdependent markets, but it also allows to examine and evaluate elements and changes of the market designs. Thus, our paper contributes to the field of electricity regulation.
In Europe, the "Electricity Balancing Guideline" states the framework conditions for a pan-European BP market with a harmonized structure (European Commission 2017a). This includes a harmonized settlement rule (uniform pricing) to incentivize truthful bidding and increase efficiency, and market flexibilization ("free energy bids") to ease the integration of volatile renewable energy sources. With the same intention, the "System Operation Guideline" sets requirements for prequalification (European Commission 2017b). In addition, there were changes in the auction design stipulated by the regulatory authority "Bundesnetzagentur" in Germany. In 2018, the scoring rule (rule for winner determination) changed to the so-called "mixed-price rule," aiming at BP cost reduction (Bundesnetzagentur 2018). While the intentions of these design changes are clear, so far, their effects on the interlinked wholesale and BP markets have not been assessed. We close this gap by applying our model to examine and evaluate these design changes.
The theoretical foundations for BP auction design are laid by Bushnell and Oren (1994) who analyze bidding strategies for different scoring rules and settlement rules. They derive conditions for efficient equilibria. Chao and Wilson (2002) show that a scoring rule consisting only of the capacity bid suffices the efficiency condition if incentive compatibility is imposed. The latter yields that suppliers energy bids reveal their variable cost, and energy bids are paid the wholesale market price. Müsgens et al. (2014) transfer this to the German BP markets. Here, the proposed scoring rule of Chao and Wilson (2002) is applied, but energy pricing is based on bids. They argue that a switch from pay-as-bid to uniform pricing ensures efficient activation. We relate to these contributions and present an equilibrium model for the wholesale and BP markets, considering different scoring and settlement rules. 2 The distinguishing feature of our model is that we consider the dependencies between the markets endogenously: prices are not exogenously given, but the result of the interplay between the markets. In contrast, Müsgens et al. (2014) assume an exogenous wholesale market price, which yields two classes of suppliers: suppliers with production units which have variable cost below the wholesale market price (inframarginal), and suppliers with variable cost above the wholesale market price (extramarginal). This distinction is crucial for the cost of BP provision: inframarginals include opportunity costs of not trading at the wholesale market, extramarginals cover their expenses by BP profits. In our model, the market interplay induces a specific assignment of the suppliers to the different markets according to their production costs and their ability to provide BP. Thus, inframarginality and extramarginality are endogenously determined. 3 Building on our market model, we prove the existence of an equilibrium and analyze its allocation, costs and prices. A central result is that all suppliers of negative BP are inframarginal, while in the positive market suppliers are inframarginal and extramarginal, depending on their production cost. We also consider empirical market data by comparing our theoretical results with German market results of 2015. Furthermore, we prove that lowering BP prequalification criteria is a promising means to reduce total costs. The mixed-price rule, however, does not impact the market equilibrium, although it may incentivize suppliers to change their bidding behavior. We also show that energy bids are not expected to foster competition, and that a switch of the settlement rule to uniform pricing does not incentivize truthful bidding in general.
The paper is structured as follows. Section 2 discusses related literature, Sect. 3 illustrates the electricity market design and Sect. 4 introduces our model. Section 5 presents equilibrium results, and Sect. 6 analyzes market design changes. Finally, Sect. 7 concludes.

Related literature
The first formalization of the economic conditions for BP markets is provided for California in the course of the electricity sector unbundling in the 1990s (e.g. Kirsch and Singh 1995;Borenstein et al. 1995;Kahn et al. 2001). Opportunity costs between markets, the effects of market power and of a change of the settlement rule are investigated. With focus on auction design, efficiency properties and incentive compatibility are examined, focusing on the scoring rule and equilibrium bidding strategies (e.g. Kamat and Oren 2002). There are several contributions dealing with regulatory aspects: increased renewables' penetration, information feedback schemes and further design variables, such as auction lead time and learning (e.g. Vandezande et al. 2010;Müsgens and Ockenfels 2011;Van der Veen and Hakvoort 2016;Borne et al. 2018;Doraszelski et al. 2018).
In Germany, there is an ongoing discussion on regulation and market imperfections since the liberalization of the auctions in 2002. Market entry barriers due to prequalification (Müller and Rammerstorfer 2008), dynamics of the reserve prices (Rammerstorfer and Wagner 2009) and strategic capacity withholding of suppliers with market power are assessed (Heim and Götz 2013). Recently, a number of market modifications were implemented, e.g. the aggregation of the four regional German BP markets to a single market and the shortening of the reservation period. Literature on BP auction design also widens, e.g. an examination of arbitrage opportunities between the wholesale market and the BP markets (Haucap et al. 2014;Just and Weber 2015;Ocker et al. 2018a). The massive subsidization of wind and solar generation yields to an investigation of renewables and the BP market. The "German Paradox" is presented: balancing reserves reduced by 15%, while wind and solar capacity tripled since 2008. The reasons are shorter intraday trading products and grid control cooperations (Hirth and Ziegenhagen 2015;Ocker et al. 2017).

Electricity wholesale market
Electric energy (henceforth energy) is traded at forward markets and at spot markets (e.g. Ströbele 2013; KU Leuven Energy Institute 2015; Zweifel et al. 2017). At forward markets trading is more long-term, whereas at spot markets the point of delivery is instantaneous. Spot markets include a "day-ahead" and an "intraday" market. The dayahead auction is considered as the central market place for electricity trading for the following day. On the contrary, intraday trading is done close to real-time and its main purpose is adjusting to changing circumstances, e.g. weather forecast or unplanned outages of units (Ocker et al. 2020).

Balancing power auctions
Traded volumes at the wholesale market usually differ from the actual production. The reason is that contracted volumes are based on predictions of supply and demand. A deviation which persists until real-time has a direct influence on the grid frequency in alternating current systems: if too much (little) energy is supplied, the grid frequency increases (decreases), which can lead to blackouts. To avoid this, system operators apply BP for stabilization (Zweifel et al. 2017).
Three different BP qualities can be distinguished: Primary BP (PBP), Secondary BP (SBP), and Tertiary BP (TBP) (e.g. ENTSO-E 2016). 4 They differ in the reaction time. 5 Each of the BP qualities comprises a separate procurement auction (Ocker et al. 2016). We focus on the weekly SBP and TBP auctions because they include the scoring rule proposed by Chao and Wilson (2002).
The deviation of the contracted production is separated in an overproduction (e.g. from wind plants during a storm) or an underproduction (e.g. from solar plants during a cloudy day). Thus, BP needs to provide an increased and decreased energy supply, which is implemented via two BP products: for the "positive product" (negative product) suppliers provide upward (downward) regulation (e.g. ENTSO-E 2016;Ocker et al. 2016). BP provision requires a prequalification, which leads to a highly invariant and limited supplier set. 6 Suppliers encounter two types of cost: "capacity cost" for reserving balancing capacity [megawatt, MW], and "energy cost" for the delivery of balancing energy [megawatt hour, MWh]. In Germany, capacity costs are part of the grid tariff, whereas energy costs are accounted to the imbalanced party (see Sect. 3.3).
Suppliers submit three-dimensional bids: a capacity offer [MW], a capacity bid [Euro/MW], and an energy bid [Euro/MWh]. 7 Based on these bid information, scores of suppliers are calculated and those with the lowest scores are awarded. Until the regulatory intervention in 2018, the scoring rule in the German SBP and TBP markets was based on the capacity bid; then, it included also the energy bid (mixed-price rule). For settlement, pay-as-bid (PaB) pricing or uniform pricing (UP) is applied. If PaB is used, awarded suppliers are paid their submitted bids, if UP is used, all awarded suppliers are paid a uniform price. In the German SBP and TBP markets PaB is applied. Ultimately, the delivery of balancing energy is activated according to a merit-order of energy bids, i.e., suppliers with the lowest energy bids are utilized first and, thus, for the longest timespan.

Imbalance settlement scheme
BP procurement is the physical means to cope with imbalances. The financial settlement is done via the imbalance price. A market party's difference between the contracted volumes and the actual physical position multiplied with the imbalance price gives her penalty payment in a certain timeframe (Koch and Hirth 2019). In Germany, the imbalance price is directly linked to the activated balancing energy bids. Thus, the German imbalance price can be interpreted as a levy for the balancing energy costs incurred to those responsible for activation (regelleistung.net 2021). To avoid the penalty payment, market parties can close open positions by trading short-term.

Basic model
There are three energy markets: a wholesale market, a positive BP market and a negative BP market. We consider a certain period (e.g. year). The average demand on the wholesale market in this period is denoted by D and measured in gigawatt (GW).
The capacity demand on the positive and negative BP market is fixed and given by There is a set of energy suppliers, each with one power plant, which have the same capacity but differ in their variable energy production cost c between c and c. We assume that UP is implemented on all considered markets, which is the common settlement rule on the wholesale market and the planned rule for future European BP auctions (European Commission 2017a). Thus, all awarded suppliers receive the marginal price. On the BP markets, the uniform price for balancing energy is repeatedly determined within a pre-defined period of time -the "Balancing Energy Pricing Period" (BEPP). That is, uniform price for each BEPP is determined by the highest activated balancing energy bid within the BEPP. The application of UP allows to assume that the suppliers reveal their cost in their bids, which is the usual assumption for the wholesale market. For the BP markets, see Sect. 6.4. Thus, the supply function S : [c, c] → R + and its inverse S −1 : R + → [c, c] are strictly increasing.
There are two different types of suppliers: BP-capable BP suppliers and non-BPcapable nBP suppliers. nBP suppliers only participate on the wholesale market, while BP suppliers can participate on the wholesale market and the BP markets. For the latter, they must run their plant on a minimal load (share of capacity) m ∈ [0, 1) and sell this energy on the wholesale market. The total supply includes both the BP and the nBP suppliers: the supply functions of BP and nBP suppliers. We assume that BP suppliers are uniformly distributed among all suppliers: at each cost level c, the BP suppliers' share of the supply S(c) is δ ∈ (0, 1] and the nBP suppliers' share is 1 − δ.
Discrepancies between demand and supply are balanced by calling BP. The functions (1) describe the cumulated relative frequency of the differences between demand and supply in the period: z − (x) is the relative frequency of an excess supply of at least x GW and, thus, refers the call of negative balancing capacity of at least x GW. Analogously, z + (y) applies to an excess demand of at least y GW and, thus, to the call of positive BP. 8 The z-functions (1) also describe the relative calling frequencies (abbrev.: rcf): are the expected total negative balancing capacity and the expected total positive balancing capacity that are needed to balance the excess supply and excess demand. Thus, γ − =B − /B − and γ + =B + /B + are the fraction of provided negative balancing capacity and of positive balancing capacity, i.e., the fraction of the balancing capacities for the delivery of balancing energy demand (withB =B + +B − ). We call the BP markets symmetric if the difference between demand and supply on the wholesale market is symmetrically distributed, i.e., On the wholesale market, suppliers constantly produce energy and are remunerated for each unit by the wholesale market price p W . On the BP markets, suppliers receive the balancing capacity price p + BC or p − BC , and, if they are called, additionally a balancing energy price p + B E (c) or p − B E (c). The balancing capacity prices p + BC and p − BC are the same for all suppliers and are determined by the highest accepted capacity bid in the positive and the negative BP market, respectively. The balancing energy price is determined by the associated costs of the highest merit-order position that is needed to cover the demand within the BEPP. Thus, the length of the BEPP influences a supplier's balancing energy prices: the longer the BEPP, the higher is the number of draws for balancing energy demand, and, thus, the higher (lower) are the cost of the last supplier on the positive (negative) market. We model the average supplier's balancing energy price in dependence of factor ϑ ∈ [0, 1] that corresponds to the length of the BEPP: The case ϑ = 1 models the longest possible BEPP, in which the balancing energy price p + B E ( p − B E ) is always determined by the highest (lowest) supplier's cost c + 1 (c − 0 ). 10 The smaller ϑ (i.e., the shorter the BEPP), the closer moves the supplier's average balancing energy price to her cost c.
Suppliers' profit per produced energy unit on the wholesale market π W , on the positive BP market π + B P , and on the negative BP market π − B P are given by 11 In the positive BP market, the suppliers' profit (7) consists of the profit from selling the minimal load on the wholesale market and the profit from providing BP, that is, balancing capacity and energy. In the negative BP market, suppliers are continuously paid the wholesale market price p W for their entire capacity, although they may decrease the load level. The provision of negative BP has no impact on their trading on the wholesale market. The first part of (8) is the margin of selling the entire capacity at the wholesale market and the second part is the BP profit.

Efficient activation of balancing power
Within the interval [ B − 1−m , B 1−m ], which applies to the positive BP market, the rcf decreases from a + max to a + min = 0, and thus, the suppliers' active capacities decrease from m +(1−m)a + max to m. Due to the condition a − max +a + max ≤ 1, 1−(1−m)a − max ≥ m + (1 − m)a + max and, thus, the curve of active capacities is strictly decreasing within [0, B 1−m ]. Applying the z-functions (1) together with (3), the curve of the BP suppliers' active capacities for providing energy for the BP markets and the wholesale market is given by the strictly decreasing function j(q): Using the maximum rcf a + max leads to The integral is the average active capacity of all BP suppliers. With (2) we get The symmetric case is given by j That is, in the symmetric case (see Fig. 2), J (m, B) is independent of the shape of the j-curve. Since the asymmetry in the German markets is small (see Fig. 1 for the SBP market), we restrict the following analysis to the symmetric case. An efficient allocation on the BP markets requires that suppliers with low variable cost have higher production volumes than those with high variable cost (see also Müsgens et al. 2014). From the j-function (10), illustrated by Fig. 2, the efficiency condition follows directly: In the negative BP market, c − 0 belongs to the supplier on the last rank in the meritorder with a − min = 0. This supplier continuously produces with her entire capacity. The supplier with c − 1 has a rcf of a − max . This supplier operates with the load m + (1 − a − max ) (1 − m) < 1. Thus, efficiency requires c − 0 < c − 1 . In the positive BP market, c +

Fig. 2 Example of a j-function in a symmetric BP market operates on the load level m+a
The supplier with c + 1 is on the last rank of the merit-order and never provides BP, but runs the plant on the load level m on the wholesale market.
Since BP suppliers only use the share 1−m of their capacities to provide BP, together they need the capacity B 1−m to cover B. To cover B − , a total capacity of B − 1−m is needed, which is provided by the BP suppliers with costs between c − 0 and c −

. The BP suppliers in the subsequent cost interval
, the rcf increases from a − min = 0 to a − max . Thus, in the negative BP market, the suppliers' (expected) active capacities for providing energy for the BP markets and the wholesale market decrease from 1 to 1 − (1 − m)a − max .

Stability and market clearing
BP suppliers either participate only in the wholesale market or simultaneously in a BP market and the wholesale market. For an allocation to be stable, the first and last position in the merit-orders are crucial. The supplier with c − 0 has to be indifferent between her last position in the merit-order of the negative BP market and a switch to the wholesale market. The supplier with c + 1 must be indifferent between her last position in the merit-order of the positive BP market and not participating at all. The suppliers with c − 1 (c + 0 ) has to be indifferent between a switch to the positive (negative) BP market. These requirements lead to the following conditions.
If one of these conditions is violated, either producing suppliers have an incentive to switch markets or non-producing suppliers have an incentive to enter the BP market.
Market clearing and energy balance require the following conditions: Demand D on the wholesale market is met by the contracted supply, provided by nBP and BP suppliers. The supply S n B P ( p W ) includes all nBP units with cost between c and p W . The supply of the BP units consists of S B P (c − 1 ), i.e., the negative BP suppliers with cost between c and c − 1 , and of m B + 1−m , i.e., the positive BP suppliers with cost between c + 0 and c + 1 who only supply m. The demand for positive and negative balancing capacity is provided by the (1 − m)th share of BP units within the interval [c + 0 , c + 1 ] and [c − 0 , c − 1 ] (S1, S2). Condition (S3) requires that total supply meets demand D also in case of deviations. Positive deviationsB + and negative deviationsB − are balanced by the BP suppliers with cost between c − 0 and c + 1 , whose active capacities are given by J (m, B).

Total system costs
Total system costs C with a linear supply function S(c) = α c + β , α ∈ R + and β ∈ R + , are with q 0 = S B P (c − 0 ) and q B P = J (m, B). Equation (14) gives the total costs for the wholesale market and the BP markets. The first integral are the costs of nBP suppliers in the interval [0, D − q B P − q 0 ], the second integral are the costs of BP suppliers in the interval [0, q 0 ], and the third integral are the costs for the BP markets, given by the costs of BP suppliers in the interval [q 0 , q 0 + B 1−m ] weighted with the average active capacity of function j(q).

Allocation, costs and prices
The propositions in this section are derived under (A0), (M0), (M1), (M2), (S0), (S1), (S2), (S3), symmetric BP markets, and a linear supply function S(c) = α c + β , α, β ∈ R + , and Proposition 1 There exists an equilibrium of the wholesale market and the BP markets with the following prices: 1. Wholesale market price: 2. Balancing capacity price in the positive market: Balancing capacity price in the negative market: p − BC = 0 See Appendix 12 for the proof. The wholesale market price p W is determined by the inverse supply function at demand D. Condition p W ≤ m c + 1 + (1 − m) c + 0 ensures stability since a higher p W induces suppliers of positive BP to switch into the wholesale market. The capacity price p + BC in the positive BP market covers the wholesale market loss of the supplier with c + 1 > p W caused by her costs of supplying the minimal load m and a rcf of zero. The capacity price p − BC is zero.

Proposition 2
In the equilibrium of the wholesale market and both BP markets the following holds for c − 0 , c + 1 , c − 0 , and c + 1 : 1. The cost c − 0 of the supplier on the last rank in the negative BP market is given by 2. The costs c − 1 and c + 0 of the suppliers on the first rank in both BP markets are given by 3. The cost c + 1 of the supplier on the last rank in the positive BP market is given by See Appendix 13 for the proof. Since the cost c − 1 of the last supplier of negative BP is determined by p W − p + BC , c − 1 ≤ p W . Thus, all suppliers of negative BP are inframarginal. This also applies to the suppliers of positive BP with low cost, beginning 12 The appendices are available in Ehrhart and Ocker (2021). 13 The appendices are available in Ehrhart and Ocker (2021). with c + 0 = c − 1 , while the suppliers with higher cost are extramarginal. According to Proposition 2 (2.), if the minimum load m converges to zero, the line between suppliers of negative BP and those of positive BP, given by c − 1 , converges to the line between inframarginals and extramarginals, given by p W . That is, c − 1 = c + 0 = p W for m = 0. Proposition 3 In the equilibrium of the wholesale market and both BP markets the following holds for the profits of the suppliers: . See Appendix 14 for the proof. On all markets, suppliers' profits are greater than zero and decrease with their variable cost. For low cost c < c − 0 the profit is higher in the wholesale market than in the BP markets, while for higher cost c > c − 0 it is the other way round.

Proposition 4
The equilibrium of the wholesale market and both BP markets ensures overall market efficiency, i.e., it minimizes the total system costs C.

Market design changes
We analyze four relevant and topical design changes. These were either already introduced (lowering BP prequalification criteria, free energy bids) or already withdrawn (mixed-price rule) or will be implemented in the near future (uniform pricing) by the regulator.

Lowering balancing power prequalification criteria
First, we consider the measure of increasing the set of BP suppliers by lowering prequalification criteria (e.g. regelleistung.net 2021). An increasing share of prequalified BP suppliers δ has the following effects: By Proposition 1, p + BC decreases, while p W and p − BC do not change. By Proposition 2, the interval [c − 0 , c + 1 ] becomes smaller because c − 0 increases and c + 1 decreases. The reduction of the interval length is caused by a higher density of BP suppliers. More precisely, the difference between p W = D−β α and c − 0 , between p W and c − 1 = c + 0 , and between c + 1 and p W decreases.

Proposition 5 The total costs (14) decrease in δ.
The proof is in the Appendix 21 . This result recommends to lower prequalification criteria.

Free energy bids
The European Commission (2017a) foresees so-called "free energy bids:" BP suppliers can submit only energy bids close to real-time (i.e., no capacity payment), if they did not participate or were not awarded in the regular (capacity bid) auction. In Germany, free energy bids were introduced on November, 03, 2020. The rationale is to increase competition and, thus, to reduce costs for BP activation (German TSOs 2020). However, our analysis does not support this hypothesis.

Proposition 6
The extension of our model by allowing suppliers, who were not allocated on a BP market, to submit free energy bids does not change the equilibrium results in Sect. 5.1.
The proof is in the Appendix 22 . The reason is that it is not profitable for suppliers to enter the BP markets or to switch between the BP markets with free energy bids. This results is supported by the market outcome in Germany: nearly no additional free energy bids are submitted and an increase in liquidity since November is not observed (Bundesnetzagentur 2020).

Switch of the scoring rule: the mixed-price rule
In Germany, the trend of increasing energy bids, particularly in 2018, caused very high imbalance prices. As a reaction, the German regulator decided to change the scoring rule to the so-called "mixed price rule" in late 2018 (Bundesnetzagentur 2018). While the prior score was determined only by the capacity bid b BC (see Sect. 3.2), the mixed price score b is determined by a combination of b BC and the energy bid b B E : The rationale behind this modification was to incentivize a changed bidding behavior to prevent extreme energy bids. However, the mixed-price rule has no effect on the equilibrium in our model.

Proposition 7
The equilibrium results in Sect. 5.1 also hold under the mixed-price rule.
See Appendix 23 for the proof. Yet, market results changed because some suppliers submitted lower energy bids and higher capacity bids (regelleistung.net 2021). Transferred to our model, where * denotes the equilibrium bids under the prior scoring rule and mp under the mixed price rule, this means In the Appendix 24 we show by means of a micro-economic model that the mixedprice rule generates incentives for (17), if the suppliers think "myopically" and do not adapt their beliefs about their opponents' bidding behavior to the new rule. However, in our model in Sect. 4, these changed bids cannot establish an equilibrium because, for example, higher capacity bids lead to higher capacity payments, which violates Condition (M1). We therefore expected that these changes would have disappeared in the course of time. Before this could have happened, after some months the regulator returned to the prior scoring rule because of a drawback of the new rule: high and frequent imbalances occurred as a result of low imbalance prices (see Sect. 3.3), which did not incentivize market parties to close open positions (Higher Regional Court Düusseldorf 2019). 25

Switch of the settlement rule: uniform pricing
The European Commission (2017a) further foresees UP because it should induce suppliers to report their true costs in their bids (e.g. Müsgens et al. 2014). 26 However, we do not attribute this property to UP. Only for a specific ϑ a bidder with cost c is incentivized to truthfully reveal her cost. Hence, incentive compatibility does not apply in general.
Proposition 8 A supplier's optimal energy bid b * B E depends positively on her cost c and negatively on the length ϑ of the BEPP. For the positive [negative] BP market the following applies: See Appendix 27 for the proof. We provide an intuitive explanation. The time a supplier delivers balancing energy depends on her merit-order position. Hence, the length of the BEPP has an impact on the suppliers' bidding strategy. The longer the BEPP, the more bids are taken into account for the determination of the price. This induces suppliers to reduce their energy bids, even below their costs, for the sake of a better merit-order position (Ocker et al. 2018a, b). In our model, the case ϑ = 1 corresponds to UP for only one BEPP, and the case ϑ ∈ (0, 1) to UP for several BEPPs. Assuming that the last accepted bid determines the uniform price, a short BEPP increases the probability that a bidder's own bid sets the price. Thus, ϑ = 0 corresponds to PaB and suppliers have an incentive to exaggerate their costs in their bids. Note that the ϑ that induces b * B E = c [−c] depends on c. Generally, a unique ϑ that induces truthful bidding to all suppliers does not exist. However, since the differences between the incentive-compatible ϑ are expected to be small, we take the freedom and assume truthful bidding in our model.
In practice, we doubt that suppliers understate their costs in their bids in case of a long BEPP. Moreover, UP may also be prone to manipulation by large suppliers submitting multiple bids, which is not captured by our model. These suppliers have an incentive to exaggerate some of their bids or even reduce their offered capacity to increase the bid that determines the uniform price and, thus, to increase their payment and profit (Haufe and Ehrhart 2018). 25 The intraday market prices were around 500 Euro/MWh and imbalance prices around 100 Euro/MWh. This resulted in an undersupply of balancing capacity of nearly 10 GW, which could not be met by the contracted SBP and TBP volumes of 3 GW. As a relieving measure, German TSOs procured additional balancing capacities, which caused capacity bid prices of over 37,000 Euro/MW (German TSOs 2019). 26 In Germany, UP will be introduced at the end of 2021 (regelleistung.net, 2021). 27 The appendices are available in Ehrhart and Ocker (2021).

Conclusion
We present an integrated market model which considers the dependencies between the wholesale and BP markets. We prove the existence of a market equilibrium, analyze its outcome and contrast this with German market data. Moreover, we show that the mixed-price rule does not impact the market equilibrium but may incentivize suppliers to change their bidding behavior in an undesirable way. Free energy bids do not foster competition, and a switch to UP does not lead to truthful bidding in general. Lowering BP prequalification criteria is a promising means to reduce costs. Section 6.3 illustrated that if the imbalance price is too low, there are insufficient incentives to close open positions. As a reaction, German TSOs changed the imbalance scheme: the imbalance price is now linked to the intraday market price, in accordance with the European target design (ACER 2021). In case of an electricity undersupply (oversupply) and the necessity for positive (negative) BP, the imbalance price will be at least as high (low) as a certain intraday index. This sets the right incentives: BP activation is more costly than self-balancing in the intraday market.
Some of our assumptions may be relaxed in an extended model: the linear supply, the same share of BP production units in the supply, or the homogeneous must-run capacities for all units. Further extensions may include suppliers with multiple plants and those with market power. Finally, experimental studies may help to test and foster our theoretical findings.