# The Uncertain Bidder Pays Principle and Its Implementation in a Simple Integrated Portfolio-Bidding Energy-Reserve Market Model

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Uncertainty Quantification

#### 2.2. Market Model of the Single Period Case

#### 2.2.1. Supplementary Reserve Demand Bids

#### 2.2.2. Minimum Surplus Conditions

#### Minimum Surplus Conditions for Uncertain Energy Supply Bids

- If the bid is entirely accepted (${y}_{j}^{ESUb}=1$), ${I}_{j}^{ESUb}$ equals the product of ${q}_{j}^{ESUb}$ and $MC{P}^{E}$ according to (8).

#### Minimum Surplus Conditions for Uncertain Energy Demand Bids

#### 2.2.3. Bid Acceptance Constraints

- ${y}_{j}^{ES}>0\to MC{P}^{E}\ge {p}_{j}^{ES}$.
- ${y}_{j}^{ES}<1\to MC{P}^{E}\le {p}_{j}^{ES}$.

- ${y}_{j}^{ESUb}<1\to MC{P}^{E}\le {p}_{j}^{ESUb}$ or ${y}_{j}^{ESUb}=0$, ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}ESUb}=0$ and ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}ESUb}=0$.

- ${y}_{j}^{ESU+}<1\to MC{P}^{E}\le {p}_{j}^{ESU+}$ or ${y}_{j}^{ESU+}=0$, and ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}ESU+}=0$.

- ${y}_{j}^{ESU-}<1\to MC{P}^{E}\le {p}_{j}^{ESU-}$ or ${y}_{j}^{ESU-}=0$, and ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}ESU-}=0$.

- ${y}_{j}^{ED}>0\to MC{P}^{E}\le {p}_{j}^{ED}$.
- ${y}_{j}^{ED}<1\to MC{P}^{E}\ge {p}_{j}^{ED}$.

- ${y}_{j}^{EDUb}<1\to MC{P}^{E}\ge {p}_{j}^{EDUb}$ or ${y}_{j}^{EDUb}=0$, ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}EDUb}=0$ and ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}EDUb}=0$.

- ${y}_{j}^{EDU+}<1\to MC{P}^{E}\le {p}_{j}^{EDU+}$ or ${y}_{j}^{EDU+}=0$, and ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}EDU+}=0$.

- ${y}_{j}^{EDU-}<1\to MC{P}^{E}\ge {p}_{j}^{EDU-}$ or ${y}_{j}^{EDU-}=0$, and ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}EDU-}=0$.

- ${y}_{j}^{RS+}>0\to MC{P}^{R+}\ge {p}_{j}^{RS+}$
- ${y}_{j}^{RS+}<1\to MC{P}^{R+}\le {p}_{j}^{RS+}$

- ${y}_{j}^{RS-}>0\to MC{P}^{R-}\ge {p}_{j}^{RS-}$
- ${y}_{j}^{RS-}<1\to MC{P}^{R-}\le {p}_{j}^{RS-}$

- ${y}_{j}^{RD+}>0\to MC{P}^{R+}\le {p}_{j}^{RD+}$
- ${y}_{j}^{RD+}<1\to MC{P}^{R+}\ge {p}_{j}^{RD+}$

- ${y}_{j}^{RD-}>0\to MC{P}^{R-}\le {p}_{j}^{RD-}$
- ${y}_{j}^{RD-}<1\to MC{P}^{R-}\ge {p}_{j}^{RD-}$

- ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}SRDB}>0\to $ Inequality (10), (12), (13) or (15) holds (depending on the type of the energy bid in the order of the SRDB) and $MC{P}^{R+}\le {p}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}SRDB}$, where $SRDB\in \{ESUb,\phantom{\rule{3.33333pt}{0ex}}ESU-,\phantom{\rule{3.33333pt}{0ex}}EDUb,\phantom{\rule{3.33333pt}{0ex}}EDU-\}$.
- ${y}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}SRDB}<1\to MC{P}^{R+}\ge {p}_{j}^{RD+\phantom{\rule{3.33333pt}{0ex}}SRDB}$ or all acceptance indicators of the respective order are 0.

- ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}SRDB}>0\to $ Inequality (10), (11), (13) or (14) holds (depending on the type of the energy bid in the order of the SRDB) and $MC{P}^{R-}\le {p}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}SRDB}$, where $SRDB\in \{ESUb,\phantom{\rule{3.33333pt}{0ex}}ESU+,\phantom{\rule{3.33333pt}{0ex}}EDUb,\phantom{\rule{3.33333pt}{0ex}}EDU+\}$.
- ${y}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}SRDB}<1\to MC{P}^{R-}\ge {p}_{j}^{RD-\phantom{\rule{3.33333pt}{0ex}}SRDB}$ or all acceptance indicators of the respective order are 0.

#### 2.2.4. Energy and Reserve Balances

#### 2.2.5. The Objective Function

## 3. Simulation Results

#### 3.1. Social Welfare of the Sub-Markets

#### 3.2. Traded Volumes

#### 3.3. Market Clearing Prices

#### 3.4. Computational Properties

## 4. Discussion

#### 4.1. Additional Possible Phenomena in the Setup

#### 4.2. Total Amount of Allocated Reserve

#### 4.3. Market Implementation

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MIC | Minimum income condition |

PPU | Price per unit |

MCP | Market clearing price |

SRDB | Supplementary reserve demand bid |

MRSBPPU | PPU of the maximal reserve supply bid |

MILP | Mixed integer linear problem |

MIQCP | Mixed integer quadratically constrained problem |

TSW | Total social welfare |

MSC | Minimum surplus condition |

EENS | Expected energy not supplied |

LGC | Load gradient condition |

H | Bid/schedule realization history |

D | Deviation vector |

u | Uncertainty indicator |

$\overline{u}$ | Uncertainty threshold |

y | Bid acceptance indicator |

q | Bid quantity |

p | Bid price per unit (PPU) |

I | Income |

$MCP$ | Market clearing price |

S | Surplus constant |

$TSW$ | Total social welfare |

ES | Energy supply |

ED | Energy demand |

ESU+ | Positively uncertain energy supply bid |

ESU− | Negatively uncertain energy supply bid |

ESUb | Bi-uncertain energy supply bid |

EDU+ | Positively uncertain energy demand bid |

EDU− | Negatively uncertain energy demand bid |

EDUb | Bi-uncertain energy demand bid |

RS+ | Positive reserve supply bid |

RS− | Negative reserve supply bid |

RD+ | Positive reserve demand bid |

RD− | Negative reserve demand bid |

RD+/RD− ESUb/ESU+/ESU− | Positive/negative reserve demand bid implied by bi-uncertain/positively uncertain/negatively uncertain energy supply bid |

RD+/RD− EDUb/EDU+/EDU− | Positive/negative reserve demand bid implied by bi-uncertain/positively uncertain/negatively uncertain energy demand bid |

## Appendix A. Structure of the Variable Vector and Formulation of Logical Constraints

${n}_{ES}$ | number of uncertainty-free energy supply bids |

${n}_{ESUb}$ | number of bi-uncertain energy supply bids |

${n}_{ESU+}$ | number of positively uncertain energy supply bids |

${n}_{ESU-}$ | number of negatively uncertain energy supply bids |

${n}_{ED}$ | number of uncertainty-free energy demand bids |

${n}_{EDUb}$ | number of bi-uncertain energy demand bids |

${n}_{EDU+}$ | number of positively uncertain energy demand bids |

${n}_{EDU-}$ | number of negatively uncertain energy demand bids |

${n}_{RS+}$ | number of non-SRDB positive reserve supply bids |

${n}_{RD+}$ | number of non-SRDB positive reserve demand bids |

${n}_{RS-}$ | number of non-SRDB negative reserve supply bids |

${n}_{RD-}$ | number of non-SRDB negative reserve demand bids |

- ${Y}^{ES}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{[0,1]}^{{n}_{ES}+{n}_{ESUb}+{n}_{ESU+}+{n}_{ESU-}}$,
- ${Y}^{ED}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{[0,1]}^{{n}_{ED}+{n}_{EDUb}+{n}_{EDU+}+{n}_{EDU-}}$,
- ${Y}^{RS+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{[0,1]}^{{n}_{RS+}}$, ${Y}^{RD+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{RD+}+{n}_{ESUb}+{n}_{EDUb}+{n}_{ESU-}+{n}_{EDU-}}$,
- ${Y}^{RS-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{[0,1]}^{{n}_{RS-}}$, ${Y}^{RD-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{RD-}+{n}_{ESUb}+{n}_{EDUb}+{n}_{ESU+}+{n}_{EDU+}}$.

- ${I}^{ESUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESUb}}$, ${I}^{ESU+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESU+}}$, ${I}^{ESU-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESU-}}$,
- ${I}^{EDUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDUb}}$, ${I}^{EDU+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDU+}}$, ${I}^{EDU-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDU-}}$,
- ${I}^{RD+\phantom{\rule{3.33333pt}{0ex}}ESUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESUb}}$, ${I}^{RD+\phantom{\rule{3.33333pt}{0ex}}EDUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDUb}}$,
- ${I}^{RD+\phantom{\rule{3.33333pt}{0ex}}ESU-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESU-}}$, ${I}^{RD+\phantom{\rule{3.33333pt}{0ex}}EDU-}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDU-}}$,
- ${I}^{RD-\phantom{\rule{3.33333pt}{0ex}}ESUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESUb}}$, ${I}^{RD-\phantom{\rule{3.33333pt}{0ex}}EDUb}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDUb}}$,
- ${I}^{RD-\phantom{\rule{3.33333pt}{0ex}}ESU+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{ESU+}}$, ${I}^{RD-\phantom{\rule{3.33333pt}{0ex}}EDU+}\phantom{\rule{3.33333pt}{0ex}}\in \phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}_{+}^{{n}_{EDU+}}$.

## Appendix B. Reference Bid Set

**Table A2.**Reference ES bid set: The columns correspond to the index of the bid (ID), quantity (q), bid price (p), positive and negative uncertainty (${u}^{+},\phantom{\rule{3.33333pt}{0ex}}{u}^{-}$), and S respectively.

ID | q [MW] | p [EUR/MW] | ${\mathit{u}}^{\mathbf{+}}$ | ${\mathit{u}}^{\mathbf{-}}$ | S [EUR] |
---|---|---|---|---|---|

1 | 32.08 | 54.04 | 0 | 0.02 | 0 |

2 | 35.76 | 67.03 | 0.16 | 0.14 | 27.06 |

3 | 72.78 | 109.6 | 0.07 | 0 | 0 |

4 | 43.2 | 83.5 | 0.01 | 0.17 | 0 |

5 | 74.77 | 82.05 | 0 | 0 | 22.3 |

6 | 75.63 | 92.84 | 0.07 | 0.02 | 0 |

7 | 76.18 | 91.3 | 0 | 0.01 | 78.34 |

8 | 28.99 | 109.2 | 0.01 | 0.12 | 17.48 |

9 | 56.92 | 67.69 | 0.03 | 0.09 | 0 |

10 | 21.34 | 56.35 | 0.05 | 0.5 | 0 |

11 | 24.86 | 59.68 | 0 | 0 | 0 |

12 | 36.13 | 52.4 | 0.01 | 0 | 16.5 |

13 | 30.41 | 61.69 | 0 | 0.07 | 6.92 |

14 | 33.66 | 101 | 0 | 0.18 | 25.98 |

15 | 46.86 | 71.59 | 0 | 0.17 | 27.26 |

16 | 45.29 | 82 | 0 | 0 | 19.06 |

17 | 55.52 | 71.47 | 0.02 | 0.12 | 0 |

18 | 75.26 | 103.2 | 0 | 0.42 | 70.78 |

19 | 51.23 | 55.33 | 0.03 | 0.03 | 34.66 |

20 | 24.94 | 72.62 | 0 | 0.09 | 22.4 |

21 | 45.05 | 100.7 | 0.02 | 0 | 19.72 |

22 | 26.92 | 55.15 | 0 | 0 | 0 |

23 | 66.69 | 51.58 | 0 | 0 | 37.82 |

24 | 21.87 | 102.6 | 0.07 | 0 | 10.32 |

25 | 62.03 | 108.9 | 0 | 0.04 | 133.3 |

26 | 43.21 | 65.7 | 0 | 0 | 0 |

27 | 67.12 | 68.3 | 0.04 | 0.07 | 0 |

28 | 57.16 | 86.29 | 0.03 | 0 | 0 |

29 | 42.01 | 98.65 | 0.11 | 0 | 0 |

30 | 70.98 | 106.4 | 0 | 0 | 98.5 |

31 | 78.39 | 104.2 | 0 | 0.03 | 0 |

32 | 22.96 | 82.1 | 0 | 0 | 0 |

33 | 48.56 | 89.24 | 0 | 0.32 | 43.88 |

34 | 54.63 | 102.8 | 0 | 0 | 0 |

35 | 69.34 | 94.12 | 0.18 | 0 | 77.9 |

36 | 37.74 | 96.92 | 0 | 0.16 | 61.82 |

37 | 47.68 | 90.75 | 0.14 | 0.01 | 76.42 |

38 | 50.2 | 50.35 | 0 | 0 | 28 |

39 | 63.17 | 63.71 | 0 | 0 | 1.22 |

40 | 76.42 | 110 | 0 | 0.05 | 81.08 |

41 | 23.56 | 77.62 | 0 | 0.01 | 9 |

42 | 34.04 | 82.91 | 0 | 0.09 | 0 |

43 | 40.81 | 106.1 | 0 | 0 | 0 |

44 | 51.88 | 62.6 | 0 | 0.07 | 0 |

45 | 74.69 | 93.82 | 0.02 | 0 | 0 |

46 | 24.02 | 67.05 | 0 | 0 | 29.88 |

47 | 55.86 | 85.02 | 0 | 0 | 20.36 |

48 | 30.22 | 70.81 | 0.08 | 0 | 0 |

49 | 74.31 | 90.97 | 0 | 0 | 0 |

50 | 75.13 | 72.05 | 0 | 0.17 | 0 |

**Table A3.**Reference ED bid set: The columns correspond to the index of the bid (ID), quantity (q), bid price (p), positive and negative uncertainty (${u}^{+},\phantom{\rule{3.33333pt}{0ex}}{u}^{-}$), and S respectively.

ID | q [MW] | p [EUR/MW] | ${\mathit{u}}^{\mathbf{+}}$ | ${\mathit{u}}^{\mathbf{-}}$ | S [EUR] |
---|---|---|---|---|---|

1 | −36.32 | 104.4 | 0 | 0 | 0 |

2 | −39.9 | 110.8 | 0.09 | 0.03 | 0 |

3 | −41.39 | 75.68 | 0 | 0 | 0 |

4 | −27.12 | 140.9 | 0 | 0 | 0 |

5 | −29.76 | 101.8 | 0.14 | 0 | 0 |

6 | −33.23 | 132.6 | 0.04 | 0 | 0 |

7 | −41.42 | 96.16 | 0.31 | 0 | 0 |

8 | −15.19 | 143.7 | 0 | 0 | 0 |

9 | −27.33 | 113.2 | 0.34 | 0 | 0 |

10 | −33.82 | 140 | 0.03 | 0.02 | 0 |

11 | −-39.86 | 77.72 | 0 | 0 | 0 |

12 | −10.38 | 134.4 | 0 | 0 | 0 |

13 | −12.27 | 85.39 | 0 | 0 | 0 |

14 | −27.64 | 94.7 | 0 | 0.1 | 0 |

15 | −25.21 | 97.9 | 0 | 0.1 | 0 |

16 | −44.12 | 79 | 0 | 0 | 0 |

17 | −21.38 | 132.9 | 0.1 | 0 | 0 |

18 | −20.38 | 144.6 | 0 | 0.07 | 0 |

19 | −28.45 | 123.9 | 0.06 | 0 | 0 |

20 | −24.55 | 76.77 | 0.09 | 0.06 | 0 |

21 | −20.66 | 80.14 | 0 | 0.11 | 0 |

22 | −14.17 | 114 | 0 | 0 | 0 |

23 | −38.62 | 135.2 | 0.15 | 0 | 0 |

24 | −22.05 | 81.56 | 0.13 | 0 | 0 |

25 | −32.56 | 103.3 | 0 | 0.06 | 0 |

26 | −25.72 | 88.34 | 0.29 | 0.32 | 0 |

27 | −23.72 | 78.41 | 0.05 | 0.03 | 0 |

28 | −35.95 | 138.4 | 0.03 | 0.04 | 0 |

29 | −48.79 | 94.67 | 0 | 0.04 | 0 |

30 | −29.24 | 140.4 | 0 | 0.16 | 0 |

31 | −34.83 | 103.1 | 0.08 | 0.03 | 0 |

32 | −34.81 | 118.1 | 0 | 0.07 | 0 |

33 | −14.32 | 134.4 | 0.04 | 0 | 0 |

34 | −13.19 | 98.94 | 0 | 0.06 | 0 |

35 | −33.59 | 97.01 | 0 | 0.2 | 0 |

36 | −44.54 | 121.2 | 0 | 0 | 0 |

37 | −40.65 | 145 | 0 | 0.3 | 0 |

38 | −39.18 | 107.4 | 0.15 | 0 | 0 |

39 | −44.97 | 118.1 | 0 | 0 | 0 |

40 | −49.31 | 142.9 | 0 | 0.06 | 0 |

41 | −49.04 | 73.14 | 0 | 0 | 0 |

42 | −24.23 | 136.6 | 0 | 0 | 0 |

43 | −34.61 | 90.51 | 0.12 | 0.14 | 0 |

44 | −15.27 | 140.7 | 0 | 0 | 0 |

45 | −29.96 | 103.9 | 0 | 0.01 | 0 |

46 | −28.88 | 129.7 | 0.04 | 0 | 0 |

47 | −27.01 | 128.7 | 0.08 | 0 | 0 |

48 | −47.28 | 114.3 | 0.15 | 0.1 | 0 |

49 | −43.24 | 129.3 | 0 | 0 | 0 |

50 | −20.66 | 124.2 | 0 | 0 | 0 |

**Table A4.**Reference (non-SRDB) RS+ bid set: The columns correspond to the index of the bid (ID), quantity (q) and bid price (p) respectively. The SRDB bids are generated according to $\overline{u}$, and the implied actual set of uncertain energy bids.

ID | q [MW] | p [EUR/MW] |
---|---|---|

1 | 7.11 | 28.34 |

2 | 4 | 28.38 |

3 | 19.94 | 51.98 |

4 | 16.2 | 63.76 |

5 | 16.89 | 67.17 |

6 | 2.64 | 31.28 |

7 | 15.06 | 64.25 |

8 | 8.73 | 34.73 |

9 | 12.37 | 26.15 |

10 | 2.78 | 67.01 |

11 | 9.23 | 60.32 |

12 | 6.36 | 52.39 |

13 | 18.08 | 53.33 |

14 | 18.64 | 69.82 |

15 | 4.7 | 26.11 |

16 | 18.41 | 31.66 |

17 | 3.61 | 57.21 |

18 | 5.15 | 32.18 |

19 | 10.62 | 67.7 |

20 | 15.34 | 55.26 |

21 | 3.26 | 38.77 |

22 | 18.87 | 63.57 |

23 | 18.04 | 51.49 |

24 | 4.92 | 32.92 |

25 | 15.46 | 67.54 |

26 | 13.85 | 49.47 |

**Table A5.**Reference (non-SRDB) RS- bid set: The columns correspond to the index of the bid (ID), quantity (q) and bid price (p) respectively. The SRDB bids are generated according to $\overline{u}$, and the implied actual set of uncertain energy bids.

ID | q [MW] | p [EUR/MW] |
---|---|---|

1 | 19.4 | 26.11 |

2 | 16 | 44.81 |

3 | 13.35 | 39.13 |

4 | 18.6 | 35.05 |

5 | 8.74 | 38.86 |

6 | 2.68 | 52.23 |

7 | 17.05 | 51.47 |

8 | 8.54 | 31.29 |

9 | 4.09 | 69.16 |

10 | 11.75 | 28.89 |

11 | 6.61 | 34.44 |

12 | 7.66 | 65.67 |

13 | 11.37 | 32.3 |

14 | 4.58 | 27.88 |

15 | 17.39 | 68.86 |

16 | 18.73 | 43.43 |

**Table A6.**Reference (non-SRDB) RD+ bid set: The columns correspond to the index of the bid (ID), quantity (q) and bid price (p) respectively. The SRDB bids are generated according to $\overline{u}$, and the implied actual set of uncertain energy bids.

ID | q [MW] | p [EUR/MW] |
---|---|---|

1 | −12.76 | 45.55 |

2 | −16.76 | 70.49 |

3 | −12.6 | 54.63 |

4 | −11.9 | 50.88 |

5 | −19.93 | 51.42 |

**Table A7.**Reference (non-SRDB) RD- bid set: The columns correspond to the index of the bid (ID), quantity (q) and bid price (p) respectively. The SRDB bids are generated according to $\overline{u}$, and the implied actual set of uncertain energy bids.

ID | q [MW] | p [EUR/MW] |
---|---|---|

1 | −18.23 | 56.8 |

2 | −10.45 | 42.55 |

3 | −3.49 | 63.49 |

4 | −7.6 | 55.99 |

5 | −5.8 | 35.06 |

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**Figure 1.**The scheme of the reserve spot market (both in the case of + and − reserve). D—demand, S—supply, $MCP$—market clearing price, $MRSBPPU$—Maximal reserve supply bid PPU, $SRDB$—Supplementary reserve demand bids. By definition, the PPU of SRDBs is equal to the PPU of the highest reserve supply bid + $\epsilon $.

**Figure 2.**(

**a**) Total social welfare of the energy sub-market ($TS{W}^{E}$) as the parameter $\overline{u}$ is decreased. (

**b**) Total social welfare of the reserve sub-markets as the parameter $\overline{u}$ is decreased. $TS{W}^{R+}$ and $TS{W}^{R-}$ denote the total social welfare of the positive and the negative reserve market respectively

**Figure 3.**(

**a**) Total traded volume of the energy sub-market as the parameter $\overline{u}$ is decreased. (

**b**) Total traded volumes of the reserve sub-markets as the parameter $\overline{u}$ is decreased.

**Figure 4.**(

**a**) Market clearing price ($MC{P}^{E}$) of the energy sub-market as the parameter $\overline{u}$ is decreased. (

**b**) Market clearing prices of reserve sub-markets ($MC{P}^{R+}/MC{P}^{R-}$) as the parameter $\overline{u}$ is decreased.

**Figure 5.**(

**a**) Required computational time as $\overline{u}$ is decreased in the case of 50–50, 100–100 and 200–200 energy supply and demand bids (with similar uncertainty parameters as in the case of the previous example detailed in Appendix A). (

**b**) Number of variables and integer variables in the 200 bids case.

**Figure 6.**Total social welfare (TSW) of the reserve sub-markets as the parameter $\overline{u}$ decreases in example 2. $TS{W}^{R+}$ and $TS{W}^{R-}$ denote the total social welfare of the positive and the negative reserve market respectively.

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## Share and Cite

**MDPI and ACS Style**

Csercsik, D.; Sleisz, Á.; Sőrés, P.M.
The Uncertain Bidder Pays Principle and Its Implementation in a Simple Integrated Portfolio-Bidding Energy-Reserve Market Model. *Energies* **2019**, *12*, 2957.
https://doi.org/10.3390/en12152957

**AMA Style**

Csercsik D, Sleisz Á, Sőrés PM.
The Uncertain Bidder Pays Principle and Its Implementation in a Simple Integrated Portfolio-Bidding Energy-Reserve Market Model. *Energies*. 2019; 12(15):2957.
https://doi.org/10.3390/en12152957

**Chicago/Turabian Style**

Csercsik, Dávid, Ádám Sleisz, and Péter Márk Sőrés.
2019. "The Uncertain Bidder Pays Principle and Its Implementation in a Simple Integrated Portfolio-Bidding Energy-Reserve Market Model" *Energies* 12, no. 15: 2957.
https://doi.org/10.3390/en12152957