# Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market

## Abstract

**:**

## 1. Introduction

## 2. Day-Ahead and Balancing Markets

## 3. Problem Formulation

#### 3.1. Imbalance Minimisation

#### 3.2. Categorical Assessment of System Length

#### 3.3. Probabilistic Assessment of System Length

#### 3.4. Risk Constrained Contracted Volume

#### 3.4.1. Additive Adjustment

#### 3.4.2. Multiplicative Adjustment

#### 3.5. Quantile Offer

## 4. Forecasting

#### 4.1. Probabilistic System Length Forecast

`R`, specifically the function

`glm`from the package

`stats`[33].

#### 4.2. Price Forecasts

`R`package

`forecast`[34].

#### 4.3. Wind Power Quantile Forecasts

## 5. Case Study and Results

#### 5.1. System Length Forecast and Evaluation

#### 5.2. Price Forecast Evaluation

#### 5.3. Offer Strategy Results

## 6. Conclusions

## Acknowledgments

## Data Statement

## Conflicts of Interest

## Nomenclature

${R}_{t+k}$ | Revenue from settlement period $t+k$ |

${T}_{t+k}$ | Imbalance cost in settlement period $t+k$ |

${\widehat{x}}_{t+k|t}$ | Forecast of ${x}_{t+k}$ made at time t |

$\mathcal{E}\left\{\xb7\right\}$ | Expectation operator |

${E}_{t+k}^{\mathrm{C}}$ | Contracted volume for settlement period $t+k$ |

${E}_{t+k}$ | Actual energy delivered in settlement period $t+k$ |

${d}_{t+k}$ | Market participant’s imbalance ${E}_{t+k}^{\mathrm{C}}-{E}_{t+k}$ |

$\overline{d}$ | Mean absolute imbalance volume |

${E}_{\mathrm{max}}$ | Maximum deliverable volume in any single settlement period |

${\pi}_{t+k}^{\mathrm{C}}$ | Price at which ${E}_{t+k}^{\mathrm{C}}$ is contracted |

${\pi}_{t+k}^{\mathrm{S}}$ | Imbalance price |

${\pi}_{t+k}^{+}$ | Imbalance price during periods of net up-regulation |

${\pi}_{t+k}^{-}$ | Imbalance price during periods of net down-regulation |

NIV${}_{t+k}$ | Net Imbalance Volume for settlement period $t+k$; a positive NIV is associated with up-regulation (‘the system is short’), negative NIV is associated with down-regulation (‘the system is long’) |

${\varphi}_{t+k}$ | Probability that the system will be short, $\mathrm{P}({\mathrm{NIV}}_{t+k}>0)$ |

$\mathrm{\Phi}$ | The ratio $\frac{{\pi}^{\mathrm{C}}-{\pi}^{-}}{{\pi}^{+}-{\pi}^{-}}$ |

${q}_{\alpha}$ | The $\alpha $-quantile forecast of energy production |

$\nu ,\phantom{\rule{3.33333pt}{0ex}}\eta ,\phantom{\rule{3.33333pt}{0ex}}{\alpha}^{\prime}$ | Parameters of the additive, multiplicative and quantile trading strategies, respectively |

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**Figure 1.**Relative operator characteristic curves for system length forecasts. The diagonal line, False Positive Rate = True Positive Rate, illustrates the performance of a random forecast, e.g., a random forecast of 70% would be expected to correctly predict 70% of all positive outcomes, and falsely predict that 70% of negative outcomes would be positive.

**Table 1.**Brier scores for probabilistic system length forecasts. The uncertainty component of the Brier score is 0.2337 in both cases.

Method | Brier Score | Reliability | Resolution |
---|---|---|---|

Empirical Proportions | 0.2318 | 0.0001 | 0.0029 |

Logistic Regression | 0.2265 | 0.0030 | 0.0102 |

**Table 2.**Normalised revenue (£/MWh) using different trading strategies. For strategies marked ${}^{*}$ “Forecast Method” refers to the type of price forecast only. The mean day-ahead price during the test period was £34.75/MWh.

Strategy | Forecast Method | ||
---|---|---|---|

Perfect | Simple | Advanced | |

Minimise Imbalance | 34.66 | n/a | 34.81 |

SL Forecast: Deterministic | 50.16 | 33.20 | 34.39 |

SL Forecast: Empirical Proportion ${}^{*}$ | 41.72 | 37.32 | 37.75 |

SL Forecast: Logistic ${}^{*}$ | 41.43 | 36.25 | 37.92 |

**Table 3.**Performance of the risk-constrained offer strategies. The case $\nu ,\phantom{\rule{3.33333pt}{0ex}}\eta =0$ is equivalent to offering a volume equal to the wind power forecast (imbalance minimisation), the additive adjustment strategy with $\nu =1$ is equivalent to offering zero/max.

Additive Adjustment | ||||

$\mathit{\nu}$ | Revenue | VaR${}_{\mathbf{1}\%}$ | CVaR${}_{\mathbf{1}\%}$ | $\tilde{\mathit{d}}$ |

0 | 34.81 | 0.55 | 1.61 | 9% |

0.1 | 35.39 | 0.72 | 2.30 | 13% |

0.5 | 37.16 | 8.22 | 15.34 | 36% |

1 | 37.92 | 14.23 | 27.51 | 49% |

Multiplicative Adjustment | ||||

$\mathit{\eta}$ | Revenue | VaR${}_{\mathbf{1}\%}$ | CVaR${}_{\mathbf{1}\%}$ | $\tilde{\mathit{d}}$ |

0 | 34.81 | 0.55 | 1.61 | 9% |

0.1 | 35.02 | 0.43 | 1.38 | 10% |

0.5 | 36.02 | 0.82 | 2.60 | 20% |

1 | 37.26 | 2.57 | 6.87 | 36% |

5 | 37.72 | 11.13 | 19.71 | 45% |

10 | 37.73 | 14.14 | 26.11 | 47% |

Quantile | ||||

${\mathit{\alpha}}^{\prime}$ | Revenue | VaR${}_{\mathbf{1}\%}$ | CVaR${}_{\mathbf{1}\%}$ | $\tilde{\mathit{d}}$ |

0.55 | 34.90 | 0.45 | 1.38 | 10% |

0.65 | 35.09 | 0.47 | 1.27 | 10% |

0.75 | 35.32 | 0.45 | 1.46 | 11% |

0.95 | 36.14 | 1.73 | 4.07 | 20% |

0.99 | 37.05 | 3.75 | 8.77 | 32% |

Units: | £/MWh | £ | % of ${E}_{\mathrm{max}}$ |

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**MDPI and ACS Style**

Browell, J. Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market. *Energies* **2018**, *11*, 1345.
https://doi.org/10.3390/en11061345

**AMA Style**

Browell J. Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market. *Energies*. 2018; 11(6):1345.
https://doi.org/10.3390/en11061345

**Chicago/Turabian Style**

Browell, Jethro. 2018. "Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market" *Energies* 11, no. 6: 1345.
https://doi.org/10.3390/en11061345