# Optimal Daily Trading of Battery Operations Using Arbitrage Spreads

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Background Research

## 3. Stylized Properties of Spread Trades

**Property**

**1.**

**Proof**

**of**

**Property**

**1.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Property**

**2.**

**Proof**

**of**

**Property**

**2.**

**Property**

**3.**

**Proof**

**of**

**Property**

**3.**

**Lemma**

**2.**

**Property**

**4.**

**Proof**

**of**

**Property**

**4.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof**

**of**

**Lemma**

**4.**

## 4. Optimization

## 5. Data and Density Forecasts for German Application

## 6. Backtested Results

#### 6.1. One Trade Per Day

#### 6.2. Two Trades Per Day

## 7. Results

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Nomenclature

## Abbreviations

DA | Day-Ahead |

PJM | Pennsylvania, Jersey, Maryland Power Pool |

GAMLSS | Generalized Additive Model for Location, Scale and Shape |

ARIMA | Autoregressive Integrated Moving Average |

PV | Photovoltaic |

SDP | Stochastic Dynamic Programming |

ADP | Approximate Dynamic Programming |

DSDP | Discretized SDP |

MDP | Markov Decision Process |

SoC | State of Charge |

SoC${}_{s}$ | Starting State of Charge |

SoC${}_{f}$ | Finishing State of Charge |

P1 / P2 / P3 / P4 | Property number $1/2/3/4$ |

VaR | Value-at-Risk |

ARA | Amsterdam-Rotterdam-Antwerp |

GPL | Germany Gaspool |

GB | Great Britain |

MWh | Megawatt Hour |

BD | Best Distribution (skew type) |

ND | Normal Distribution |

## Nomenclature

i | Hourly index, 0 … 23, corresponding to the buy trade |

j | Hourly index, 0 … 23, corresponding to the sell trade |

${Y}^{(i,j)}$ | Random variable (r.v.) representing individual spread price, element of a matrix $\in {\mathbb{R}}^{24\times 24}$ (element of row i, col j) |

s | Spread number (flattened $24\times 24$ hourly matrix), element of a vector $\in {\mathbb{R}}^{276}$ |

${Y}_{t}^{\left(s\right)}$ | Shorthand notation for r.v. ${Y}^{(i,j)}$ i.e., spread price for spread s at time step t |

${y}_{t}^{\left(s\right)}$ | Realized spread price for spread s at time step t |

$\mathit{Y}$ | Full dataset of spread prices $\in {\mathbb{R}}^{1917\times 276}$, last 383 observations used for out-of-sample |

$E\left({Y}_{t}^{\left(s\right)}\right)$ | Expected price of spread s at time t (forecasted / fitted spread price value) |

${q}_{05}^{(i,j)}$ | 5th quantile of the spread density for ${Y}^{(i,j)}$ |

${q}_{95}^{(i,j)}$ | 95th quantile of the spread density for ${Y}^{(i,j)}$ |

$\eta $ | Battery efficiency for roundtrip spread trade |

c | Transaction costs for roundtrip spread trade |

b | Starting/finishing state of charge of the battery level, $\in \{0,1\}$ |

${\mathsf{b}}_{i}$ | Amount of energy, ≥0, to be bought by the battery at hr i (long position) |

${\mathsf{s}}_{j}$ | Amount of energy, ≥0, that the battery intends to sell at hr j (short position) |

${\mathsf{bs}}_{ij}$ | Element of indicator matrix that tracks timing of spread trades (row i, col j) |

${\mathsf{b}}_{i}^{*}$ | Optimal value of ${\mathsf{b}}_{i}$ |

${\mathsf{s}}_{j}^{*}$ | Optimal value of ${\mathsf{s}}_{j}$ |

${\mathsf{bs}}_{ij}^{*}$ | Optimal value of ${\mathsf{bs}}_{ij}$ |

N | Number of hours in a day available for the battery to trade, $N=24$ |

n | Number of spread trades per day |

${n}_{1}$ | Number of days with 1 spread trade per day, from 383 out-of-sample data |

${n}_{2}$ | Number of days with 2 spread trades per day, from 383 out-of-sample data |

${n}_{l}$ | Number of loss days after costs are taken into account, from 383 out-of-sample data |

${\widehat{M}}_{t}^{\left(s\right)}$ | Learnt model for spread s at t specifying the best distribution and its parameters |

${\widehat{\theta}}_{t}^{\left(s\right)}$ | Estimated parameters of the best skew-type of distribution for spread s at time step t |

$E\left({\pi}_{t}^{\left(s\right)}\right)$ | Expected net payoff for spread s of day t, ∈$\mathbb{R}$ |

$E\left({\pi}_{t}^{\left(s\right)}\right)$ | Expected net payoff of day t, ∈$\mathbb{R}$ |

${\pi}_{t}$ | Realized net payoff for spread s of day t (actual profit / loss), $\in \mathbb{R}$ |

$\pi $ | Realized net payoff vector, ∈${\mathbb{R}}^{383}$ |

$E\left(\mathbf{\Pi}\right)$ | Expected payoffs matrix single trade scenario, ∈${\mathbb{R}}^{276\times 383}$ |

$E\left({\mathbf{\Pi}}_{t}\right)$ | Vector of expected payoffs from $E\left(\mathbf{\Pi}\right)$ at time t, (can be reshaped into ∈${\mathbb{R}}^{24\times 24}$) |

$\mathbf{\Pi}$ | Actual matrix of all possible payoffs, single trade scenario, ∈${\mathbb{R}}^{276\times 383}$ |

${\mathbf{\Pi}}_{t}$ | Vector of realized strategy payoffs, single trade scenario, ∈${\mathbb{R}}^{276}$ at t |

${\pi}_{t}^{\left(s\right)}$ | Element of actual payoff vector, ${\mathbf{\Pi}}_{t}$, i.e., payoff at t for spread trade s, ∈$\mathbb{R}$ |

${\pi}_{total}$ | Total realized strategy payoff over the backtest period, ∈$\mathbb{R}$ |

$\overline{\pi}$ | Average realized strategy payoff, of 383 out-of-sample data, ∈$\mathbb{R}$ |

${s}^{\pi}$ | Standard error of the realized strategy payoffs, of 383 out-of-sample data, ∈$\mathbb{R}$ |

$s{d}_{\pi}$ | Sample std. dev. obtained from payoff vector, $\pi $, of 383 out-of-sample data, ∈$\mathbb{R}$ |

l | Total monetary value resulting from loss days, from 383 out-of-sample data, ∈$\mathbb{R}$ |

$\overline{l}$ | Average loss value resulting from loss days, from 383 out-of-sample data, ∈$\mathbb{R}$ |

$E\left({\mathbf{\Pi}}^{tot}\right)$ | Expected total payoff matrix, 2 trade scenario, ∈${\mathbb{R}}^{276\times 383}$ |

$E\left({\mathbf{\Pi}}_{t}^{tot}\right)$ | Expected total payoff vector at time step t, 2 trade scenario, ∈${\mathbb{R}}^{276}$ |

${\mathbf{\Pi}}^{tot}$ | Actual total payoff matrix, 2 trade scenario, ∈${\mathbb{R}}^{276\times 383}$ |

$E\left({\pi}_{t}^{tot,\left(s\right)}\right)$ | Element of $E\left({\mathbf{\Pi}}^{tot}\right)$ for spread s at time t, ∈$\mathbb{R}$ |

${\pi}_{t}^{tot,\left(s\right)}$ | Element of ${\mathbf{\Pi}}^{tot}$ for spread s at time t, ∈$\mathbb{R}$ |

${\pi}^{\mathit{tot}}$ | Realized total payoff vector, $\in {\mathbb{R}}^{383}$ |

## Appendix A. Algorithms

Algorithm A1. One Trade Forecasted Strategy Payoff (Battery Level b and Risk Criterion c) |

1: Init strategy expected payoff matrix, $E\left(\mathbf{\Pi}\right)\leftarrow \mathbf{0}\in {\mathbb{R}}^{276\times 383}$ |

2: for each time step (i.e., test point) $t=1,\dots ,383$ do |

3: for each hour spread $s=1,\dots ,276$ do |

4: Use model ${\widehat{M}}_{t}^{\left(s\right)}$ to extract forecasted parameters ${\widehat{\theta}}_{t}^{\left(s\right)}={[{\widehat{\mu}}_{t}^{\left(s\right)},{\widehat{\sigma}}_{t}^{\left(s\right)},{\widehat{\nu}}_{t}^{\left(s\right)},{\widehat{\tau}}_{t}^{\left(s\right)}]}^{T}$ |

5: if distribution parameters ${\widehat{\theta}}_{t}^{\left(s\right)}$ were successfully forecasted then |

6: Extract calculated expected value of the spread, $E\left({Y}_{t}^{\left(s\right)}\right)$ |

7: if $E\left({Y}_{t}^{\left(s\right)}\right)>0$ then |

8: Extract 5th quantile (i.e., 95% of dist. is on the right), ${\widehat{q}}_{05,t}^{\left(s\right)}$, using ${\widehat{\theta}}_{t}^{\left(s\right)}$ |

9: else |

10: Extract 95th quantile (i.e., 95% of dist. is on the left), ${\widehat{q}}_{95,t}^{\left(s\right)}$, using ${\widehat{\theta}}_{t}^{\left(s\right)}$ |

11: if $\eta |{\widehat{q}}_{xx,t}^{\left(s\right)}|>c$ then |

12: if $E\left({Y}_{t}^{\left(s\right)}\right)>0$ then |

13: $E\left({\pi}_{t}^{\left(s\right)}\right)\leftarrow \left(\eta E\left({Y}_{t}^{\left(s\right)}\right)-c\right)b$, expected payoff for spread s at t |

14: else |

15: $E\left({\pi}_{t}^{\left(s\right)}\right)\leftarrow \left(\eta |E\left({Y}_{t}^{\left(s\right)}\right)|-c\right)(1-b)$, expected payoff for spread s at t |

Algorithm A2. One Trade Actual Strategy Payoff (Battery Level b and Risk Criterion c) |

1: Init strategy actual payoff matrix $\mathbf{\Pi}\leftarrow \mathbf{0}\in {\mathbb{R}}^{276\times 383}$ payoff for each spread s at each t |

2: Init strategy realized payoff vector, $\pi \leftarrow \mathbf{0}\in {\mathbb{R}}^{383}$ |

3: for each time step (i.e., test point) $t=1,\dots ,383$ do |

4: for each hour spread $s=1,\dots ,276$ do |

5: Extract actual (observed) spread price ${y}_{t}^{\left(s\right)}$ |

6: if ${y}_{t}^{\left(s\right)}>0$ then |

7: ${\pi}_{t}^{\left(s\right)}\leftarrow (\eta {y}_{t}^{\left(s\right)}-c)b$, element of ${\mathbf{\Pi}}_{t}$ |

8: else |

9: ${\pi}_{t}^{\left(s\right)}\leftarrow \left(\eta |{y}_{t}^{\left(s\right)}|-c\right)(1-b)$, element of ${\mathbf{\Pi}}_{t}$ |

10: $maxVal\leftarrow ma{x}_{s}E\left({\mathbf{\Pi}}_{t}\right)$, extract max value from vector of expected payoffs at t |

11: if $maxVal\ne 0$ then |

12: ${s}^{\prime}\leftarrow argma{x}_{s}{\mathbf{\Pi}}_{t}$, extract spread hour associated with max expected payoff at t |

13: ${\pi}_{t}\leftarrow {\pi}_{t}^{\left({s}^{\prime}\right)}$ extract payoff from ${\mathbf{\Pi}}_{t}$ at ${s}^{\prime}$, t; assigned as elem of realised payoff $\pi $ |

Algorithm A3. Two-Trade Forecasted Strategy Payoff (Battery Level b and Risk Criterion c) |

1: Init strategy expected total payoff matrix, $E\left({\mathbf{\Pi}}^{tot}\right)\in {\mathbb{R}}^{276\times 383}$ |

2: Init strategy actual total payoff matrix, ${\mathbf{\Pi}}^{tot}\in {\mathbb{R}}^{276\times 383}$ |

3: for each time step (i.e., test point) $t=1,\dots ,383$ do |

4: for each spread hour $s=1,\dots ,276$ do |

5: if $E\left({\pi}_{t}^{\left(s\right)}\right)>0$, if exp. payoff for 1st spread s at t is profitable then |

6: Init $\mathbf{S}\leftarrow E\left({\mathbf{\Pi}}_{t}\right)\in {\mathbb{R}}^{24\times 24}$ expected payoffs for 2nd trade; copy of 1st trade |

7: Set $\mathbf{S}$[ii-1:jj,:] $\leftarrow \mathbf{0}$; and $\mathbf{S}$[1:ii-1, ii:24] $\leftarrow \mathbf{0}$, only allow feasible 2nd trades |

8: found ← False |

9: while not found do |

10: $maxVal\leftarrow {max}_{s}\mathbf{S}\in {\mathbb{R}}_{0}^{+}$, max payoff value over all possible 2nd trades |

11: if $maxVal=0$ then |

12: $E\left({\pi}_{t}^{tot,\left(s\right)}\right)\leftarrow E\left({\pi}_{t}^{\left(s\right)}\right)+0$, expected total payoff of 1 trade only |

13: ${\pi}_{t}^{tot,\left(s\right)}\leftarrow {\pi}_{t}^{\left(s\right)}+0$, actual total payoff comprised of 1 trade only |

14: trade found ← True |

15: else Calculate total two trade payoff |

16: ${s}^{\prime}\leftarrow {argmax}_{s}\mathbf{S}$ spread hour ${i}^{\prime}-{j}^{\prime}$ corresp. to max expected payoff |

17: $E\left({\pi}_{t}^{tot,\left(s\right)}\right)\leftarrow E\left({\pi}_{t}^{\left(s\right)}\right)+{\mathbf{S}}^{\left({s}^{\prime}\right)}$ |

18: ${\pi}_{t}^{tot,\left(s\right)}\leftarrow {\pi}_{t}^{\left(s\right)}+{\pi}_{t}^{\left({s}^{\prime}\right)}$ |

19: trade found ← True |

Algorithm A4. Two-Trade Realized Strategy Payoff (Battery Level b and Risk Criterion c) |

1: Init strategy realised total payoff vector, ${\pi}^{tot}\leftarrow \mathbf{0}\in {\mathbb{R}}^{383}$ |

2: Init $n\leftarrow 0$ number of loss days |

3: Init $L\leftarrow \left[\phantom{\rule{3.33333pt}{0ex}}\right]$ array to store monetary value of each loss day |

4: for each test point $t=1,\dots ,383$ do |

5: $maxVal\leftarrow {max}_{s}E\left({\mathbf{\Pi}}_{t}^{tot}\right)$, select largest expected total payoff for all spreads s at t |

6: if $maxVal\ne 0$ then |

7: ${s}^{\prime}\leftarrow {argmax}_{s}E\left({\mathbf{\Pi}}_{t}^{tot}\right)$ extract spread hour associated with max expected payoff |

8: ${\pi}_{t}^{tot}\leftarrow {\pi}_{t}^{tot,\left({s}^{\prime}\right)}$ extract actual total payoff for state ${s}^{\prime}$ at t |

9: if ${\pi}_{t}^{tot}\phantom{\rule{4pt}{0ex}}<0$ then |

10: $n\leftarrow n+1$ |

11: ${L}_{n}\leftarrow {\pi}_{t}^{tot}$ |

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**Figure 4.**Price and battery profiles for test time step $t=56,b=0,c=5,\eta =0.8$. Single trade selected by both algorithms, the spread between hours 14–21.

**Figure 5.**Price and battery profiles for test time step $t=317,b=0,c=5,\eta =0.8$ (

**a**) two trades using spreads 4–9, 12–18; (

**b**) single trade using spread 00–18.

**Figure 6.**Key results of single trade and double trade algorithms under initial battery level $b=0$, transaction costs, $c=\{5,10\}$ and normal/skew-type distributions; (

**a**) total monetary loss across the backtest horizon, (

**b**) total payoff accumulated over the backtest horizon.

**Table 1.**Single trade scenario, cost $c=5$ Euro/MWh, battery efficiency $\eta =0.8$. Results for models estimated under best distribution vs. the benchmark normal.

Best Dist. | Normal Dist. | ||||
---|---|---|---|---|---|

b | 0 | 1 | b | 0 | 1 |

${\pi}_{total}$ | $4853.8$ | $3186.4$ | ${\pi}_{total}$ | $4828.8$ | $2942.4$ |

$\overline{\pi}$ | $12.94$ | $11.54$ | $\overline{\pi}$ | $12.84$ | $10.55$ |

${s}^{\pi}$ | $0.61$ | $0.72$ | ${s}^{\pi}$ | $0.65$ | $0.79$ |

${n}_{l}$ | 3 | 8 | ${n}_{l}$ | 15 | 31 |

l | $-2.2$ | $-6.4$ | l | $-33.4$ | $-66.2$ |

$\overline{l}$ | $-0.73$ | $-0.80$ | $\overline{l}$ | $-2.23$ | $-2.14$ |

**Table 2.**Single trade scenario, cost $c=10$ Euro/MWh, battery efficiency $\eta =0.8$. Results for models estimated under best distribution vs. the benchmark normal.

Best Dist. | Normal Dist. | ||||
---|---|---|---|---|---|

b | 0 | 1 | b | 0 | 1 |

${\pi}_{total}$ | 2702 | $1318.4$ | ${\pi}_{total}$ | $2037.6$ | $664.8$ |

$\overline{\pi}$ | $11.40$ | $14.33$ | $\overline{\pi}$ | $10.40$ | $11.46$ |

${s}^{\pi}$ | $0.825$ | $1.66$ | ${s}^{\pi}$ | $1.04$ | $1.60$ |

${n}_{l}$ | 15 | 2 | ${n}_{l}$ | 25 | 2 |

l | $-14$ | $-3.2$ | l | $-82$ | $-5.6$ |

$\overline{l}$ | $-0.93$ | $-1.6$ | $\overline{l}$ | $-3.28$ | $-2.8$ |

**Table 3.**Two-trade scenario, cost $c=5$ Euro/MWh, battery efficiency $\eta =0.8$. Results for models estimated under best distribution vs. the benchmark normal.

Best Dist. | Normal Dist. | ||||
---|---|---|---|---|---|

b | 0 | 1 | b | 0 | 1 |

${\pi}_{total}$ | $5200.8$ | $3357.8$ | ${\pi}_{total}$ | $5039.4$ | $3089.2$ |

$\overline{\pi}$ | $13.87$ | $12.17$ | $\overline{\pi}$ | $13.4$ | $11.1$ |

${s}^{\pi}$ | $0.63$ | $0.74$ | ${s}^{\pi}$ | $0.67$ | $0.79$ |

${n}_{l}$ | 3 | 8 | ${n}_{l}$ | 14 | 31 |

l | $-2.2$ | $-6.4$ | l | $-33.2$ | $-66.2$ |

$\overline{l}$ | $-0.73$ | $-0.80$ | $\overline{l}$ | $-2.37$ | $-2.13$ |

${n}_{1}$ | 288 | 245 | ${n}_{1}$ | 321 | 249 |

${n}_{2}$ | 87 | 31 | ${n}_{2}$ | 55 | 30 |

**Table 4.**Two-trade scenario, cost $c=10$ Euro/MWh, battery efficiency $\eta =0.8$. Results for models estimated under best distribution vs. the benchmark normal.

Best Dist. | Normal Dist. | ||||
---|---|---|---|---|---|

b | 0 | 1 | b | 0 | 1 |

${\pi}_{total}$ | 2696 | $1318.4$ | ${\pi}_{total}$ | $2039.2$ | $664.8$ |

$\overline{\pi}$ | $11.38$ | $14.33$ | $\overline{\pi}$ | $10.40$ | $11.46$ |

${s}^{\pi}$ | $0.825$ | $1.66$ | ${s}^{\pi}$ | $1.06$ | $1.60$ |

${n}_{l}$ | 15 | 2 | ${n}_{l}$ | 25 | 2 |

l | $-14$ | $-3.2$ | l | $-82$ | $-5.6$ |

$\overline{l}$ | $-0.93$ | $-1.6$ | $\overline{l}$ | $-3.28$ | $-2.8$ |

${n}_{1}$ | 236 | 92 | ${n}_{1}$ | 194 | 58 |

${n}_{2}$ | 1 | 0 | ${n}_{2}$ | 2 | 0 |

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**MDPI and ACS Style**

Abramova, E.; Bunn, D.
Optimal Daily Trading of Battery Operations Using Arbitrage Spreads. *Energies* **2021**, *14*, 4931.
https://doi.org/10.3390/en14164931

**AMA Style**

Abramova E, Bunn D.
Optimal Daily Trading of Battery Operations Using Arbitrage Spreads. *Energies*. 2021; 14(16):4931.
https://doi.org/10.3390/en14164931

**Chicago/Turabian Style**

Abramova, Ekaterina, and Derek Bunn.
2021. "Optimal Daily Trading of Battery Operations Using Arbitrage Spreads" *Energies* 14, no. 16: 4931.
https://doi.org/10.3390/en14164931