# Optimal Daily Trading of Battery Operations Using Arbitrage Spreads

^{*}

^{†}

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

## References

- Denholm, P.; O’Connell, M.; Brinkman, G.; Jorgenson, J. Overgeneration from Solar Energy in California. A Field Guide to the Duck Chart; Technical Report; National Renewable Energy Lab. (NREL): Golden, CO, USA, 2015. [Google Scholar]
- Connolly, D.; Lund, H.; Finn, P.; Mathiesen, B.V.; Leahy, M. Practical operation strategies for pumped hydroelectric energy storage (PHES) utilising electricity price arbitrage. Energy Policy
**2011**, 39, 4189–4196. [Google Scholar] [CrossRef][Green Version] - Qi, W.; Liang, Y.; Shen, Z.J.M. Joint planning of energy storage and transmission for wind energy generation. Oper. Res.
**2015**, 63, 1280–1293. [Google Scholar] [CrossRef][Green Version] - Lucas, R. Throughput v Revenues: Making the Most from Battery Storage. 2019. Available online: https://www.current-news.co.uk/blogs/throughput-vs-revenues-making-the-most-from-battery-storage (accessed on 15 February 2021).
- Elexon. Electricity Storage in the GB Market. 2015. Available online: https://www.elexon.co.uk/wp-content/uploads/2015/03/Electricity_storage_in_the_GB_market_March2015.pdf (accessed on 15 February 2021).
- Eunomia. Investing in UK Electricity Storage. 2016. Available online: https://www.eunomia.co.uk/investing-in-uk-electricity-storage/ (accessed on 15 February 2021).
- Marchgraber, J.; Gawlik, W. Dynamic Prioritization of Functions during Real-Time Multi-Use Operation of Battery Energy Storage Systems. Energies
**2021**, 14, 655. [Google Scholar] [CrossRef] - Staffell, I.; Rustomji, M. Maximising the value of electricity storage. J. Energy Storage
**2016**, 8, 212–225. [Google Scholar] [CrossRef][Green Version] - Balardy, C. An Empirical Analysis of the Bid-Ask Spread in the German Power Continuous Market. 2018. Available online: http://www.ceem-dauphine.org/assets/dropbox/0918-CEEM_Working_Paper_35_Clara_BALARDY.pdf (accessed on 15 February 2021).
- Elexon. MIDS Consultation. 2017. Available online: https://www.elexon.co.uk/wp-content/uploads/2017/08/MIDS-Review-2017-Consultation-v1.0.pdf (accessed on 15 February 2021).
- Arup. Electricity Storage Technologies. 2018. Available online: https://www.arup.com/perspectives/publications/research/section/five-minute-guide-to-electricity-storage (accessed on 15 February 2021).
- Foster, W. Understanding the “Arbitrage” Value of a Large Battery in SA. 2017. Available online: https://reneweconomy.com.au/understanding-the-arbitrage-value-of-a-large-battery-in-sa-16154/ (accessed on 15 February 2021).
- Mohsenian-Rad, H. Optimal bidding, scheduling, and deployment of battery systems in California day-ahead energy market. IEEE Trans. Power Syst.
**2015**, 31, 442–453. [Google Scholar] [CrossRef][Green Version] - Salles, M.; Aziz, M.J.; Hogan, W.W. Potential arbitrage revenue of energy storage systems in PJM during 2014. In Proceedings of the 2016 IEEE Power and Energy Society General Meeting (PESGM), Boston, MA, USA, 17–21 July 2016; pp. 1–5. [Google Scholar]
- Nowotarski, J.; Weron, R. Computing electricity spot price prediction intervals using quantile regression and forecast averaging. Comput. Stat.
**2015**, 30, 791–803. [Google Scholar] [CrossRef][Green Version] - Garcia-Martos, C.; Rodrıguez, J.; Sanchez, M. Forecasting electricity prices by extracting dynamic common factors: Application to the Iberian market. IET Gener. Transm. Distrib.
**2012**, 6, 11–20. [Google Scholar] [CrossRef] - Karakatsani, N.V.; Bunn, D.W. Fundamental and behavioural drivers of electricity price volatility. Stud. Nonlinear Dyn. Econom.
**2010**, 14. [Google Scholar] [CrossRef] - Weron, R. Electricity price forecasting: A review of the state-of-the-art with a look into the future. Int. J. Forecast.
**2014**, 30, 1030–1081. [Google Scholar] [CrossRef][Green Version] - Nowotarski, J.; Weron, R. Recent advances in electricity price forecasting: A review of probabilistic forecasting. Renew. Sustain. Energy Rev.
**2018**, 81, 1548–1568. [Google Scholar] [CrossRef] - Gianfreda, A.; Bunn, D. A stochastic latent moment model for electricity price formation. Oper. Res.
**2018**, 66, 1189–1203. [Google Scholar] [CrossRef][Green Version] - Stasinopoulos, D.M.; Rigby, R.A. Generalized additive models for location scale and shape (GAMLSS) in R. J. Stat. Softw.
**2007**, 23, 1–46. [Google Scholar] [CrossRef][Green Version] - Abramova, E.; Bunn, D. Forecasting the Intra-Day Spread Densities of Electricity Prices. Energies
**2020**, 13, 687. [Google Scholar] [CrossRef][Green Version] - Hagfors, L.I.; Bunn, D.; Kristoffersen, E.; Staver, T.T.; Westgaard, S. Modeling the UK electricity price distributions using quantile regression. Energy
**2016**, 102, 231–243. [Google Scholar] [CrossRef] - Hu, S.; Souza, G.C.; Ferguson, M.E.; Wang, W. Capacity investment in renewable energy technology with supply intermittency: Data granularity matters! Manuf. Serv. Oper. Manag.
**2015**, 17, 480–494. [Google Scholar] [CrossRef] - Aflaki, S.; Netessine, S. Strategic investment in renewable energy sources: The effect of supply intermittency. Manuf. Serv. Oper. Manag.
**2017**, 19, 489–507. [Google Scholar] [CrossRef] - Wu, O.Q.; Kapuscinski, R. Curtailing intermittent generation in electrical systems. Manuf. Serv. Oper. Manag.
**2013**, 15, 578–595. [Google Scholar] [CrossRef][Green Version] - Al-Gwaiz, M.; Chao, X.; Wu, O.Q. Understanding how generation flexibility and renewable energy affect power market competition. Manuf. Serv. Oper. Manag.
**2016**, 19, 114–131. [Google Scholar] [CrossRef] - Kim, J.H.; Powell, W.B. Optimal energy commitments with storage and intermittent supply. Oper. Res.
**2011**, 59, 1347–1360. [Google Scholar] [CrossRef][Green Version] - Zhou, Y.; Scheller-Wolf, A.; Secomandi, N.; Smith, S. Electricity trading and negative prices: Storage vs. disposal. Manag. Sci.
**2015**, 62, 880–898. [Google Scholar] [CrossRef] - Broneske, G.; Wozabal, D. How do contract parameters influence the economics of vehicle-to-grid? Manuf. Serv. Oper. Manag.
**2017**, 19, 150–164. [Google Scholar] [CrossRef] - Kahlen, M.T.; Ketter, W.; van Dalen, J. Electric vehicle virtual power plant dilemma: Grid balancing versus customer mobility. Prod. Oper. Manag.
**2018**, 27, 2054–2070. [Google Scholar] [CrossRef] - White, C.; Thompson, B.; Swan, L.G. Comparative performance study of electric vehicle batteries repurposed for electricity grid energy arbitrage. Appl. Energy
**2021**, 288, 116637. [Google Scholar] [CrossRef] - Wu, O.Q.; Wang, D.D.; Qin, Z. Seasonal energy storage operations with limited flexibility: The price-adjusted rolling intrinsic policy. Manuf. Serv. Oper. Manag.
**2012**, 14, 455–471. [Google Scholar] [CrossRef] - Secomandi, N. Optimal commodity trading with a capacitated storage asset. Manag. Sci.
**2010**, 56, 449–467. [Google Scholar] [CrossRef][Green Version] - Nadarajah, S.; Secomandi, N. Merchant energy trading in a network. Oper. Res.
**2018**, 66, 1304–1320. [Google Scholar] [CrossRef][Green Version] - Boogert, A.; De Jong, C. Gas storage valuation using a multifactor price process. J. Energy Mark.
**2011**, 4, 29–52. [Google Scholar] [CrossRef][Green Version] - Secomandi, N. An improved basket of spread options heuristic for merchant energy storage. IISE Trans.
**2018**, 50, 645–653. [Google Scholar] [CrossRef] - Anjos, M.F.; Conejo, A.J. Unit commitment in electric energy systems. Found. Trends® Electr. Energy Syst.
**2017**, 1, 220–310. [Google Scholar] [CrossRef] - McDaniel, G.; Gabrielle, A. Dispatching pumped storage hydro. IEEE Trans. Power Appar. Syst.
**1966**, PAS-85, 465–471. [Google Scholar] - Pandžić, H.; Kuzle, I. Energy storage operation in the day-ahead electricity market. In Proceedings of the 2015 12th International Conference on the European Energy Market (EEM), Lisbon, Portugal, 19–22 May 2015; pp. 1–6. [Google Scholar]
- Mohsenian-Rad, H. Coordinated price-maker operation of large energy storage units in nodal energy markets. IEEE Trans. Power Syst.
**2015**, 31, 786–797. [Google Scholar] [CrossRef][Green Version] - Nasrolahpour, E.; Kazempour, J.; Zareipour, H.; Rosehart, W.D. Impacts of ramping inflexibility of conventional generators on strategic operation of energy storage facilities. IEEE Trans. Smart Grid
**2016**, 9, 1334–1344. [Google Scholar] [CrossRef][Green Version] - Krishnamurthy, D.; Uckun, C.; Zhou, Z.; Thimmapuram, P.R.; Botterud, A. Energy storage arbitrage under day-ahead and real-time price uncertainty. IEEE Trans. Power Syst.
**2017**, 33, 84–93. [Google Scholar] [CrossRef] - Tohidi, Y.; Gibescu, M. Stochastic optimisation for investment analysis of flow battery storage systems. IET Renew. Power Gener.
**2018**, 13, 555–562. [Google Scholar] [CrossRef] - Yu, N.; Foggo, B. Stochastic valuation of energy storage in wholesale power markets. Energy Econ.
**2017**, 64, 177–185. [Google Scholar] [CrossRef][Green Version] - Ding, H.; Hu, Z.; Song, Y. Rolling optimization of wind farm and energy storage system in electricity markets. IEEE Trans. Power Syst.
**2014**, 30, 2676–2684. [Google Scholar] [CrossRef] - Wang, Y.; Dvorkin, Y.; Fernandez-Blanco, R.; Xu, B.; Qiu, T.; Kirschen, D.S. Look-ahead bidding strategy for energy storage. IEEE Trans. Sustain. Energy
**2017**, 8, 1106–1117. [Google Scholar] [CrossRef] - Aliasghari, P.; Zamani-Gargari, M.; Mohammadi-Ivatloo, B. Look-ahead risk-constrained scheduling of wind power integrated system with compressed air energy storage (CAES) plant. Energy
**2018**, 160, 668–677. [Google Scholar] [CrossRef] - Jiang, D.R.; Powell, W.B. Optimal hour-ahead bidding in the real-time electricity market with battery storage using approximate dynamic programming. INFORMS J. Comput.
**2015**, 27, 525–543. [Google Scholar] [CrossRef][Green Version] - Shu, Z.; Jirutitijaroen, P. Optimal operation strategy of energy storage system for grid-connected wind power plants. IEEE Trans. Sustain. Energy
**2013**, 5, 190–199. [Google Scholar] [CrossRef] - Jiang, D.R.; Powell, W.B. Risk-averse approximate dynamic programming with quantile-based risk measures. Math. Oper. Res.
**2018**, 43, 554–579. [Google Scholar] [CrossRef][Green Version] - Durante, J.; Nascimento, J.; Powell, W.B. Backward approximate dynamic programming with hidden semi-markov stochastic models in energy storage optimization. arXiv
**2017**, arXiv:1710.03914. [Google Scholar] - Salas, D.F.; Powell, W.B. Benchmarking a Scalable Approximate Dynamic Programming Algorithm for Stochastic Control of Multidimensional Energy Storage Problems; Department of Operations Research and Financial Engineering. 2013. Available online: https://castlelab.princeton.edu/html/Papers/SalasPowell-BenchmarkingADPformultidimensionalenergystorageproblems.pdf (accessed on 11 August 2021).
- Xi, X.; Sioshansi, R.; Marano, V. A stochastic dynamic programming model for co-optimization of distributed energy storage. Energy Syst.
**2014**, 5, 475–505. [Google Scholar] [CrossRef] - Jiang, D.R.; Powell, W.B. Practicality of nested risk measures for dynamic electric vehicle charging. arXiv
**2016**, arXiv:1605.02848. [Google Scholar]

**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