# Energy Storage Scheduling in Distribution Systems Considering Wind and Photovoltaic Generation Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. State of the Art of Multi-Period Optimal Power Flow for Distribution Systems with Energy Storage

## 3. Methodology

#### 3.1. Multi-Period Optimal Power Flow Model for Distribution System with Energy Storage

#### 3.2. Expected Future Value Function for Stored Energy

#### 3.3. Determining the Value Function

Algorithm 1: Determining Value Function for Stored Energy |

Input: Grid data for the distribution system model in Section 3.1, historic DG resource (wind speed or solar irradiance) time series data, selection of a discrete set S_{x} of DG state variable values $x$; selection of a discrete set ${S}_{E}$ of initial stored energy values $E$Output: Estimates of value function parameters $\gamma \left(x\right)$ and $\beta \left(x\right)$ for all DG state variables $x\in {S}_{x}$1: Initialize value function parameters $\gamma \left(x\right)=0$, $\beta \left(x\right)=1$ for all $x\in {S}_{x}$ 2: Generate ${k}_{\mathrm{max}}$ synthetic time series y_{k} for the stochastic DG resource variables for each value of $x\in {S}_{x}$ 3: while $\beta $ and $\gamma $ not converged for all $x\in {S}_{x}$4: for ${j}_{x}=1\mathrm{to}\left|{S}_{x}\right|$ do5: Set DG state variable $x$ to the ${j}_{x}$th value in ${S}_{x}$ 6: for ${j}_{E}=1\mathrm{to}\left|{S}_{E}\right|$ do7: Set initial stored energy ${E}_{0,p+1}$ to the ${j}_{E}$th value in ${S}_{E}$ 8: for $k=1\mathrm{to}{k}_{\mathrm{max}}$ do9: Use DG resource time series y_{k} for state variable value $x$10: Solve second-stage problem for planning horizon $p$ ($t=1,2,\dots ,T$) with value function parameterized by $\gamma \left(x\right)$ and $\beta \left(x\right)$ for initial stored energy ${E}_{0,p+1}$ and DG resource time series y_{k}11: Evaluate dual value π(x, E _{0,p+1})12: end for13: end for14: Fit dual values $\pi \left(x,E\right)$ to a linear function of $E$ 15: Determine updated values of $\gamma \left(x\right)$ and $\beta \left(x\right)$ 16: end for17: end while |

#### 3.4. Modeling Stochasticity of Wind Power Generation

#### 3.5. Modeling Stochasticity of Solar PV Power Generation

_{t}can be classified as belonging to one of three clearness regimes $r$: overcast $\left(r=1\right)$, partly cloudy $\left(r=2\right),$ or sunny $\left(r=3\right)$. For time series for ${w}_{t}$ within each regime, we have, following Reference [38], assumed that the stochasticity can be described by a Markov chain model for the clearness ${c}_{t}$, defined as

_{clr}values ${\left\{{c}_{i}=\frac{i}{{n}_{\mathrm{clr}}-1}\right\}}_{i=0}^{{n}_{\mathrm{clr}}}$. As the daily expected irradiance time series ${s}_{t}$ may vary substantially from one month to the next, transition matrices for the regime-switching process between days and the Markov process for clearness within days are estimated separately for each month.

## 4. Case Study

#### 4.1. Distribution System Model

#### 4.2. Results for the Expected Future Value of Stored Energy

#### 4.3. Evaluation of Energy Storage Scheduling Considering the Value of Stored Energy

## 5. Conclusions and Further Work

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Schematic of distribution grid considered in the case study, prepared using the visualization techniques of [63].

_{DG}and a unity power factor for the PV inverters. These parameter choices correspond to a maximum total DG power output of 119.8 kW for the maximum solar irradiance of $w=795\text{}\mathrm{W}/{\mathrm{m}}^{2}$ observed in July 2010–2015. The hourly solar irradiance time series (in units $\mathrm{W}/{\mathrm{m}}^{2}$) are available in the Supplementary Materials (Tables S3–S8).

_{α}= 0.5 and ${\tau}_{\mathrm{overcast}}=0.135$ were chosen. To discretize the clearness states, the same value ${n}_{\mathrm{clr}}=14$ is used as in Reference [38]. The synthetic time series ${\mathit{y}}_{k}$ reproduce the general autocorrelation characteristics of the historic time series, with autocorrelation times around 14.4 h for wind speed and 8–9 h for clearness when averaged over sets $\left\{{\mathit{y}}_{k}\right\}.$

^{®}, Natick, MA, USA), and tolerances (decision variables and objective value) set to 10

^{−6}. Based on tests for $T\in \left[12,48\right]$, the computation time increases with the number of time steps $T$ approximately according to $O\left({T}^{1.4}\right).$

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**Figure 1.**Illustration of iterative solution approach to an infinite-horizon multi-stage decision problem solved in determining the future value function.

**Figure 2.**Schematic illustration of the model of stochasticity in wind speed beyond the planning horizon $t=T$.

**Figure 3.**Schematic illustration of the model of stochasticity in clearness (above) and solar irradiance (below) beyond the planning horizon $t=T$.

**Figure 4.**Convergence of value function parameters for different wind speeds for the case of wind power (

**a**) and for different cloud cover conditions (clearness regimes) for the case of PV (

**b**).

**Figure 5.**Marginal value of stored energy at the end of the planning horizon as a function of the amount of stored energy for different wind speeds for the case of wind power (

**a**) and for different cloud cover conditions (clearness regimes) for the case of PV (

**b**).

**Figure 6.**Estimated future value of stored energy as a function of the amount of stored energy for different wind speeds for the case of wind power (

**a**) and for different cloud cover conditions (clearness regimes) for the case of PV (

**b**).

**Figure 7.**Comparison of approaches to considering the expected value of stored energy at the end of the planning horizon for distributed wind power (

**a**) and PV (

**b**) generation in terms of the total cost of the distribution system over the simulation period.

Reference | OPF Model and Application | Handling of Stored Energy At The End Of Planning Horizon |
---|---|---|

[14] | AC MPOPF with small voltage angle approximation (convex problem), formulated as a finite-horizon optimal control problem | Linear penalty function in the objective function (proportional to the deviation of the amount of stored energy at the end of the planning horizon from the maximal energy capacity); 24 h horizon |

[15] | AC MPOPF; applied to power system with wind power | Rolling horizon (24 h look-ahead horizon) |

[16] | AC MPOPF for optimal charging of EVs | Requiring that all EVs are charged at the end of 10 h planning horizon |

[17] | AC MPOPF (coupled real-reactive) | ${E}_{T}={E}_{0}$ (24 h horizon or 120 h horizon) |

[18] | Semidefinite programming relaxation of AC MPOPF | ${E}_{T}\ge {E}_{T}^{\mathrm{min}}$ (8 h horizon) |

[19] | AC MPOPF; applied to power system with wind power | Rolling horizon (10 × 5 min look-ahead horizon) |

[20] | Scheduling of ESS (not including power flow constraints) solved by dynamic programming; genetic algorithm for sizing and siting problem as an outer loop (including checking of power flow constraints) | ${E}_{T}={E}_{0}$ (24 h horizon) |

[21] | AC MPOPF with linearized power flow constraints; genetic algorithm for sizing and siting problem as an outer loop | ${E}_{T}={E}_{0}$ (24 h horizon) |

[22] | Stochastic security-constrained AC MPOPF; implemented in the MATPOWER Optimal Scheduling Tool [23] | Linear penalty function (that is a linear combination of charged and discharged power for all time steps) |

[24] | AC MPOPF | Rolling horizon (24 h look-ahead horizon) |

[25] | Dynamic programming search in the time domain combined with conventional PF solver in the network domain; grid-connected microgrid with DG | n/a (72 h horizon) |

[26] | Combined ESS scheduling and sizing problem for distribution system with PV; no power flow constraints but including linearized voltage constraints from base case power flow sensitivities | ${E}_{T}\ge {E}_{T}^{\mathrm{min}}$ (16 week horizon) |

[27] | AC MPOPF; applied to distribution system with DG | ${E}_{T}={E}_{0}$ (24 h horizon) |

[28] | AC MPOPF with second-order cone programming relaxation | Rolling horizon (72 h look-ahead horizon) |

[29] | AC MPOPF for unbalanced 3-phase distribution network | ${E}_{T}={E}_{0}$ (24 h horizon) |

[30] | AC MPOPF for distribution system with wind power | Linear penalty function in the objective function for each time step (proportional to the deviation of the amount of stored energy from the maximal energy capacity); 24 h horizon |

[31,32] | AC MPOPF; applied to distribution system with wind power | Linear penalty function in the objective function for each time step proportional to energy stored, and implicitly through rolling horizon (24 h look-ahead horizon) |

[33] | AC MPOPF | n/a (72 h horizon) |

[34] | AC MPOPF (comparing with solving each time step in isolation) | ${E}_{T}={E}_{0}$ (24 h horizon) |

[9] | AC MPOPF; applied to distribution system with wind power (which is treated as stochastic for the next planning horizon) | Explicit valuation of the future value of stored energy at the end of 24 h planning horizon |

[35] | MPOPF with linearized AC power flow equations for radial distribution grids; compared with full AC power flow | n/a (up to 744 h horizon) |

[36] | Robust MPOPF with linearized AC power flow equations for radial distribution grids; applied to LV grid with high PV penetration | Rolling horizon (24 h look-ahead horizon) combined with real-time control within each hour |

[37] | AC MPOPF for radial distribution systems based on convex relaxation; optimizing EV charging | Requiring fully charged EV at the end of the 24 h planning horizon [31] |

[38] | Finite-horizon optimal policy problem for Markov decision process for distribution system with PV generation, solved by stochastic dynamic programming, explicitly checking for violations of power flow constraints | Taken into account within each daily planning horizon through stochastic dynamic programming approach (not explicitly discussing the storage level at the end of the planning horizon) |

[39] | AC-Quadratic Programming MPOPF | A quadratic penalty function penalizing the deviation from a reference storage level (with penalty coefficient and reference storage level varying over the day); up to 8 h horizons |

[40] | AC MPOPF; optimal scheduling of EVs in distribution system with PV and wind power | Requiring fully charged EV at the end of the 33 h planning horizon |

[41] | AC MPOPF, applied to minimizing generation costs | n/a (2 h horizon) |

[42] | Conditionally exact convex MPOPF embedded in model for optimal sizing and siting with stochastic load, electricity prices and PV; applied to distribution system with PV | ${E}_{T}={E}_{0}$ (24 h horizons for separate days with time series for the stochastic variables) |

[43] | AC MPOPF | ${E}_{T}={E}_{0}$ (up to 2880 time steps) |

[44] | Robust AC MPOPF for unbalanced 3-phase distribution network, applied to EV charging scheduling | Requiring fully charged EV at the end of the 24 h planning horizon |

[45] | Chance-constrained AC MPOPF for radial distribution systems | n/a (24 h planning horizon) |

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**MDPI and ACS Style**

Sperstad, I.B.; Korpås, M. Energy Storage Scheduling in Distribution Systems Considering Wind and Photovoltaic Generation Uncertainties. *Energies* **2019**, *12*, 1231.
https://doi.org/10.3390/en12071231

**AMA Style**

Sperstad IB, Korpås M. Energy Storage Scheduling in Distribution Systems Considering Wind and Photovoltaic Generation Uncertainties. *Energies*. 2019; 12(7):1231.
https://doi.org/10.3390/en12071231

**Chicago/Turabian Style**

Sperstad, Iver Bakken, and Magnus Korpås. 2019. "Energy Storage Scheduling in Distribution Systems Considering Wind and Photovoltaic Generation Uncertainties" *Energies* 12, no. 7: 1231.
https://doi.org/10.3390/en12071231