# Optimal Power Management Strategy for Energy Storage with Stochastic Loads

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## Abstract

**:**

## 1. Introduction

## 2. System Topology

#### 2.1. The Primary Source

#### 2.2. The Load

#### 2.3. The Energy Storage

## 3. Optimal Power Management Strategy

#### 3.1. Constraints

#### 3.2. Numerical Calculation

#### 3.3. The Output

## 4. Simulations and Results

#### 4.1. Numerical Calculation of Optimal Values

#### 4.2. Distribution of Lift Durations

#### 4.3. Model of the RTG Crane

#### 4.3.1. Primary Source

#### 4.3.2. The Hoist Motor

#### 4.3.3. The Flywheel Energy Storage System

#### 4.4. Test Cycle

- The empty headblock is lowered over a container: a small amount of energy is regenerated;
- The load is hoisted to a height decided by the crane operator: a large amount of energy is consumed;
- The load is lowered in place: a large amount of energy is regenerated;
- The headblock is hoisted back into the starting position: a small amount of energy is consumed.

- No ESS: In this scenario, no storage is installed, and all of the recovered energy is dissipated through the brake resistors;
- Constant power: The ESS uses a set-point control strategy where the ESS output is limited to a value that is the average load power, i.e., ${p}_{set}\left(t\right)=max\{72\phantom{\rule{4.pt}{0ex}}\text{kW},{P}_{L}\}$;
- Proposed PMS: An ESS with the optimal control strategy proposed in this paper;
- Infinite capacity: An ideal ESS with unlimited energy capacity and set to absorb or generate energy with a power limit of 150 kW with no time limitations, similarly to the second scenario, but with no capacity constraints and with the highest power limit.

#### 4.5. Results of the Simulation and Analysis

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PMS | Power Management Strategy |

RTG | Rubber tyre gantry, a type of container crane |

ESS | Energy storage system |

FESS | Flywheel energy storage system |

CDF | Cumulative distribution function |

PCHIP | Piecewise cubic Hermite interpolating polynomial |

SR | Switched reluctance (electric motor) |

## Nomenclature

${p}_{L}\left(t\right)$ | Power demand from the load |

${p}_{g}\left(t\right)$ | Power from the primary source (e.g., diesel generator) |

${p}_{s}\left(t\right)$ | Power from the storage system |

${p}_{s}$ | Power rating of the storage system |

${D}_{tot}$ | Total cost associated with the energy production |

$D\left(cdot\right)$ | Cost function associated with energy production |

${P}_{L}$ | Constant power demand of the load |

${W}_{s}\left(t\right)$ | Energy stored in the storage system at time t |

${W}_{max}$ | Energy capacity of the storage system |

${\mathsf{\eta}}_{1},{\mathsf{\eta}}_{2}$ | Constants defining the dynamic properties of the storage system |

${f}_{L}\left(t\right)$ | Probability that an event occurs at time t when defined by a distribution L |

${F}_{L}\left(t\right)$ | Cumulative distribution function associated with the distribution L |

$\mathsf{\alpha},\mathsf{\beta}$ | constant parameters that define a Gamma distribution |

## Appendix A

**Theorem 1.**

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**Figure 4.**Temporal distribution of 10 sample points of a piecewise cubic Hermite interpolating polynomial (PCHIP) interpolant superimposed on the CDF. The vertical position of the PCHIP points indicates the distribution in the $[0,1]$ region.

**Figure 5.**Examples of control points calculated in the minimisation. (

**a**) The control points calculated for a single pair of initial conditions. Three different interpolations are also shown (PCHIP, spline and linear). Notice how the spline interpolation does not maintain monotonicity. (

**b**) A range of optimal control strategies calculated for 0.72 MJ < ${W}_{s}\left(0\right)$ < 3.00 MJ and $Pl$ = 100 kW. The colour bar on the right shows the power output of the storage expressed in kW.

**Figure 6.**Histogram of the lift durations superimposed on the Gamma distribution that fits the data.

**Figure 9.**Results from the test cycle. (

**a**) The total energy from the primary source for the four scenarios; (

**b**) the percentage of reduction of energy consumption for the three storage scenarios with respect to the first scenario.

**Figure 10.**Example extracted from the simulation. (

**a**) The power flows of the three main elements in the model; when lowering, the ESS power is equal to the hoist power; (

**b**) the profile of the energy stored in the ESS.

Parameter | Value |
---|---|

${P}_{s}$ | 150 kW |

${W}_{max}$ | 3.6 MJ |

${\mathsf{\eta}}_{1}$ | 1% |

${\mathsf{\eta}}_{2}$ | 1 kW |

Parameter | Value |
---|---|

${P}_{s}$ | 150 kW |

T | 70 s |

$\Delta {P}_{L}$ | 10kW |

${W}_{s}\left(0\right)$ (range) | [720, 3470] kJ |

$\Delta {W}_{s}\left(0\right)$ | 101.8 kJ |

Parameter | Value |
---|---|

α | 5.0292 |

β | 4.3923 |

Duration | 1 h |
---|---|

Number of lifts (container and empty headblock) | 89 |

Energy consumed | 18.24 kWh |

Average load weight (container plus headblock) | 19.09 t |

Average hoist power (when lifting) | 72.74 kW |

**Table 5.**Percentage of time that the primary source power output is over 150 and 200 kW. PMS, power management strategy.

Scenario | Percentage of Time over 150 kW | Percentage of Time over 200 kW |
---|---|---|

No ESS | 3.997% | 0.0437% |

Constant power | 0.902% | 0.0167% |

Proposed PMS | 1.365% | 0.0028% |

Infinite capacity | 1.856% | 0.0139% |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Pietrosanti, S.; Holderbaum, W.; Becerra, V.M. Optimal Power Management Strategy for Energy Storage with Stochastic Loads. *Energies* **2016**, *9*, 175.
https://doi.org/10.3390/en9030175

**AMA Style**

Pietrosanti S, Holderbaum W, Becerra VM. Optimal Power Management Strategy for Energy Storage with Stochastic Loads. *Energies*. 2016; 9(3):175.
https://doi.org/10.3390/en9030175

**Chicago/Turabian Style**

Pietrosanti, Stefano, William Holderbaum, and Victor M. Becerra. 2016. "Optimal Power Management Strategy for Energy Storage with Stochastic Loads" *Energies* 9, no. 3: 175.
https://doi.org/10.3390/en9030175