# Iterative Dynamic Programming—An Efficient Method for the Validation of Power Flow Control Strategies

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## Abstract

**:**

## 1. Introduction

- Primary control (microseconds): Local supervision, voltage and current control, power sharing control.
- Secondary control (milliseconds): Voltage/frequency control restoration, voltage unbalance, harmonic compensation.
- Tertiary control (minutes, hourly): Economic dispatching and optimization.

- Application of the DP on a multistate, heterogenous BESS for the benchmarking of PFCS;
- Development of the iDP for the efficient computation of the multistate optimization problem;
- Analysis of the computation times and comparison against conventional DP for the metaparameters of the iDP;
- Benchmarking of conventional PFCS through iDP and discussion of the optimal trajectories for different use cases.

## 2. System Description

#### 2.1. Model Design

#### 2.2. Power Flow Control Strategies

**equal power share**considers an equal power split to all battery systems. As shown in

**rated energy**. Furthermore, the dynamic PFCS method of $SoC$ balancing mentioned in [3,16,44,45,46] can be transferred to the

**SoE**

**balancing**to calculate the power sharing factor

#### 2.3. Target Indicator

#### 2.4. Load Profiles

## 3. Dynamic Programming

## 4. Iterative Dynamic Programming Approach

- 1
- Perform a standard DP for the selected load profile as a pre-loop of the iDP with a defined coarse discretization, using the optimal states as input signal for the iDP;
- 2
- Determine the calculation boundaries with a defined bandwidth based on the optimal states of the previous DP (or iDP). The bandwidth size can be chosen as a variable, which is a multiple of the state-space discretization. The bandwidth directly influences the globality and computing time of the solution. A detailed discussion of the influences is given in Section 5;
- 3
- Constrain the time step-dependent state space through the upper and lower values of the calculation boundaries. In terms of batteries, a range of the $SoE$ is given;
- 4
- Discretize more finely the constrained, time step-dependent state space. In this work, the discretization of power is chosen to be twice as high for each iteration. According to Equation (23), the discrete state space is also changed;
- 5
- Check whether the termination criterion is met. For example, the number of iteration steps, the change of the value of the objective function, or the calculation time can be used for this purpose. If the criterion is met, the algorithm is stopped and the results are transferred. If this is not the case, it continues with step 6;
- 6
- Perform a DP with the previously defined constraints on the state space. Subsequently, the algorithm jumps back to step 2.

## 5. Verification and Computing Speed

#### 5.1. Verification

#### 5.2. Computing Speed

## 6. Validation of Power Flow Control Strategies

#### 6.1. Results

#### 6.2. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BESS | Battery energy storage system |

DP | Dynamic programming |

ECM | Equivalent circuit model |

EMS | Energy management system |

iDP | Iterative dynamic programming |

LCOE | Levelized cost of electricity |

LCOS | Levelized cost of storage |

PCC | Point of common coupling |

PFCS | Power flow control strategies |

SoC | State of charge |

SoE | State of energy |

## Appendix A. Paramters and Efficiency Maps

**Figure A1.**Chosen Parameters of the ECM for an NCA 18650 cell. Electrolyte and line resistance (

**a**), charge transfer resistance (

**b**), time constant of the charge transfer (

**c**), and open circuit voltage (

**d**).

Parameter | Determination |
---|---|

Open circuit voltage | Incremental method at 25 ${}^{\circ}\mathrm{C}$ inside a climate chamber |

ECM Parameters | Puls test with a current of 2C and a pulse time of
10
$\mathrm{s}$. Determination of the parameters by the voltage relaxation and a lsqnonlin fit to the ECM. |

**Figure A2.**Efficiency map considering the charge parameters with a discretization of the power of 1 W and a discretization of the $SoE$ of 1%.

**Figure A3.**Efficiency map considering the discharge parameters with a discretization of the power of 1 W and a discretization of the $SoE$ of 1%.

## Appendix B. Boundary Value Test

**Figure A4.**Optimal decision variables for the discharging (

**a**) and charging (

**b**) cases, as well as optimal states of the discharging- (

**c**) and charging (

**d**) cases of the boundary value verification.

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**Figure 1.**Schematic of the BESS with the differentiation of the clamp and open circuit values of the batteries.

**Figure 2.**Unidirectional load profile I (

**a**) and bidirectional load profile II (

**b**). Powers are normalized to ${P}_{\mathrm{LP}}$.

**Figure 3.**Program structure of the iDP approach (

**a**), schematic representation of pre-loop with discrete bandwidth of 2 (

**b**), and discretization of the first iteration loop (

**c**).

**Figure 4.**Analyzed load profile, normalized to ${P}_{\mathrm{LP}}$ (

**a**), the influence of bandwidth of the first iteration and discretization of the pre-loop on the optimal total loss (

**b**), differences in power share of DP and iDP, normalized to ${P}_{\mathrm{max}}$ (

**c**).

**Figure 5.**Time step-specific computing speed of the iDP on the bandwidth (1 to 10 in a step size of 1) (

**a**) and the total computing speed of the first iteration depending on the discretization of the pre-loop as a function of bandwidth (

**b**).

**Table 1.**Computing speed of the iDP for different bandwidths. The mean values and standard deviation are referred per time step.

Bandwidth | Total Time | Mean Value $\mathit{\mu}$ | Standard Deviation $\mathit{\sigma}$ |
---|---|---|---|

1 | 2$\mathrm{s}$ | $0.1$$\mathrm{s}$ | $0.1$$\mathrm{s}$ |

2 | $5.5$$\mathrm{s}$ | $0.4$$\mathrm{s}$ | $0.3$$\mathrm{s}$ |

3 | $11.6$$\mathrm{s}$ | $0.8$$\mathrm{s}$ | $0.4$$\mathrm{s}$ |

4 | $22.7$$\mathrm{s}$ | $1.5$$\mathrm{s}$ | $0.9$$\mathrm{s}$ |

5 | $42.6$$\mathrm{s}$ | $2.8$$\mathrm{s}$ | $1.8$$\mathrm{s}$ |

6 | $64.2$$\mathrm{s}$ | $4.3$$\mathrm{s}$ | $3.1$$\mathrm{s}$ |

7 | $91.4$$\mathrm{s}$ | $6.1$$\mathrm{s}$ | $4.8$$\mathrm{s}$ |

8 | $121.5$$\mathrm{s}$ | $8.1$$\mathrm{s}$ | $6.6$$\mathrm{s}$ |

9 | $154.5$$\mathrm{s}$ | $10.3$$\mathrm{s}$ | $8.8$$\mathrm{s}$ |

10 | $184.7$$\mathrm{s}$ | $12.3$$\mathrm{s}$ | $10.8$$\mathrm{s}$ |

**Table 2.**Rated energies of the batteries normalized to ${E}_{0}$ in different system configurations.

System I | System II | System III | System IV | |
---|---|---|---|---|

Battery 1 | 1 p.u. | 0.8 p.u. | 1 p.u. | 0.8 p.u. |

Battery 2 | 1 p.u. | 1 p.u. | 1.25 p.u. | 1.25 p.u. |

Battery 3 | 1 p.u. | 1 p.u. | 1.5 p.u. | 1.5 p.u. |

**Table 3.**Cumulative loss of energy of different power split methods for load profile I. Energy losses are given relative to the total energy of the system.

PFCS | System I | System II | System III | System IV |
---|---|---|---|---|

Equal power share | $9.35$‰ | $42.50$‰ | $7.49$‰ | $32.6$‰ |

Rated energy | $9.35$‰ | $36.89$‰ | $7.68$‰ | $24.94$‰ |

SoE balancing | $9.35$‰ | $38.22$‰ | $7.5$‰ | $29.25$‰ |

iDP | $9.33$‰ | $15.52$‰ | $7.47$‰ | $12.07$‰ |

**Table 4.**Cumulative loss of energy of different power split methods for load profile II. Energy losses are given relative to the total energy of the system.

PFCS | System I | System II | System III | System IV |
---|---|---|---|---|

Equal power share | 10‰ | $39.24$‰ | $8.07$‰ | $31.02$‰ |

Rated energy | 10‰ | $35.22$‰ | $8.23$‰ | $24.11$‰ |

SoE balancing | 10‰ | $37.6$‰ | $8.07$‰ | $31.02$‰ |

iDP | $9.97$‰ | $15.51$‰ | $8.04$‰ | $12.41$‰ |

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## Share and Cite

**MDPI and ACS Style**

Rüther, T.; Mößle, P.; Mühlbauer, M.; Bohlen, O.; Danzer, M.A.
Iterative Dynamic Programming—An Efficient Method for the Validation of Power Flow Control Strategies. *Electricity* **2022**, *3*, 542-562.
https://doi.org/10.3390/electricity3040027

**AMA Style**

Rüther T, Mößle P, Mühlbauer M, Bohlen O, Danzer MA.
Iterative Dynamic Programming—An Efficient Method for the Validation of Power Flow Control Strategies. *Electricity*. 2022; 3(4):542-562.
https://doi.org/10.3390/electricity3040027

**Chicago/Turabian Style**

Rüther, Tom, Patrick Mößle, Markus Mühlbauer, Oliver Bohlen, and Michael A. Danzer.
2022. "Iterative Dynamic Programming—An Efficient Method for the Validation of Power Flow Control Strategies" *Electricity* 3, no. 4: 542-562.
https://doi.org/10.3390/electricity3040027