# Analysis of a Multi-Timescale Framework for the Voltage Control of Active Distribution Grids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Impact of DG Installation on the Voltage Profile

- PV: with ${\mathbf{Q}}_{\mathbf{DG}}$ the controlled reactive power power injection and ${\mathbf{P}}_{\mathbf{DG}}^{\mathbf{curt}}$ the controlled active power curtailment.
- ESS: with ${\mathbf{P}}_{\mathbf{ESS}}$ the controlled active power injection, whereas no reactive power control was considered.

## 3. Multi-Timescale Framework

#### 3.1. Scheduling of Energy Storage Systems

- ${\mathbf{V}}_{\mathbf{K}}$ is the full vector of voltage magnitudes calculated at iteration $\mathbf{K}$
- ${\mathbf{V}}_{\mathbf{nom}}=\mathbf{1}{V}_{0}$ is the vector of nominal voltage values.
- ${\mathbf{P}}_{\mathbf{ESS},\mathbf{K}}$ is the vector of active power set-points assigned to ESSs calculated at iteration $\mathbf{K}$.
- ${\mathbf{W}}_{\mathbf{V}},{\mathbf{W}}_{\mathbf{ESS}}$ are the ${N}_{BUS}$-dimensional symmetric matrices of weights associated with each variable.

- ${\mathbf{P}}_{\mathbf{DG},\mathbf{K}}^{\mathbf{max}}$ represents the maximum available DGs’ forecasted active power at the instant $\mathbf{K}$.
- ${\mathbf{P}}_{\mathbf{Load},\mathbf{K}}$ represents the active power load forecast at the node where the DGs are installed for the instant $\mathbf{K}$.

#### 3.2. Model Predictive Control

- $\mathsf{\Delta}{\mathbf{P}}_{\mathbf{DG},\mathbf{k}}^{\mathbf{curt}}$ is the vector of active power set-points assigned to the PVs calculated at iteration $\mathbf{k}$.
- $\mathsf{\Delta}{\mathbf{P}}_{\mathbf{ESS},\mathbf{k}}$ is the vector of active power set-points assigned to the ESSs calculated at iteration $\mathbf{k}$.
- $\mathsf{\Delta}{\mathbf{Q}}_{\mathbf{DG},\mathbf{k}}$ is the vector of reactive power set-points assigned to the PVs calculated at iteration $\mathbf{k}$.

- ${\mathbf{SoC}}_{\mathbf{k}}$ is the SoC calculated at time-step $\mathbf{k}$.
- ${\mathbf{W}}_{\mathbf{P}},{\mathbf{W}}_{\mathbf{Q}},{\mathbf{W}}_{\mathbf{SoC}}$ are the N-dimensional symmetric weighting matrices linked to the variables ${\mathbf{P}}_{\mathbf{DG}}^{\mathbf{curt}}$, ${\mathbf{Q}}_{\mathbf{DG}}$, and $\mathbf{SoC}$, respectively.

- ${\mathbf{P}}_{\mathbf{DG},\mathbf{MPC}}^{\mathbf{curt}}={\mathbf{P}}_{\mathbf{DG},\mathbf{meas}}^{\mathbf{curt}}+\mathsf{\Delta}{\mathbf{P}}_{\mathbf{DG},\mathbf{k}=\mathbf{1}}^{\mathbf{curt}}$
- ${\mathbf{Q}}_{\mathbf{DG},\mathbf{MPC}}={\mathbf{Q}}_{\mathbf{DG},\mathbf{meas}}+\mathsf{\Delta}{\mathbf{Q}}_{\mathbf{DG},\mathbf{k}=\mathbf{1}}$
- ${\mathbf{P}}_{\mathbf{ESS},\mathbf{MPC}}={\mathbf{P}}_{\mathbf{ESS},\mathbf{meas}}+\mathsf{\Delta}{\mathbf{P}}_{\mathbf{ESS},\mathbf{k}=\mathbf{1}}$

#### 3.3. Online Feedback Control for Fast Dynamics

- ${\mathbf{P}}_{\mathbf{DG},\mathbf{TOT}}^{\mathbf{curt}}={\mathbf{P}}_{\mathbf{DG},\mathbf{MPC}}^{\mathbf{curt}}+\mathsf{\Delta}{\mathbf{p}}_{\mathbf{DG}}^{\mathbf{curt}}$
- ${\mathbf{Q}}_{\mathbf{DG},\mathbf{TOT}}={\mathbf{Q}}_{\mathbf{DG},\mathbf{MPC}}+\mathsf{\Delta}{\mathbf{q}}_{\mathbf{DG}}$
- ${\mathbf{P}}_{\mathbf{ESS},\mathbf{TOT}}={\mathbf{P}}_{\mathbf{ESS},\mathbf{MPC}}+\mathsf{\Delta}{\mathbf{p}}_{\mathbf{ESS}}$

## 4. Simulation Setup

#### 4.1. Simulation Description

- Test 1, framework without the scheduler: In this test, the full framework was compared with a case where the scheduler was removed from the framework, to demonstrate that a framework consisting only of the MPC and online control can lead the voltage close to the nominal value, but does not optimally control the SoC of the ESSs.
- Test 2, framework without the MPC: In this test, the full framework was compared with a case where the MPC was removed, to show the importance of the MPC in creating set-points for PVs and ESSs that are able to compensate for the forecast errors in the scheduling. In this case, the scheduler calculates the reference values for the PVs and ESSs, which are used directly in the online feedback control.
- Test 3, framework without the online control: In this test, the online feedback control was removed. The purpose of this test was to demonstrate the capability of the online control to solve voltage violations due to rapid variations of the load and generation profiles. Thus, the full framework was compared with a structure without the online control. For this test, the MPC transmits the power set-points every time-step ${\mathbf{T}}_{M}$ without being modified by the low-level control.

- Loading (W): This indicator was applied in Test 1 and Test 2 and calculated the total loading index as the sum of the power flowing though the branches (${\mathbf{P}}_{\mathbf{BR}}$) calculated for the full day:$${\mathbf{P}}_{\mathbf{loading}}=\frac{{\sum}_{\mathbf{t}=1,\phantom{\rule{0.277778em}{0ex}}\mathbf{t}\in {\mathbf{T}}_{C}}^{{\mathbf{N}}_{tot}}{\sum}_{b\in {N}_{BR}}\left|{\mathbf{P}}_{\mathbf{BR},\mathbf{b}}\right|}{{\mathbf{N}}_{tot}}$$
- Voltage variation (%): This indicator was used in Test 1 and Test 2 and calculated the sum of the percentage voltage variation on all nodes ${N}_{BUS}$ for the whole day, compared to the nominal value.$$\mathsf{\Delta}{\mathbf{V}}_{\mathbf{tot}}=\frac{{\sum}_{\mathbf{t}=1,\phantom{\rule{0.277778em}{0ex}}\mathbf{t}\in {\mathbf{T}}_{C}}^{{\mathbf{N}}_{tot}}{\sum}_{h\in {N}_{BUS}}|{V}_{h}-{V}_{0}|}{{\mathbf{N}}_{tot}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{V}_{0}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}100$$
- $\mathsf{\Delta}{\mathbf{SoC}}_{\mathbf{K}=\mathbf{N}}$: This indicator verified the achievement of the condition on the SoC, and it was applied in Test 1 and Test 2. The indicator is defined as:$$\mathsf{\Delta}{\mathbf{SoC}}_{\mathbf{K}=\mathbf{N}}=|{\mathbf{SoC}}_{\mathbf{K}=\mathbf{N}}-{\mathbf{SoC}}_{\mathbf{K}=\mathbf{0}}|$$
- Voltage outside the limits (%): This was the only indicator used in Test 3 and calculated the percentage of time that the voltage remained outside the two limits compared to the full day, calculated as:$$\mathsf{\Delta}{\mathbf{N}}_{{\mathbf{V}}_{\mathbf{out}}}=\frac{{\sum}_{h\in {N}_{BUS}}{\mathbf{N}}_{viol,h}}{{\mathbf{N}}_{tot}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{N}_{BUS}}$$

#### 4.2. Grid Data

#### 4.3. Load Data

#### 4.4. Generation Data

#### 4.5. Data for Test 3

## 5. Simulation Results

#### 5.1. Complete Multi-Timescale Framework

#### 5.2. Test 1: Framework without the Scheduler

- ${\mathbf{W}}_{\mathbf{ESS}}$ is the ${N}_{BUS}$-dimensional symmetric weighting matrix linked to the variable ${\mathbf{P}}_{\mathbf{ESS}}$.
- The results of the MPC optimization are the set-points values ${\mathbf{P}}_{\mathbf{DG},\mathbf{MPC}}^{\mathbf{curt}}$, ${\mathbf{Q}}_{\mathbf{DG},\mathbf{MPC}}$, and ${\mathbf{P}}_{\mathbf{ESS},\mathbf{MPC}}$, which are defined as the sum of the last measured value and the result of the first iteration of the MPC.

#### 5.3. Test 2: Framework without the MPC

#### 5.4. Test 3: Framework without the Online Control

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Generation profiles. (

**a**) Forecasts of the generation profiles; (

**b**) measurements of the generation profiles.

**Figure 7.**Results without the online control #1. (

**a**) Detail of the generation profile; (

**b**) detail of the load profile.

**Figure 8.**Result of the schedule. (

**a**) Output of the schedule for active power ESSs; (

**b**) utput of the schedule for the SoC.

**Figure 9.**Results of the hierarchical control #1. (

**a**) Voltage with no control applied; (

**b**) after the MPC and online control.

**Figure 10.**Results of the hierarchical control #2. (

**a**) Active power curtailment PV set-points; (

**b**) reactive power PV set-points.

**Figure 15.**Results without the MPC. (

**a**) Voltage with no MPC applied; (

**b**) active power ESS set-points.

**Figure 17.**Results without the online control 2. (

**a**) Voltage with no online control applied; (

**b**) voltage with online control.

**Figure 18.**Results without the online control 3. (

**a**) Active power ESSs with no online control applied; (

**b**) active power ESSs with online control.

Start | End | P Unit Resistance | Per Unit Reactance | Start | End | P Unit Resistance | Per Unit Reactance |
---|---|---|---|---|---|---|---|

1 | 2 | 4.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.17 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 17 | 18 | 6.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

2 | 3 | 1.08 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 18 | 19 | 5.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

3 | 4 | 4.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 19 | 20 | 8.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

4 | 5 | 8.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 20 | 21 | 3.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

5 | 6 | 9.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 21 | 22 | 2.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 9.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

3 | 7 | 1.35 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 22 | 23 | 4.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

7 | 8 | 3.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 23 | 24 | 4.20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

7 | 9 | 6.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 24 | 25 | 8.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

7 | 10 | 9.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 25 | 26 | 9.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

7 | 11 | 7.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 23 | 27 | 1.35 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

10 | 12 | 9.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 27 | 28 | 3.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

10 | 13 | 7.20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 27 | 29 | 6.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

13 | 14 | 4.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.50 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 27 | 30 | 9.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

11 | 15 | 9.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 27 | 31 | 7.10 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.60 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

15 | 16 | 8.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.00 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 30 | 32 | 9.80 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 3.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

15 | 17 | 3.40 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 30 | 33 | 7.20 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

Node | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

Customers | 0 | 2 | 1 | 1 | 1 | 2 | 6 | 1 | 3 | 3 | 2 | 4 | 2 | 1 | 2 | 3 | 1 | 2 |

Node | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | |||

Customers | 5 | 2 | 4 | 4 | 5 | 1 | 1 | 2 | 6 | 1 | 3 | 3 | 2 | 4 | 2 |

Indicators | Uncontrolled | Full Controlled | Test 1 | Test 2 |
---|---|---|---|---|

Loading (W) | 3922 | 830 | 340 | 1067 |

Voltage variation (%) | 10.9 | 4.1 | 2.5 | 6.7 |

$\mathsf{\Delta}{\mathbf{SoC}}_{\mathbf{K}=\mathbf{N}}$ | 0.0 | 5.2 | 116 | 7.3 |

Indicators | With Online Control | Without Online Control |
---|---|---|

Voltage outside the limits (%) | 0.45 | 1.5 |

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## Share and Cite

**MDPI and ACS Style**

De Din, E.; Bigalke, F.; Pau, M.; Ponci, F.; Monti, A.
Analysis of a Multi-Timescale Framework for the Voltage Control of Active Distribution Grids. *Energies* **2021**, *14*, 1965.
https://doi.org/10.3390/en14071965

**AMA Style**

De Din E, Bigalke F, Pau M, Ponci F, Monti A.
Analysis of a Multi-Timescale Framework for the Voltage Control of Active Distribution Grids. *Energies*. 2021; 14(7):1965.
https://doi.org/10.3390/en14071965

**Chicago/Turabian Style**

De Din, Edoardo, Fabian Bigalke, Marco Pau, Ferdinanda Ponci, and Antonello Monti.
2021. "Analysis of a Multi-Timescale Framework for the Voltage Control of Active Distribution Grids" *Energies* 14, no. 7: 1965.
https://doi.org/10.3390/en14071965