# Black Start Restoration of Islanded Droop-Controlled Microgrids

^{*}

## Abstract

**:**

## 1. Introduction

- Propose a systematic black start formulation for the sequential restoration of islanded droop-controlled microgrids. To the best of the authors’ knowledge, this is the first systematic black start restoration formulation considering droop as the primary control.
- Show how to coordinate multiple grid-supporting (master/droop) and grid following (or PQ) DGs to dynamically configure microgrid instead of the multiple microgrids commonly considered in other systematic formulations. Coordination of the grid-supporting DGs to form a single microgrid, when possible, can help improve load balancing, resilience, redundancy, and better utilization of the DGs.
- Propose a zero-dispatch synchronization scheme for the grid-supporting DGs during the sequential build-up of the islanded microgrid and show how this can be mathematically composed into the restoration formulation.
- Validate the restoration approach through detailed electromagnetic transient program (EMTP) simulation in PSCAD and highlight some of the observations from the simulation such as restoration modeling limitations and suggestions on how these limitations can be solved.

## 2. Droop Control Basics and Reference Operation

## 3. Description of Islanded Microgrid to be Restored

## 4. Formulation of Black Start Restoration for Droop-Controlled Microgrid

#### 4.1. Overview of the Black Start Process

#### 4.1.1. Step 1, 2, and 3

#### 4.1.2. Step 4 and 5

#### 4.2. Objective Function

#### 4.3. Initial Sequencing Constraints

#### 4.4. Connectivity Constraints

- Whenever a DG is energized, then that DG’s node must have been energized in the same or previous time step as the DG energization time step (Equation (7))
- Once any DG, branch, or load is energized, it cannot be de-energized (EQUATIONS (8), (11), and (14) respectively)
- Both nodes of an energized switchable branch must be energized (Equation (9)) and the status of non-switchable branches are the same with its two nodes (Equation (10))
- A switchable load can only be energized once its node is energized (Equation (12)) and a non-switchable load is automatically energized when its node is energized (Equation (13))

#### 4.5. Synchronization Enhancing Constraints

- At most one droop-controlled DG can be newly connected to the system per time step (Equation (15))
- “Freeze” the status/settings of every other element at any time step that a droop-controlled DG is synchronized to the system. This means that:
- No additional load is restored at a synchronization time step (Equation (16)).
- The status and dispatch settings of PQ DGs should remain the same as the previous time step just before the synchronization step (Equations (17)–(19)).
- The active and reactive power reference settings of all droop DGs should remain the same as the previous time step just before the synchronization step, and the DG that is about to be synchronized to the system should do at zero power reference settings (Equations (20) and (21)). Equations (20) and (21) also ensure that the synchronized DG is connected with a zero reference power since its reference power at the previous step would be set to zero according to the DG operation constraints.
- The status of all branches should remain the same as the previous time step just before the synchronization step except for one branch that can connect the synchronizing DG to the system, that is, only the branch that connects the DG to the system is allowed to change from “OFF” to “ON” (Equation (22)) if it was not already energized.

#### 4.6. Power Flow Constraints

#### 4.6.1. Kirchhoff’s Current Law (KCL) at each Node

#### 4.6.2. Current Injection at Each Node

#### 4.6.3. Current Injection at Droop Nodes

#### 4.6.4. Current Injection at Load Node

#### 4.6.5. Current Injection at PQ DG Nodes

#### 4.7. DG and System Operating Constraints

#### 4.7.1. Phase Voltage Unbalance Rate Constraint (PVUR)

#### 4.7.2. Voltage Limit Constraint

#### 4.7.3. DG Power Unbalance Constraints

#### 4.7.4. Nominal System Load Unbalance Index (NSLUI) Constraints

#### 4.7.5. DG Output Constraints

#### 4.7.6. Ramp Rate Constraints

#### 4.8. Topology and Sequencing Constraints

## 5. Example Case Study and Discussions

#### 5.1. Description of the Test System

#### 5.2. Restoration Solution

#### 5.3. Solution Verification Using PSCAD Simulation as Benchmark

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Droop graph for (

**a**) frequency droop (

**b**) voltage droop, with reference/desired operation points.

**Figure 2.**Islanded microgrid representation with loads, droop controlled distributed generators (DGs) and PQ DGs.

**Figure 6.**Parallel restoration with master DG and problem with synchronization and interconnecting the restored microgrids.

**Figure 12.**(

**a**) Droop three-phase active power reference (

**b**) Droop three-phase reactive power reference.

**Figure 15.**Droop reference active power and output active power for DGs at nodes (

**a**) 2671 and (

**b**) 2632 as obtained from PSCAD simulation.

**Figure 17.**Droop reference reactive power and output reactive power for DGs at nodes (

**a**) 2671 and (

**b**) 2632 as obtained from PSCAD simulation.

Sets | |

$n\left(A\right)$ | The number of elements in set A |

$B,\text{}{B}^{S},\text{}{B}^{F},\text{}C$ | Set of branches, switchable and damaged branches, set of switchable branches between bus blocks |

$G,\text{}{G}^{BS},\text{}{G}^{Dr},\text{}{G}^{F},\text{}{G}^{PQ}$ | Set of all DGs, subset of black start DGs, subset of droop-controlled DGs, subset of damaged DGs, subset of PQ DGs |

$L,\text{}{L}^{S},\text{}{L}^{F}$ | Set of loads, subset of switchable loads, and subset of damaged load |

$T$ | Set of time steps $\left\{1,\text{}2,\text{}\dots ,\text{}{N}_{T}\right\}$ and $n\left(T\right)={N}_{T}$ |

${N}_{p},\text{}N,\text{}K$ | Set of phase nodes, nodes, bus blocks, $n\left({N}_{p}\right)\ge n\left(N\right)\ge n\left(K\right)$ |

Binary Decision Variables (1–Energized, 0–Not Energized) | |

${\widehat{x}}_{i,t}^{N},\text{}{\widehat{x}}_{j,t}^{K}$ | Energization status of node $i$ at time step $t$, energization status of bus block $j$ at time step $t$ |

${\widehat{x}}_{g,t}^{G}$ | Energization status of DG $g$ at time step $t$ |

${\widehat{x}}_{ij,t}^{BR},\text{}{\widehat{x}}_{ij,t}^{K}$ | Energization status of line $\left(i,j\right)$ at time step, $t$, where $\left(i,j\right)\in B,C$ respectively |

${\widehat{x}}_{l,t}^{L}$ | Energization status of load $l$ at time step $t$ |

Continuous Decision Variables | |

$a,\text{}b,\text{}or\text{}c$ | Used as subscript or superscript to denote variable or parameter in phase a, b or c respectively |

${\widehat{P}}_{ph,k,t}^{dg}$, ${\widehat{Q}}_{ph,k,t}^{dg}$ | Active and reactive power output of PQ DG $k$ at step $t$ and phase $ph\in \left\{a,b,c\right\}$ |

${\widehat{V}}_{n,t}^{re}$, ${\widehat{V}}_{n,t}^{im}$ | Real and imaginary part of three-phase nodal voltage vector of node $n$ at step $t$ |

${\widehat{V}}_{n,t}^{re,ph}$, ${\widehat{V}}_{n,t}^{im,ph}$ | Real and imaginary part of nodal voltage of node $n$ at step $t$ and phase $ph$ |

${n}_{v,g,t}$ | Voltage droop co-efficient of DG $g$, $g\in {G}^{Dr}$ at step $t$ |

${n}_{f,g,t}$ | Frequency droop co-efficient of DG $g$, $g\in {G}^{Dr}$ at step $t$ |

${\widehat{P}}_{ph,k,t}^{ref},\text{}{\widehat{Q}}_{ph,k,t}^{ref}$ | Droop reference active and reactive power output of DG $k$ at step $t$, $k\in {G}^{Dr}$ |

${\widehat{P}}_{ph,l,t},\text{}{\widehat{Q}}_{ph,l,t}$ | Nominal active and reactive power demand of load $l$, phase $ph$, at time step $t$ |

Parameters | |

$M$ | A large number chosen deliberately to manipulate the constraint equations |

$\Delta t$ | time interval between restoration steps and is assumed to be a constant value for all intervals |

${P}_{g}^{G,ramp},\text{}{Q}_{g}^{G,ramp}$ | Maximum absolute value of differential active and reactive power output of DG $g$ for each time step (DG ramp rate) |

${P}_{g}^{min},\text{}{P}_{g}^{max}$ | Minimum and maximum active power output of DG $g$ |

${Q}_{g}^{min},\text{}{Q}_{g}^{max}$ | Minimum and maximum reactive power output of DG $g$ |

${P}_{ph,l,t},\text{}{Q}_{ph,l,t}$ | Nominal active and reactive power value of load $l$, phase $ph$, at time step $t$ (fixed to the same value for all time steps and is independent of whether the load has been restored or not) |

${z}_{i,j}={r}_{i,j}+j{x}_{i,j}$ | Impedance of line between nodes $i$ and $j$, and ${y}_{i,j}=\frac{1}{{z}_{i,j}}={g}_{i,j}+j{b}_{i,j}$ |

${y}_{i,j}^{sh}={g}_{i,j}^{sh}+j{b}_{i,j}^{sh}$ | Shunt admittance between nodes $i$ and $j$ |

Label | Node | Type | ${\mathit{f}}^{\mathit{r}\mathit{e}\mathit{f}}$ (Hz) | Per Phase BaseMVA | Per Phase baseKV | Pu Coupling X | Pmax (KW) | Pmin (KW) | Qmax (KVAR) | Qmin (KVAR) | Phase | Status | Blackstart | Ramp Rate % |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

DG1 | 2671 | Droop | 60 | 1 | 2.4018 | 0.5 | 600 | 0 | 300 | −50 | ABC | 1 | 1 | 50 |

DG2 | 2632 | Droop | 60 | 1 | 2.4018 | 0.5 | 600 | 0 | 300 | −50 | ABC | 1 | 1 | 50 |

DG3 | 671 | PQ | NA | NA | 2.4018 | NA | 200 | 0 | 120 | 0 | C | 1 | 0 | 50 |

DG4 | 675 | PQ | NA | NA | 2.4018 | NA | 200 | 0 | 120 | 0 | A | 1 | 0 | 50 |

DG5 | 632 | PQ | NA | NA | 2.4018 | NA | 200 | 0 | 120 | 0 | B | 1 | 0 | 50 |

Node | Config | P(a/B/C) KW | Q(A/B/C) KVAR | Turn-On Step |
---|---|---|---|---|

611 | Y | 0/0/85 | 0/0/40 | 4 |

634 | Y | 83/60/30 | 55/45/15 | 4 |

645 | Y | 0/82/0 | 0/62.5/0 | 4 |

646 | D | 0/115/0 | 0/66/0 | 5 |

652 | Y | 100/0/0 | 55/0/0 | 5 |

671 | D | 110/90/90 | 60/50/50 | 4 |

671 | Y | 2.835/11/19.5 | 1.665/6.335/11.35 | 2 |

675 | Y | 100/34/70 | 45/30/40 | 5 |

692 | D | 0/0/85 | 0/0/75.5 | Off |

5632 | Y | 5.665/22/39 | 3.335/12.665/22.65 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Bassey, O.; Butler-Purry, K.L.
Black Start Restoration of Islanded Droop-Controlled Microgrids. *Energies* **2020**, *13*, 5996.
https://doi.org/10.3390/en13225996

**AMA Style**

Bassey O, Butler-Purry KL.
Black Start Restoration of Islanded Droop-Controlled Microgrids. *Energies*. 2020; 13(22):5996.
https://doi.org/10.3390/en13225996

**Chicago/Turabian Style**

Bassey, Ogbonnaya, and Karen L. Butler-Purry.
2020. "Black Start Restoration of Islanded Droop-Controlled Microgrids" *Energies* 13, no. 22: 5996.
https://doi.org/10.3390/en13225996